Course detail

# Mathematics 1

FEKT-BPC-MA1AAcad. year: 2024/2025

Basic mathematical notions. Sets, operations with sets, concept of a 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's 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's theorem, Taylor series.

Language of instruction

Number of ECTS credits

Mode of study

Guarantor

Department

Entry knowledge

Rules for evaluation and completion of the course

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.

Aims

After completing the course, the student 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.

Study aids

Prerequisites and corequisites

Basic literature

Kolářová, E: Matematika 1 - Sbírka úloh. (CS)

Krupková, V., Fuchs, P., Matematika 1. (CS)

Recommended reading

Fong, Y., Wang, Y., Calculus, Springer, 2000. (EN)

Goldstein, L.J., Lay, D.C., Schneider, D.I., Asmar, N.H., Calculus & Its Applications, Pearson, 2017. (EN)

Hoffmann, L., Bradley, G., Sobecki, D., Price, M., Applied Calculus for Business, Economics, and the Social and Life Sciences, Expanded Edition, McGraw-Hill Education, 2012. (EN)

Kolářová, E: Maple. (CS)

Lial, M.L., Greenwell, R.N., Ritchey, N.P., Calculus with Applications, Pearson, 2015. (EN)

Ross, K.A., Elementary analysis: The Theory of Calculus, Springer, 2000. (EN)

Švarc, S. a kol., Matematická analýza I, PC DIR, Brno, 1997. (CS)

Thomas, G.B., Finney, R.L., Calculus and Analytic Geometry, Addison-Wesley Publ. Comp., 1994. (EN)

Elearning

**eLearning:**currently opened course

Classification of course in study plans

- Programme BPC-BTB Bachelor's 1 year of study, winter semester, compulsory

#### Type of course unit

Lecture

Teacher / Lecturer

Syllabus

2. Vector - combination, dependence and independence of vectors, base and dimension of a vector space.

3. Matrices and determinants.

4. Systems of linear equations and their solution.

5. Differential calculus of one variable. Limit, continuity, derivative of a function.

6. Derivatives of higher order, Taylor theorem.

7. L'Hospital rule, behaviour of a function.

8. Integral calculus of functions of one variable, primitive function, indefinite integral. Methods of direct integration.

9. Per partes method and substitution method. Integration of some elementary functions.

10. Definite integral and its applications.

11. Improper integral.

12. Infinite number series, convergence criteria.

13. Power series, Taylor theorem, Taylor series.

Fundamentals seminar

Teacher / Lecturer

Syllabus

2. Matice, determinanty.

3. Řešení soustav lineárních rovnic.

4. Derivace funkce jedné proměnné.

5. Průběh funkce.

6. Výpočet neurčitého a určitého integrálu.

7. Nekonečné řady.

Computer-assisted exercise

Teacher / Lecturer

Syllabus

2. Matrices, determinants.

3. Solving a system of linear equations.

4. Derivative of a function of one variable.

5. Behaviour of a function.

6. Calculation of indefinite and definite integrals.

7. Series.

Project

Teacher / Lecturer

Syllabus

1. Basic mathematical notions. Function of one variable.

2. Vector spaces, basis, dimension, operations with vectors.

3. Matrices and determinants, Systems of linear equations.

4. Calculation of limits and derivatives of a function of one variable.

5. Calculation of indefinite and definite integrals.

6. Series.

Elearning

**eLearning:**currently opened course