Course detail

# Mathematics 1

Basic mathematical notions. Function, inverse function, sequences. Linear algebra and geometry. Vector spaces, basic notions,linear combination of vectors, linear dependence, independence vectors, base, dimension of a vector space. Matrices and determinants. Systems of linear equations and their solution. Differential calculus of one variable, limit, continuity, derivative of a function. Derivatives of higher orders, l´Hospital rule, behavior of a function. Integral calculus of fuctions of one variable, antiderivatives, indefinite integral. Methods of a direct integration. Integration by parts, substitution methods, integration of some elementary functions. Definite integral and its applications. Improper integral. Infinite number series, convergence criteria. Power series, Taylor theorem, Taylor series.

Learning outcomes of the course unit

After completing the course, students should be able to:

- decide whether vectors are linearly independent and whether they form a basis of a vector space;
- add and multiply matrices, compute the determinant of a square matrix to the 4x4 type, compute the rank and the inverse of a matrix;
- solve a system of linear equations;
- estimate the domains and sketch the grafs of elementary functions;
- compute limits and asymptots for the functions of one variable, use the L’Hospital rule to evaluate limits;
- differentiate and find the tangent to the graph of a function, find the Taylor ploynomial of a function near a given point;
- sketch the graph of a function including extrema, points of inflection and asymptotes;
- integrate using technics of integration, such as substitution, partial fractions and integration by parts;
- evaluate a definite integral including integration by parts and by a substitution for the definite integral;
- compute the area of a region using the definite integral, evaluate the inmproper integral;
- discuss the convergence of the number series, find the
set of the convergence for the power series.

Prerequisites

Students should be able to work with expressions and elementary functions within the scope of standard secondary school requirements; in particular, they shoud be able to transform and simplify expressions, solve basic equations and inequalities, and find the domain and the range of a function.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Krupková, V., Fuchs, P.,: Matematika 1 (CS)
Fong, Y., Wang, Y., Calculus, Springer, 2000. (EN)
Small, D.B., Hosack, J.M., Calculus (An Integrated Approach), Mc Graw-Hill Publ. Comp., 1990. (EN)
Švarc, S. a kol., Matematická analýza I, PC DIR, Brno, 1997. (CS)
Kolářová, E: Maple (CS)
Kolářová, E: Matematika 1 - Sbírka úloh (CS)

Planned learning activities and teaching methods

Teaching methods include lectures, computer exercise and other activities.

Assesment methods and criteria linked to learning outcomes

Maximum 30 points during the semester (for one project and two tests). To get the course-unit credit, student must gain at least 10 points, from which at least 7 points he must get for the tests (from max.25).

The exam is only written exam for maximum 70 points.

Language of instruction

Czech

Work placements

Not applicable.

Course curriculum

1. Sets, functions and the inverse function.
2. Vectors and matrices.
3. Determinants, systems of linear equations.
4. Limits and the continuity of the functions of one variable.
5. The derivative of the functions of one variable.
6. The Taylor polynom and the l'Hospitalovo rule.
7. Graphing a function.
8. Antiderivatives, the per partes method and the substitution technic.
9. Integration of the rational functions.
10. Definite integral.
11. The aplications of the definite integral and the improper integral.
12. Series.
13. Power series and Taylor series.

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 electrical engineering. The practical aspects of application of these methods and their use in solving concrete problems (including the application of contemporary mathematical software) are emphasized.

Specification of controlled education, way of implementation and compensation for absences

The content and forms of instruction in the evaluated course are specified by a regulation issued by the lecturer responsible for the course and updated for every academic year.

Classification of course in study plans

• Programme AUDIO-J Bachelor's

branch J-AUD , 1. year of study, winter semester, 7 credits, compulsory

• Programme EEKR-B Bachelor's

branch B-AMT , 1. year of study, winter semester, 7 credits, compulsory
branch B-EST , 1. year of study, winter semester, 7 credits, compulsory
branch B-MET , 1. year of study, winter semester, 7 credits, compulsory
branch B-SEE , 1. year of study, winter semester, 7 credits, compulsory
branch B-TLI , 1. year of study, winter semester, 7 credits, compulsory

• Programme IBEP-T Bachelor's

branch T-IBP , 1. year of study, winter semester, 7 credits, compulsory

#### Type of course unit

Lecture

52 hours, optionally

Teacher / Lecturer

Computer-assisted exercise

20 hours, compulsory

Teacher / Lecturer

The other activities

6 hours, compulsory

Teacher / Lecturer