Course detail

# Mathematics 1

FAST-BAA001Acad. year: 2023/2024

Real function of one real variable. Sequences, limit of a function, continuous functions. Derivative of a function, its geometric and physical applications, basic theorems on derivatives, higher-order derivatives, differential of a function, Taylor expansion of a function, sketching the graph of a function.

Linear algebra (basics of the matrix calculus, rank of a matrix, Gauss elimination method, inverse to a matrix, determinants and their applications). Eigenvalues and eigenvectors of a matrix. Basics of vectors, vector spaces. Linear spaces. Analytic geometry (dot, cross and mixed product of vectors, affine and metric problems for linear bodies in 3D).

The basic problems in numerical mathematics (interpolation, solving nonlinear equation and systems of linear equations, numerical differentiation).

Language of instruction

Number of ECTS credits

Mode of study

Guarantor

Department

Offered to foreign students

Entry knowledge

Rules for evaluation and completion of the course

Aims

They should know how to perform operations with matrices, elementary transactions, calculate determinants, solve systems of algebraic equations using Gauss elimination method.

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.

Study aids

Prerequisites and corequisites

Basic literature

DLOUHÝ, O., TRYHUK, V.: Diferenciální počet I. CERM, 2009. CZ 2009

NOVOTNÝ, J.: Základy lineární algebry. CERM, 2004. CZ 2004

DANĚČEK, J. a kolektiv: Sbírka příkladů z matematiky I. CERM, 2003. CZ 2003

Recommended reading

BUDÍNSKÝ, B. - CHARVÁT, J.: Matematika I. Praha, SNTL, 1987. CZ 1987

STEIN, S. K: Calculus and analytic geometry. New York, 1989. EN 1989

LARSON, R.- HOSTETLER, R.P.- EDWARDS, B.H.: Calculus (with Analytic Geometry). Brooks Cole, 2005. EN 2005

eLearning

**eLearning:**currently opened course

Classification of course in study plans

- Programme BPC-SI Bachelor's
specialization VS , 1. year of study, winter semester, compulsory

- Programme BPA-SI Bachelor's, 1. year of study, winter semester, compulsory
- Programme BKC-SI Bachelor's, 1. year of study, winter semester, compulsory
- Programme BPC-MI Bachelor's, 1. year of study, winter semester, compulsory

#### Type of course unit

Lecture

Teacher / Lecturer

Syllabus

Exercise

Teacher / Lecturer

Syllabus

1. Absolute value of a function. Quadratic equations in complex field. Conics. Graphs of selected elementary functions. Basic properties of functions. 2. Composite function and inverse to a function (inverse trigonometric functions, logarithmic functions). Numerical solutions of equations by bisection and regula falsi method. 3. Polynomial, sign of a polynomial. Lagrange and Newton interpolation polynomial. 4. Rational function, sign of a rational function, decomposition into partial fractions. 5. Limit of a function. Derivative of a function (basic calculation) and its geometric applications, basic formulas and rules for differentiating. 6. Derivative of an inverse function. Basic differentiation formulas and rules. Numerical differentiation. 7. Test I. Higher-order derivatives. Taylor theorem. L` Hospital's rule. Approximation of solutions of one equation in one variable by the Newton method. 8. Asymptotes of the graph of a function. Sketching the graph of a function. 9. Basic operations with matrices. Elementary transformations of a matrix, rank of a matrix, solutions to systems of linear algebraic equations by Gauss elimination method. Numerical solutions of systems of linear equations. 10. Calculating determinants using Laplace expansion and rules for calculating with determinants. Calculating the inverse to a matrix using Jordan's method. Solutions of systems of linear equations by iteration. 11. Test II. Matrix equations. The discrete least square method. Eigenvalues and eigenvectors of a matrix. 12. Using dot and cross products in solving problems in 3D analytic geometry. 13. Mixed product. Seminar evaluation.

eLearning

**eLearning:**currently opened course