# 151-0502/02 – Mathematics B (MathB)

 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 English Year of introduction 2015/2016 Year of cancellation Intended for the faculties EKF Intended for study types Bachelor
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LoginNameTuitorTeacher giving lectures
GEN02 Mgr. Marian Genčev, Ph.D.  Extent of instruction for forms of study
Form of studyWay of compl.Extent
Full-time Credit and Examination 2+2

### 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) - recognize the type of a basic ordinary first-order differential equation, - explain the existence and the solution form of the first- and second-order differential equations, - 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, - explain the solution form of homogeneous 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
Other activities

### Summary

Aims of the subject are... - to get acquainted with further basic concepts of caluclus, - to develop the logical thinking and argumentation skills, - 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:

Stewart J.S.: Calculus - Concepts and Contexts. Cengage Learning (2010) Canuto C., Tabacco A.: Mathematical Analysis I. Springer Verlag (2008) Luderer B., Nollau V., Vetters K.: Mathematical Formulas for Economists. Springer Verlag, 3rd ed. (2007) 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) - max. 40 pts, - min. 20 pts Examination (written and oral) - written exam (written test) - max. 42 pts, - min. 22 pts - oral exam (open-book) - max. 18 pts, - min. 9 pts

### E-learning

http://lms.vsb.cz/

### Other requirements

- active participation on seminars, - maximal one unexcused absence, - understanding the basic concepts and methods presented at the lecture, - completion of all tasks assigned by the lecturer

### Prerequisities

Subject codeAbbreviationTitleRequirement
151-0500 Math A 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, - use of systems of linear equations for determining the mutual position of - two planes in E3, - two lines in E2 and E3, - plane and a line in E3 - 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 (general, particular, singular, extraordinary), - basic types of first-order ODE's - separated, - separable, - linear first-order DE (variation of constants), - second-order linear DE 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 (general, particular) - first- and second-order ODifE with constant coefficients and special right-hand side (undetermined coefficients) - basic applications in economics

### Conditions for subject completion

Full-time form (validity from: 2016/2017 Winter semester)
Task nameType of taskMax. number of points
(act. for subtasks)
Min. number of points
Credit and Examination Credit and Examination 100 (100) 51
Credit Credit 40 (40) 20
Písemka Written test 40  20
Examination Examination 60 (60) 31
Písemná zkouška Written examination 42  22
Ústní zkouška Oral examination 18  9
Mandatory attendence parzicipation:

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### Occurrence in study plans

Academic yearProgrammeField of studySpec.ZaměřeníFormStudy language Tut. centreYearWSType of duty 2021/2022 (B0311A050005) Applied Economics (S01) Economic Development P English Ostrava 1 Compulsory study plan
2021/2022 (B0311A050005) Applied Economics (S02) International Economic Relations P English Ostrava 1 Compulsory study plan
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2020/2021 (B0311A050005) Applied Economics (S02) International Economic Relations P English Ostrava 1 Compulsory study plan
2020/2021 (B0311A050005) Applied Economics (S01) Economic Development P English Ostrava 1 Compulsory study plan
2020/2021 (B0412A050006) Finance P English Ostrava 1 Compulsory study plan
2019/2020 (B0311A050005) Applied Economics (S02) International Economic Relations P English Ostrava 1 Compulsory study plan
2019/2020 (B0311A050005) Applied Economics (S01) Economic Development P English Ostrava 1 Compulsory study plan
2019/2020 (B0412A050006) Finance P English Ostrava 1 Compulsory study plan
2018/2019 (B6202) Economic Policy and Administration (6202R010) Finance P English Ostrava 1 Compulsory study plan
2017/2018 (B6202) Economic Policy and Administration (6202R010) Finance P English Ostrava 1 Compulsory study plan
2016/2017 (B6202) Economic Policy and Administration (6202R010) Finance P English Ostrava 1 Compulsory study plan
2015/2016 (B6202) Economic Policy and Administration (6202R010) Finance P English Ostrava 1 Compulsory study plan

### Occurrence in special blocks

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