Mathematical analysis I.1
General data
Course ID:  1000111bAM1a  Erasmus code / ISCED:  11.101 / (0541) Mathematics 
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 view allocation of credits 

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 BolzanoWeierstrass 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 nth roots of positive numbers. Basic techniques for calculating limits, the definition of e. CesaroStolz theorem. Subsequences, BolzanoWeierstrass 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 nonnegative 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 12, 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, OxfordEdinburghNew 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. McGrawHill Book Co., New YorkAucklandDüsseldorf, 1976. 8. P. Strzelecki, Analiza Matematyczna I (lecture notes in Polish), http://dydmat.mimuw.edu.pl/sites/default/files/wyklady/analizamatematycznai.pdf Addition to the lecture notes (auth. M. Jóźwikowski, S. Kolasiński), http://dydmat.mimuw.edu.pl/sites/default/files/wyklady/analizamatematycznaizadania.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 2021/22" (past)
Time span:  20211001  20220220 
see course schedule 
Type of class: 
Class, 60 hours more information Lecture, 60 hours more information 

Coordinators:  Marek Bodnar, Marta Szumańska  
Group instructors:  Witold Bednorz, Bartosz Bieganowski, Marek Bodnar, Zofia Grochulska, Agnieszka Kałamajska, Sławomir Kolasiński, Michał Krych, Marcin Moszyński, Piotr Nayar, Przemysław Ohrysko, Marta Szumańska  
Course homepage:  https://moodle.mimuw.edu.pl/course/view.php?id=1096  
Students list:  (inaccessible to you)  
Examination: 
Course 
Examination
Lecture  Grading 
Classes in period "Winter semester 2022/23" (future)
Time span:  20221001  20230129 
see course schedule 
Type of class: 
Class, 60 hours more information Lecture, 60 hours more information 

Coordinators:  Michał Jóźwikowski, Anna ZatorskaGoldstein  
Group instructors:  Marek Bodnar, Michał Jóźwikowski, Agnieszka Kałamajska, Tomasz Kochanek, Marcin Moszyński, Piotr Nayar, Mikołaj Rotkiewicz, Marta Szumańska, Anna ZatorskaGoldstein  
Students list:  (inaccessible to you)  
Examination: 
Course 
Examination
Lecture  Grading 
Copyright by University of Warsaw.