Course detail

Mathematics 1

FAST-BAA001Acad. year: 2022/2023

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

Czech

Number of ECTS credits

7

Mode of study

Not applicable.

Guarantor

doc. Mgr. Irena Hinterleitner, Ph.D.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Offered to foreign students

Of all faculties

Learning outcomes of the course unit

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.

Prerequisites

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 2D analytic geometry. Identifying the the types and basic properties of conics, sketching graphs of conics).

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Not applicable.

Course curriculum

1. Real function of one real variable, explicit and parametric definition of a function. Composite function and inverse to a function.
2. Some elementary functions, inverse trigonometric functions. Hyperbolic functions. Polynomial and the basic properties of its roots, decomposition of a polynomial in the field of real numbers.
3. Rational functions. Sequence and its limit.
4. Limit of a function, continuous functions, basic theorems. Derivative of a function, its geometric and physical applications, differentiating rules.
5. Derivatives of composite and inverse functions. Differential of a function. Rolle and Lagrange theorem.
6. Higher-order derivatives, higher-order differentials. Taylor theorem.
7. L`Hospital's rule. Asymptotes of the graph of a function. Sketching the graph of a function.
8. Basics of matrix calculus, elementary transformations of a matrix, rank of a matrix. Solutions to systems of linear algebraic equations by Gauss elimination method.
9. Second-order determinants. Higher-order determinants calculated by Laplace expansion. Rules for calculating with determinants. Cramer's rule of solving a system of linear algebraic equations.
10. Inverse to a matrix. Jordan's method of calculation. Matrix equations. Real linear space, base and dimension of a linear space. Linear spaces of arithmetic and geometric vectors.
11. Eigenvalues and eigenvectors of a matrix. Coordinates of a vector. Dot and cross product of vectors, calculating with coordinates.
12. Mixed product of vectors. Plane and straight line in 3D, positional problems.
13. Metric problems. Surfaces.

Work placements

Not applicable.

Aims

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.

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

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

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

DANĚČEK, J. a kolektiv: Sbírka příkladů z matematiky I. CERM, 2003. CZ 2003
DLOUHÝ, O., TRYHUK, V.: Diferenciální počet I. CERM, 2009. CZ 2009
NOVOTNÝ, J.: Základy lineární algebry. CERM, 2004. CZ 2004

Recommended reading

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

Classification of course in study plans

  • Programme BPC-SI Bachelor's

    specialization VS , 1 year of study, winter semester, compulsory

  • Programme BPC-MI Bachelor's 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

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

RNDr. Karel Mikulášek, Ph.D.
RNDr. Helena Koutková, CSc.
Mgr. Jana Bulantová, Ph.D.
Mgr. Blanka Morávková, Ph.D.
Mgr. Darina Brothánková, Ph.D.

Syllabus

1. Real function of one real variable, explicit and parametric definition of a function. Composite function and inverse to a function. 2. Some elementary functions, inverse trigonometric functions. Hyperbolic functions. Polynomial and the basic properties of its roots, decomposition of a polynomial in the field of real numbers. 3. Rational functions. Sequence and its limit. 4. Limit of a function, continuous functions, basic theorems. Derivative of a function, its geometric and physical applications, differentiating rules. 5. Derivatives of composite and inverse functions. Differential of a function. Rolle and Lagrange theorem. 6. Higher-order derivatives, higher-order differentials. Taylor theorem. 7. L`Hospital's rule. Asymptotes of the graph of a function. Sketching the graph of a function. 8. Basics of matrix calculus, elementary transformations of a matrix, rank of a matrix. Solutions to systems of linear algebraic equations by Gauss elimination method. 9. Second-order determinants. Higher-order determinants calculated by Laplace expansion. Rules for calculating with determinants. Cramer's rule of solving a system of linear algebraic equations. 10. Inverse to a matrix. Jordan's method of calculation. Matrix equations. Real linear space, base and dimension of a linear space. Linear spaces of arithmetic and geometric vectors. 11. Eigenvalues and eigenvectors of a matrix. Coordinates of a vector. Dot and cross product of vectors, calculating with coordinates. 12. Mixed product of vectors. Plane and straight line in 3D, positional problems. 13. Metric problems. Surfaces.

Exercise

39 hod., compulsory

Teacher / Lecturer

Ing. Pavel Špaček, Dr.
RNDr. Oto Přibyl
Mgr. Jana Bulantová, Ph.D.
Mgr. Šárka Pechancová, Ph.D.
doc. Mgr. Ing. Miroslav Trcala, Ph.D.
RNDr. Jan Vondra, Ph.D.
Mgr. Blanka Morávková, Ph.D.
Dr. RNDr. Karel Lepka, Ph.D.
Mgr. Eva Jansová, Ph.D.
Mgr. Darina Brothánková, Ph.D.
Ing. Anna Derevianko, Ph.D.
Mgr. Marie Hartmanová

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.