# Mathematical Analysis

- Credits: 6
- Ending: Examination
- Range: 2P + 2C
- Semester: summer
- Year: 1
- Faculty of Business Economics with seat in Košice

## Teachers

## Included in study programs

**Teaching results**

The aim of the course is to expand the student's knowledge of mathematical analysis by integral calculus, numerical and power series, the function of several variables and differential equations. The student will learn the theoretical foundations and principles of solving different types of problems from given sub-areas of mathematical analysis. After completing the course, he will be able to solve simpler and more complex tasks that he will encounter during further study, both on quantitatively oriented subjects and on other subjects with an economic focus.

Knowledge:

The student will master the theoretical basic definitions and statements of integral calculus, the principles of integration of rational, irrational and trigonometric functions, the definitions of definite and improper integrals, the definitions and criteria of convergence of numerical and power series. The student will be able to define the function of several variables, partial derivation, limit and continuity of the function of several variables, local extrema and bound local extrema of the function of several variables. He will be able to define the basic types of first-order differential equations and methods of their solution, special types of higher-order differential equations and methods of their solution.

Skills:

The student will be able to solve simpler and more complex problems from mathematical analysis. It will master the decomposition of a rational function into partial fractions, he will be able to calculate the integral of rational, irrational and trigonometric functions for the case of indefinite, to definite and improper integral, to investigate the convergence of numerical and power series. The student will be able to calculate the domain of a function of several variables, to find partial derivatives, local extrema and bound local extrema of a function of several variables. The student will be able to solve the basic types of differential equations of the first and higher order.

Competences:

After completing the course, the student is able to solve simpler and more complex problems in mathematical analysis. The student is ready to solve various assignments by their converting into a mathematical problem. He is able to apply his knowledge to real problems of a quantitative nature which he will encounter in further study. The knowledge that the student acquires in this subject will be used in various subjects of quantitative or economic nature.

**Indicative content**

Lectures:

1. Integral calculus: integration of rational function.

2. Integration of irrational function.

3. Integration of goniometric function.

4. Definite integral.

5. Improper integral.

6. Numerical series.

7. Power series.

8. Functions of several variables: the concept of a function of several variables.

9. Differential calculus of a function of several variables, partial derivatives of a function of two or more variables.

10. Local extrema and saddle points of a function of two or more variables.

11. First order differential equations.

12. Higher order differential equations.

13. Higher order differential equations.

Seminars:

1. Indefinite integral. Integration of rational function by decomposition into partial fractions.

2. Integration of irrational function by substitution method and Ostrogradsky method by indefinite coefficients.

3. Integration of goniometric function.

4. Definite integral by per partes method and substitution method.

5. Improper integral.

6. Numerical series. Convergence of numerical series.

7. Power series. Convergence of power series.

8. Function of several real variables: domain of function of several real variables, partial derivatives.

9. Local extrema and saddle points of a function of two or more variables.

10. Test.

11. First order differential equations.

12. Homogeneous higher order differential equations.

13. Higher order differential equations with right hand side.

**Support literature**

1. KRBÁLEK, M. 2017. Funkce více promněnných. CVUT Praha, 2017. ISBN: 978-8-001-06154-1

2. KRBÁLEK, M. 2019. Matematická analýza III. CVUT Praha, 2018. ISBN: 978-8-001-06663-8

3. LUCKÁ, M. 2016. Úvod do matematickej analýzy. STU, 2016. ISBN: 978-8-022-74489-8

4. NAGY, J. – NAVRÁTIL, O. 2017. Matematická analýza. CVUT Praha, 2017. ISBN: 978-8-001-06142-8

5. PLETANOVÁ, E. – VONDRÁČKOVÁ, J. 2018. Matematická analýza. CVUT, 2018. ISBN: 978-8-001-06441-2

6. SÝKOROVÁ, I. – KLUFA, J. 2018. Matematika 2. Professional Publishing, 2018. ISBN: 978-8-088-26006-6

Supplementary literature:

7. BRANNAN, D. 2021. A first course in mathematical analysis. Cambridge University Press, 2021. ISBN: 978-0-521-68424-8

8. HENNINGS, M. 2017. Cambridge Pre-U Mathematics Coursebook. Cambridge University Press, 2017. ISBN: 978-1-316-63575-9

9. SYDSAETER, K. – HAMMOND, P. – STROM, A. – CARVAJAL, A. 2016. Essential Mathematics for Economics Analysis, 5th edition, Pearson, 2016, ISBN: 978-1-292-07461-0

**Syllabus**

Lectures: 1. Integral calculus: integration of rational function. 2. Integration of irrational function. 3. Integration of goniometric function. 4. Definite integral. 5. Improper integral. 6. Numerical series. 7. Power series. 8. Functions of several variables: the concept of a function of several variables. 9. Differential calculus of a function of several variables, partial derivatives of a function of two or more variables. 10. Local extrema and saddle points of a function of two or more variables. 11. First order differential equations. 12. Higher order differential equations. 13. Higher order differential equations. Seminars: 1. Indefinite integral. Integration of rational function by decomposition into partial fractions. 2. Integration of irrational function by substitution method and Ostrogradsky method by indefinite coefficients. 3. Integration of goniometric function. 4. Definite integral by per partes method and substitution method. 5. Improper integral. 6. Numerical series. Convergence of numerical series. 7. Power series. Convergence of power series. 8. Function of several real variables: domain of function of several real variables, partial derivatives. 9. Local extrema and saddle points of a function of two or more variables. 10. Test. 11. First order differential equations. 12. Homogeneous higher order differential equations. 13. Higher order differential equations with right hand side.

**Requirements to complete the course**

individual work, test

combined exam

• test - 40%

• combined exam - 60%

**Student workload**

• participation in lectures - 26 hours

• participation in seminars - 26 hours

• preparation for seminars - 26 hours

• preparation for the semester test - 26 hours

• preparation for the exam - 52 hours

Total: 156 hours

**Language whose command is required to complete the course**

Slovak

Date of approval: 06.03.2024

Date of the latest change: 25.01.2022