BSc: Mathematical Analysis I

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.

- 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?

|a|c| & Yes/No
Development of individual parts of software product code & 1
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. $a_{n}=n-{\sqrt {n^{2}-70n+1400}}$;
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

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?

|a|c| & Yes/No
Development of individual parts of software product code & 1
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 $x(t)=-{\frac {t^{2}+4t+8}{t+2}}$, ${\textstyle y(t)={\frac {t^{2}+9t+22}{t+6}}}$. Find
   1. the asymptotes of this curve;
   2. the derivative ${\textstyle y'_{x}}$.
2. Derive the Maclaurin expansion for ${\textstyle f(x)={\sqrt[{3}]{1+e^{-2x}}}}$ up to ${\textstyle o\left(x^{3}\right)}$.

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 ${\textstyle y^{(n)}(x)}$ 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:
   1. ${\textstyle f(x)=\log _{|\sin x|}{\sqrt[{6}]{x^{2}+6}}}$;
   2. ${\textstyle y(x)}$ that is given implicitly by ${\textstyle x^{3}+5xy+y^{3}=0}$.

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?

|a|c| & Yes/No
Development of individual parts of software product code & 1
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. Find the indefinite integral ${\textstyle \displaystyle \int x\ln \left(x+{\sqrt {x^{2}-1}}\right)\,dx}$.
2. Find the length of a curve given by ${\textstyle y=\ln \sin x}$, ${\textstyle {\frac {\pi }{4}}\leqslant x\leqslant {\frac {\pi }{2}}}$.
3. 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.

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. ${\textstyle \int {\frac {{\sqrt {4+x^{2}}}+2{\sqrt {4-x^{2}}}}{\sqrt {16-x^{4}}}}\,dx}$;

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. Use Cauchy criterion to prove that the series ${\textstyle \sum \limits _{k=1}^{\infty }{\frac {k+1}{k^{2}+3}}}$ is divergent.

7. Find the sums of the following series:

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

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