raw/raw_bsc__mathematical_analysis_i.md

BSc: Mathematical Analysis I
============================






  




Contents
--------


* [1 Mathematical Analysis I](#Mathematical_Analysis_I)
	+ [1.1 Course Characteristics](#Course_Characteristics)
		- [1.1.1 Key concepts of the class](#Key_concepts_of_the_class)
		- [1.1.2 What is the purpose of this course?](#What_is_the_purpose_of_this_course.3F)
	+ [1.2 Course Objectives Based on Bloom’s Taxonomy](#Course_Objectives_Based_on_Bloom.E2.80.99s_Taxonomy)
		- [1.2.1 What should a student remember at the end of the course?](#What_should_a_student_remember_at_the_end_of_the_course.3F)
		- [1.2.2 What should a student be able to understand at the end of the course?](#What_should_a_student_be_able_to_understand_at_the_end_of_the_course.3F)
		- [1.2.3 What should a student be able to apply at the end of the course?](#What_should_a_student_be_able_to_apply_at_the_end_of_the_course.3F)
		- [1.2.4 Course evaluation](#Course_evaluation)
		- [1.2.5 Grades range](#Grades_range)
		- [1.2.6 Resources and reference material](#Resources_and_reference_material)
	+ [1.3 Course Sections](#Course_Sections)
		- [1.3.1 Section 1](#Section_1)
			* [1.3.1.1 Section title:](#Section_title:)
		- [1.3.2 Topics covered in this section:](#Topics_covered_in_this_section:)
		- [1.3.3 What forms of evaluation were used to test students’ performance in this section?](#What_forms_of_evaluation_were_used_to_test_students.E2.80.99_performance_in_this_section.3F)
		- [1.3.4 Typical questions for ongoing performance evaluation within this section](#Typical_questions_for_ongoing_performance_evaluation_within_this_section)
		- [1.3.5 Typical questions for seminar classes (labs) within this section](#Typical_questions_for_seminar_classes_.28labs.29_within_this_section)
		- [1.3.6 Test questions for final assessment in this section](#Test_questions_for_final_assessment_in_this_section)
		- [1.3.7 Section 2](#Section_2)
			* [1.3.7.1 Section title:](#Section_title:_2)
		- [1.3.8 Topics covered in this section:](#Topics_covered_in_this_section:_2)
		- [1.3.9 What forms of evaluation were used to test students’ performance in this section?](#What_forms_of_evaluation_were_used_to_test_students.E2.80.99_performance_in_this_section.3F_2)
		- [1.3.10 Typical questions for ongoing performance evaluation within this section](#Typical_questions_for_ongoing_performance_evaluation_within_this_section_2)
		- [1.3.11 Typical questions for seminar classes (labs) within this section](#Typical_questions_for_seminar_classes_.28labs.29_within_this_section_2)
		- [1.3.12 Test questions for final assessment in this section](#Test_questions_for_final_assessment_in_this_section_2)
		- [1.3.13 Section 3](#Section_3)
			* [1.3.13.1 Section title:](#Section_title:_3)
			* [1.3.13.2 Topics covered in this section:](#Topics_covered_in_this_section:_3)
		- [1.3.14 What forms of evaluation were used to test students’ performance in this section?](#What_forms_of_evaluation_were_used_to_test_students.E2.80.99_performance_in_this_section.3F_3)
		- [1.3.15 Typical questions for ongoing performance evaluation within this section](#Typical_questions_for_ongoing_performance_evaluation_within_this_section_3)
			* [1.3.15.1 Typical questions for seminar classes (labs) within this section](#Typical_questions_for_seminar_classes_.28labs.29_within_this_section_3)
			* [1.3.15.2 Test questions for final assessment in this section](#Test_questions_for_final_assessment_in_this_section_3)



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


Course Characteristics
----------------------


### Key concepts of the class


* Differentiation
* Integration
* Series


### What is the purpose of this course?


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.


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.


Course Objectives Based on Bloom’s Taxonomy
-------------------------------------------


### What should a student remember at the end of the course?


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



* Derivative. Differential. Applications
* Indefinite integral. Definite integral. Applications
* Sequences. Series. Convergence. Power Series


### What should a student be able to understand at the end of the course?


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



* Derivative. Differential. Applications
* Indefinite integral. Definite integral. Applications
* Sequences. Series. Convergence. Power Series
* Taylor Series


### What should a student be able to apply at the end of the course?


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


### Course evaluation




Course grade breakdown
|  |  | **Proposed points** |
| --- | --- | --- |
| Labs/seminar classes
 | 20
 |  |
| Interim performance assessment
 | 30
 |  |
| Exams
 | 50
 |  |


If necessary, please indicate freely your course’s features in terms of students’ performance assessment.



### Grades range




Course grading range
|  |  | **Proposed range** |
| --- | --- | --- |
| A. Excellent
 | 90-100
 |  |
| B. Good
 | 75-89
 |  |
| C. Satisfactory
 | 60-74
 |  |
| D. Poor
 | 0-59
 |  |


If necessary, please indicate freely your course’s grading features.



### Resources and reference material


* Zorich, V. A. “Mathematical Analysis I, Translator: Cooke R.” (2004)
* 
* 


Course Sections
---------------


The main sections of the course and approximate hour distribution between them is as follows:





Course Sections
| **Section** | **Section Title** | **Teaching Hours** |
| --- | --- | --- |
| 1
 | Sequences and Limits
 | 28
 |
| 2
 | Differentiation
 | 24
 |
| 3
 | Integration and Series
 | 28
 |


### Section 1


#### Section title:


Sequences and Limits



### Topics covered in this section:


* Sequences. Limits of sequences
* Limits of sequences. Limits of functions
* Limits of functions. Continuity. Hyperbolic functions


### What forms of evaluation were used to test students’ performance in this section?




|  | **Yes/No** |
| --- | --- |
| Development of individual parts of software product code
 | 0
 |
| Homework and group projects
 | 1
 |
| Midterm evaluation
 | 1
 |
| Testing (written or computer based)
 | 1
 |
| Reports
 | 0
 |
| Essays
 | 0
 |
| Oral polls
 | 0
 |
| Discussions
 | 1
 |


### Typical questions for ongoing performance evaluation within this section


1. A sequence, limiting value
2. Limit of a sequence, convergent and divergent sequences
3. Increasing and decreasing sequences, monotonic sequences
4. Bounded sequences. Properties of limits
5. Theorem about bounded and monotonic sequences.
6. Cauchy sequence. The Cauchy Theorem (criterion).
7. Limit of a function. Properties of limits.
8. The first remarkable limit.
9. The Cauchy criterion for the existence of a limit of a function.
10. Second remarkable limit.


### Typical questions for seminar classes (labs) within this section


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


### Test questions for final assessment in this section


1. Find limits of the following sequences or prove that they do not exist:
2. 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);
3. 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);
4. 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


#### Section title:


Differentiation



### Topics covered in this section:


* Derivatives. Differentials
* Mean-Value Theorems
* l’Hopital’s rule
* Taylor Formula with Lagrange and Peano remainders
* Taylor formula and limits
* Increasing / decreasing functions. Concave / convex functions


### What forms of evaluation were used to test students’ performance in this section?




|  | **Yes/No** |
| --- | --- |
| Development of individual parts of software product code
 | 0
 |
| Homework and group projects
 | 1
 |
| Midterm evaluation
 | 1
 |
| Testing (written or computer based)
 | 1
 |
| Reports
 | 0
 |
| Essays
 | 0
 |
| Oral polls
 | 0
 |
| Discussions
 | 1
 |


### Typical questions for ongoing performance evaluation within this section


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
	1. the asymptotes of this curve;
	2. the derivative 
	
	
	
	
	y
	
	x
	
	′
	
	
	
	{\textstyle y'\_{x}}
	
	![{\textstyle y'_{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef2ed2d5f61e3172938257665534af01f608117f).
2. Derive the Maclaurin expansion for 



f
(
x
)
=



1
+

e

−
2
x




3





{\textstyle f(x)={\sqrt[{3}]{1+e^{-2x}}}}

![{\textstyle f(x)={\sqrt[{3}]{1+e^{-2x}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14bfa69ed7bcfd7de396a3360c031c64292828b3) up to 



o

(

x

3


)



{\textstyle o\left(x^{3}\right)}

![{\textstyle o\left(x^{3}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e23f841ca36ac3c4d7ccc3920dbe12d69f5b304).


### Typical questions for seminar classes (labs) within this section


1. Differentiation techniques: inverse, implicit, parametric etc.
2. Find a derivative of a function
3. Apply Leibniz formula
4. Draw graphs of functions
5. Find asymptotes of a parametric function


### Test questions for final assessment in this section


1. Find a derivative of a (implicit/inverse) function
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).
3. Draw graphs of functions
4. Find asymptotes
5. Apply l’Hopital’s rule
6. Find the derivatives of the following functions:
	1. 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);
	2. 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


#### Section title:


Integration and Series



#### Topics covered in this section:


* Antiderivative. Indefinite integral
* Definite integral
* The Fundamental Theorem of Calculus
* Improper Integrals
* Convergence tests. Dirichlet’s test
* Series. Convergence tests
* Absolute / Conditional convergence
* Power Series. Radius of convergence
* Functional series. Uniform convergence


### What forms of evaluation were used to test students’ performance in this section?




|  | **Yes/No** |
| --- | --- |
| Development of individual parts of software product code
 | 0
 |
| Homework and group projects
 | 1
 |
| Midterm evaluation
 | 0
 |
| Testing (written or computer based)
 | 1
 |
| Reports
 | 0
 |
| Essays
 | 0
 |
| Oral polls
 | 0
 |
| Discussions
 | 1
 |


### Typical questions for ongoing performance evaluation within this section


1. 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).
2. 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).
3. Find all values of parameter 



α


{\textstyle \alpha }

![{\textstyle \alpha }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d86dbd6183264b2f8569da1751380b173c7b185) such that series 





∑

k
=
1


+
∞




(



3
k
+
2


2
k
+
1



)


k



α

k





{\textstyle \displaystyle \sum \limits \_{k=1}^{+\infty }\left({\frac {3k+2}{2k+1}}\right)^{k}\alpha ^{k}}

![{\textstyle \displaystyle \sum \limits _{k=1}^{+\infty }\left({\frac {3k+2}{2k+1}}\right)^{k}\alpha ^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f6dabe061cfa1e87b7fc4629f85418142fa7b1d) converges.


#### Typical questions for seminar classes (labs) within this section


1. Integration techniques
2. Integration by parts
3. Calculation of areas, lengths, volumes
4. Application of convergence tests
5. Calculation of Radius of convergence


#### Test questions for final assessment in this section


1. Find the following integrals:
2. ∫





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);
3. ∫

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);
4. ∫



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).
5. Use comparison test to determine if the following series converge.







∑

k
=
1


∞





3
+
(
−
1

)

k




k

2






{\textstyle \sum \limits \_{k=1}^{\infty }{\frac {3+(-1)^{k}}{k^{2}}}}

![{\textstyle \sum \limits _{k=1}^{\infty }{\frac {3+(-1)^{k}}{k^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e87788ec917e8c1f759186838a02cfc24cdd912);
6. Use Cauchy criterion to prove that the series 




∑

k
=
1


∞





k
+
1



k

2


+
3





{\textstyle \sum \limits \_{k=1}^{\infty }{\frac {k+1}{k^{2}+3}}}

![{\textstyle \sum \limits _{k=1}^{\infty }{\frac {k+1}{k^{2}+3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1db1b7d2a764a327ba2530374b1f9452eabe866c) is divergent.
7. Find the sums of the following series:
8. ∑

k
=
1


∞




1

16

k

2


−
8
k
−
3





{\textstyle \sum \limits \_{k=1}^{\infty }{\frac {1}{16k^{2}-8k-3}}}

![{\textstyle \sum \limits _{k=1}^{\infty }{\frac {1}{16k^{2}-8k-3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/912efbbc162ea622ba6e3d18f8e519149ef054b7);
9. ∑

k
=
1


∞





k
−



k

2


−
1





k

2


+
k





{\textstyle \sum \limits \_{k=1}^{\infty }{\frac {k-{\sqrt {k^{2}-1}}}{\sqrt {k^{2}+k}}}}

![{\textstyle \sum \limits _{k=1}^{\infty }{\frac {k-{\sqrt {k^{2}-1}}}{\sqrt {k^{2}+k}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ce35076e06e2b6f92f5e7adf17507591f658f0d).







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