# 151-0342/03 – Mathematics G (Mat G)

 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 winter Study language Czech Year of introduction 2010/2011 Year of cancellation 2021/2022 Intended for the faculties EKF Intended for study types Bachelor
Instruction secured by
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

The student will be able ... • to interpret correctly the concept of the real function (one variable), • to find the domain of real functions of one real variable (by means of solving systems of non-linear inequations), • to characterize the basic properties of continuous functions, • to explain the behavior of certain discontinuous functions, • to compute and to interpret correctly the concept of limits of functions, • to define and compute the derivative of a function, • to interpret graphically the value of 1st and 2nd derivative of a function at a fixed point, • to find and to determine the local extrema, points of inflextion, asymptotes and to interpret these concepts graphical and from the practival point of view, • to control the basic rules for computing the antiderivative of a function, • to explain the concept of definite integral (Darboux approach), • to describe certain phenomenons with the help of the matrix algebra, • to solve the systems of linear equations by means of Gauß' elimination technique.

Lectures
Tutorials

### Summary

The course Mathematics G extends the basic concepts of mathematics and introduces the most important concepts of higher mathematics, i.e., the concept of limit of a function at a fixed point and the derivative of a function (as a special case of limit). Similarly, the course introduces the concept of integration (indefinite and definite integration). All this techniques have extensive applications in economic theories. Moreover, the course discuss also the solvability of systems of linear equations which forms the starting point for description of many problems from economic branche. Besides this facts, the student in the course should learn and fix the accuracy when arguing.

### 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. The 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

Requirements (100 points, minimum 51 points): Credit (45 points, minimal 23 points) 1. passing out the tests (at least 50 %), 2. sucesfull completion of all tasks until the end of the winter term (see the valid harmonogram). Exam (55 points, minimal 28 points) - combined 1. written part (max. 50 points, min. 26 points), 2. spoken part (max. 10 points, min. 5 points). Remark For passing out the spoken part of the exam, the absolute starting-popint is the precise accomplishment of all problems submitted by the teacher, especially theoretic problems and problems related to economic praxis. Submission of the problems, detailed instructions for elaborating and the relation of the submitted problems to PDF-versions of all lectures will be specified in the first seminar and also in the web-environment Moodle.

### Other requirements

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

### Prerequisities

Subject has no prerequisities.

### Co-requisities

Subject has no co-requisities.

### Subject syllabus:

------------------------------------------ Part 1 Differential calculus of one real variable ------------------------------------------ 1. Real functions of one real variable (number of lectures: 2) - domian and range of real functions, graphs of real functions, graphical interpretation of the graph of a function, - basic properties of real functions (odd and even functions, monotonicity, boundedness, one-to-one maps), - compositions of functions, - inverse functions 2. Continuity and limits of real functions (number of lectures: 2) - delta-neighborhood of a real point, left and right delta-neighborhoods, - continuity of real functions at real points and on closed intervals, properties of continuous functions on closed intervals (Weierstraß' theorem with consequences), - improper points and their arithmetic, reduced delta-neighborhood of a point, limits of functions at proper and improper points, the algebra of limits, 3. The derivative of a function (number of lectures: 1) - the possibilities of measuring the slope of a curve at the point x=x_0, the transition from a secant of a curve to the tangent at x=x_0, the meaning of the indeterminate term [0/0] and of the theory of limits for computing the slope of the curve at x=x_0, - the definition of the derivative of a function with the help of the derivative, - general derivatives of elementary functions, basic rules for computing derivatives 4. The course of the graph of a function (number of lectures: 2) - investigation of monotonicity with the help of the derivative sign, - local extremes of a function and their characteristics, investigation of extremes with the help of derivatives and in certain special cases also with the help of the definition, - convexity and concavity of a function, points of inflection, mathematical and practical meaning, - asymptotes, graphical and practical meaning ------------------------------------------------------- Part 2 Integral calculus of real functions of one ral variable ------------------------------------------------------- 5. The indefinite integral of a real function (number of lectures: 2) - basic concepts - basic integration rules and techniques, - integration by substitution, - integration by parts, - decomposition of rational functions into partial fractions 6. Volume of a plane area, construction of the definite integral (number of lectures: 1+) - construction of the upper and lower estimations of an plane area, - definition of an plane area by limiting procedure, - sketch of the proof of the formula for computation of the volume of an plane area (optional), Newton-Leibniz formula, - basic applications in the microeconomics ------------------------------------------ Part 3 Linear algebra ------------------------------------------ 9. Introduction to matrix algebra (number of lectures: 1) - definition of real matrices and related concepts, - basic classification of matrices by their type and by their values, - basics of matrix algebra (addition, subtraction, scalar multiplication, multiplication of matrices, power of a matrix, transposition), - stochastic matrices and their applications in the preference model 10. Number characteristics of real matrices, linear matrix equations, inverse matrices (number of lectures: 2) - rank of matrix, transformation to Gauß' form, related concepts, - definition and computation of determinants of orders n=2,3,4, Sarrus' rule, - properties of determinants, - Laplace's expansion, - basic geometric application of the determinant value, - definition and computation of the inverse matrix, adjoint matrix and other related concepts, - matrix equations of the form A+k*X=B, A*X=B, X*A=B, 11. Systems of linear equations a their applications in economics (number of lectures: 1) - definitions and basic concepts, - matrix notation, - Gauss' elimination and Frobenius' theorem, - systems of linear equations involving real parameters, - network analysis, polynomial curve fitting, Leontief input-output model (optional).

### Conditions for subject completion

Full-time form (validity from: 2010/2011 Winter semester, validity until: 2013/2014 Summer semester)
Min. number of pointsMax. počet pokusů
Exercises evaluation and Examination Credit and Examination 100 (100) 51
Exercises evaluation Credit 45 (45) 23 2
Písemka Written test 45  23 2
Examination Examination 55 (55) 28 3
Písemná zkouška Written examination 55  28 3
Mandatory attendence participation:

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Conditions for subject completion and attendance at the exercises within ISP:

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### Occurrence in study plans

Academic yearProgrammeField of studySpec.ZaměřeníFormStudy language Tut. centreYearWSType of duty 2016/2017 (B6202) Economic Policy and Administration (7202R020) Economics Journalism P Czech Ostrava 1 Compulsory study plan
2015/2016 (B6202) Economic Policy and Administration (7202R020) Economics Journalism P Czech Ostrava 1 Compulsory study plan
2014/2015 (B6202) Economic Policy and Administration (7202R020) Economics Journalism P Czech Ostrava 1 Compulsory study plan
2013/2014 (B6202) Economic Policy and Administration P Czech Ostrava 1 Compulsory study plan
2013/2014 (B6202) Economic Policy and Administration (7202R020) Economics Journalism P Czech Ostrava 1 Compulsory study plan
2012/2013 (B6202) Economic Policy and Administration P Czech Ostrava 1 Compulsory study plan
2011/2012 (B6202) Economic Policy and Administration P Czech Ostrava 1 Compulsory study plan
2010/2011 (B6202) Economic Policy and Administration P Czech Ostrava 1 Compulsory study plan
2010/2011 (B6202) Economic Policy and Administration (7202R020) Economics Journalism P Czech Ostrava 1 Compulsory study plan

### Occurrence in special blocks

Block nameAcademic yearForm of studyStudy language YearWSType of blockBlock owner Subject block without study plan - EKF - P - cs 2018/2019 Full-time Czech  Optional EKF - Faculty of Economics stu. block
Subject block without study plan - EKF - P - cs 2017/2018 Full-time Czech  Optional EKF - Faculty of Economics stu. block