Course detail

# Mathematical Analysis

FIT-IMAAcad. year: 2018/2019

Limit and continuity, derivative of a function. Partial derivatives. Basic differentiation rules. Elementary functions. Extrema for functions (of one and of several variables). Indefinite integral. Techniques of integration. The Riemann (definite) integral. Multiple integrals. Applications of integrals. Infinite sequences and infinite series. Taylor polynomials.

Supervisor

Department

Learning outcomes of the course unit

The ability of orientation in the basic problems of higher mathematics and the ability to apply the basic methods. Solving problems in the areas cited in the annotation above by using basic rules. Solving these problems by using modern mathematical software.

Prerequisites

Secondary school mathematics and the kowledge from Discrete Mathematics course.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

- Brabec, B., Hrůza, B., Matematická analýza II, SNTL, Praha, 1986.
- Švarc, S., kol., Matematická analýza I, PC DIR, Brno, 1997.
- Krupková, V. Matematická analýza pro FIT, elektronický učební text, 2007.

- Edwards, C.H., Penney, D.E., Calculus with Analytic Geometry, Prentice Hall, 1993.
- Fong, Y., Wang, Y., Calculus, Springer, 2000.
- Ross, K.A., Elementary analysis: The Theory of Calculus, Springer, 2000.
- Small, D.B., Hosack, J.M., Calculus (An Integrated Approach), Mc Graw-Hill Publ. Comp., 1990.
- Thomas, G.B., Finney, R.L., Calculus and Analytic Geometry, Addison-Wesley Publ. Comp., 1994.
- Zill, D.G., A First Course in Differential Equations, PWS-Kent Publ. Comp., 1992.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Practice tasks: 28 points.

Homeworks: 12 points.

Semestral examination: 60 points.

Language of instruction

Czech

Work placements

Not applicable.

Course curriculum

- Function of one variable, limit, continuity.
- Differential calculus of functions of one variable I: derivative, differential, Taylor theorem.
- Differential calculus of functions of one variable II: maximum, minimum, behaviour of the function.
- Integral calculus of functions of one variable I: indefinite integral, basic methods of integration.
- Integral calculus of functions of one variable II: definite Riemann integral and its application.
- Infinite number and power series.
- Taylor series.
- Functions of two and three variables, geometry and mappings in three-dimensional space.
- Differential calculus of functions of more variables I: directional and partial derivatives, Taylor theorem.
- Differential calculus of functions of more variables II: funcional extrema, absolute and bound extrema.
- Integral calculus of functions of more variables I: two and three-dimensional integrals.
- Integral calculus of functions of more variables II: method of substitution in two and three-dimensional integrals.
- Limit, continuity and derivative of a function. Partial derivative. Derivative of a composite function.
- Differential of function of one and several variables. L'Hospital's rule. Behaviour of continuous and differentiable function. Extrema of functions of one and several variables.
- Primitive function and undefinite integral. Basic methods of integration. Definite one-dimensional and multidimensional integral.
- Methods for solution of definite integrals (Newton-Leibnitz formula, Fubini theorem).
- Indefinite number series. Convergence of series. Sequences and series of functions. Taylor theorem. Power series.

**Syllabus of lectures:**

**Syllabus of numerical exercises:**

The class work is prepared in accordance with the lecture.

**Syllabus of computer exercises:**

Trained tasks are prepared to follow and complete study matter from lectures and seminar practice.

**Syllabus - others, projects and individual work of students:**

Aims

The main goal of the calculus course is to explain the basic principles and methods of higher mathematics that are necessary for the study of computer science. The practical aspects of applications of these methods and their use in solving concrete problems (including the application of contemporary mathematical software in the laboratories) are emphasized.

#### Type of course unit

Lecture

39 hours, optionally

Teacher / Lecturer

Syllabus

- Function of one variable, limit, continuity.

- Differential calculus of functions of one variable I: derivative, differential, Taylor theorem.

- Differential calculus of functions of one variable II: maximum, minimum, behaviour of the function.

- Integral calculus of functions of one variable I: indefinite integral, basic methods of integration.

- Integral calculus of functions of one variable II: definite Riemann integral and its application.

- Infinite number and power series.

- Taylor series.

- Functions of two and three variables, geometry and mappings in three-dimensional space.

- Differential calculus of functions of more variables I: directional and partial derivatives, Taylor theorem.

- Differential calculus of functions of more variables II: funcional extrema, absolute and bound extrema.

- Integral calculus of functions of more variables I: two and three-dimensional integrals.

- Integral calculus of functions of more variables II: method of substitution in two and three-dimensional integrals.

Computer-assisted exercise

20 hours, compulsory

Teacher / Lecturer

Syllabus

Trained tasks are prepared to follow and complete study matter from lectures and seminar practice.

Project

6 hours, compulsory

Teacher / Lecturer

Syllabus

- Limit, continuity and derivative of a function. Partial derivative. Derivative of a composite function.

- Differential of function of one and several variables. L'Hospital's rule. Behaviour of continuous and differentiable function. Extrema of functions of one and several variables.

- Primitive function and undefinite integral. Basic methods of integration. Definite one-dimensional and multidimensional integral.

- Methods for solution of definite integrals (Newton-Leibnitz formula, Fubini theorem).

- Indefinite number series. Convergence of series. Sequences and series of functions. Taylor theorem. Power series.

eLearning

**eLearning:** currently opened course