Mathematical analysis I.1
General data
Course ID: | 1000-111bAM1a |
Erasmus code / ISCED: |
11.101
|
Course title: | Mathematical analysis I.1 |
Name in Polish: | Analiza matematyczna I.1 (potok I) |
Organizational unit: | Faculty of Mathematics, Informatics, and Mechanics |
Course groups: |
Obligatory courses for 1st grade JSEM Obligatory courses for 1st grade JSIM Obligatory courses for 1st grade Mathematics |
ECTS credit allocation (and other scores): |
10.00
|
Language: | Polish |
Main fields of studies for MISMaP: | mathematics |
Type of course: | obligatory courses |
Short description: |
The course introduces the fundamental notions of differential calculus in one variable. It covers properties of real and rational numbers, mathematical induction, limits of sequences (including the Bolzano-Weierstrass theorem), convergence of series (from basic criteria to the Cauchy multiplication of series), limits and continuity of a function of one variable, properties of continuous functions (intermediate value property, Weierstrass' theorem), convex functions and the notion of a derivative. |
Full description: |
1. Real numbers, infima and suprema of sets, Dedekind's axiom. Natural, integer, rational and irrational numbers, mathematical induction and its applications. 2. Limits of sequences (also infinite), Cauchy's condition, existence of limits of monotone sequences. Existence of n-th roots of positive numbers. Basic techniques for calculating limits, the definition of e. Cesaro-Stolz theorem. Subsequences, Bolzano-Weierstrass theorem. 3. Series of real and complex numbers, convergence of a series. Geometric series, expanding real numbers in numeral systems with different bases. Cauchy's condition for series. Series of non-negative numbers, comparison tests, Cauchy's condensation test, d'Alembert's ratio test, Cauchy's root test. Series of arbitrary numbers - dependence of the series' sum on the order of summation. Alternating series and Leibniz' test. Absolutely convergent series. Abel's and Dirichlet's tests. Cauchy's product of series and its convergence. Irrationality of e. 4. Limit of a function at a point, continuity of a function (Heine's and Cauchy's conditions), intermediate value property. Continuity of the inverse function. Weierstrass' extreme value theorem. Uniform continuity of a continuous function defined on a closed interval. Exponential, logarythmic, trigonometric and cyclometric functions. 5. Convex functions and their geometric interpretation. Jensen's inequality and its consequences: the inequality between arythmetic and geometric means, Schwartz' inequality. Derivative of a function and its basic properties, tangent to the graph of a function. Characterizing convexity in terms of difference quotients and the derivative. |
Bibliography: |
1. A. Birkholc, Analiza matematyczna dla nauczycieli. PWN, Warszawa 1977. 2. B. P. Demidowicz, Zbiór zadań z analizy matematycznej, Naukowa Książka, Lublin 1992 (t. I) i 1993 (t. II i III). Baranenkov, G. S.; Demidovich, B.; Efimenko, V. A.; Kogan, S. M.; Lunts, G.; Porshneva, E.; Sycheva, E.; Frolov, S. V.; Shostak, R.; Yanpolsky, A., Problems in mathematical analysis. (Under the editorship of B. Demidovich). Mir Publishers, Moscow, 1976. 3. G. M. Fichtenholz, Rachunek różniczkowy i całkowy. Tom 1-2, PWN, Warszawa 2007. 4. W. Kaczor, M. Nowak, Zadania z Analizy Matematycznej 1. Liczby rzeczywiste, ciągi i szeregi liczbowe, PWN, Warszawa 2005. W. Kaczor, M. Nowak, Problems in Mathematical Analysis I, AMS Student Mathematical Library (Book 4), 2000. 5. K. Kuratowski, Rachunek różniczkowy i całkowy, PWN, Warszawa 1979. K. Kuratowski, Introduction to calculus. International Series of Monographs in Pure and Applied Mathematics, Vol. 17. Pergamon Press, Oxford-Edinburgh-New York; PWN—Polish Scientific Publishers, Warsaw, 1969 6. W. Pusz, A. Strasburger, Zbiór zadań z analizy matematycznej Wydział Fizyki UW, Warszawa 1982. 7. W. Rudin, Podstawy analizy matematycznej, PWN, Warszawa 2000. W. Rudin, Principles of mathematical analysis. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976. 8. P. Strzelecki, Analiza Matematyczna I (lecture notes in Polish), http://dydmat.mimuw.edu.pl/sites/default/files/wyklady/analiza-matematyczna-i.pdf Addition to the lecture notes (auth. M. Jóźwikowski, S. Kolasiński), http://dydmat.mimuw.edu.pl/sites/default/files/wyklady/analiza-matematyczna-i-zadania.pdf |
Learning outcomes: |
The student
|
Assessment methods and assessment criteria: |
Credit based on the sum of points from exercises, and tests. |
Classes in period "Winter semester 2023/24" (past)
Time span: | 2023-10-01 - 2024-01-28 |
Navigate to timetable
MO TU WYK
WYK
CW
CW
CW
CW
CW
CW
CW
CW
CW
CW
CW
CW
W CW
TH WYK
WYK
CW
CW
CW
CW
FR CW
CW
CW
CW
CW
CW
CW
|
Type of class: |
Classes, 60 hours
Lecture, 60 hours
|
|
Coordinators: | Marcin Bobieński, Marek Bodnar | |
Group instructors: | Witold Bednorz, Marcin Bobieński, Marek Bodnar, Tomasz Cieśla, Agnieszka Kałamajska, Tomasz Kochanek, Michał Kotowski, Rafał Martynek, Marcin Moszyński, Mikołaj Rotkiewicz, Jacek Sadowski, Michał Strzelecki | |
Students list: | (inaccessible to you) | |
Examination: |
Course -
Examination
Lecture - Grading |
Classes in period "Winter semester 2024/25" (future)
Time span: | 2024-10-01 - 2025-01-26 |
Navigate to timetable
MO TU W TH FR |
Type of class: |
Classes, 60 hours
Lecture, 60 hours
|
|
Coordinators: | Marta Szumańska, Anna Zatorska-Goldstein | |
Group instructors: | Witold Bednorz, Stanisław Cichomski, Galina Filipuk, Michał Jóźwikowski, Tomasz Kochanek, Sławomir Kolasiński, Rafał Martynek, Piotr Mormul, Marcin Moszyński, Piotr Nayar, Marta Szumańska, Anna Zatorska-Goldstein | |
Students list: | (inaccessible to you) | |
Examination: |
Course -
Examination
Lecture - Grading |
Copyright by University of Warsaw.