Datasets:

l3lab

/

ntp-mathlib

like 2

case e_a 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E n : ℕ s : Set ℝ x₀ x : ℝ ⊢ (x - x₀) ^ (n + 1) • (↑(n + 1)!)⁻¹ • iteratedDerivWithin (n + 1) f s x₀ = ((↑n !)⁻¹ * (↑n + 1)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin]

rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev]

@[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin]

Mathlib.Analysis.Calculus.Taylor.83_0.INXnr4jrmq9RIjK

@[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E s : Set ℝ x₀ x : ℝ ⊢ taylorWithinEval f 0 s x₀ x = f x₀

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by

dsimp only [taylorWithinEval]

/-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by

Mathlib.Analysis.Calculus.Taylor.96_0.INXnr4jrmq9RIjK

/-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E s : Set ℝ x₀ x : ℝ ⊢ (PolynomialModule.eval x) (taylorWithin f 0 s x₀) = f x₀

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval]

dsimp only [taylorWithin]

/-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval]

Mathlib.Analysis.Calculus.Taylor.96_0.INXnr4jrmq9RIjK

/-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E s : Set ℝ x₀ x : ℝ ⊢ (PolynomialModule.eval x) (∑ k in Finset.range (0 + 1), (PolynomialModule.comp (Polynomial.X - Polynomial.C x₀)) ((PolynomialModule.single ℝ k) (taylorCoeffWithin f k s x₀))) = f x₀

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin]

dsimp only [taylorCoeffWithin]

/-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin]

Mathlib.Analysis.Calculus.Taylor.96_0.INXnr4jrmq9RIjK

/-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E s : Set ℝ x₀ x : ℝ ⊢ (PolynomialModule.eval x) (∑ k in Finset.range (0 + 1), (PolynomialModule.comp (Polynomial.X - Polynomial.C x₀)) ((PolynomialModule.single ℝ k) ((↑k !)⁻¹ • iteratedDerivWithin k f s x₀))) = f x₀

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin]

simp

/-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin]

Mathlib.Analysis.Calculus.Taylor.96_0.INXnr4jrmq9RIjK

/-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E n : ℕ s : Set ℝ x₀ : ℝ ⊢ taylorWithinEval f n s x₀ x₀ = f x₀

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by

induction' n with k hk

/-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by

Mathlib.Analysis.Calculus.Taylor.106_0.INXnr4jrmq9RIjK

/-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀

Mathlib_Analysis_Calculus_Taylor

case zero 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E s : Set ℝ x₀ : ℝ ⊢ taylorWithinEval f Nat.zero s x₀ x₀ = f x₀

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk ·

exact taylor_within_zero_eval _ _ _ _

/-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk ·

Mathlib.Analysis.Calculus.Taylor.106_0.INXnr4jrmq9RIjK

/-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀

Mathlib_Analysis_Calculus_Taylor

case succ 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E s : Set ℝ x₀ : ℝ k : ℕ hk : taylorWithinEval f k s x₀ x₀ = f x₀ ⊢ taylorWithinEval f (Nat.succ k) s x₀ x₀ = f x₀

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _

simp [hk]

/-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _

Mathlib.Analysis.Calculus.Taylor.106_0.INXnr4jrmq9RIjK

/-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E n : ℕ s : Set ℝ x₀ x : ℝ ⊢ taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((↑k !)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by

induction' n with k hk

theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by

Mathlib.Analysis.Calculus.Taylor.115_0.INXnr4jrmq9RIjK

theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀

Mathlib_Analysis_Calculus_Taylor

case zero 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E s : Set ℝ x₀ x : ℝ ⊢ taylorWithinEval f Nat.zero s x₀ x = ∑ k in Finset.range (Nat.zero + 1), ((↑k !)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk ·

simp

theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk ·

Mathlib.Analysis.Calculus.Taylor.115_0.INXnr4jrmq9RIjK

theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀

Mathlib_Analysis_Calculus_Taylor

case succ 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E s : Set ℝ x₀ x : ℝ k : ℕ hk : taylorWithinEval f k s x₀ x = ∑ k in Finset.range (k + 1), ((↑k !)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ ⊢ taylorWithinEval f (Nat.succ k) s x₀ x = ∑ k in Finset.range (Nat.succ k + 1), ((↑k !)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp

rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]

theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp

Mathlib.Analysis.Calculus.Taylor.115_0.INXnr4jrmq9RIjK

theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀

Mathlib_Analysis_Calculus_Taylor

case succ 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E s : Set ℝ x₀ x : ℝ k : ℕ hk : taylorWithinEval f k s x₀ x = ∑ k in Finset.range (k + 1), ((↑k !)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ ⊢ ∑ k in Finset.range (k + 1), ((↑k !)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ + (((↑k + 1) * ↑k !)⁻¹ * (x - x₀) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s x₀ = ∑ x_1 in Finset.range (k + 1), ((↑x_1 !)⁻¹ * (x - x₀) ^ x_1) • iteratedDerivWithin x_1 f s x₀ + ((↑(k + 1)!)⁻¹ * (x - x₀) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s x₀

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]

simp [Nat.factorial]

theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]

Mathlib.Analysis.Calculus.Taylor.115_0.INXnr4jrmq9RIjK

theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x : ℝ n : ℕ s : Set ℝ hs : UniqueDiffOn ℝ s hf : ContDiffOn ℝ (↑n) f s ⊢ ContinuousOn (fun t => taylorWithinEval f n s t x) s

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by

simp_rw [taylor_within_apply]

/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by

Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK

/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x : ℝ n : ℕ s : Set ℝ hs : UniqueDiffOn ℝ s hf : ContDiffOn ℝ (↑n) f s ⊢ ContinuousOn (fun t => ∑ k in Finset.range (n + 1), ((↑k !)⁻¹ * (x - t) ^ k) • iteratedDerivWithin k f s t) s

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply]

refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _

/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply]

Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK

/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x : ℝ n : ℕ s : Set ℝ hs : UniqueDiffOn ℝ s hf : ContDiffOn ℝ (↑n) f s i : ℕ hi : i ∈ Finset.range (n + 1) ⊢ ContinuousOn (fun t => ((↑i !)⁻¹ * (x - t) ^ i) • iteratedDerivWithin i f s t) s

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _

refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _

/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _

Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK

/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x : ℝ n : ℕ s : Set ℝ hs : UniqueDiffOn ℝ s hf : ContDiffOn ℝ (↑n) f s i : ℕ hi : i ∈ Finset.range (n + 1) ⊢ ContinuousOn (fun t => iteratedDerivWithin i f s t) s

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _

rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf

/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _

Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK

/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x : ℝ n : ℕ s : Set ℝ hs : UniqueDiffOn ℝ s hf : (∀ (m : ℕ), ↑m ≤ ↑n → ContinuousOn (iteratedDerivWithin m f s) s) ∧ ∀ (m : ℕ), ↑m < ↑n → DifferentiableOn ℝ (iteratedDerivWithin m f s) s i : ℕ hi : i ∈ Finset.range (n + 1) ⊢ ContinuousOn (fun t => iteratedDerivWithin i f s t) s

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf

cases' hf with hf_left

/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf

Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK

/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s

Mathlib_Analysis_Calculus_Taylor

case intro 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x : ℝ n : ℕ s : Set ℝ hs : UniqueDiffOn ℝ s i : ℕ hi : i ∈ Finset.range (n + 1) hf_left : ∀ (m : ℕ), ↑m ≤ ↑n → ContinuousOn (iteratedDerivWithin m f s) s right✝ : ∀ (m : ℕ), ↑m < ↑n → DifferentiableOn ℝ (iteratedDerivWithin m f s) s ⊢ ContinuousOn (fun t => iteratedDerivWithin i f s t) s

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left

specialize hf_left i

/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left

Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK

/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s

Mathlib_Analysis_Calculus_Taylor

case intro 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x : ℝ n : ℕ s : Set ℝ hs : UniqueDiffOn ℝ s i : ℕ hi : i ∈ Finset.range (n + 1) right✝ : ∀ (m : ℕ), ↑m < ↑n → DifferentiableOn ℝ (iteratedDerivWithin m f s) s hf_left : ↑i ≤ ↑n → ContinuousOn (iteratedDerivWithin i f s) s ⊢ ContinuousOn (fun t => iteratedDerivWithin i f s t) s

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i

simp only [Finset.mem_range] at hi

/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i

Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK

/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s

Mathlib_Analysis_Calculus_Taylor

case intro 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x : ℝ n : ℕ s : Set ℝ hs : UniqueDiffOn ℝ s i : ℕ right✝ : ∀ (m : ℕ), ↑m < ↑n → DifferentiableOn ℝ (iteratedDerivWithin m f s) s hf_left : ↑i ≤ ↑n → ContinuousOn (iteratedDerivWithin i f s) s hi : i < n + 1 ⊢ ContinuousOn (fun t => iteratedDerivWithin i f s t) s

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi

refine' hf_left _

/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi

Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK

/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s

Mathlib_Analysis_Calculus_Taylor

case intro 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x : ℝ n : ℕ s : Set ℝ hs : UniqueDiffOn ℝ s i : ℕ right✝ : ∀ (m : ℕ), ↑m < ↑n → DifferentiableOn ℝ (iteratedDerivWithin m f s) s hf_left : ↑i ≤ ↑n → ContinuousOn (iteratedDerivWithin i f s) s hi : i < n + 1 ⊢ ↑i ≤ ↑n

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _

simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi]

/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _

Mathlib.Analysis.Calculus.Taylor.124_0.INXnr4jrmq9RIjK

/-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E t x : ℝ n : ℕ ⊢ HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(↑n + 1) * (x - t) ^ n) t

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by

simp_rw [sub_eq_neg_add]

/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by

Mathlib.Analysis.Calculus.Taylor.140_0.INXnr4jrmq9RIjK

/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E t x : ℝ n : ℕ ⊢ HasDerivAt (fun y => (-y + x) ^ (n + 1)) (-(↑n + 1) * (-t + x) ^ n) t

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add]

rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc]

/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add]

Mathlib.Analysis.Calculus.Taylor.140_0.INXnr4jrmq9RIjK

/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E t x : ℝ n : ℕ ⊢ HasDerivAt (fun y => (-y + x) ^ (n + 1)) ((↑n + 1) * (-t + x) ^ n * -1) t

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc]

convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)

/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc]

Mathlib.Analysis.Calculus.Taylor.140_0.INXnr4jrmq9RIjK

/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t

Mathlib_Analysis_Calculus_Taylor

case h.e'_7.h.e'_5.h.e'_5 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E t x : ℝ n : ℕ ⊢ ↑n + 1 = ↑(n + 1)

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)

simp only [Nat.cast_add, Nat.cast_one]

/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)

Mathlib.Analysis.Calculus.Taylor.140_0.INXnr4jrmq9RIjK

/-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ k : ℕ s t : Set ℝ ht : UniqueDiffWithinAt ℝ t y hs : s ∈ 𝓝[t] y hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y ⊢ HasDerivWithinAt (fun z => (((↑k + 1) * ↑k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by

replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by

Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ k : ℕ s t : Set ℝ ht : UniqueDiffWithinAt ℝ t y hs : s ∈ 𝓝[t] y hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y ⊢ HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by

convert (hf.mono_of_mem hs).hasDerivWithinAt using 1

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by

Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y

Mathlib_Analysis_Calculus_Taylor

case h.e'_7 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ k : ℕ s t : Set ℝ ht : UniqueDiffWithinAt ℝ t y hs : s ∈ 𝓝[t] y hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y ⊢ iteratedDerivWithin (k + 2) f s y = derivWithin (iteratedDerivWithin (k + 1) f s) t y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1

rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1

Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y

Mathlib_Analysis_Calculus_Taylor

case h.e'_7 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ k : ℕ s t : Set ℝ ht : UniqueDiffWithinAt ℝ t y hs : s ∈ 𝓝[t] y hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y ⊢ derivWithin (iteratedDerivWithin (k + 1) f s) s y = derivWithin (iteratedDerivWithin (k + 1) f s) t y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]

exact (derivWithin_of_mem hs ht hf).symm

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]

Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ k : ℕ s t : Set ℝ ht : UniqueDiffWithinAt ℝ t y hs : s ∈ 𝓝[t] y hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y ⊢ HasDerivWithinAt (fun z => (((↑k + 1) * ↑k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm

have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm

Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ k : ℕ s t : Set ℝ ht : UniqueDiffWithinAt ℝ t y hs : s ∈ 𝓝[t] y hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y ⊢ HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (-((↑k !)⁻¹ * (x - y) ^ k)) t y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors:

have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors:

Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ k : ℕ s t : Set ℝ ht : UniqueDiffWithinAt ℝ t y hs : s ∈ 𝓝[t] y hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y ⊢ -((↑k !)⁻¹ * (x - y) ^ k) = ((↑k + 1) * ↑k !)⁻¹ * (-(↑k + 1) * (x - y) ^ k)

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by

field_simp

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by

Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ k : ℕ s t : Set ℝ ht : UniqueDiffWithinAt ℝ t y hs : s ∈ 𝓝[t] y hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y ⊢ -((x - y) ^ k * ((↑k + 1) * ↑k !)) = (-1 + -↑k) * (x - y) ^ k * ↑k !

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp;

ring

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp;

Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ k : ℕ s t : Set ℝ ht : UniqueDiffWithinAt ℝ t y hs : s ∈ 𝓝[t] y hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y this : -((↑k !)⁻¹ * (x - y) ^ k) = ((↑k + 1) * ↑k !)⁻¹ * (-(↑k + 1) * (x - y) ^ k) ⊢ HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (-((↑k !)⁻¹ * (x - y) ^ k)) t y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring

rw [this]

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring

Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ k : ℕ s t : Set ℝ ht : UniqueDiffWithinAt ℝ t y hs : s ∈ 𝓝[t] y hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y this : -((↑k !)⁻¹ * (x - y) ^ k) = ((↑k + 1) * ↑k !)⁻¹ * (-(↑k + 1) * (x - y) ^ k) ⊢ HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (((↑k + 1) * ↑k !)⁻¹ * (-(↑k + 1) * (x - y) ^ k)) t y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this]

exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this]

Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ k : ℕ s t : Set ℝ ht : UniqueDiffWithinAt ℝ t y hs : s ∈ 𝓝[t] y hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y this : HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (-((↑k !)⁻¹ * (x - y) ^ k)) t y ⊢ HasDerivWithinAt (fun z => (((↑k + 1) * ↑k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _

convert this.smul hf using 1

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _

Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y

Mathlib_Analysis_Calculus_Taylor

case h.e'_7 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ k : ℕ s t : Set ℝ ht : UniqueDiffWithinAt ℝ t y hs : s ∈ 𝓝[t] y hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y this : HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (-((↑k !)⁻¹ * (x - y) ^ k)) t y ⊢ (((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y = (((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y + -((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1

field_simp

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1

Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y

Mathlib_Analysis_Calculus_Taylor

case h.e'_7 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ k : ℕ s t : Set ℝ ht : UniqueDiffWithinAt ℝ t y hs : s ∈ 𝓝[t] y hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y this : HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (-((↑k !)⁻¹ * (x - y) ^ k)) t y ⊢ ((x - y) ^ (k + 1) / ((↑k + 1) * ↑k !)) • iteratedDerivWithin (k + 2) f s y - ((x - y) ^ k / ↑k !) • iteratedDerivWithin (k + 1) f s y = ((x - y) ^ (k + 1) / ((↑k + 1) * ↑k !)) • iteratedDerivWithin (k + 2) f s y + (-(x - y) ^ k / ↑k !) • iteratedDerivWithin (k + 1) f s y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp

rw [neg_div, neg_smul, sub_eq_add_neg]

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp

Mathlib.Analysis.Calculus.Taylor.149_0.INXnr4jrmq9RIjK

theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ n : ℕ s s' : Set ℝ hs'_unique : UniqueDiffWithinAt ℝ s' y hs_unique : UniqueDiffOn ℝ s hs' : s' ∈ 𝓝[s] y hy : y ∈ s' h : s' ⊆ s hf : ContDiffOn ℝ (↑n) f s hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y ⊢ HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by

induction' n with k hk

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by

Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y

Mathlib_Analysis_Calculus_Taylor

case zero 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ s s' : Set ℝ hs'_unique : UniqueDiffWithinAt ℝ s' y hs_unique : UniqueDiffOn ℝ s hs' : s' ∈ 𝓝[s] y hy : y ∈ s' h : s' ⊆ s hf : ContDiffOn ℝ (↑Nat.zero) f s hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin Nat.zero f s) s y ⊢ HasDerivWithinAt (fun t => taylorWithinEval f Nat.zero s t x) (((↑Nat.zero !)⁻¹ * (x - y) ^ Nat.zero) • iteratedDerivWithin (Nat.zero + 1) f s y) s' y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk ·

simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul]

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk ·

Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y

Mathlib_Analysis_Calculus_Taylor

case zero 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ s s' : Set ℝ hs'_unique : UniqueDiffWithinAt ℝ s' y hs_unique : UniqueDiffOn ℝ s hs' : s' ∈ 𝓝[s] y hy : y ∈ s' h : s' ⊆ s hf : ContDiffOn ℝ (↑Nat.zero) f s hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin Nat.zero f s) s y ⊢ HasDerivWithinAt (fun t => f t) (iteratedDerivWithin (Nat.zero + 1) f s y) s' y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul]

simp only [iteratedDerivWithin_zero] at hf'

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul]

Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y

Mathlib_Analysis_Calculus_Taylor

case zero 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ s s' : Set ℝ hs'_unique : UniqueDiffWithinAt ℝ s' y hs_unique : UniqueDiffOn ℝ s hs' : s' ∈ 𝓝[s] y hy : y ∈ s' h : s' ⊆ s hf : ContDiffOn ℝ (↑Nat.zero) f s hf' : DifferentiableWithinAt ℝ f s y ⊢ HasDerivWithinAt (fun t => f t) (iteratedDerivWithin (Nat.zero + 1) f s y) s' y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf'

rw [iteratedDerivWithin_one (hs_unique _ (h hy))]

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf'

Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y

Mathlib_Analysis_Calculus_Taylor

case zero 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ s s' : Set ℝ hs'_unique : UniqueDiffWithinAt ℝ s' y hs_unique : UniqueDiffOn ℝ s hs' : s' ∈ 𝓝[s] y hy : y ∈ s' h : s' ⊆ s hf : ContDiffOn ℝ (↑Nat.zero) f s hf' : DifferentiableWithinAt ℝ f s y ⊢ HasDerivWithinAt (fun t => f t) (derivWithin f s y) s' y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))]

norm_num

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))]

Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y

Mathlib_Analysis_Calculus_Taylor

case zero 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ s s' : Set ℝ hs'_unique : UniqueDiffWithinAt ℝ s' y hs_unique : UniqueDiffOn ℝ s hs' : s' ∈ 𝓝[s] y hy : y ∈ s' h : s' ⊆ s hf : ContDiffOn ℝ (↑Nat.zero) f s hf' : DifferentiableWithinAt ℝ f s y ⊢ HasDerivWithinAt (fun t => f t) (derivWithin f s y) s' y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num

exact hf'.hasDerivWithinAt.mono h

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num

Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y

Mathlib_Analysis_Calculus_Taylor

case succ 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ s s' : Set ℝ hs'_unique : UniqueDiffWithinAt ℝ s' y hs_unique : UniqueDiffOn ℝ s hs' : s' ∈ 𝓝[s] y hy : y ∈ s' h : s' ⊆ s k : ℕ hk : ContDiffOn ℝ (↑k) f s → DifferentiableWithinAt ℝ (iteratedDerivWithin k f s) s y → HasDerivWithinAt (fun t => taylorWithinEval f k s t x) (((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) s' y hf : ContDiffOn ℝ (↑(Nat.succ k)) f s hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin (Nat.succ k) f s) s y ⊢ HasDerivWithinAt (fun t => taylorWithinEval f (Nat.succ k) s t x) (((↑(Nat.succ k)!)⁻¹ * (x - y) ^ Nat.succ k) • iteratedDerivWithin (Nat.succ k + 1) f s y) s' y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h

simp_rw [Nat.add_succ, taylorWithinEval_succ]

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h

Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y

Mathlib_Analysis_Calculus_Taylor

case succ 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ s s' : Set ℝ hs'_unique : UniqueDiffWithinAt ℝ s' y hs_unique : UniqueDiffOn ℝ s hs' : s' ∈ 𝓝[s] y hy : y ∈ s' h : s' ⊆ s k : ℕ hk : ContDiffOn ℝ (↑k) f s → DifferentiableWithinAt ℝ (iteratedDerivWithin k f s) s y → HasDerivWithinAt (fun t => taylorWithinEval f k s t x) (((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) s' y hf : ContDiffOn ℝ (↑(Nat.succ k)) f s hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin (Nat.succ k) f s) s y ⊢ HasDerivWithinAt (fun t => taylorWithinEval f k s t x + (((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s t) (((↑(Nat.succ k)!)⁻¹ * (x - y) ^ Nat.succ k) • iteratedDerivWithin (Nat.succ (Nat.succ k + 0)) f s y) s' y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ]

simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ]

Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y

Mathlib_Analysis_Calculus_Taylor

case succ 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ s s' : Set ℝ hs'_unique : UniqueDiffWithinAt ℝ s' y hs_unique : UniqueDiffOn ℝ s hs' : s' ∈ 𝓝[s] y hy : y ∈ s' h : s' ⊆ s k : ℕ hk : ContDiffOn ℝ (↑k) f s → DifferentiableWithinAt ℝ (iteratedDerivWithin k f s) s y → HasDerivWithinAt (fun t => taylorWithinEval f k s t x) (((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) s' y hf : ContDiffOn ℝ (↑(Nat.succ k)) f s hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin (Nat.succ k) f s) s y ⊢ HasDerivWithinAt (fun t => taylorWithinEval f k s t x + (((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s t) ((((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ Nat.succ k) • iteratedDerivWithin (Nat.succ (Nat.succ k)) f s y) s' y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]

have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one]

Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y

Mathlib_Analysis_Calculus_Taylor

case succ 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ s s' : Set ℝ hs'_unique : UniqueDiffWithinAt ℝ s' y hs_unique : UniqueDiffOn ℝ s hs' : s' ∈ 𝓝[s] y hy : y ∈ s' h : s' ⊆ s k : ℕ hk : ContDiffOn ℝ (↑k) f s → DifferentiableWithinAt ℝ (iteratedDerivWithin k f s) s y → HasDerivWithinAt (fun t => taylorWithinEval f k s t x) (((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) s' y hf : ContDiffOn ℝ (↑(Nat.succ k)) f s hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin (Nat.succ k) f s) s y coe_lt_succ : ↑k < ↑(Nat.succ k) ⊢ HasDerivWithinAt (fun t => taylorWithinEval f k s t x + (((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s t) ((((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ Nat.succ k) • iteratedDerivWithin (Nat.succ (Nat.succ k)) f s y) s' y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self

have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self

Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y

Mathlib_Analysis_Calculus_Taylor

case succ 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ s s' : Set ℝ hs'_unique : UniqueDiffWithinAt ℝ s' y hs_unique : UniqueDiffOn ℝ s hs' : s' ∈ 𝓝[s] y hy : y ∈ s' h : s' ⊆ s k : ℕ hk : ContDiffOn ℝ (↑k) f s → DifferentiableWithinAt ℝ (iteratedDerivWithin k f s) s y → HasDerivWithinAt (fun t => taylorWithinEval f k s t x) (((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) s' y hf : ContDiffOn ℝ (↑(Nat.succ k)) f s hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin (Nat.succ k) f s) s y coe_lt_succ : ↑k < ↑(Nat.succ k) hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' ⊢ HasDerivWithinAt (fun t => taylorWithinEval f k s t x + (((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s t) ((((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ Nat.succ k) • iteratedDerivWithin (Nat.succ (Nat.succ k)) f s y) s' y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h

specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h

Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y

Mathlib_Analysis_Calculus_Taylor

case succ 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ s s' : Set ℝ hs'_unique : UniqueDiffWithinAt ℝ s' y hs_unique : UniqueDiffOn ℝ s hs' : s' ∈ 𝓝[s] y hy : y ∈ s' h : s' ⊆ s k : ℕ hf : ContDiffOn ℝ (↑(Nat.succ k)) f s hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin (Nat.succ k) f s) s y coe_lt_succ : ↑k < ↑(Nat.succ k) hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' hk : HasDerivWithinAt (fun t => taylorWithinEval f k s t x) (((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) s' y ⊢ HasDerivWithinAt (fun t => taylorWithinEval f k s t x + (((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s t) ((((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ Nat.succ k) • iteratedDerivWithin (Nat.succ (Nat.succ k)) f s y) s' y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')

convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs')

Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y

Mathlib_Analysis_Calculus_Taylor

case h.e'_7 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E x y : ℝ s s' : Set ℝ hs'_unique : UniqueDiffWithinAt ℝ s' y hs_unique : UniqueDiffOn ℝ s hs' : s' ∈ 𝓝[s] y hy : y ∈ s' h : s' ⊆ s k : ℕ hf : ContDiffOn ℝ (↑(Nat.succ k)) f s hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin (Nat.succ k) f s) s y coe_lt_succ : ↑k < ↑(Nat.succ k) hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' hk : HasDerivWithinAt (fun t => taylorWithinEval f k s t x) (((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) s' y ⊢ (((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ Nat.succ k) • iteratedDerivWithin (Nat.succ (Nat.succ k)) f s y = ((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y + ((((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y)

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1

exact (add_sub_cancel'_right _ _).symm

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1

Mathlib.Analysis.Calculus.Taylor.173_0.INXnr4jrmq9RIjK

/-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f g g' : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn g (Icc x₀ x) gdiff : ∀ x_1 ∈ Ioo x₀ x, HasDerivAt g (g' x_1) x_1 g'_ne : ∀ x_1 ∈ Ioo x₀ x, g' x_1 ≠ 0 ⊢ ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / ↑n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x'

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem

rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩

/-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem

Mathlib.Analysis.Calculus.Taylor.229_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x'

Mathlib_Analysis_Calculus_Taylor

case intro.intro 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f g g' : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn g (Icc x₀ x) gdiff : ∀ x_1 ∈ Ioo x₀ x, HasDerivAt g (g' x_1) x_1 g'_ne : ∀ x_1 ∈ Ioo x₀ x, g' x_1 ≠ 0 y : ℝ hy : y ∈ Ioo x₀ x h : (g x - g x₀) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) y = (taylorWithinEval f n (Icc x₀ x) x x - taylorWithinEval f n (Icc x₀ x) x₀ x) * g' y ⊢ ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / ↑n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x'

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩

use y, hy

/-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩

Mathlib.Analysis.Calculus.Taylor.229_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x'

Mathlib_Analysis_Calculus_Taylor

case right 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f g g' : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn g (Icc x₀ x) gdiff : ∀ x_1 ∈ Ioo x₀ x, HasDerivAt g (g' x_1) x_1 g'_ne : ∀ x_1 ∈ Ioo x₀ x, g' x_1 ≠ 0 y : ℝ hy : y ∈ Ioo x₀ x h : (g x - g x₀) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) y = (taylorWithinEval f n (Icc x₀ x) x x - taylorWithinEval f n (Icc x₀ x) x₀ x) * g' y ⊢ f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - y) ^ n / ↑n ! * (g x - g x₀) / g' y) • iteratedDerivWithin (n + 1) f (Icc x₀ x) y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations

simp only [taylorWithinEval_self] at h

/-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations

Mathlib.Analysis.Calculus.Taylor.229_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x'

Mathlib_Analysis_Calculus_Taylor

case right 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f g g' : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn g (Icc x₀ x) gdiff : ∀ x_1 ∈ Ioo x₀ x, HasDerivAt g (g' x_1) x_1 g'_ne : ∀ x_1 ∈ Ioo x₀ x, g' x_1 ≠ 0 y : ℝ hy : y ∈ Ioo x₀ x h : (g x - g x₀) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) y = (f x - taylorWithinEval f n (Icc x₀ x) x₀ x) * g' y ⊢ f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - y) ^ n / ↑n ! * (g x - g x₀) / g' y) • iteratedDerivWithin (n + 1) f (Icc x₀ x) y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h

rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h

/-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h

Mathlib.Analysis.Calculus.Taylor.229_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x'

Mathlib_Analysis_Calculus_Taylor

case right 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f g g' : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn g (Icc x₀ x) gdiff : ∀ x_1 ∈ Ioo x₀ x, HasDerivAt g (g' x_1) x_1 g'_ne : ∀ x_1 ∈ Ioo x₀ x, g' x_1 ≠ 0 y : ℝ hy : y ∈ Ioo x₀ x h : ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (g x - g x₀) / g' y = f x - taylorWithinEval f n (Icc x₀ x) x₀ x ⊢ f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - y) ^ n / ↑n ! * (g x - g x₀) / g' y) • iteratedDerivWithin (n + 1) f (Icc x₀ x) y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h

rw [← h]

/-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h

Mathlib.Analysis.Calculus.Taylor.229_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x'

Mathlib_Analysis_Calculus_Taylor

case right 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f g g' : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn g (Icc x₀ x) gdiff : ∀ x_1 ∈ Ioo x₀ x, HasDerivAt g (g' x_1) x_1 g'_ne : ∀ x_1 ∈ Ioo x₀ x, g' x_1 ≠ 0 y : ℝ hy : y ∈ Ioo x₀ x h : ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (g x - g x₀) / g' y = f x - taylorWithinEval f n (Icc x₀ x) x₀ x ⊢ ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (g x - g x₀) / g' y = ((x - y) ^ n / ↑n ! * (g x - g x₀) / g' y) • iteratedDerivWithin (n + 1) f (Icc x₀ x) y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h]

field_simp [g'_ne y hy]

/-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h]

Mathlib.Analysis.Calculus.Taylor.229_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x'

Mathlib_Analysis_Calculus_Taylor

case right 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f g g' : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn g (Icc x₀ x) gdiff : ∀ x_1 ∈ Ioo x₀ x, HasDerivAt g (g' x_1) x_1 g'_ne : ∀ x_1 ∈ Ioo x₀ x, g' x_1 ≠ 0 y : ℝ hy : y ∈ Ioo x₀ x h : ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (g x - g x₀) / g' y = f x - taylorWithinEval f n (Icc x₀ x) x₀ x ⊢ (x - y) ^ n * iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (g x - g x₀) = (x - y) ^ n * (g x - g x₀) * iteratedDerivWithin (n + 1) f (Icc x₀ x) y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy]

ring

/-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy]

Mathlib.Analysis.Calculus.Taylor.229_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x'

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) ⊢ ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)!

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by

have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by

Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)!

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) ⊢ ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by

refine' Continuous.continuousOn _

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by

Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)!

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) ⊢ Continuous fun t => (x - t) ^ (n + 1)

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _

exact (continuous_const.sub continuous_id').pow _

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _

Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)!

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x) ⊢ ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)!

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`

have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne'

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity`

Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)!

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x) ⊢ ∀ y ∈ Ioo x₀ x, (x - y) ^ n ≠ 0

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by

intro y hy

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by

Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)!

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x) y : ℝ hy : y ∈ Ioo x₀ x ⊢ (x - y) ^ n ≠ 0

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy

refine' pow_ne_zero _ _

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy

Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)!

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x) y : ℝ hy : y ∈ Ioo x₀ x ⊢ x - y ≠ 0

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _

rw [mem_Ioo] at hy

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _

Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)!

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x) y : ℝ hy : x₀ < y ∧ y < x ⊢ x - y ≠ 0

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy

rw [sub_ne_zero]

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy

Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)!

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x) y : ℝ hy : x₀ < y ∧ y < x ⊢ x ≠ y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero]

exact hy.2.ne'

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero]

Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)!

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x) xy_ne : ∀ y ∈ Ioo x₀ x, (x - y) ^ n ≠ 0 ⊢ ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)!

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne'

have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne'

Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)!

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x) xy_ne : ∀ y ∈ Ioo x₀ x, (x - y) ^ n ≠ 0 hg' : ∀ y ∈ Ioo x₀ x, -(↑n + 1) * (x - y) ^ n ≠ 0 ⊢ ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)!

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1)

rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1)

Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)!

Mathlib_Analysis_Calculus_Taylor

case intro.intro 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x) xy_ne : ∀ y ∈ Ioo x₀ x, (x - y) ^ n ≠ 0 hg' : ∀ y ∈ Ioo x₀ x, -(↑n + 1) * (x - y) ^ n ≠ 0 y : ℝ hy : y ∈ Ioo x₀ x h : f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) • iteratedDerivWithin (n + 1) f (Icc x₀ x) y ⊢ ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)!

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩

use y, hy

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩

Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)!

Mathlib_Analysis_Calculus_Taylor

case right 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x) xy_ne : ∀ y ∈ Ioo x₀ x, (x - y) ^ n ≠ 0 hg' : ∀ y ∈ Ioo x₀ x, -(↑n + 1) * (x - y) ^ n ≠ 0 y : ℝ hy : y ∈ Ioo x₀ x h : f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) • iteratedDerivWithin (n + 1) f (Icc x₀ x) y ⊢ f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (x - x₀) ^ (n + 1) / ↑(n + 1)!

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy

simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy

Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)!

Mathlib_Analysis_Calculus_Taylor

case right 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x) xy_ne : ∀ y ∈ Ioo x₀ x, (x - y) ^ n ≠ 0 hg' : ∀ y ∈ Ioo x₀ x, -(↑n + 1) * (x - y) ^ n ≠ 0 y : ℝ hy : y ∈ Ioo x₀ x h : f x - taylorWithinEval f n (Icc x₀ x) x₀ x = (-((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) • iteratedDerivWithin (n + 1) f (Icc x₀ x) y ⊢ f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (x - x₀) ^ (n + 1) / ↑(n + 1)!

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h

rw [h, neg_div, ← div_neg, neg_mul, neg_neg]

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h

Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)!

Mathlib_Analysis_Calculus_Taylor

case right 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x) xy_ne : ∀ y ∈ Ioo x₀ x, (x - y) ^ n ≠ 0 hg' : ∀ y ∈ Ioo x₀ x, -(↑n + 1) * (x - y) ^ n ≠ 0 y : ℝ hy : y ∈ Ioo x₀ x h : f x - taylorWithinEval f n (Icc x₀ x) x₀ x = (-((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) • iteratedDerivWithin (n + 1) f (Icc x₀ x) y ⊢ ((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1) / ((↑n + 1) * (x - y) ^ n)) • iteratedDerivWithin (n + 1) f (Icc x₀ x) y = iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (x - x₀) ^ (n + 1) / ↑(n + 1)!

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg]

field_simp [xy_ne y hy, Nat.factorial]

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg]

Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)!

Mathlib_Analysis_Calculus_Taylor

case right 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x) xy_ne : ∀ y ∈ Ioo x₀ x, (x - y) ^ n ≠ 0 hg' : ∀ y ∈ Ioo x₀ x, -(↑n + 1) * (x - y) ^ n ≠ 0 y : ℝ hy : y ∈ Ioo x₀ x h : f x - taylorWithinEval f n (Icc x₀ x) x₀ x = (-((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) • iteratedDerivWithin (n + 1) f (Icc x₀ x) y ⊢ (x - y) ^ n * (x - x₀) ^ (n + 1) * iteratedDerivWithin (n + 1) f (Icc x₀ x) y * ((↑n + 1) * ↑n !) = iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (x - x₀) ^ (n + 1) * (↑n ! * ((↑n + 1) * (x - y) ^ n))

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial];

ring

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial];

Mathlib.Analysis.Calculus.Taylor.258_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)!

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) ⊢ ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / ↑n ! * (x - x₀)

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by

have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity)

/-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by

Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀)

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) ⊢ Continuous id

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by

continuity

/-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by

Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀)

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn id (Icc x₀ x) ⊢ ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / ↑n ! * (x - x₀)

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity)

have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _

/-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity)

Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀)

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn id (Icc x₀ x) gdiff : ∀ x_1 ∈ Ioo x₀ x, HasDerivAt id ((fun x => 1) x_1) x_1 ⊢ ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / ↑n ! * (x - x₀)

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id

rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩

/-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id

Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀)

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn id (Icc x₀ x) gdiff : ∀ x_1 ∈ Ioo x₀ x, HasDerivAt id ((fun x => 1) x_1) x_1 x✝¹ : ℝ x✝ : x✝¹ ∈ Ioo x₀ x ⊢ (fun x => 1) x✝¹ ≠ 0

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by

simp

/-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by

Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀)

Mathlib_Analysis_Calculus_Taylor

case intro.intro 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn id (Icc x₀ x) gdiff : ∀ x_1 ∈ Ioo x₀ x, HasDerivAt id ((fun x => 1) x_1) x_1 y : ℝ hy : y ∈ Ioo x₀ x h : f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - y) ^ n / ↑n ! * (id x - id x₀) / (fun x => 1) y) • iteratedDerivWithin (n + 1) f (Icc x₀ x) y ⊢ ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / ↑n ! * (x - x₀)

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩

use y, hy

/-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩

Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀)

Mathlib_Analysis_Calculus_Taylor

case right 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn id (Icc x₀ x) gdiff : ∀ x_1 ∈ Ioo x₀ x, HasDerivAt id ((fun x => 1) x_1) x_1 y : ℝ hy : y ∈ Ioo x₀ x h : f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - y) ^ n / ↑n ! * (id x - id x₀) / (fun x => 1) y) • iteratedDerivWithin (n + 1) f (Icc x₀ x) y ⊢ f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (x - y) ^ n / ↑n ! * (x - x₀)

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩ use y, hy

rw [h]

/-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩ use y, hy

Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀)

Mathlib_Analysis_Calculus_Taylor

case right 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn id (Icc x₀ x) gdiff : ∀ x_1 ∈ Ioo x₀ x, HasDerivAt id ((fun x => 1) x_1) x_1 y : ℝ hy : y ∈ Ioo x₀ x h : f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - y) ^ n / ↑n ! * (id x - id x₀) / (fun x => 1) y) • iteratedDerivWithin (n + 1) f (Icc x₀ x) y ⊢ ((x - y) ^ n / ↑n ! * (id x - id x₀) / (fun x => 1) y) • iteratedDerivWithin (n + 1) f (Icc x₀ x) y = iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (x - y) ^ n / ↑n ! * (x - x₀)

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩ use y, hy rw [h]

field_simp [n.factorial_ne_zero]

/-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩ use y, hy rw [h]

Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀)

Mathlib_Analysis_Calculus_Taylor

case right 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → ℝ x x₀ : ℝ n : ℕ hx : x₀ < x hf : ContDiffOn ℝ (↑n) f (Icc x₀ x) hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x) gcont : ContinuousOn id (Icc x₀ x) gdiff : ∀ x_1 ∈ Ioo x₀ x, HasDerivAt id ((fun x => 1) x_1) x_1 y : ℝ hy : y ∈ Ioo x₀ x h : f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - y) ^ n / ↑n ! * (id x - id x₀) / (fun x => 1) y) • iteratedDerivWithin (n + 1) f (Icc x₀ x) y ⊢ (x - y) ^ n * (x - x₀) * iteratedDerivWithin (n + 1) f (Icc x₀ x) y = iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (x - y) ^ n * (x - x₀)

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩ use y, hy rw [h] field_simp [n.factorial_ne_zero]

ring

/-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩ use y, hy rw [h] field_simp [n.factorial_ne_zero]

Mathlib.Analysis.Calculus.Taylor.290_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀)

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E a b C x : ℝ n : ℕ hab : a ≤ b hf : ContDiffOn ℝ (↑n + 1) f (Icc a b) hx : x ∈ Icc a b hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C ⊢ ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / ↑n !

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩ use y, hy rw [h] field_simp [n.factorial_ne_zero] ring #align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy /-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by

rcases eq_or_lt_of_le hab with (rfl | h)

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by

Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n !

Mathlib_Analysis_Calculus_Taylor

case inl 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E a C x : ℝ n : ℕ hab : a ≤ a hf : ContDiffOn ℝ (↑n + 1) f (Icc a a) hx : x ∈ Icc a a hC : ∀ y ∈ Icc a a, ‖iteratedDerivWithin (n + 1) f (Icc a a) y‖ ≤ C ⊢ ‖f x - taylorWithinEval f n (Icc a a) a x‖ ≤ C * (x - a) ^ (n + 1) / ↑n !

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩ use y, hy rw [h] field_simp [n.factorial_ne_zero] ring #align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy /-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) ·

rw [Icc_self, mem_singleton_iff] at hx

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) ·

Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n !

Mathlib_Analysis_Calculus_Taylor

case inl 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E a C x : ℝ n : ℕ hab : a ≤ a hf : ContDiffOn ℝ (↑n + 1) f (Icc a a) hx : x = a hC : ∀ y ∈ Icc a a, ‖iteratedDerivWithin (n + 1) f (Icc a a) y‖ ≤ C ⊢ ‖f x - taylorWithinEval f n (Icc a a) a x‖ ≤ C * (x - a) ^ (n + 1) / ↑n !

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩ use y, hy rw [h] field_simp [n.factorial_ne_zero] ring #align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy /-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx

simp [hx]

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx

Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n !

Mathlib_Analysis_Calculus_Taylor

case inr 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E a b C x : ℝ n : ℕ hab : a ≤ b hf : ContDiffOn ℝ (↑n + 1) f (Icc a b) hx : x ∈ Icc a b hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C h : a < b ⊢ ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / ↑n !

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩ use y, hy rw [h] field_simp [n.factorial_ne_zero] ring #align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy /-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable

have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h)

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable

Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n !

Mathlib_Analysis_Calculus_Taylor

case inr 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E a b C x : ℝ n : ℕ hab : a ≤ b hf : ContDiffOn ℝ (↑n + 1) f (Icc a b) hx : x ∈ Icc a b hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C h : a < b hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) ⊢ ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / ↑n !

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩ use y, hy rw [h] field_simp [n.factorial_ne_zero] ring #align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy /-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial

have h' : ∀ y ∈ Ico a x, ‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (n ! : ℝ)⁻¹ * |x - a| ^ n * C := by rintro y ⟨hay, hyx⟩ rw [norm_smul, Real.norm_eq_abs] gcongr · rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast] gcongr rw [abs_of_nonneg, abs_of_nonneg] <;> linarith -- Estimate the iterated derivative by `C` · exact hC y ⟨hay, hyx.le.trans hx.2⟩

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial

Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n !

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E a b C x : ℝ n : ℕ hab : a ≤ b hf : ContDiffOn ℝ (↑n + 1) f (Icc a b) hx : x ∈ Icc a b hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C h : a < b hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) ⊢ ∀ y ∈ Ico a x, ‖((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (↑n !)⁻¹ * |x - a| ^ n * C

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩ use y, hy rw [h] field_simp [n.factorial_ne_zero] ring #align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy /-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial have h' : ∀ y ∈ Ico a x, ‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (n ! : ℝ)⁻¹ * |x - a| ^ n * C := by

rintro y ⟨hay, hyx⟩

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial have h' : ∀ y ∈ Ico a x, ‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (n ! : ℝ)⁻¹ * |x - a| ^ n * C := by

Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n !

Mathlib_Analysis_Calculus_Taylor

case intro 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E a b C x : ℝ n : ℕ hab : a ≤ b hf : ContDiffOn ℝ (↑n + 1) f (Icc a b) hx : x ∈ Icc a b hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C h : a < b hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) y : ℝ hay : a ≤ y hyx : y < x ⊢ ‖((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (↑n !)⁻¹ * |x - a| ^ n * C

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩ use y, hy rw [h] field_simp [n.factorial_ne_zero] ring #align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy /-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial have h' : ∀ y ∈ Ico a x, ‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (n ! : ℝ)⁻¹ * |x - a| ^ n * C := by rintro y ⟨hay, hyx⟩

rw [norm_smul, Real.norm_eq_abs]

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial have h' : ∀ y ∈ Ico a x, ‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (n ! : ℝ)⁻¹ * |x - a| ^ n * C := by rintro y ⟨hay, hyx⟩

Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n !

Mathlib_Analysis_Calculus_Taylor

case intro 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E a b C x : ℝ n : ℕ hab : a ≤ b hf : ContDiffOn ℝ (↑n + 1) f (Icc a b) hx : x ∈ Icc a b hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C h : a < b hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) y : ℝ hay : a ≤ y hyx : y < x ⊢ |(↑n !)⁻¹ * (x - y) ^ n| * ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (↑n !)⁻¹ * |x - a| ^ n * C

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩ use y, hy rw [h] field_simp [n.factorial_ne_zero] ring #align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy /-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial have h' : ∀ y ∈ Ico a x, ‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (n ! : ℝ)⁻¹ * |x - a| ^ n * C := by rintro y ⟨hay, hyx⟩ rw [norm_smul, Real.norm_eq_abs]

gcongr

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial have h' : ∀ y ∈ Ico a x, ‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (n ! : ℝ)⁻¹ * |x - a| ^ n * C := by rintro y ⟨hay, hyx⟩ rw [norm_smul, Real.norm_eq_abs]

Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n !

Mathlib_Analysis_Calculus_Taylor

case intro.h₁ 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E a b C x : ℝ n : ℕ hab : a ≤ b hf : ContDiffOn ℝ (↑n + 1) f (Icc a b) hx : x ∈ Icc a b hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C h : a < b hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) y : ℝ hay : a ≤ y hyx : y < x ⊢ |(↑n !)⁻¹ * (x - y) ^ n| ≤ (↑n !)⁻¹ * |x - a| ^ n

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩ use y, hy rw [h] field_simp [n.factorial_ne_zero] ring #align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy /-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial have h' : ∀ y ∈ Ico a x, ‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (n ! : ℝ)⁻¹ * |x - a| ^ n * C := by rintro y ⟨hay, hyx⟩ rw [norm_smul, Real.norm_eq_abs] gcongr ·

rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast]

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial have h' : ∀ y ∈ Ico a x, ‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (n ! : ℝ)⁻¹ * |x - a| ^ n * C := by rintro y ⟨hay, hyx⟩ rw [norm_smul, Real.norm_eq_abs] gcongr ·

Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n !

Mathlib_Analysis_Calculus_Taylor

case intro.h₁ 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E a b C x : ℝ n : ℕ hab : a ≤ b hf : ContDiffOn ℝ (↑n + 1) f (Icc a b) hx : x ∈ Icc a b hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C h : a < b hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) y : ℝ hay : a ≤ y hyx : y < x ⊢ (↑n !)⁻¹ * |x - y| ^ n ≤ (↑n !)⁻¹ * |x - a| ^ n

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩ use y, hy rw [h] field_simp [n.factorial_ne_zero] ring #align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy /-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial have h' : ∀ y ∈ Ico a x, ‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (n ! : ℝ)⁻¹ * |x - a| ^ n * C := by rintro y ⟨hay, hyx⟩ rw [norm_smul, Real.norm_eq_abs] gcongr · rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast]

gcongr

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial have h' : ∀ y ∈ Ico a x, ‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (n ! : ℝ)⁻¹ * |x - a| ^ n * C := by rintro y ⟨hay, hyx⟩ rw [norm_smul, Real.norm_eq_abs] gcongr · rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast]

Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n !

Mathlib_Analysis_Calculus_Taylor

case intro.h₁.h.hab 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E a b C x : ℝ n : ℕ hab : a ≤ b hf : ContDiffOn ℝ (↑n + 1) f (Icc a b) hx : x ∈ Icc a b hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C h : a < b hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) y : ℝ hay : a ≤ y hyx : y < x ⊢ |x - y| ≤ |x - a|

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩ use y, hy rw [h] field_simp [n.factorial_ne_zero] ring #align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy /-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial have h' : ∀ y ∈ Ico a x, ‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (n ! : ℝ)⁻¹ * |x - a| ^ n * C := by rintro y ⟨hay, hyx⟩ rw [norm_smul, Real.norm_eq_abs] gcongr · rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast] gcongr

rw [abs_of_nonneg, abs_of_nonneg]

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial have h' : ∀ y ∈ Ico a x, ‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (n ! : ℝ)⁻¹ * |x - a| ^ n * C := by rintro y ⟨hay, hyx⟩ rw [norm_smul, Real.norm_eq_abs] gcongr · rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast] gcongr

Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n !

Mathlib_Analysis_Calculus_Taylor

case intro.h₁.h.hab 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E a b C x : ℝ n : ℕ hab : a ≤ b hf : ContDiffOn ℝ (↑n + 1) f (Icc a b) hx : x ∈ Icc a b hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C h : a < b hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) y : ℝ hay : a ≤ y hyx : y < x ⊢ x - y ≤ x - a

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩ use y, hy rw [h] field_simp [n.factorial_ne_zero] ring #align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy /-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial have h' : ∀ y ∈ Ico a x, ‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (n ! : ℝ)⁻¹ * |x - a| ^ n * C := by rintro y ⟨hay, hyx⟩ rw [norm_smul, Real.norm_eq_abs] gcongr · rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast] gcongr rw [abs_of_nonneg, abs_of_nonneg] <;>

linarith

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial have h' : ∀ y ∈ Ico a x, ‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (n ! : ℝ)⁻¹ * |x - a| ^ n * C := by rintro y ⟨hay, hyx⟩ rw [norm_smul, Real.norm_eq_abs] gcongr · rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast] gcongr rw [abs_of_nonneg, abs_of_nonneg] <;>

Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n !

Mathlib_Analysis_Calculus_Taylor

case intro.h₁.h.hab 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E a b C x : ℝ n : ℕ hab : a ≤ b hf : ContDiffOn ℝ (↑n + 1) f (Icc a b) hx : x ∈ Icc a b hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C h : a < b hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) y : ℝ hay : a ≤ y hyx : y < x ⊢ 0 ≤ x - a

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩ use y, hy rw [h] field_simp [n.factorial_ne_zero] ring #align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy /-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial have h' : ∀ y ∈ Ico a x, ‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (n ! : ℝ)⁻¹ * |x - a| ^ n * C := by rintro y ⟨hay, hyx⟩ rw [norm_smul, Real.norm_eq_abs] gcongr · rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast] gcongr rw [abs_of_nonneg, abs_of_nonneg] <;>

linarith

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial have h' : ∀ y ∈ Ico a x, ‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (n ! : ℝ)⁻¹ * |x - a| ^ n * C := by rintro y ⟨hay, hyx⟩ rw [norm_smul, Real.norm_eq_abs] gcongr · rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast] gcongr rw [abs_of_nonneg, abs_of_nonneg] <;>

Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n !

Mathlib_Analysis_Calculus_Taylor

case intro.h₁.h.hab 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E a b C x : ℝ n : ℕ hab : a ≤ b hf : ContDiffOn ℝ (↑n + 1) f (Icc a b) hx : x ∈ Icc a b hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C h : a < b hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) y : ℝ hay : a ≤ y hyx : y < x ⊢ 0 ≤ x - y

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩ use y, hy rw [h] field_simp [n.factorial_ne_zero] ring #align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy /-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial have h' : ∀ y ∈ Ico a x, ‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (n ! : ℝ)⁻¹ * |x - a| ^ n * C := by rintro y ⟨hay, hyx⟩ rw [norm_smul, Real.norm_eq_abs] gcongr · rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast] gcongr rw [abs_of_nonneg, abs_of_nonneg] <;>

linarith

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial have h' : ∀ y ∈ Ico a x, ‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (n ! : ℝ)⁻¹ * |x - a| ^ n * C := by rintro y ⟨hay, hyx⟩ rw [norm_smul, Real.norm_eq_abs] gcongr · rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast] gcongr rw [abs_of_nonneg, abs_of_nonneg] <;>

Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n !

Mathlib_Analysis_Calculus_Taylor

case intro.h₂ 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E a b C x : ℝ n : ℕ hab : a ≤ b hf : ContDiffOn ℝ (↑n + 1) f (Icc a b) hx : x ∈ Icc a b hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C h : a < b hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) y : ℝ hay : a ≤ y hyx : y < x ⊢ ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩ use y, hy rw [h] field_simp [n.factorial_ne_zero] ring #align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy /-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial have h' : ∀ y ∈ Ico a x, ‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (n ! : ℝ)⁻¹ * |x - a| ^ n * C := by rintro y ⟨hay, hyx⟩ rw [norm_smul, Real.norm_eq_abs] gcongr · rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast] gcongr rw [abs_of_nonneg, abs_of_nonneg] <;> linarith -- Estimate the iterated derivative by `C` ·

exact hC y ⟨hay, hyx.le.trans hx.2⟩

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial have h' : ∀ y ∈ Ico a x, ‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (n ! : ℝ)⁻¹ * |x - a| ^ n * C := by rintro y ⟨hay, hyx⟩ rw [norm_smul, Real.norm_eq_abs] gcongr · rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast] gcongr rw [abs_of_nonneg, abs_of_nonneg] <;> linarith -- Estimate the iterated derivative by `C` ·

Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n !

Mathlib_Analysis_Calculus_Taylor

case inr 𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E a b C x : ℝ n : ℕ hab : a ≤ b hf : ContDiffOn ℝ (↑n + 1) f (Icc a b) hx : x ∈ Icc a b hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C h : a < b hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) h' : ∀ y ∈ Ico a x, ‖((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (↑n !)⁻¹ * |x - a| ^ n * C ⊢ ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / ↑n !

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩ use y, hy rw [h] field_simp [n.factorial_ne_zero] ring #align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy /-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial have h' : ∀ y ∈ Ico a x, ‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (n ! : ℝ)⁻¹ * |x - a| ^ n * C := by rintro y ⟨hay, hyx⟩ rw [norm_smul, Real.norm_eq_abs] gcongr · rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast] gcongr rw [abs_of_nonneg, abs_of_nonneg] <;> linarith -- Estimate the iterated derivative by `C` · exact hC y ⟨hay, hyx.le.trans hx.2⟩ -- Apply the mean value theorem for vector valued functions:

have A : ∀ t ∈ Icc a x, HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((↑n !)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a x) t := by intro t ht have I : Icc a x ⊆ Icc a b := Icc_subset_Icc_right hx.2 exact (has_deriv_within_taylorWithinEval_at_Icc x h (I ht) hf.of_succ hf').mono I

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial have h' : ∀ y ∈ Ico a x, ‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (n ! : ℝ)⁻¹ * |x - a| ^ n * C := by rintro y ⟨hay, hyx⟩ rw [norm_smul, Real.norm_eq_abs] gcongr · rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast] gcongr rw [abs_of_nonneg, abs_of_nonneg] <;> linarith -- Estimate the iterated derivative by `C` · exact hC y ⟨hay, hyx.le.trans hx.2⟩ -- Apply the mean value theorem for vector valued functions:

Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n !

Mathlib_Analysis_Calculus_Taylor

𝕜 : Type u_1 E : Type u_2 F : Type u_3 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace ℝ E f : ℝ → E a b C x : ℝ n : ℕ hab : a ≤ b hf : ContDiffOn ℝ (↑n + 1) f (Icc a b) hx : x ∈ Icc a b hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C h : a < b hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) h' : ∀ y ∈ Ico a x, ‖((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (↑n !)⁻¹ * |x - a| ^ n * C ⊢ ∀ t ∈ Icc a x, HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((↑n !)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a x) t

/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import Mathlib.Analysis.Calculus.Deriv.Pow import Mathlib.Analysis.Calculus.IteratedDeriv import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Data.Polynomial.Module #align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14" /-! # Taylor's theorem This file defines the Taylor polynomial of a real function `f : ℝ → E`, where `E` is a normed vector space over `ℝ` and proves Taylor's theorem, which states that if `f` is sufficiently smooth, then `f` can be approximated by the Taylor polynomial up to an explicit error term. ## Main definitions * `taylorCoeffWithin`: the Taylor coefficient using `iteratedDerivWithin` * `taylorWithin`: the Taylor polynomial using `iteratedDerivWithin` ## Main statements * `taylor_mean_remainder`: Taylor's theorem with the general form of the remainder term * `taylor_mean_remainder_lagrange`: Taylor's theorem with the Lagrange remainder * `taylor_mean_remainder_cauchy`: Taylor's theorem with the Cauchy remainder * `exists_taylor_mean_remainder_bound`: Taylor's theorem for vector valued functions with a polynomial bound on the remainder ## TODO * the Peano form of the remainder * the integral form of the remainder * Generalization to higher dimensions ## Tags Taylor polynomial, Taylor's theorem -/ open scoped BigOperators Interval Topology Nat open Set variable {𝕜 E F : Type*} variable [NormedAddCommGroup E] [NormedSpace ℝ E] /-- The `k`th coefficient of the Taylor polynomial. -/ noncomputable def taylorCoeffWithin (f : ℝ → E) (k : ℕ) (s : Set ℝ) (x₀ : ℝ) : E := (k ! : ℝ)⁻¹ • iteratedDerivWithin k f s x₀ #align taylor_coeff_within taylorCoeffWithin /-- The Taylor polynomial with derivatives inside of a set `s`. The Taylor polynomial is given by $$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$ where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. -/ noncomputable def taylorWithin (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : PolynomialModule ℝ E := (Finset.range (n + 1)).sum fun k => PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ k (taylorCoeffWithin f k s x₀)) #align taylor_within taylorWithin /-- The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E`-/ noncomputable def taylorWithinEval (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : E := PolynomialModule.eval x (taylorWithin f n s x₀) #align taylor_within_eval taylorWithinEval theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ + PolynomialModule.comp (Polynomial.X - Polynomial.C x₀) (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by dsimp only [taylorWithin] rw [Finset.sum_range_succ] #align taylor_within_succ taylorWithin_succ @[simp] theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x + (((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval] congr simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, PolynomialModule.eval_single, mul_inv_rev] dsimp only [taylorCoeffWithin] rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one, mul_inv_rev] #align taylor_within_eval_succ taylorWithinEval_succ /-- The Taylor polynomial of order zero evaluates to `f x`. -/ @[simp] theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f 0 s x₀ x = f x₀ := by dsimp only [taylorWithinEval] dsimp only [taylorWithin] dsimp only [taylorCoeffWithin] simp #align taylor_within_zero_eval taylor_within_zero_eval /-- Evaluating the Taylor polynomial at `x = x₀` yields `f x`. -/ @[simp] theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) : taylorWithinEval f n s x₀ x₀ = f x₀ := by induction' n with k hk · exact taylor_within_zero_eval _ _ _ _ simp [hk] #align taylor_within_eval_self taylorWithinEval_self theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) : taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by induction' n with k hk · simp rw [taylorWithinEval_succ, Finset.sum_range_succ, hk] simp [Nat.factorial] #align taylor_within_apply taylor_within_apply /-- If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial `taylorWithinEval f n s x₀ x` is continuous in `x₀`. -/ theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ} (hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) : ContinuousOn (fun t => taylorWithinEval f n s t x) s := by simp_rw [taylor_within_apply] refine' continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => _ refine' (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul _ rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf cases' hf with hf_left specialize hf_left i simp only [Finset.mem_range] at hi refine' hf_left _ simp only [WithTop.coe_le_coe, Nat.cast_le, Nat.lt_succ_iff.mp hi] #align continuous_on_taylor_within_eval continuousOn_taylorWithinEval /-- Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions.-/ theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) : HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by simp_rw [sub_eq_neg_add] rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x) simp only [Nat.cast_add, Nat.cast_one] #align monomial_has_deriv_aux monomial_has_deriv_aux theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ} (ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y) (hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) : HasDerivWithinAt (fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z) ((((k + 1 : ℝ) * k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y - ((k ! : ℝ)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y) t y := by replace hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by convert (hf.mono_of_mem hs).hasDerivWithinAt using 1 rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))] exact (derivWithin_of_mem hs ht hf).symm have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1)) (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by -- Commuting the factors: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by field_simp; ring rw [this] exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _ convert this.smul hf using 1 field_simp rw [neg_div, neg_smul, sub_eq_add_neg] #align has_deriv_within_at_taylor_coeff_within hasDerivWithinAt_taylor_coeff_within /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for arbitrary sets -/ theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ} (hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s) (hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) : HasDerivWithinAt (fun t => taylorWithinEval f n s t x) (((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f s y) s' y := by induction' n with k hk · simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero, mul_one, zero_add, one_smul] simp only [iteratedDerivWithin_zero] at hf' rw [iteratedDerivWithin_one (hs_unique _ (h hy))] norm_num exact hf'.hasDerivWithinAt.mono h simp_rw [Nat.add_succ, taylorWithinEval_succ] simp only [add_zero, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] have coe_lt_succ : (k : WithTop ℕ) < k.succ := Nat.cast_lt.2 k.lt_succ_self have hdiff : DifferentiableOn ℝ (iteratedDerivWithin k f s) s' := (hf.differentiableOn_iteratedDerivWithin coe_lt_succ hs_unique).mono h specialize hk hf.of_succ ((hdiff y hy).mono_of_mem hs') convert hk.add (hasDerivWithinAt_taylor_coeff_within hs'_unique (nhdsWithin_mono _ h self_mem_nhdsWithin) hf') using 1 exact (add_sub_cancel'_right _ _).symm #align has_deriv_within_at_taylor_within_eval hasDerivWithinAt_taylorWithinEval /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for open intervals -/ theorem taylorWithinEval_hasDerivAt_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Ioo a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)) : HasDerivAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) t := have h_nhds : Ioo a b ∈ 𝓝 t := isOpen_Ioo.mem_nhds ht have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhdsWithin_le_nhds h_nhds (hasDerivWithinAt_taylorWithinEval (uniqueDiffWithinAt_Ioo ht) (uniqueDiffOn_Icc hx) h_nhds' ht Ioo_subset_Icc_self hf <| (hf' t ht).mono_of_mem h_nhds').hasDerivAt h_nhds #align taylor_within_eval_has_deriv_at_Ioo taylorWithinEval_hasDerivAt_Ioo /-- Calculate the derivative of the Taylor polynomial with respect to `x₀`. Version for closed intervals -/ theorem has_deriv_within_taylorWithinEval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ} (hx : a < b) (ht : t ∈ Icc a b) (hf : ContDiffOn ℝ n f (Icc a b)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b)) : HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a b) t := hasDerivWithinAt_taylorWithinEval (uniqueDiffOn_Icc hx t ht) (uniqueDiffOn_Icc hx) self_mem_nhdsWithin ht rfl.subset hf (hf' t ht) #align has_deriv_within_taylor_within_eval_at_Icc has_deriv_within_taylorWithinEval_at_Icc /-! ### Taylor's theorem with mean value type remainder estimate -/ /-- **Taylor's theorem** with the general mean value form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on `Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{(x - x')^n}{n!} \frac{g(x) - g(x₀)}{g' x'},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$. -/ theorem taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) (gcont : ContinuousOn g (Icc x₀ x)) (gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt g (g' x_1) x_1) (g'_ne : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → g' x_1 ≠ 0) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = ((x - x') ^ n / n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' := by -- We apply the mean value theorem rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x₀ x) t x) (fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x₀ x) t) hx (continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx) hf) (fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx hy hf hf') g g' gcont gdiff with ⟨y, hy, h⟩ use y, hy -- The rest is simplifications and trivial calculations simp only [taylorWithinEval_self] at h rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h rw [← h] field_simp [g'_ne y hy] ring #align taylor_mean_remainder taylor_mean_remainder /-- **Taylor's theorem** with the Lagrange form of the remainder. We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}{(n+1)!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)! := by have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by refine' Continuous.continuousOn _ exact (continuous_const.sub continuous_id').pow _ -- Porting note: was `continuity` have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by intro y hy refine' pow_ne_zero _ _ rw [mem_Ioo] at hy rw [sub_ne_zero] exact hy.2.ne' have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy => mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) -- We apply the general theorem with g(t) = (x - t)^(n+1) rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with ⟨y, hy, h⟩ use y, hy simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h rw [h, neg_div, ← div_neg, neg_mul, neg_neg] field_simp [xy_ne y hy, Nat.factorial]; ring #align taylor_mean_remainder_lagrange taylor_mean_remainder_lagrange /-- **Taylor's theorem** with the Cauchy form of the remainder. We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and `n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists an `x' ∈ Ioo x₀ x` such that $$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)}{n!},$$ where $P_n f$ denotes the Taylor polynomial of degree $n$ and $f^{(n+1)}$ is the $n+1$-th iterated derivative. -/ theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / n ! * (x - x₀) := by have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity) have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 := fun _ _ => hasDerivAt_id _ -- We apply the general theorem with g = id rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => by simp with ⟨y, hy, h⟩ use y, hy rw [h] field_simp [n.factorial_ne_zero] ring #align taylor_mean_remainder_cauchy taylor_mean_remainder_cauchy /-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial have h' : ∀ y ∈ Ico a x, ‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (n ! : ℝ)⁻¹ * |x - a| ^ n * C := by rintro y ⟨hay, hyx⟩ rw [norm_smul, Real.norm_eq_abs] gcongr · rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast] gcongr rw [abs_of_nonneg, abs_of_nonneg] <;> linarith -- Estimate the iterated derivative by `C` · exact hC y ⟨hay, hyx.le.trans hx.2⟩ -- Apply the mean value theorem for vector valued functions: have A : ∀ t ∈ Icc a x, HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((↑n !)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a x) t := by

intro t ht

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n ! := by rcases eq_or_lt_of_le hab with (rfl | h) · rw [Icc_self, mem_singleton_iff] at hx simp [hx] -- The nth iterated derivative is differentiable have hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Icc a b) := hf.differentiableOn_iteratedDerivWithin (WithTop.coe_lt_coe.mpr n.lt_succ_self) (uniqueDiffOn_Icc h) -- We can uniformly bound the derivative of the Taylor polynomial have h' : ∀ y ∈ Ico a x, ‖((n ! : ℝ)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ (n ! : ℝ)⁻¹ * |x - a| ^ n * C := by rintro y ⟨hay, hyx⟩ rw [norm_smul, Real.norm_eq_abs] gcongr · rw [abs_mul, abs_pow, abs_inv, Nat.abs_cast] gcongr rw [abs_of_nonneg, abs_of_nonneg] <;> linarith -- Estimate the iterated derivative by `C` · exact hC y ⟨hay, hyx.le.trans hx.2⟩ -- Apply the mean value theorem for vector valued functions: have A : ∀ t ∈ Icc a x, HasDerivWithinAt (fun y => taylorWithinEval f n (Icc a b) y x) (((↑n !)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) (Icc a x) t := by

Mathlib.Analysis.Calculus.Taylor.313_0.INXnr4jrmq9RIjK

/-- **Taylor's theorem** with a polynomial bound on the remainder We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`. The difference of `f` and its `n`-th Taylor polynomial can be estimated by `C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. -/ theorem taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ} (hab : a ≤ b) (hf : ContDiffOn ℝ (n + 1) f (Icc a b)) (hx : x ∈ Icc a b) (hC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C) : ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1) / n !

Mathlib_Analysis_Calculus_Taylor