SubjectsSubjects(version: 901)
Course, academic year 2021/2022
Mathematical Analysis 1a - NMAA001
Title: Matematická analýza 1a
Guaranteed by: Department of Mathematical Analysis (32-KMA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2017
Semester: winter
E-Credits: 8
Hours per week, examination: winter s.:4/2 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
Virtual mobility / capacity: no
State of the course: cancelled
Language: Czech
Teaching methods: full-time
Classification: Mathematics > Real and Complex Analysis
Incompatibility : NMAA071, NMAF033, NMAI008, NUMP001
Interchangeability : NHIU076, NMAA071, NMAF033, NMAI008, NMMA101, NUMP001
Is incompatible with: NMAA007, NMMA101, NMAA071, NMAA171, NMAI008, NMAA008, NMAI046
Is interchangeable with: NMAA171, NMAI046, NMAA071, NMMA101, NMAI008
In complex pre-requisite: NMAA003, NMAA004, NMAA069, NMAA070, NMAA074
Annotation -
Last update: T_KMA (23.05.2003)
Real numbers. Limits of sequences. Basic theory of series. Basic transcendental functions. Calculus of functions of a real variable.
Literature -
Last update: T_KMA (22.05.2008)

V. Jarník: Diferenciální počet I, Academia 1984

V. Jarník: Diferenciální počet II, Academia 1984

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

J. Milota: Matematická analýza I, 1. a 2. část (skriptum), MFF UK 1978

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


J. Čerych a kol.: Příklady z matematické analýzy V (skriptum), MFF UK 1983

P. Holický, O. Kalenda: Metody řešení vybraných úloh z matematické analýzy pro 2.-4. semestr, Matfyzpress 2006

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

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

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

Syllabus -
Last update: T_KMA (23.05.2003)
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

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