raw/raw_bsc__mathematical_analysis_i.f22.md

BSc: Mathematical Analysis I.f22
================================






Contents
--------


* [1 Mathematical Analysis I](#Mathematical_Analysis_I)
	+ [1.1 Short Description](#Short_Description)
	+ [1.2 Course Topics](#Course_Topics)
	+ [1.3 Intended Learning Outcomes (ILOs)](#Intended_Learning_Outcomes_.28ILOs.29)
		- [1.3.1 What is the main purpose of this course?](#What_is_the_main_purpose_of_this_course.3F)
		- [1.3.2 ILOs defined at three levels](#ILOs_defined_at_three_levels)
			* [1.3.2.1 Level 1: What concepts should a student know/remember/explain?](#Level_1:_What_concepts_should_a_student_know.2Fremember.2Fexplain.3F)
			* [1.3.2.2 Level 2: What basic practical skills should a student be able to perform?](#Level_2:_What_basic_practical_skills_should_a_student_be_able_to_perform.3F)
			* [1.3.2.3 Level 3: What complex comprehensive skills should a student be able to apply in real-life scenarios?](#Level_3:_What_complex_comprehensive_skills_should_a_student_be_able_to_apply_in_real-life_scenarios.3F)
	+ [1.4 Grading](#Grading)
		- [1.4.1 Course grading range](#Course_grading_range)
		- [1.4.2 Course activities and grading breakdown](#Course_activities_and_grading_breakdown)
		- [1.4.3 Recommendations for students on how to succeed in the course](#Recommendations_for_students_on_how_to_succeed_in_the_course)
	+ [1.5 Resources, literature and reference materials](#Resources.2C_literature_and_reference_materials)
		- [1.5.1 Open access resources](#Open_access_resources)
* [2 Teaching Methodology: Methods, techniques, & activities](#Teaching_Methodology:_Methods.2C_techniques.2C_.26_activities)
	+ [2.1 Activities and Teaching Methods](#Activities_and_Teaching_Methods)
	+ [2.2 Formative Assessment and Course Activities](#Formative_Assessment_and_Course_Activities)
		- [2.2.1 Ongoing performance assessment](#Ongoing_performance_assessment)
			* [2.2.1.1 Section 1](#Section_1)
			* [2.2.1.2 Section 2](#Section_2)
			* [2.2.1.3 Section 3](#Section_3)
		- [2.2.2 Final assessment](#Final_assessment)
			* [2.2.2.1 Section 1](#Section_1_2)
			* [2.2.2.2 Section 2](#Section_2_2)
			* [2.2.2.3 Section 3](#Section_3_2)
		- [2.2.3 The retake exam](#The_retake_exam)



Mathematical Analysis I
=======================


* **Course name**: Mathematical Analysis I
* **Code discipline**: CSE201
* **Subject area**: Math


Short Description
-----------------


This calculus course covers differentiation and integration of functions of one variable, with applications. The basic objective of Calculus is to relate small-scale (differential) quantities to large-scale (integrated) quantities. This is accomplished by means of the Fundamental Theorem of Calculus. Should be understanding of the integral as a cumulative sum, of the derivative as a rate of change, and of the inverse relationship between integration and differentiation.



Course Topics
-------------




Course Sections and Topics
| Section | Topics within the section
 |
| --- | --- |
| Limits | 1. Limits of Sequences
2. Newton's Method
3. Limits of Functions
 |
| Derivatives | 1. Derivative as a Limit
2. Leibniz Notation
3. Rates of Change
4. The Chain Rule
5. Fractional Powers and Implicit Differentiation
6. Related Rates and Parametric Curves
7. Inverse Functions and Differentiation
8. Differentiation of the Trigonometric, Exponential and Logarithmic Functions
9. L'Hopital's Rule
10. Increasing and Decreasing Functions
11. The Second Derivative and Concavity
12. Maximum-Minimum Problems
13. Graphing
 |
| Integrals | 1. Sums and Areas
2. The Fundamental Theorem of Calculus
3. Definite and Indefinite Integrals
4. Integration by Substitution
5. Changing Variables in the Definite Integral
6. Integration by Parts
7. Trigonometric Integrals
8. Partial Fractions
9. Parametric Curves
10. Applications of the integrals
11. Improper integrals
 |


Intended Learning Outcomes (ILOs)
---------------------------------


### What is the main purpose of this course?


This calculus course will provide an opportunity for participants to:



* understand key principles involved in differentiation and integration of functions
* solve problems that connect small-scale (differential) quantities to large-scale (integrated) quantities
* become familiar with the fundamental theorems of Calculus
* get hands-on experience with the integral and derivative applications and of the inverse relationship between integration and differentiation.


### ILOs defined at three levels


We specify the intended learning outcomes at three levels: conceptual knowledge, practical skills, and comprehensive skills.



#### Level 1: What concepts should a student know/remember/explain?


By the end of the course, the students should be able to ...



* remember the differentiation techniques
* remember the integration techniques
* remember how to work with sequences and series


#### Level 2: What basic practical skills should a student be able to perform?


By the end of the course, the students should be able to ...



* apply the derivatives to analyse the functions
* integrate
* understand the basics of approximation


#### Level 3: What complex comprehensive skills should a student be able to apply in real-life scenarios?


By the end of the course, the students should be able to ...



* Take derivatives of various type functions and of various orders
* Integrate
* Apply definite integral
* Expand functions into Taylor series
* Apply convergence tests


Grading
-------


### Course grading range





| Grade | Range | Description of performance
 |
| --- | --- | --- |
| A. Excellent | 85-100 | -
 |
| B. Good | 70-84 | -
 |
| C. Satisfactory | 50-69 | -
 |
| D. Fail | 0-49 | -
 |


### Course activities and grading breakdown





| Activity Type | Percentage of the overall course grade
 |
| --- | --- |
| Midterm | 20
 |
| Tests | 28 (14 for each)
 |
| Final exam | 50
 |
| In-class participation | 7 (including 5 extras)
 |


### Recommendations for students on how to succeed in the course


* Participation is important. Attending lectures is the key to success in this course.
* Review lecture materials before classes to do well.
* Reading the recommended literature is obligatory, and will give you a deeper understanding of the material.


Resources, literature and reference materials
---------------------------------------------


### Open access resources


* Jerrold E. Marsden and Alan Weinstein, Calculus I, II, and II. Springer-Verlag, Second Edition 1985 [link](https://www.cds.caltech.edu/~marsden/volume/Calculus/)
* Zorich, V. A. Mathematical Analysis I, Translator: Cooke R. (2004)


Teaching Methodology: Methods, techniques, & activities
=======================================================


Activities and Teaching Methods
-------------------------------




Teaching and Learning Methods within each section
| Teaching Techniques | Section 1 | Section 2 | Section 3
 |
| --- | --- | --- | --- |
| Problem-based learning (students learn by solving open-ended problems without a strictly-defined solution) | 1 | 1 | 1
 |
| Project-based learning (students work on a project) | 0 | 0 | 0
 |
| Modular learning (facilitated self-study) | 0 | 0 | 0
 |
| Differentiated learning (provide tasks and activities at several levels of difficulty to fit students needs and level) | 1 | 1 | 1
 |
| Contextual learning (activities and tasks are connected to the real world to make it easier for students to relate to them) | 0 | 0 | 0
 |
| Business game (learn by playing a game that incorporates the principles of the material covered within the course) | 0 | 0 | 0
 |
| Inquiry-based learning | 0 | 0 | 0
 |
| Just-in-time teaching | 0 | 0 | 0
 |
| Process oriented guided inquiry learning (POGIL) | 0 | 0 | 0
 |
| Studio-based learning | 0 | 0 | 0
 |
| Universal design for learning | 0 | 0 | 0
 |
| Task-based learning | 0 | 0 | 0
 |




Activities within each section
| Learning Activities | Section 1 | Section 2 | Section 3
 |
| --- | --- | --- | --- |
| Lectures | 1 | 1 | 1
 |
| Interactive Lectures | 1 | 1 | 1
 |
| Lab exercises | 1 | 1 | 1
 |
| Experiments | 0 | 0 | 0
 |
| Modeling | 0 | 0 | 0
 |
| Cases studies | 0 | 0 | 0
 |
| Development of individual parts of software product code | 0 | 0 | 0
 |
| Individual Projects | 0 | 0 | 0
 |
| Group projects | 0 | 0 | 0
 |
| Flipped classroom | 0 | 0 | 0
 |
| Quizzes (written or computer based) | 1 | 1 | 1
 |
| Peer Review | 0 | 0 | 0
 |
| Discussions | 1 | 1 | 1
 |
| Presentations by students | 0 | 0 | 0
 |
| Written reports | 0 | 0 | 0
 |
| Simulations and role-plays | 0 | 0 | 0
 |
| Essays | 0 | 0 | 0
 |
| Oral Reports | 0 | 0 | 0
 |


Formative Assessment and Course Activities
------------------------------------------


### Ongoing performance assessment


#### Section 1


1. Find limits of the following sequences or prove that they do not exist:
	* a
	
	n
	
	
	=
	n
	−
	
	
	
	n
	
	2
	
	
	−
	70
	n
	+
	1400
	
	
	
	
	{\displaystyle a\_{n}=n-{\sqrt {n^{2}-70n+1400}}}
	
	![{\displaystyle a_{n}=n-{\sqrt {n^{2}-70n+1400}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aeca3ea0fc01bed1f98eb199a9819614e88e793f);
	* d
	
	n
	
	
	=
	
	
	(
	
	
	
	2
	n
	−
	4
	
	
	2
	n
	+
	1
	
	
	
	)
	
	
	n
	
	
	
	
	{\textstyle d\_{n}=\left({\frac {2n-4}{2n+1}}\right)^{n}}
	
	![{\textstyle d_{n}=\left({\frac {2n-4}{2n+1}}\right)^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/389a3735f2899205a00b490e482acbcfa39b3edb);
	* x
	
	n
	
	
	=
	
	
	
	
	
	(
	
	2
	
	n
	
	2
	
	
	+
	1
	
	)
	
	
	6
	
	
	(
	n
	−
	1
	
	)
	
	2
	
	
	
	
	
	(
	
	
	n
	
	7
	
	
	+
	1000
	
	n
	
	6
	
	
	−
	3
	
	)
	
	
	2
	
	
	
	
	
	
	{\textstyle x\_{n}={\frac {\left(2n^{2}+1\right)^{6}(n-1)^{2}}{\left(n^{7}+1000n^{6}-3\right)^{2}}}}
	
	![{\textstyle x_{n}={\frac {\left(2n^{2}+1\right)^{6}(n-1)^{2}}{\left(n^{7}+1000n^{6}-3\right)^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f2e2a41471540c1cb5f3f8b2ad3d96ef91cf7e9).


#### Section 2


1. A plane curve is given by 



x
(
t
)
=
−




t

2


+
4
t
+
8


t
+
2





{\displaystyle x(t)=-{\frac {t^{2}+4t+8}{t+2}}}

![{\displaystyle x(t)=-{\frac {t^{2}+4t+8}{t+2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c87d1bcdd432a16c052b35df5eeafde36d5f1c), 



y
(
t
)
=




t

2


+
9
t
+
22


t
+
6





{\textstyle y(t)={\frac {t^{2}+9t+22}{t+6}}}

![{\textstyle y(t)={\frac {t^{2}+9t+22}{t+6}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5053f85973bbe307983c1751cf8555915e010966). Find
the asymptotes of this curve;
the derivative 




y

x

′



{\textstyle y'\_{x}}

![{\textstyle y'_{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef2ed2d5f61e3172938257665534af01f608117f).
2. Apply Leibniz formula Find 




y

(
n
)


(
x
)


{\textstyle y^{(n)}(x)}

![{\textstyle y^{(n)}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7589798b5f12dd4045984596bdeef45c97ebbb2) if 



y
(
x
)
=

(


x

2


−
2

)

cos
⁡
2
x
sin
⁡
3
x


{\textstyle y(x)=\left(x^{2}-2\right)\cos 2x\sin 3x}

![{\textstyle y(x)=\left(x^{2}-2\right)\cos 2x\sin 3x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec58b46b813170ce96a00bff16a41f464508272a).
Draw graphs of functions
Find asymptotes
3. Find the derivatives of the following functions:
	* f
	(
	x
	)
	=
	
	log
	
	
	|
	
	sin
	⁡
	x
	
	|
	
	
	
	⁡
	
	
	
	
	x
	
	2
	
	
	+
	6
	
	
	6
	
	
	
	
	
	{\textstyle f(x)=\log \_{|\sin x|}{\sqrt[{6}]{x^{2}+6}}}
	
	![{\textstyle f(x)=\log _{|\sin x|}{\sqrt[{6}]{x^{2}+6}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4dd86bbb69d9a98691da7b8c676178d856cbd6f);
	* y
	(
	x
	)
	
	
	{\textstyle y(x)}
	
	![{\textstyle y(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4639e7d86a9d2274f64a48570e7fe4ef17f7efa) that is given implicitly by 
	
	
	
	
	x
	
	3
	
	
	+
	5
	x
	y
	+
	
	y
	
	3
	
	
	=
	0
	
	
	{\textstyle x^{3}+5xy+y^{3}=0}
	
	![{\textstyle x^{3}+5xy+y^{3}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ee197bafd124ae61986c15d077d9183dbcd3cc8).


#### Section 3


1. Find the following integrals:
	* ∫
	
	
	
	
	
	4
	+
	
	x
	
	2
	
	
	
	
	+
	2
	
	
	4
	−
	
	x
	
	2
	
	
	
	
	
	
	16
	−
	
	x
	
	4
	
	
	
	
	
	
	d
	x
	
	
	{\textstyle \int {\frac {{\sqrt {4+x^{2}}}+2{\sqrt {4-x^{2}}}}{\sqrt {16-x^{4}}}}\,dx}
	
	![{\textstyle \int {\frac {{\sqrt {4+x^{2}}}+2{\sqrt {4-x^{2}}}}{\sqrt {16-x^{4}}}}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a336f5054ecdaaf721dd26521ceea178d0538e7f);
	* ∫
	
	2
	
	2
	x
	
	
	
	e
	
	x
	
	
	
	d
	x
	
	
	{\textstyle \int 2^{2x}e^{x}\,dx}
	
	![{\textstyle \int 2^{2x}e^{x}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e670c4f2b17ea6c370f8a531d165d79245d3dc45);
	* ∫
	
	
	
	d
	x
	
	
	3
	
	x
	
	2
	
	
	−
	
	x
	
	4
	
	
	
	
	
	
	
	{\textstyle \int {\frac {dx}{3x^{2}-x^{4}}}}
	
	![{\textstyle \int {\frac {dx}{3x^{2}-x^{4}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfc1a209cc0faf2f924c6f8af0e31d8024eb8769).
2. Find the indefinite integral 




∫
x
ln
⁡

(

x
+



x

2


−
1



)


d
x



{\textstyle \displaystyle \int x\ln \left(x+{\sqrt {x^{2}-1}}\right)\,dx}

![{\textstyle \displaystyle \int x\ln \left(x+{\sqrt {x^{2}-1}}\right)\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cecb2d4bbc0d780bd1a3833c2dcb3a512f8a745d).
3. Find the length of a curve given by 



y
=
ln
⁡
sin
⁡
x


{\textstyle y=\ln \sin x}

![{\textstyle y=\ln \sin x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9f39b93580dbed62ccb3b5560e4e2fa35b8900b), 





π
4


⩽
x
⩽


π
2




{\textstyle {\frac {\pi }{4}}\leqslant x\leqslant {\frac {\pi }{2}}}

![{\textstyle {\frac {\pi }{4}}\leqslant x\leqslant {\frac {\pi }{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a70476c552255c896a29ea67b3eea049324922f0).


### Final assessment


#### Section 1


1. Find a limit of a sequence
2. Find a limit of a function


#### Section 2


1. Apply the appropriate differentiation technique to a given problem.
2. Find a derivative of a function
3. Apply Leibniz formula
4. Draw graphs of functions
5. Find asymptotes of a parametric function


#### Section 3


1. Apply the appropriate integration technique to the given problem
2. Find the value of the devinite integral
3. Calculate the area of the domain or the length of the curve


### The retake exam


Retakes will be run as a comprehensive exam, where the student will be assessed the acquired knowledge coming from the textbooks, the lectures, the labs, and the additional required reading material, as supplied by the instructor. During such comprehensive oral/written the student could be asked to solve exercises and to explain theoretical and practical aspects of the course.