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Mathematical Analysis I

• Course name: Mathematical Analysis I
• Code discipline:
• Subject area: ['Differentiation', 'Integration', 'Series']

Course Topics

Course Sections and Topics

Section	Topics within the section
Sequences and Limits	Sequences. Limits of sequences Limits of sequences. Limits of functions Limits of functions. Continuity. Hyperbolic functions
Differentiation	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
Integration and Series	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

Intended Learning Outcomes (ILOs)

What is the main purpose of this course?

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

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

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

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 ...

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

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	90-100	-
B. Good	75-89	-
C. Satisfactory	60-74	-
D. Poor	0-59	-

Course activities and grading breakdown

Activity Type	Percentage of the overall course grade
Labs/seminar classes	20
Interim performance assessment	30
Exams	50

Resources, literature and reference materials

Open access resources

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

Teaching Methodology: Methods, techniques, & activities

Activities and Teaching Methods

Activities within each section

Learning Activities	Section 1	Section 2	Section 3
Homework and group projects	1	1	1
Midterm evaluation	1	1	0
Testing (written or computer based)	1	1	1
Discussions	1	1	1

Formative Assessment and Course Activities

Ongoing performance assessment

Section 1

Activity Type	Content	Is Graded?
Question	A sequence, limiting value	1
Question	Limit of a sequence, convergent and divergent sequences	1
Question	Increasing and decreasing sequences, monotonic sequences	1
Question	Bounded sequences Properties of limits	1
Question	Theorem about bounded and monotonic sequences	1
Question	Cauchy sequence The Cauchy Theorem (criterion)	1
Question	Limit of a function Properties of limits	1
Question	The first remarkable limit	1
Question	The Cauchy criterion for the existence of a limit of a function	1
Question	Second remarkable limit	1
Question	Find a limit of a sequence	0
Question	Find a limit of a function	0

Section 2

Activity Type	Content	Is Graded?
Question	A plane curve is given by ${\displaystyle x(t)=-{\frac {t^{2}+4t+8}{t+2}}}$ , ${\textstyle y(t)={\frac {t^{2}+9t+22}{t+6}}}$ Find ${\textstyle y'_{x}}$	1
Question	the asymptotes of this curve;	1
Question	the derivative ${\textstyle y'_{x}}$	1
Question	Derive the Maclaurin expansion for ${\textstyle f(x)={\sqrt[{3}]{1+e^{-2x}}}}$ up to ${\textstyle o\left(x^{3}\right)}$	1
Question	Differentiation techniques: inverse, implicit, parametric etc	0
Question	Find a derivative of a function	0
Question	Apply Leibniz formula	0
Question	Draw graphs of functions	0
Question	Find asymptotes of a parametric function	0

Section 3

Activity Type	Content	Is Graded?
Question	Find the indefinite integral ${\textstyle \displaystyle \int x\ln \left(x+{\sqrt {x^{2}-1}}\right)\,dx}$	1
Question	Find the length of a curve given by ${\textstyle y=\ln \sin x}$ , ${\textstyle {\frac {\pi }{4}}\leqslant x\leqslant {\frac {\pi }{2}}}$	1
Question	Find all values of parameter ${\textstyle \alpha }$ such that series ${\textstyle \displaystyle \sum \limits _{k=1}^{+\infty }\left({\frac {3k+2}{2k+1}}\right)^{k}\alpha ^{k}}$ converges	1
Question	Integration techniques	0
Question	Integration by parts	0
Question	Calculation of areas, lengths, volumes	0
Question	Application of convergence tests	0
Question	Calculation of Radius of convergence	0

Final assessment

Section 1

1. Find limits of the following sequences or prove that they do not exist:
2. ${\displaystyle a_{n}=n-{\sqrt {n^{2}-70n+1400}}}$ ;
3. ${\textstyle d_{n}=\left({\frac {2n-4}{2n+1}}\right)^{n}}$ ;
4. ${\textstyle x_{n}={\frac {\left(2n^{2}+1\right)^{6}(n-1)^{2}}{\left(n^{7}+1000n^{6}-3\right)^{2}}}}$

Section 2

1. Find a derivative of a (implicit/inverse) function
2. Apply Leibniz formula Find ${\textstyle y^{(n)}(x)}$ if ${\textstyle y(x)=\left(x^{2}-2\right)\cos 2x\sin 3x}$
3. Draw graphs of functions
4. Find asymptotes
5. Apply l’Hopital’s rule
6. Find the derivatives of the following functions: ${\textstyle f(x)=\log _{|\sin x|}{\sqrt[{6}]{x^{2}+6}}}$
7. ${\textstyle f(x)=\log _{|\sin x|}{\sqrt[{6}]{x^{2}+6}}}$ ;
8. ${\textstyle y(x)}$ that is given implicitly by ${\textstyle x^{3}+5xy+y^{3}=0}$

Section 3

1. Find the following integrals:
2. ${\textstyle \int {\frac {{\sqrt {4+x^{2}}}+2{\sqrt {4-x^{2}}}}{\sqrt {16-x^{4}}}}\,dx}$
3. ${\textstyle \int 2^{2x}e^{x}\,dx}$
4. ${\textstyle \int {\frac {dx}{3x^{2}-x^{4}}}}$
5. Use comparison test to determine if the following series converge ${\textstyle \sum \limits _{k=1}^{\infty }{\frac {3+(-1)^{k}}{k^{2}}}}$
6. ${\textstyle \sum \limits _{k=1}^{\infty }{\frac {k+1}{k^{2}+3}}}$
7. Find the sums of the following series:
8. ${\textstyle \sum \limits _{k=1}^{\infty }{\frac {1}{16k^{2}-8k-3}}}$
9. ${\textstyle \sum \limits _{k=1}^{\infty }{\frac {k-{\sqrt {k^{2}-1}}}{\sqrt {k^{2}+k}}}}$

The retake exam

Section 1

Section 2

Section 3