## Mathematics B

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Course Unit Code Number of ECTS Credits Allocated Type of Course Unit * Level of Course Unit * Year of Study * 151-0301/04 5 ECTS credits Compulsory First Cycle First Year Summer Semester Face-to-face Czech, English, German, Spanish 151-0300 Mathematics A ARE30 Ing. Orlando Arencibia Montero, Ph.D. S1A20 prof. RNDr. Dana Šalounová, Ph.D. FUN01 Mgr. Taťána Funioková, Ph.D. RUC05 RNDr. Pavel Rucki, Ph.D. KOZ214 Ing. Mgr. Petr Kozel, Ph.D. ZAH0001 Marek Zahradníček GEN02 Mgr. Marian Genčev, Ph.D. The aim of the subject is to get acquainted with the basic knowledge of advanced mathematics, 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 various practical problems. Knowledge, comprehension The student will be able... - to solve the systems of linear equations (including simple parameterized coefficients), 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, to compute the definite integral with the help of Newton-Leibniz formula, to outline the proof 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 (mainly constant, linear and Cobb-Douglass function), to explain the concept of homogenous functions of order 's', to present geometrical point of view 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 (method of substitution, 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, to explain the connection between summation and difference, to find closed form of basic sums by means of 1st order difference equations (1) Linear algebra ============================== Systems of linear equations - introduction and definitions, solvability, methods of solving, Frobenius theorem, system consistence in case of parameterized coefficients. Analytical geometry in 2- and 3-dimensional Euclidean space - basic concepts, mutual position of plains and/or lines in 2- and 3-dimensional space, the distance between points, lines and planes. Appliacations of systems of linear equations in economics. (2) Indefinite integral - introduction ============================== Indefinite integral of functions of one variable - basic definitions and rules, per-partes technique, substitution, integration of rational functions. Definite integral - basic definitions and properties, Newton-Leibniz formula, principle of its proof, geometric and economic applications. Generalized and improper integral - definitions and basic properties. (3) Functions of two real variables ============================== Basic definitions, domain determination and its graphic representation, homogenous functions of order 's'. Partial functions, partial derivatives, tangent plane and normal line to a surface, partial and total differential. Local extremes (un-/constrained), method of substitution and Lagrange's function. Applications in economics. (4) Ordinary differential equations ============================== Ordinary differential equations (ODE), order of ODE, general, particular and singular solution, basic types of ODE of the 1st and 2nd order (separable, linear), the method of undetermined coefficients. Applications in economics. (5) Differential calculus ============================== Introduction to the calculus of differences - difference of order 'k', the signum of the difference as an indicator of sequence monotonicity. Difference equations - its order and solution, linear difference equations solved by undetermined coefficients. Summation and its connection to difference calculus, closed form of finite sums solved by means of 1st order difference equations. Applications in economics. (6) Final considerations ============================== [1] Larson R., Falvo C.D. Elementary Linear Algebra. Houghton Mifflin, Boston, New York, 2008. [2] Tan T.S. Calculus: Multivariable Calculus. Brooks/Cole Cengage Learning, Belmont, 2010. [3] Hoy M., Livernois J., McKenna Ch., Rees R., Stengos T. Mathematics for Economics. The MIT Press, London, 2011. [1] Genčev M., Rucki P. Cvičebnice z matematiky nejen pro ekonomy. SOT, Ostrava, 2017. [2] Genčev M., Hrubá J., Pulcerová S., Rucki P. Matematika A. SOT, Ostrava, 2013. [3] Genčev M., Hrubá J., Pulcerová S., Rucki P. Matematika B. SOT, Ostrava, 2013. [4] Genčev M. Cvičebnice ke kurzu Matematika B. SOT, Ostrava, 2013. [1] Stewart J.S. Calculus - Concepts and Contexts. Cengage Learning, 2010. [2] Canuto C., Tabacco A. Mathematical Analysis I. Springer Verlag, 2008. [3] Luderer B., Nollau V., Vetters K. Mathematical Formulas for Economists. Springer Verlag, 3rd ed., 2007. [4] Tan T.S. Calculus: Early Transcendentals. Brooks/Cole Cengage Learning, Belmont, 2011. [1] Šalounová D., Poloučková A. Úvod do lineární algebry. VŠB-TU, Ostrava, 2002. [2] Genčev M. Cvičebnice ke kurzu Matematika A. SOT, Ostrava, 2013. [3] Moučka J., Rádl P. Matematika pro studenty ekonomie. Grada, Praha, 2010. [4] Poloučková A., Ošťádalová E. Diferenciální a diferenční rovnice. VŠB-TU, Ostrava, 2003. [5] Ošťádalová E., Ulmannová V. Integrální počet (cvičení pro 1. ročník EkF, VŠB-TU Ostrava). VŠB-TU, Ostrava, 2000. Lectures, Individual consultations, Tutorials, Other activities 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