Mathematical Analysis 2 - NMMA102
Title in English: Matematická analýza 2
Guaranteed by: Department of Mathematical Analysis (32-KMA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2018 to 2018
Semester: summer
E-Credits: 10
Hours per week, examination: summer s.:4/4 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech
Teaching methods: full-time
Guarantor: prof. RNDr. Luboš Pick, CSc., DSc.
Class: M Bc. MMIB
M Bc. MMIB > Povinné
M Bc. MMIB > 1. ročník
M Bc. MMIT
M Bc. MMIT > Povinné
M Bc. OM
M Bc. OM > Povinné
M Bc. OM > 1. ročník
Classification: Mathematics > Real and Complex Analysis
Co-requisite : NMMA101
Incompatibility : NMAA002
Interchangeability : NMAA002
In complex pre-requisite: NMAG204, NMFM205, NMMA201, NMMA202, NMMA203, NMNM201, NMSA336
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Annotation -
Last update: G_M (16.05.2012)
The second part of a four-semester course in mathematical analysis for bachelor's programs General Mathematics and Information Security.
Course completion requirements - Czech
Last update: prof. RNDr. Luboš Pick, CSc., DSc. (16.02.2019)

Podrobné požadavky k zápočtu a ke zkoušce jsou uvedeny na webové stránce přednášejícího http://www.karlin.mff.cuni.cz/~pick/

Literature -
Last update: prof. RNDr. Luboš Pick, CSc., DSc. (16.02.2019)
BASIC LITERATURE

lecture notes

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

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

COMPLEMENTARY READING

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

Syllabus -
Last update: prof. RNDr. Luboš Pick, CSc., DSc. (16.02.2019)
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