# 151-0502/02 – Mathematics B (MathB)

 Gurantor department Department of Mathematical Methods in Economics Credits 5 Subject guarantor prof. RNDr. Dana Šalounová, 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
Instruction secured by
GEN02 Mgr. Marian Genčev, Ph.D.
RUC05 RNDr. Pavel Rucki, Ph.D.
S1A20 prof. RNDr. Dana Šalounová, 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, to control basic terminology and related applications - to explain the concept of the primitive function and indefinite integral, to control the basic rules, formulas and techniques of integration - to define the definite integral (Darboux construction), to compute the definite integral with the help of Newton-Leibniz formula, to control the related basic geometric and economic applications - to define real functions of two real variables, to find the domain of functions of two variables and its visualization, to give the overview of basic functions of two variables used in economics, to explain the concept of homogenous functions of order 's' and to give the connections to economics - to define and to compute the partial derivatives with the help of their definitions and with the help of rules and formulas, to apply the partial derivatives for determining of local extremes (Hessian matrix), to define and interpret local extrema in a correct way, to discuss local extrema by means of their definition (inequality-type conditions, i.e., without the Hessian matrix), to find constrained extremes (Lagrange's multiplier) - to distinguish and to solve the basic types of differential and difference equations of 1st and 2nd order, to state the basic application possibilities in economics - to control the principles of difference calculus in connection with the monotonicity and dynamics of real sequences

Lectures
Tutorials

### Summary

The aim of the subject is to get acquainted with the basic knowledge of advanced mathematics (linear algebra, in-/definite integral, functions of two variables, differential and difference equations), which is necessary for further studies of quantitative methods in economics. The subject’s structure and nature themselves have their importance as they help to develop logical thinking as well as the ability to enunciate thoughts accurately and to give clear argumentation when solving practical problems.

### Compulsory literature:

[1] Larson R., Falvo C.D. Elementary Linear Algebra. Houghton Mifflin, Boston, New York, 2008. [2] Larson R., Edwards B. Calculus. Brooks/Cole Cengage Learning, Belmont, 2014. [3] Tan T.S. Calculus: Multivariable Calculus. Brooks/Cole Cengage Learning, Belmont, 2010. [4] Hoy M., Livernois J., McKenna Ch., Rees R., Stengos T. Mathematics for Economics. The MIT Press, London, 2011.

### Recommended literature:

[1] Stewart J.S. Calculus - Concepts and Contexts. Cengage Learning, 2010. [2] Canuto C., Tabacco A. Mathematical Analysis I. Springer Verlag, 2008. [4] Luderer B., Nollau V., Vetters K. Mathematical Formulas for Economists. Springer Verlag, 3rd ed., 2007. [5] Tan T.S. Calculus: Early Transcendentals. Brooks/Cole Cengage Learning, Belmont, 2011.

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

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

### E-learning

http://lms.vsb.cz/

Requirements for the successful closure of the course - valuation: max. 100 points, min. 51 points. Credit (max. 40 points, min. 20 points) The credit is successfully accomplished if the following conditions are satisfied: 1. Successfully accomplished written test, i.e., at least 50 %. The test will consist of problems presented on the seminars and verifies the computational skills of the student. 2. Completion of all problems submitted by the teacher. Exam (max. 60 points, min. 31 points) 1. The exam is combined and consist of two parts: (a) the written part, (b) the verbal part. 2. The goal of both parts of the exam is the total verification of student's knowledge within the course Mathematics B. In accordance with valid study prescriptions, the student is also requested to apply his mathematical knowledge in practical economic situations. Hence, the written part of the exam will consist of at least one problem associated with applications. Similarly conditions holds for the verbal part. Other concrete requirements will be established by the teacher. 3. The exam is successfully accomplished if the following valuations holds true: (a) written exam: max. 42 points, min. 22 points, (b) verbal exam: max. 18 points, min. 9 points.

### Prerequisities

Subject codeAbbreviationTitleRequirement
151-0500 Math A Mathematics A Compulsory

### Co-requisities

Subject has no co-requisities.

### Subject syllabus:

Syllabus 1. Systems of linear equations a their applications in economics - 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) 2. Introduction to analytical geometry in two- and three-dimensional spaces - arithmetic affine spaces, - affine line and plane equations, - intersections of affine lines and planes, - Euclidean geometry, - distances, orthogonality, angles 3. Indefinite integral of functions of one variable I - basic concepts - basic integration rules and techniques, - integration by substitution 4. Indefinite integral of functions of one variable II - integration by parts, - decomposition of rational functions into partial fractions 5. Volume of a plane area, construction of the definite integral - 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 calculation of the volume of an plane area, Newton-Leibniz formula, - basic applications in microeconomics 6. Generalized definite and improper integral - definitions of the generalized concepts, - computation techniques by using limits 7. Real functions of two real variables - definitions of basic concepts - constant and linear functions and their graphs, Cobb-Douglas production function, - homogenous functions of the degree 's', basic examples, 8. Partial derivatives and differentials of the function of two variables - geometric and economic meaning of the partial functions 'f(x_0,y)', resp. 'f(x,y_0)', - partial derivatives, - partial differentials and total differential of a function, geometric and numeric meaning, applications 9. Local extremes of functions of two variables - 'delta'-neighborhood of a point, definitions of strict and non-strict local extremes, - necessary, resp. sufficient condition for the existence of a local extreme, - constrained local extremes (substitution, Lagrange multipliers), basic applications in the economics 10. Ordinary differential equations I - introduction to DE of the 1st order, their connection to economics - separate and separable DE, - linear DE (Lagrange constant variation) 11. Ordinary differential equations II - introduction to DE of the 2nd order, their connection to economics, - linear DE of 2nd order with constant coefficients and special right-hand side (estimation of the solution form - method of undetermined coefficients) 12. Difference calculus I - the difference of a sequence, graphic and economic interpretation, - the proofs of selected rules used by difference computation, - sign of the difference, connection to the monotonicity, - differences of higher orders 13. Difference calculus II - introduction to ordinary difference equations, applications in economics or finantial mathematics, - the general solution of the equations 'Delta(a_n)=f(n)', 'a_{n+1}-r*a_n=f(n)' by means of summation, - linear difference equations of the 1st and 2nd order with constant coefficients and special right-hand side (estimation of the solution form - undetermined coefficients)

### Conditions for subject completion

Full-time form (validity from: 2016/2017 Winter semester)
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 special blocks

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