COURSE DESCRIPTION

Course code

MAP1142

Kind of studies

common to all departments;

Course title

Mathematical Analysis 1A

First name, surname and title of the lecturer/supervisor

Komisja programowa Instytutu Matematyki i Informatyki

First name, surname and title of the team's members

Lecturers of the Institute of Mathematics and Computer Science

Course structure

Form of class Lecture Problems classes Laboratory Project Seminar Number of points
Number of hours/week 2 2 5+3
Course grade based on Examination Test

Prerequisites

High school mathematics

Course description

Review of basic elementary functions. Limits. Continuity of functions of one variable. Derivative. Examination of a function. Applications of differential calculus in physics and technics. Indefinite integral.

Lectures

Contents of particular hours Number of hours
1. Elements of mathematical logic and set theory. Quantifiers. Sets on the real line.2
2. Composition of functions. Injective functions. Inverse function and its graph. Power and exponential functions and their inverses.Składanie funkcji.2
3. Trigonometric functions. Trigonometric identities and reduction formulas. Inverse trigonometric functions.2
4. Finite limits of a sequence. Basic theorems. The number e. Infinite limit of a sequence. Calculation of infinite limits. Indeterminate expressions.3
5. Finite and infinite limit of a function at a point. One-sided limits. Methods of calculation of limits. Limits of basic indeterminate expressions. Asymptotes of functions.4
6. Continuity of a function at a point and on an interval. One-sided continuity. Discontinuity points and their classification. Basic theorems on continuous functions on closed interval (Weierstrass and Darboux theorems) and applications. Approximate solution of equations.3
7. Derivative of a function at a point. One-sided and infinite derivatives. Calculation of derivatives of basic elementary functions. Rules of differentiation. Higher order derivatives.2
8. Geometric and physical interpretation of derivative. Tangent line. Differential and its applications. Maximum and minimum value of a function on a closed interval. Applications to geometry, physics and technics.3
9. Mean-value theorems (Rolle’s, Lagrange’s). Consequences of Lagrange’s theorem. Taylor and Maclaurin formulas and their applications. L’Hosplital rule.2
10. Intervals of monotonicity of a function. Local extrema. Necessary and sufficient conditions for existence of local extrema. Convex and concave functions and inflection points. Examination of a function.3
11. Indefinite integrals, basic properties. Integration by parts. Integration by substitution. Integration of rational and trigonometric functions.4

Problems classes

Contents of particular hours Number of hours
1. Exercises illustrating the material presented during the lectures.30

Material for self preparation

Basic literature

1. G. Decewicz, W. Żakowski, Matematyka, Cz. 1, WNT, Warszawa 2007.
2. M. Gewert, Z. Skoczylas, Analiza matematyczna 1. Przykłady i zadania, Oficyna Wydawnicza GiS, Wrocław 2005.
3. W. Krysicki, L. Włodarski, Analiza matematyczna w zadaniach, Cz. I, PWN, Warszawa 2006.

Additional literature

1. G. M. Fichtenholz, Rachunek różniczkowy i całkowy, T. I-II, PWN, Warszawa 2007.
2. M. Gewert, Z. Skoczylas, Analiza matematyczna 1. Definicje, twierdzenia, wzory, Oficyna Wydawnicza GiS, Wrocław 2005.
3. R. Leitner, Zarys matematyki wyższej dla studiów technicznych, Cz. 1-2, WNT, Warszawa 2006.
4. F. Leja, Rachunek różniczkowy i całkowy ze wstępem do równań różniczkowych, PWN, Warszawa 2008.
5. H. i J. Musielakowie, Analiza matematyczna, T. I, Cz. 1-2, Wydawnictwo Naukowe UAM, Poznań 1993.
6. R. Nowakowski, Elementy matematyki wyższej, T. I, Wydawnictwo Naukowo Oświatowe ALEF, Wrocław 2000.
7. W. Stankiewicz, Zadania z matematyki dla wyższych uczelni technicznych, Cz. B, PWN, Warszawa 2003.

Conditions required for a student to pass the course

Positive result of the written test (for problems classes) and of the written exam (for the lecture).