Course detail

Mathematics I

FCH-BC_MAT1Acad. year: 2024/2025

Basics of calculus of functions of one real variable. Basics of linear algebra.

Language of instruction

Czech

Number of ECTS credits

Mode of study

Not applicable.

Guarantor

doc. Ing. Vladislav Kozák, CSc.

Department

Institute of Physical and Applied Chemistry (ÚFSCH)

Entry knowledge

Elementary knowledge of mathematics on the level of the secondary school. Linear and quadratic equations, inequalities, elements of the geometry of lines and planes.

Rules for evaluation and completion of the course

Students must first obtain the credit from seminars. Compulsory attendance at seminars. In the exercises are included 2 tests (each at most 12 points).. In total the exercises can receive a maximum of 24 points. A student has to obtain at least 6 points from each test. (Students are allowed to undergo corrective control work. Evaluation of corrective labor inspection is final.)

The exam is written. Students do not use any electronic devies during the exam, however they can use written preparation in the range of two A4 sheets.
The compulsory attendance at seminars. In the exercises are included 2 tests (each at most 12 points). In total the exercises can receive a maximum of 24 points. A student has to obtain at least 6 points from each test.

Aims

The aim of the course is making acquitance with the basic concepts of mathematics necessary for managing the following courses of physics, chemistry and engineering disciplines. Another claim is obtaining the basic principles of mathematical thinking and skills and applying them in the above mentioned courses.
The knowledge and skills will appear on the following fields
1. Students will manage successfully a work with matrices.
2. Students will be endowed with the knowledge of elementary functions and their properties. Students are expected to manage the concept of a limit and derivative and comprehend their meaning.They master their computation applying basic rules including the L´Hospital rule. Students will also be able to investgate the course of a function of one variable.
3. Students will be endowed with the knowledge of the indefinite and definite integral including the improper integral. They learn the basic methods of integral computations and be aquaitanced with the basic applications.
4. Students obtain the ability of solving simple tasks of the physical character and tasks occuring in the advanced courses.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Matematika online, http://mathonline.fme.vutbr.cz/ (CS)
Thomas G. B.: Calculus, Addison Wesley (EN)
Thomas G.B., Finney R.L.: Calculus and Analytic Geometry, Addison Wesley (EN)

Type of course unit

Lecture

26 hod., optionally

Teacher / Lecturer

doc. Ing. Vladislav Kozák, CSc.

Syllabus

1. Number sets, vectors, matrices. Matrix operations.
Sem. The recapitulation of selected themes of secondary schools. Introduction to matrices.
2. Linear independence, rank of matrices, determinants.
Sem. Matrix operations. Elementary row operations, rank.
3. Systems of linear equations. Frobenius theorem, Gaussian elimination, Cramer's rule.
Sem. Determinants to order 3. Systems of linear equations.
4. Geometry in E2 and E3: inner, outer and vector products. Lines and planes.
Sem. Systems of linear equations. Application of the products.
5. Geometry in E2 and E3: the role of angles and distances. Conics.
Sem. Parametric and general equations of lines and planes. Classification of conic sections and quadrics without mixed member (filling into a square).
6. Functions of one real variable. Basic features, graph. Inverse function.
Sem. TEST 1: 1) Matrix multiplication 2) Determinant 3) The system of linear equations 4) The geometry of lines and planes 5) Classification of conic sections and quadrics
7. Elementary functions: polynomials, rational functions, power functions, exponential and logarithmic functions, trigonometric functions.
Sem. Domains of elementary functions.
8. Derivative, geometric and physical meaning, calculation, chemical applications.
Sem.. Calculations of derivatives.
9. Calculations limits using of derivative (L'Hospital's rule). Taylor polynomial.
Sem. Taylor polynomial (briefly). Calculations of limits.
10. The determination of functions properties (with emphasis on the extremes).
Sem. Functions properties.
11. The method of least squares.
Cv. The method of least squares.
12. Interpolation polynomials and splines.
Cv. TEST 2: 1) Domain of functions 2) Derivative 3) [six-point example] Graphing functions
13. Summarizing lecture, discussion.
Cv. Interpolation polynomials and splines. Evaluation of seminars, granting credits.

Exercise

26 hod., compulsory

Teacher / Lecturer

Ing. Pavel Špaček, Dr.
doc. Ing. Vladislav Kozák, CSc.
Ing. Anna Derevianko, Ph.D.
Ing. Dušan Navrátil
Mgr. Hana Boháčková, Ph.D.
Ing. Petra Rozehnalová, Ph.D.

Syllabus

1. Stručné opakování vybraných témat středoškolské látky. Úvod do matic.
2. Operace s maticemi. Elementární úpravy, hodnost.
3. Determinant. Determinant stačí do řádu 3. Soustavy lineárních rovnic.
4. Soustavy lineárních rovnic – dokončení. Aplikace součinů.
5. Parametrické a obecné rovnice přímek a rovin. Klasifikace kuželoseček a kvadrik bez smíšeného členu (doplňování na čtverec).
6. TEST 1: 1) Násobení matic 2) Determinant 3) Soustava lineárních rovnic 4) Geometrie přímek a rovin 5) Klasifikace kuželoseček a kvadrik
7. Definiční obory elementárních funkcí.
8. Výpočty derivací.
9. Taylorův polynom (stručně). Výpočty limit.
10. Průběh funkce.
11. Metoda nejmenších čtverců.
12. TEST 2: 1) Definiční obor 2) Derivace 3) [šestibodový příklad] Průběh funkce
13. Interpolační polynomy a splajny. Vyhodnocení cvičení, udělení zápočtů.