B_MaA_2 Mathematics A 2

University of Finance and Administration
Summer 2013
Extent and Intensity
2/2. 5 credit(s). Type of Completion: zk (examination).
Teacher(s)
doc. RNDr. Slavomír Burýšek, CSc. (seminar tutor)
RNDr. Eva Ulrychová, Ph.D. (seminar tutor)
Guaranteed by
doc. RNDr. Slavomír Burýšek, CSc.
Department of Computer Science and Mathematics – Departments – University of Finance and Administration
Contact Person: Dagmar Medová, DiS.
Timetable of Seminar Groups
B_MaA_2/cAPH: Wed 12:15–12:59 E304, Wed 13:00–13:45 E304, S. Burýšek
B_MaA_2/pAPH: Wed 10:30–11:14 E304, Wed 11:15–12:00 E304, S. Burýšek
Prerequisites
Mathematics B_Ma_A_1
Course Enrolment Limitations
The course is offered to students of any study field.
Course objectives
Know necessary and sufficient conditions for the existence of local maximum and minimum of a given function of many variables and to know how to compute the extremes. They should be able to know the notion of the constrained extreme, to know Jacobi’s method, the method of Lagrange’s multipliers and to know how to compute global extremes of a given function on a compact set. They should be able to compute the indefinite and definite integrals by parts and by the substitution. They should be able to decide about the convergence or divergence of numerical and functional series, to know the domain of convergence of the power series and can express a given function by Taylor’s series.
1. To gain basic knowledge from theory of functions of many variables, particularly functions of two variables, to determine the domain of continuity, computation of partial derivatives and the total differential of a given function, to gain knowledge on solving problems of local, constrained and global extremes of functions.
2. To gain basic knowledge from theory of indefinite integral and methods of its computation, the notion of Riemann’s definite integral and to be able to apply this knowledge on specific problems.
3. To gain basic knowledge from theory of infinite numerical and functional series, specially power and Taylor’s series.
Syllabus
  • 1. Basic topological notions in Euclidean space E_n: metric, neighbourhood of a point, open , closed bounded sets, compact set.
  • 2. Function of many variables, domain, graph, partial derivatives and the gradient of a function.
  • 3. Total differential of a function, partial derivatives of higher order, Hess’s matrix.
  • 4. Local extremes of functions of many variables.
  • 5. Constrained extremes, Jacobi’s method and method of Lagrange’s multipliers.
  • 6. Weierstrass’s theorem. Global extremes of functions on a compact sets
  • 7. Definition and basic properties of indefinite integral, integration by parts.
  • 8. Integration by substitution, integration of rational functions.
  • 9. Definition and properties of Riemann’s definite integral, Newton-Leibniz formula. The improper (infinite) integral.
  • 10. Some applications of the definite integral (area of a surface, length of a curve, volume of a rotating body). Numerical series, criteria of the absolute convergence, Leibniz’s criterion for alternating series.
  • 11. Power series, radius and domain of convergence, operations with power series.
  • 12. Expansion of a function in a power series, Taylor’s series and its applications.
Literature
    required literature
  • BUDINSKÝ, Petr a Ivan HAVLÍČEK. Matematika pro vysoké školy ekonomického a technického zaměření. EUPRESS, Praha 2005, ISBN 80-886754-45-6.
  • BUDINSKÝ, Petr a Ivan HAVLÍČEK. Sbírka příkladů z matematiky pro vysoké školy ekonomického a technického zaměření. 1. vyd. Praha: Vysoká škola finanční a správní, 2005, 121 s. ISBN 80-867-5452-9.
  • KLŮFA, Jindřich. Matematika pro studenty VŠE. Vyd. 1. Praha: Ekopress, 2011, 188 s. ISBN 978-808-6929-743.
    recommended literature
  • KAŇKA, Miloš, Jan COUFAL a Jindřich KLŮFA. Učebnice matematiky pro ekonomy. 1. vyd. Praha: Ekopress, 2007, 198 s. ISBN 978-80-86929-24-8.
Teaching methods
Lectures and seminars in full-time study; tutorials in part-time study; compulsory seminar participation is 75% in full-time study, compulsory tutorial participation is 50% in part-time study. Students with lower than required participation have to fulfill additional study duties.
Assessment methods
The course ends with an exam. The exam is based on active participation on lectures and seminars, in creation and presentation of a final seminar study and on specific written test (5 out of 10 points required) and subsequent verbal exam (answer one of the 15 topic questions correctly is required) to pass the exam.“
Language of instruction
Czech
Further comments (probably available only in Czech)
The course can also be completed outside the examination period.
General note: Aa1.
Information on the extent and intensity of the course: 10 hodin KS/semestr.
The course is also listed under the following terms Winter 2007, Summer 2008, Winter 2008, Summer 2009, Winter 2009, Summer 2010, Winter 2010, Summer 2011, Winter 2011, summer 2012, Winter 2012, Summer 2014, Summer 2015, Summer 2016, Summer 2017, Summer 2018, Summer 2019.
  • Enrolment Statistics (Summer 2013, recent)
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