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Course, academic year 2023/2024
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Calculus 1 - NMMA111
Title: Kalkulus 1
Guaranteed by: Department of Mathematical Analysis (32-KMA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2020
Semester: winter
E-Credits: 8
Hours per week, examination: winter s.:4/2, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: not taught
Language: Czech
Teaching methods: full-time
Teaching methods: full-time
Class: M Bc. FM
M Bc. FM > Povinné
M Bc. FM > 1. ročník
Classification: Mathematics > Real and Complex Analysis
Incompatibility : NMAA071, NMMA101, NMTM101
Interchangeability : NMAA071, NMMA101, NMTM101
Is co-requisite for: NMMA122, NMMA112
Is incompatible with: NMMA101
Is interchangeable with: NMAA071
In complex pre-requisite: NMFM204, NMFM205, NMMA211, NMMA212, NMMA221, NMNM211, NMSA336
Annotation -
The first part of a four-semester course in calculus for bachelor's program Financial Mathematics.
Last update: G_M (16.05.2012)
Course completion requirements - Czech

PODMÍNKY PRO SEMESTR 2018/19 jsou k dispozici na adrese

http://matematika.cuni.cz/pyrih-kalkulus.html

Last update: Pyrih Pavel, doc. RNDr., CSc. (24.09.2018)
Literature - Czech

ZÁKLADNÍ LITERATURA

M. Hušek, P. Pyrih: Matematická analýza, online http://matematika.cuni.cz/dl/analyza/

I. Černý : Inteligentní kalkulus, online http://matematika.cuni.cz/ikalkulus.html

KLASICKÁ LITERATURA

V. Jarník: Diferenciální počet I, Academia 1984, online http://matematika.cuni.cz/jarnik-all.html

L. Zajíček: Vybrané úlohy z matematické analýzy pro 1. a 2. ročník, Matfyzpress 2006

B. P. Děmidovič: Sbírka úloh a cvičení z matematické analýzy, Fragment 2003

W. Rudin: Principles of mathematical analysis, McGraw-Hill 1976

G. B. Thomas, M. D. Weir, J. Hass: Thomas' Calculus, Addison Wesley 2009

DOPLŇKOVÁ LITERATURA

J. Lukeš a kol.: Problémy z matematické analýzy, MFF UK 1982

I. Netuka, J. Veselý: Příklady z matematické analýzy III, MFF UK 1977

ODKAZY NA DALŠÍ LITERATURU

http://matematika.cuni.cz/BC-MA.html

Last update: Pyrih Pavel, doc. RNDr., CSc. (22.09.2012)
Teaching methods - Czech

Informace pro studující jsou k dispozici na adrese

http://matematika.cuni.cz/pyrih-kalkulus.html

Last update: Pyrih Pavel, doc. RNDr., CSc. (24.09.2018)
Syllabus -
1. Basic notions

a) Sets, relations, mappings

b) Axiomatics of real numbers, infimum and supremum

2. Limits of sequences

a) Limits and arithmetic operations, limits and inequalities, extension of reals

b) Limits of monotone sequences, Cantor nested interval theorem, Bolzano-Cauchy condition

c) Borel covering theorem. Cluster points of a sequence, lim sup

3. Series of real numbers

a) Convergent series, absolutely convergent series

b) Cauchy's root and ratio tests, Leibniz's test.

4. Limits and continuity of functions

a) Theorems on limits, Heine's approach to limits of functions. Bolzano-Cauchy condition for the convergence of functions

b) Limits and continuity, limit of a composition of functions, continuity of the inverse function

c) Properties of continuous functions on a closed interval. Intermediate value property, extrems, uniform continuity

5. Elementary transcendental functions

a) Polynomials, rational functions, n-th root

b) Exponential function, logarithm, power function

c) Trigonometric and hyperbolic functions, cyclometric functions

6. Derivative of function

a) Definition, derivative as a function, applications

b) Derivatives and arithmetic operations, derivative of composed and inverse function (chain rule)

c) Higher derivatives, Leibniz's formula

7. Properties of functions

a) Theorems of Rolle, Lagrange and Cauchy (mean value theorems)

b) Relation between derivative and monotonicity (convexity).

c) Extreme values, points of inflection, asymptots

Last update: G_M (27.04.2012)
Entry requirements -

Basic high school knowledge in mathematics.

Last update: Pyrih Pavel, doc. RNDr., CSc. (07.05.2018)
 
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