Course detail

# Mathematics I

FSI-1MAcad. year: 2020/2021

Basic concepts of the set theory and mathematical logic.

Linear algebra: matrices, determinants, systems of linear equations.

Vector calculus and analytic geometry.

Differential calculus of functions of one variable: basic elementary functions, limits, derivative and its applications.

Integral calculus of functions of one variable: primitive function, proper integral and its applications.

Language of instruction

Number of ECTS credits

Mode of study

Department

Learning outcomes of the course unit

Prerequisites

Co-requisites

Planned learning activities and teaching methods

Assesment methods and criteria linked to learning outcomes

FORM OF EXAMINATIONS:

The exam has a written part (at most 75 points) and an oral part (at most 25 points)

WRITTEN PART OF EXAMINATION (at most 75 points)

In a 120-minute written test, students have to solve the following four problems:

Problem 1: Functions and their properties: domains, graphs (at most 10 points)

Problem 2: In linear algebra, analytic geometry (at most 20 points)

Problem 3: In differential calculus (at most 20 points)

Problem 4: In integral calculus (at most 25 points)

The above problems can also contain a theoretical question.

ORAL PART OF EXAMINATION (max 25 points)

• Discussion based on the written test: students have to explain how they solved each problem. Should the student fail to explain it sufficiently, the test results will not be accepted and will be classified by 0 points.

• Possible theoretic question.

• Possible simple problem to be solved straight away.

• The results achieved in the written tests in seminars may be taken into account within the oral examination.

FINAL CLASSIFICATION:

0-49 points: F

50-59 points: E

60-69 points: D

70-79 points: C

80-89 points: B

90-100 points: A

Course curriculum

Work placements

Aims

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

Recommended optional programme components

Prerequisites and corequisites

Basic literature

Thomas G.B., Finney R.L.: Calculus and Analytic Geometry (7th edition)

Sneall D.B., Hosack J.M.: Calculus, An Integrated Approach

Rektorys K. a spol.: Přehled užité matematiky I,II (SNTL, 1988)

Howard, A.A.: Elementary Linear Algebra, Wiley 2002

Satunino, L.S., Hille, E., Etgen, J.G.: Calculus: One and Several Variables, Wiley 2002

Recommended reading

Nedoma J.: Matematika I. Část druhá. Diferenciální a integrální počet funkcí jedné proměnné (skriptum VUT)

Děmidovič B. P.: Sbírka úloh a cvičení z matematické analýzy

Eliaš J., Horváth J., Kajan J.: Zbierka úloh z vyššej matematiky I, II, III, IV (Alfa Bratislava, 1985)

Nedoma J.: Matematika I. Část třetí, Integrální počet funkcí jedné proměnné (skriptum VUT)

Thomas G.B., Finney R.L.: Calculus and Analytic Geometry (7th edition)

eLearning

**eLearning:**currently opened course

Classification of course in study plans

- Programme B-ENE-P Bachelor's, 1. year of study, winter semester, compulsory
- Programme B-STR-P Bachelor's
specialization STR , 1. year of study, winter semester, compulsory

- Programme B3S-P Bachelor's
branch B-PRP , 1. year of study, winter semester, compulsory

- Programme B-ZSI-P Bachelor's
specialization STI , 1. year of study, winter semester, compulsory

specialization MTI , 1. year of study, winter semester, compulsory - Programme B-PDS-P Bachelor's, 1. year of study, winter semester, compulsory
- Programme B-MET-P Bachelor's, 1. year of study, winter semester, compulsory
- Programme B-VTE-P Bachelor's, 1. year of study, winter semester, compulsory
- Programme B-PRP-P Bachelor's, 1. year of study, winter semester, compulsory

#### Type of course unit

Lecture

Teacher / Lecturer

Syllabus

Week 2: Matrices and determinants (determinants and their properties, regular and singular matrices, inverse to a matrix, calculating the inverse to a matrix using determinants), systems of linear algebraic equations (Cramer's rule, Gauss elimination method).

Week 3: More about systems of linear algebraic equations (Frobenius theorem, calculating the inverse to a matrix using the elimination method), vector calculus (operations with vectors, scalar (dot) product, vector (cross) product, scalar triple (box) product).

Week 4: Analytic geometry in 3D (problems involving straight lines and planes, classification of conics and quadratic surfaces), the notion of a function (domain and range, bounded functions, even and odd functions, periodic functions, monotonous functions, composite functions, one-to-one functions, inverse functions).

Week 5: Basic elementary functions (exponential, logarithm, general power, trigonometric functions and cyclometric (inverse to trigonometric functions), polynomials (root of a polynomial, the fundamental theorem of algebra, multiplicity of a root, product breakdown of a polynomial), introducing the notion of a rational function.

Week 6: Sequences and their limits, limit of a function, continuous functions.

Week 7: Derivative of a function (basic problem of differential calculus, notion of derivative, calculating derivatives, geometric applications of derivatives), calculating the limit of a function using L' Hospital rule.

Week 8: Monotonous functions, maxima and minima of functions, points of inflection, convex and concave functions, asymptotes, sketching the graph of a function.

Week 9: Differential of a function, Taylor polynomial, parametric and polar definitions of curves and functions (parametric definition of a derivative, transforming parametric definitions into polar ones and vice versa).

Week 10: Primitive function (antiderivative) (definition, properties and basic formulas), integrating by parts, method of substitution.

Week 11: Integrating rational functions (no complex roots in the denominator), calculating a primitive function by the method of substitution in some of the elementary functions.

Week 12: Riemann integral (basic problem of integral calculus, definition and properties of the Riemann integral), calculating the Riemann integral (Newton' s formula).

Week 13: Applications of the definite integral (surface area of a plane figure, length of a curve, volume and lateral surface area of a rotational body), improper integral.

Exercise

Teacher / Lecturer

Ing. Matouš Cabalka

Ing. Anna Derevianko

Ing. Matej Dolník

Ing. Jindřich Dospiva

Ing. Ivan Eryganov, Ph.D.

Ing. Lucie Fedorková

doc. Mgr. Jaroslav Hrdina, Ph.D.

doc. RNDr. Jiří Klaška, Dr.

Ing. Tereza Konečná

Michael Joseph Lieberman, Ph.D.

Ing. Mgr. Eva Mrázková, Ph.D.

doc. Mgr. et Mgr. Aleš Návrat, Ph.D.

Mgr. Jan Pavlík, Ph.D.

Mgr. Jana Procházková, Ph.D.

Ing. Ondřej Resl, Ph.D.

Ing. Marek Stodola

Mgr. Viera Štoudková Růžičková, Ph.D.

doc. RNDr. Jiří Tomáš, Dr.

Ing. Jana Vechetová

Mgr. Jitka Zatočilová, Ph.D.

Syllabus

Computer-assisted exercise

Teacher / Lecturer

Syllabus

eLearning

**eLearning:**currently opened course