Mathematical analysis I.2

mathematics
physics

obligatory courses

(in Polish) Oczekuje się dobrej znajomości zagadnień ujętych w sylabusie przedmiotu Analiza matematyczna I.1.

The course is a continuation of Analiza matematyczna I.1. It covers the differential and integral calculus of one variable, starting from the notion of a derivative and its applications (de l'Hospital's rule, Taylor polynomials), through sequences and series of functions and their convergence (Weierstrass' test for absolute convergence, Arzela-Ascoli theorem), properties of power series, to the theory of the Riemann integral, improper integrals and their applications (length of C¹ curves, Euler's Γ function, Wallis' formula).

1. Algebraic properties of differentiation (derivative of sum, difference, product and ratio of two functions), chain rule and the derivative of the inverse function. Mean value theorems of Rolle, Lagrange and Cauchy. Criteria for monotonicity of differentiable functions. De l'Hospital's rule. Local extrema. Second- and higher order derivatives, Taylor's formula with remainder in the forms of Peano, Lagrange and Cauchy. Taylor polynomials of exponential and logarythmic functions, of sin, cos, arc sin and arc tan functions. Inflection points. Sufficient conditions for the existence of an extremum and an inflection point. C^k functions.
3. Sequences and series of functions, their pointwise and uniform convergence. Uniform Cauchy's condition, Weierstrass' test for uniform convergence. Continuity of a limit of a uniformly convergent sequence of continuous function. Weierstrass' theorem on approximation of continuous functions by polynomials (e.g., through Bernstein polynomials). Arzela-Ascoli theorem. Example of a continuous, nowhere differentiable function.
5. Power series, its radius of convergence (Cauchy-Hadamard's formula). Uniform and absolute convergence of a power series. Continuity of the sum of a power series, Abel's theorem. Power series expansions of elementary functions.

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 12), 2001.

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

Student:

1. Can justify the correctness of his reasoning. He operates with examples.

2. Knows methods of calculating derivatives and the most important theorems of differential calculus of functions of one real variable, including Lagrange's theorem on mean value, Taylor formula and de l'Hospital rule. Uses typical tools of differential calculus of functions of one variable, incl. determines local extremes, intervals of monotonicity and convexity as well as limits of real variable functions, and also solves optimization tasks based on the study of extremes. Uses the Taylor formula to calculate limits.

3. Knows the concept of point and uniform convergence of a sequence and a functional series, Weierstrass's criterion of uniform convergence, theorem on the continuity of the limit of a uniformly convergent sequence / series of continuous functions and the theorem on differentiation of functional sequences. He can investigate uniform convergence of functional sequences and prove the continuity or differentiability of the limits of such sequences.

4. Knows the concept of a power series and the most important functional properties of the sum of such a series. He knows the Cauchy-Hadamard formula. Specifies the radius of convergence of the power series; is able to use the theorem on the differentiability of functional series to sum up specific series.

5. knows the concept of a primary function and an indefinite integral; can integrate by parts and by substitution.

6. Knows the concept of definite integral, the definition of the Riemann integral and its geometric interpretation. He knows the relationship between the definite and indefinite integral. Uses the tools of integral calculus in tasks of a geometrical nature. Calculates the area under the graph and the curve length.

7. Knows the concept of indefinite integral and examples of functions defined by such integrals. Using various methods, he/she studies the convergence of indefinite integrals.

Final mark based on the sum of points from exercises, the two tests and exam.