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Real numbers. Limits of sequences. Basic theory of series. Basic transcendental functions.
Calculus of functions of a real variable.
Last update: G_M (13.05.2010)
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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: T_KMA (18.09.2012)
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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 (13.05.2010)
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