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). |