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The first part of a four-semester course in mathematical analysis for bachelor's programs General Mathematics
and Information Security.
Last update: G_M (16.05.2012)
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Detailed requirements will be posted on the instructor's personal website. Last update: Pick Luboš, prof. RNDr., CSc., DSc. (28.09.2022)
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BASIC LITERATURE
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 COMPLEMENTARY READING 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 Last update: G_M (24.04.2012)
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Přednáška i cvičení probíhají presenčně. Přednášky budou nahrávány, avšak nebudou streamovány. Last update: Pick Luboš, prof. RNDr., CSc., DSc. (26.09.2024)
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1. Introduction
Logic, mappings, countable sets, real numbers, supremum property, complex numbers. 2. Limit of sequences Convergence of sequences, the limit of a monotone sequence, values of accumulation, limsup, liminf, Bolzano-Weierstrass theorem, Cantor principle, Bolzano-Cauchy condition. 3. Limit of functions and continuity Basic notions, limit, the neighborhood of a point, limit and continuity at a point (one-sided versions included), theorems on limits and arithmetics, ordering, the limit of a composed function, Heine theorem, the limit of a monotone function, continuous functions on an interval (intermediate value theorem, continuous image, boundedness, attaining of extrema, continuity of an inverse function. 4. Elementary functions Exponential, logarithmic, goniometric and cyclometric, and power function (without proofs). 5. Derivative Definition and basic properties, arithmetics of derivatives, the derivative of a composed function, derivative of an inverse function, derivative of elementary functions, Rolle theorem, Lagrange theorem, Cauchy theorem, l'Hospital rule, the limit of a derivative at a point, monotonicity and sign of a derivative, convex and concave functions, inflection point, derivative and convexity, asymptote, investigation of a function. 6. Taylor polynomial Taylor polynomial, Peano theorem, Lagrange theorem, Cauchy theorem, the symbol "little o" and its properties, Taylor polynomial of elementary functions. Last update: Pick Luboš, prof. RNDr., CSc., DSc. (30.07.2022)
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