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Course, academic year 2022/2023
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Calculus II - OPBM2M111A
Title: Matematická analýza II
Guaranteed by: Katedra matematiky a didaktiky matematiky (41-KMDM)
Faculty: Faculty of Education
Actual: from 2022
Semester: summer
E-Credits: 4
Examination process: summer s.:
Hours per week, examination: summer s.:2/1, Ex [HT]
Capacity: unknown / unknown (unknown)
Min. number of students: unlimited
Virtual mobility / capacity: no
State of the course: taught
Language: Czech
Teaching methods: full-time
Teaching methods: full-time
Is provided by: OKBM2M111A
Note: course can be enrolled in outside the study plan
enabled for web enrollment
priority enrollment if the course is part of the study plan
Guarantor: prof. RNDr. Ladislav Kvasz, DSc., Dr.
Pre-requisite : OPBM2M103A
Annotation -
Last update: Mgr. Derek Pilous, Ph.D. (03.02.2019)
Basics of differential and integral calculus - derivative, antiderivative, definite integral and their use.
Literature - Czech
Last update: Mgr. Derek Pilous, Ph.D. (03.02.2019)

§ Ross, K. A.: Elementary Analysis: The Tudory of Calculus. Undergraduate texts in Mathematics, Springer Verlag New York-Heidelberg-Berlin 1980

§ Fischer, E.: Intermediate Real Analysis. 1983

§ Jarník, V.: Diferenciální počet I, II. Academia, Praha 1984

§ Jarník, V.: Integrální počet I, II. Academia, Praha 1984

§ Veselý, J.: Matematická analýza pro učitele I, II. Matfyzpress, Praha 1997

§ Děmidovič, B. P.: Sbírka úloh a cvičení z matematické analýzy. Fragment, Praha 2004

Syllabus -
Last update: Mgr. Derek Pilous, Ph.D. (03.02.2019)

Weierstrass extreme value theorem and its applications.

Derivative -- definition, geometric interpretation. Existence and finiteness of derivative, examples. One-sided differentiability. Derivative as a function.

Derivative and continuity, relations and counterexamples. Derivative of continuous function.

Derivative of arithmetic operations, linearity of derivative.

Derivative of inverse and composite function.

Derivative of elementary functions. Calculation of derivative, limit of derivative theorem.

Mean value theorems (two-sided and one-sided versions), their geometric interpretation and applications.

Derivative and local/global monotonicity, isolated points and endpoints.

Derivative a convexity/concavity, isolated points and endpoints.

L'Hospital's rule and its use.

Taylor polynomials -- polynomial approximation of function, algebraic form, Lagrange remainder, applications.

Antiderivative -- definition, uniqueness, properties.

Newton integral and methods of its calculation.

Partitions of interval, lower nad upper Darboux sums, refinement of partition, relations.

Riemann integral (Darboux approach) -- definiton, equivalent condition of existence, examples of existence and nonexistence, improper integral.

Properties of Riemann integral -- linearity, monotonicity, additivity. Extension for lower limit being greater than upper one.

Fundamental theorem of calculus. Existence of antiderivative of function.

Course completion requirements - Czech
Last update: Mgr. Derek Pilous, Ph.D. (18.02.2022)

Písemný test nebo ústní zkouška z úloh (vyšetření průběhu funkce, určitý a neurčitý integrál) a ústní zkouška z teorie (podle sylabu). Zkouška z teorie je podmíněna úspěšným absolvováním testu nebo zkoušky z úloh.

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