# 151-0502/01 – Mathematics B (MathB)

 Gurantor department Department of Mathematical Methods in Economics Credits 4 Subject guarantor Mgr. Marian Genčev, Ph.D. Subject version guarantor Mgr. Marian Genčev, Ph.D. Study level undergraduate or graduate Requirement Compulsory Year 2 Semester summer Study language English Year of introduction 1999/2000 Year of cancellation 2014/2015 Intended for the faculties EKF Intended for study types Bachelor
Instruction secured by
GEN02 Mgr. Marian Genčev, Ph.D.
MAJ40 PaedDr. Renata Majovská, PhD.
SOB33 RNDr. Simona Pulcerová, Ph.D., MBA
Extent of instruction for forms of study
Form of studyWay of compl.Extent
Full-time Credit and Examination 1+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

active participation on seminars, maximal three unexcused absences, understanding the basic concepts and methods presented at the lecture

### Other requirements

no further requirements

### Prerequisities

Subject codeAbbreviationTitleRequirement
151-0500 Math A Mathematics A Compulsory

### Co-requisities

Subject has no co-requisities.

### Subject syllabus:

Part 1 - Linear Algebra ========================= * Systems of linear equations - basic concepts, equivalent systems of equations, Gauß elimination, Frobenius Theorem, the solvability of systems of linear equations; analytic geometry in affine spaces A2, A3 and Euclidean spaces E2, E3 - basic affine concepts, line and plane equations; vector spaces with scalar multiplication, norm of general vectors, mutual position of planes, lines and their combinations; the distance of a point from a line or plane in E2 and E3. Part 2 - Introduction to integral calculus ========================= * Antiderivative of functions - definitions and basic concepts, rules of integration, integrations by parts. * Antiderivative of functions - integration by substitution (transformations of integrals), integration of basic rational, irrational and goniometric functions. * The definite integral, properties of the Riemann integral, Newton-Leibniz' formula, geometric application of Riemann integral (computation of area-largeness). * The definite integral - definition of improper integral, basic properties. Part 3 - Introduction to differential calculus of two variables ========================= * Functions of two variables - basic concepts, the domain of real functions of two real variables and their visualisation; graph of a function and its visualisation. * Functions of two variables - partial derivatives of first and higher orders; tangent plane, total differential of a function and its basic applications. * Functions of two variables - local extrema and basic optimization methods (unconstrained optimization). * Functions of two variables - constrained optimization (elimination of variables, the method of Lagrange multiplier). Part 4 - Ordinary differential equations (ODE) ========================= * 1st-order ODE - basic concepts, general solution, particular solution; solving ODE by separation and integration; linear differential equations with non-constant coefficients, variation of constant. * 2nd-order ODE - 2nd-order linear differential equations with constant coefficients and special RHS, solution estimation. Part 5 - Ordinary difference equations ========================= * Introduction to difference calculus - basic concepts, general and particular solution, 1st-order linear difference equation with constant coefficients and special RHS, solution estimation. * Introduction to difference calculus - 2nd-order linear difference equation with constant coefficients and special RHS, solution estimation. Part 6 (formally) ========================= Revision of the previous concepts and their interrelationship.

### Conditions for subject completion

Full-time form (validity from: 1960/1961 Summer semester, validity until: 2014/2015 Summer semester)
Min. number of points
Exercises evaluation and Examination Credit and Examination 100 (100) 51
Exercises evaluation Credit 40 (40) 20
Examination Examination 60 (60) 31
Written examination Written examination 50  26
Ústní zkouška Oral examination 10  5
Mandatory attendence parzicipation:

Show history

### Occurrence in study plans

Academic yearProgrammeField of studySpec.ZaměřeníFormStudy language Tut. centreYearWSType of duty
2006/2007 (B6202) Economic Policy and Administration (6202R010) Finance (01) Finance P Czech Ostrava 1 Compulsory study plan
2005/2006 (M6202) Economic Policy and Administration (6201T004) Economics P Czech Ostrava 2 Compulsory study plan
2004/2005 (M6202) Economic Policy and Administration (6201T004) Economics P Czech Ostrava 2 Compulsory study plan
2004/2005 (M6208) Business and Management (6201T004) Economics P Czech Ostrava 2 Compulsory study plan
2003/2004 (M6202) Economic Policy and Administration (6201T004) Economics P Czech Ostrava 2 Compulsory study plan
2003/2004 (B6202) Economic Policy and Administration (6201R004) Economics P Czech Ostrava 2 Compulsory study plan
2003/2004 (B6208) Economics and Management (6201R004) Economics P Czech Ostrava 2 Compulsory study plan

### Occurrence in special blocks

Block nameAcademic yearForm of studyStudy language YearWSType of blockBlock owner