Course details

# Mathematical Analysis

IMA Acad. year 2010/2011 Summer semester 5 credits

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. Fourier series.

Guarantor

Language of instruction

Completion

Time span

Assessment points

Department

Lecturer

Subject specific learning outcomes and competences

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.

Learning objectives

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.

Prerequisite kwnowledge and skills

Secondary shool mathematics and the kowledge from Discrete Mathematics course.

Study literature

- Brabec B., Hrůza,B., Matematická analýza II, SNTL, Praha, 1986.
- Diblík, J., Baštinec, J., Matematika III, ES VUT, Brno, 1991.
- 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.
- Švarc, S. a kol., Matematická analýza I, PC DIR, Brno, 1997.
- 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.

Fundamental literature

- Brabec, B., Hrůza, B., Matematická analýza II, SNTL, Praha, 1986.
- Diblík, J., Baštinec, J., Matematika III, ES VUT, Brno, 1991.
- 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.
- Švarc, S., kol., Matematická analýza I, PC DIR, Brno, 1997.
- 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.

Syllabus of lectures

- 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.
- 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.

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

- 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.

Progress assessment

Computer practice tasks: 25 points.

Homeworks: 15 points.

Semestral examination: 60 points.

Course inclusion in study plans