Czech

Not applicable.

Institute of Mathematics and Descriptive Geometry (MAT)

Students will achieve the subject's main objectives. They will get the understanding of the basics of differential and integral calculus of functions of one variable and the geometric interpretations of some of the concepts. They will master differentiating and sketching the graph of a function.
They will be able to perform operations with matrices and elementary transactions, to calculate determinants and solve systems of algebraic equations (using Gauss elimination method, Cramer's rule, and the inverse of the system matrix). They will get acquainted with applications of the vector calculus to solving problems of 3D analytic geometry.

Basic secondary-school mathematics. Graphs of essential elementary functions (powers and roots, quadratic function, direct and indirect proportion, absolute value, trigonometric functions) and the basic properties of such functions. Simplification of algebraic expressions. Definition of a geometric vector and basics of 3D analytic geometry (parametric equations of a straight line, dot product of vectors and its applications to metric and positional problems). Identifying the the types and basic properties of conics, sketching graphs of conics).

Not applicable.

Not applicable.

Requirements for successful completion of the subject are specified by guarantor’s regulation updated for every academic year.

1. Real function of one real variable, explicit and parametric definition of a function. (50 min.) – Composite function and inverse to a function. (50 min.)
Sem. Absolute value of a function. Quadratic equations in complex field. Conics. (50 min.) - Graphs of selected elementary functions. Basic properties of functions. (50 min.)

2. Some elementary functions, inverse trigonometric functions. Hyperbolic functions. (50 min.) – Polynomial and the basic properties of its roots, decomposition of a polynomial in the field of real numbers. (50 min.)
Sem. Composite function and inverse to a function (inverse trigonometric functions, logarithmic functions). (50 min.) Polynomial. (50 min.)

3. Rational functions. (50 min.) – Sequence and its limit. (50 min.)
Sem. Initial test. (50 min.) – Polynomial, sign of a polynomial. (50 min.)

4. Limit of a function, continuous functions, basic theorems. (50 min.) – Derivative of a function, its geometric and physical applications, differentiating rules. (50 min.)
Sem. Rational function, sign of a rational function, decomposition into partial fractions. (2 times 50 min.)

5. Derivatives of composite and inverse functions. (50 min.) – Differential of a function. Rolle and Lagrange theorem. (50 min.)
Sem. Limit of a function. Derivative of a function (basic calculation) and its geometric applications, basic formulas and rules for differentiating. (2 times 50 min.)

6. Higher-order derivatives, higher-order differentials. (50 min.) – Taylor theorem. (50 min.)
Sem. Derivative of an inverse function. Basic differentiation formulas and rules, simplification of the results of differentiation. (2 times 50 min.)

7. L`Hospital's rule. Asymptotes of the graph of a function. Sketching the graph of a function. (2 times 50 min.)
Sem. Higher-order derivatives. Taylor theorem. (50 min.) L` Hospital's rule. (50 min.)

8. Basics of matrix calculus, elementary transformations of a matrix, rank of a matrix. (50 min.) – Solutions to systems of linear algebraic equations by Gauss elimination method. (50 min.)
Sem. Test I. (20 min.) Asymptotes of the graph of a function. Sketching the graph of a function. (80 min.)

9. Second-order determinants. Higher-order determinants calculated by Laplace expansion. (50 min.) – Rules for calculating with determinants. Cramer's rule of solving a system of linear algebraic equations. (50 min.)
Sem. Basic operations with matrices. Elementary transformations of a matrix, rank of a matrix, solutions to systems of linear algebraic equations by Gauss elimination method. (2 times 50 min.)

10. Inverse to a matrix. Jordan's method of calculation. Matrix equations. (50 min.) - Real linear space, base and dimension of a linear space. Linear spaces of arithmetic and geometric vectors. (50 min.)
Sem. Calculating determinants using Laplace expansion and rules for calculating with determinants. (50 min.) Calculating the inverse to a matrix for A(2,2), A(3,3) matric

Not applicable.

After the course, students should understand the basics of calculus of functions of one variable and the basic interpretations of some of the concepts. They should master differentiating and sketching the graph of a function.
They should know how to perform operations with matrices, elementary transactions, calculate determinants, solve systems of algebraic equations using Gauss elimination method.

Extent and forms are specified by guarantor’s regulation updated for every academic year.

Not applicable.

Not applicable.

BUDÍNSKÝ, B. - CHARVÁT, J.: Matematika I. Praha, SNTL, 1987. (CS)
LARSON, R.- HOSTETLER, R.P.- EDWARDS, B.H.: Calculus (with Analytic Geometry). Brooks Cole, 2005. (EN)
STEIN, S. K: Calculus and analytic geometry. New York, 1989. (EN)

DANĚČEK, J. a kolektiv: Sbírka příkladů z matematiky I. CERM, 2003. (CS)
DLOUHÝ, O., TRYHUK, V.: Diferenciální počet I. CERM, 2009. (CS)
kolektiv: Elektronické studijní opory. FAST VUT, 2004. [https://intranet.fce.vutbr.cz/pedagog/predmety/opory.asp] (CS)
NOVOTNÝ, J.: Základy lineární algebry. CERM, 2004. (CS)
TRYHUK, V. - DLOUHÝ, O.: Modul GA01_M01 studijních opor předmětu GA01. FAST VUT, Brno, 2004. [https://intranet.fce.vutbr.cz/pedagog/predmety/opory.asp] (CS)