# 151-0401/05 – Mathematics B (MBKS)

 Gurantor department Department of Mathematical Methods in Economics Credits 5 Subject guarantor Mgr. Marian Genčev, Ph.D. Subject version guarantor Mgr. Marian Genčev, Ph.D. Study level undergraduate or graduate Requirement Compulsory Year 1 Semester summer Study language Czech Year of introduction 2019/2020 Year of cancellation Intended for the faculties EKF Intended for study types Bachelor
Instruction secured by
ARE30 Ing. Orlando Arencibia Montero, Ph.D.
FUN01 Mgr. Taťána Funioková, Ph.D.
GEN02 Mgr. Marian Genčev, Ph.D.
KUB33 Mgr. Aleš Kubíček
S1A20 prof. RNDr. Dana Šalounová, Ph.D.
Extent of instruction for forms of study
Form of studyWay of compl.Extent
Part-time Credit and Examination 6+8

### Subject aims expressed by acquired skills and competences

Knowledge, comprehension The student will be able to: - solve the systems of linear equations by using the Gaussian elimination and Cramer's rule, - control basic terminology and related applications in economics, - write the systems of linear equations with the help of the matrix notation - explain the concept of the primitive function and the indefinite integral, - explain and to control the basic rules, formulas and techniques of integration - define and to calculate the definite integral with the help of the Newton-Leibniz formula, - explain the validity of the geometric application (quadrature only), - present at least one application of the definite integral in economics - define the real function of two real variables, - give examples of basic functions of two variables (especially the constant, linear and Cobb-Douglass function), - give examples of using the functions of two variables in economics, - find the domain of the functions of two variables and its graphical visualization, - find the level curves of basic functions of two variables and to know the economic interpretation, - explain the concept of the homogenous functions of order 's' and to present the geometric and economic interpretation - define and to calculate the partial derivatives with the help of rules and formulas, - define the local extremes of functions of two variables, - interpret the local extremes in economics, - apply the partial derivatives for determining the existence and the nature of local extremes, - discuss the existence and nature of local extremes by means of their definition, - find the constrained extremes (the method of substitution, Lagrange's multiplier) - determine the type of a basic ordinary first-order differential equation, - solve basic types of first-order differential equations with the help of the direct integration, the constant variation and with the method of undetermined coefficients, - solve the basic second-order linear differential equations with the constant coefficients and with the special right-hand side by means of the method of undetermined coefficients, - outline at least one basic interpretation of the first- and second-order differential equations in economics - control and to explain the basic rules and formulas of the difference calculus, - explain the connection of the first- and second-order difference sign in connection with monotonicity and its dynamics, - determine the monotonicity of sequences with the help of the first- and second-order difference, - know the relationship between the summation and difference, - define the first- and second-order linear difference equations, - explain the existence of general first- and second-order linear difference equations with constant coefficients, - solve the first- and second-order linear difference equations with constant coefficients and with the special right-hand side, - find the closed form of basic finite sums with the help of the first-order linear difference equations, - present basic applications of difference equations in economics

### Teaching methods

Lectures
Individual consultations
Tutorials

### Summary

Aims of the subject are... - to get acquainted with further basic concepts of caluclus, - to develop the logical thinking and argumentation skills, - to develop the analytical and critical thinking, - to point out the basic application context of mathematics and economics.

### Compulsory literature:

Larson R., Falvo C.D.: Elementary Linear Algebra. Houghton Mifflin, Boston, New York (2008) Tan T.S.: Calculus: Multivariable Calculus. Brooks/Cole Cengage Learning, Belmont (2010) Hoy M., Livernois J., McKenna Ch., Rees R., Stengos T.: Mathematics for Economics. MIT Press, London (2011)

### Recommended literature:

Šalounová D., Poloučková A.: Úvod do lineární algebry. VŠB-TU, Ostrava (2002) Genčev M.: Cvičebnice ke kurzu Matematika A. SOT, Ostrava (2013) Moučka J., Rádl P.: Matematika pro studenty ekonomie. Grada, Praha (2010) Tan T.S.: Calculus: Early Transcendentals. Brooks/Cole Cengage Learning, Belmont (2011)

### Way of continuous check of knowledge in the course of semester

Credit (written form - test) ---------------------------- - Conditions for granting the credit: - passing the written test (success rate: 50 %), - active participation at seminars according to the teacher's instructions, - at most one unexcused abcsence. - In the Edison system, the redit will be confirmed with 40 pts if all the above conditions are met. Exam (obligatory written part, resp. optional oral part) -------------------------------------------------------- - written part (obligatory) - max. 30 pts, - min. 15 pts, - success rate: 50 %, - oral part (optional) - max. 30 pts, - min. 0 pts. - The oral part is conducted as an open-book discussion. - The subject of the exam will be the discussion of the material from the lecture according to the specification of the guarantee.

### Other requirements

According to teacher's requests in accoradnce with official conditions discussed with the department head.

### Prerequisities

Subject codeAbbreviationTitleRequirement
151-0400 MatKomb Mathematics A Compulsory

### Co-requisities

Subject has no co-requisities.

### Subject syllabus:

I. Systems of linear equations and analytic geometry ---------------------------------------------------- - basic concepts, - Gaussian elimination, Frobenius' theorem, - Cramer's rule, - basic applications in economics II. Integral calculus --------------------- Indefinite integral - definition and properties, - basic integration formulas and rules, - per partes, substitution, - integration of rational functions (partial fractions), - basic applications in economics Definite integral - the problem of calculating the area of a region bounded by continuous curves - definitions a properties of the definite integral, - Newton-Leibniz' formula, - basic applications in economics Generalized and improper integral - improper integral of the first and second kind, - Gaussian integral (for information only), - calculating improper integrals by limits, - generalized definite integrals (the case of discontinuous functions), - basic applications in economics and connection with statistics III. Functions of two real variables ------------------------------------ - definitions of basic concepts, - domain and its visualization, - homogeneous functions of order 's', - partial derivatives and their geometric interpretation, - tangent plane, - total differential, differentiable functions, approximations of number expressions, - local extremes, - constrained local extremes - method of substitution, - Lagrange's multiplier, - basic applications in economics IV. Ordinary differential equations (ODE) ----------------------------------------- - definition of ODE, - order of ODE, - solution of ODE, - basic types of first-order ODE's - separated, - separable, - linear first-order ODE (variation of constants), - second-order linear ODE with constant coefficients and special right-hand side (undetermined coefficients), - basic applications in economics V. Difference calculus and difference equations ----------------------------------------------- Introduction to difference calculus - difference of order 'k', - basic formulas and rules for calculating the differences, - the sign of the first-order difference as the indicator of the sequence monotonicity, - the sign of the second-order difference as the indicator of the sequence monotonicity dynamics, - relation of summation and difference Ordinary difference equations (ODifE) - definition of the ODifE, - order of the ODifE, - solution of the ODifE, - first- and second-order ODifE with constant coefficients and special right-hand side (undetermined coefficients), - basic applications in economics

### Conditions for subject completion

Part-time form (validity from: 2021/2022 Winter semester)
Min. number of points
Credit and Examination Credit and Examination 100 (100) 51
Credit Credit 40 (40) 40
Písemka Written test 40  40
Examination Examination 60 (60) 15
Písemná zkouška Written examination 30  15
Ústní zkouška Oral examination 30  0
Mandatory attendence parzicipation: One unexcused absence at seminars.

Show history

### Occurrence in study plans

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2021/2022 (B0312A050001) Public Economics and Administration K Czech Valašské Meziříčí 1 Compulsory study plan
2021/2022 (B0312A050001) Public Economics and Administration K Czech Ostrava 1 Compulsory study plan
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2021/2022 (B0413A050012) Economics and Management (S02) Business Administration K Czech Ostrava 1 Compulsory study plan
2021/2022 (B0413A050012) Economics and Management (S03) Management K Czech Ostrava 1 Compulsory study plan
2021/2022 (B0311A050004) Applied Economics (S01) International Economic Relations K Czech Ostrava 1 Compulsory study plan
2021/2022 (B0311A050004) Applied Economics (S02) Economic Development K Czech Ostrava 1 Compulsory study plan
2020/2021 (B0312A050001) Public Economics and Administration K Czech Ostrava 1 Compulsory study plan
2020/2021 (B0312A050001) Public Economics and Administration K Czech Šumperk 1 Compulsory study plan
2020/2021 (B0312A050001) Public Economics and Administration K Czech Valašské Meziříčí 1 Compulsory study plan
2020/2021 (B0413A050012) Economics and Management (S02) Business Administration K Czech Ostrava 1 Compulsory study plan
2020/2021 (B0413A050012) Economics and Management (S03) Management K Czech Ostrava 1 Compulsory study plan
2020/2021 (B0311A050004) Applied Economics (S01) International Economic Relations K Czech Ostrava 1 Compulsory study plan
2020/2021 (B0311A050004) Applied Economics (S02) Economic Development K Czech Ostrava 1 Compulsory study plan
2019/2020 (B0312A050001) Public Economics and Administration K Czech Ostrava 1 Compulsory study plan
2019/2020 (B0312A050001) Public Economics and Administration K Czech Šumperk 1 Compulsory study plan
2019/2020 (B0312A050001) Public Economics and Administration K Czech Valašské Meziříčí 1 Compulsory study plan
2019/2020 (B0413A050012) Economics and Management (S02) Business Administration K Czech Ostrava 1 Compulsory study plan
2019/2020 (B0413A050012) Economics and Management (S03) Management K Czech Ostrava 1 Compulsory study plan
2019/2020 (B0311A050004) Applied Economics (S01) International Economic Relations K Czech Ostrava 1 Compulsory study plan
2019/2020 (B0311A050004) Applied Economics (S02) Economic Development K Czech Ostrava 1 Compulsory study plan

### Occurrence in special blocks

Block nameAcademic yearForm of studyStudy language YearWSType of blockBlock owner