Calculus

obligatory courses

Learning the basic concepts, theorems and methods of mathematical analysis, with particular emphasis on the differential and integral calculus of functions of one variable. Applications of these methods to problems in natural sciences.

Course content:

Elements of mathematical logic and set theory; supplementing knowledge from school mathematics: polynomials and the Bezout theorem, rational functions and elementary functions (exponential function, logarithm, trigonometric and cyclometric functions).

Number sequences: boundedness, upper and lower bounds, limits, methods of calculating limits, the three sequence theorem.

Numerical series: basic tests of convergence (comparison, quotient, d'Alembert, Cauchy, Leibniz), absolute convergence, radius of convergence of power series.

Limit and continuity of functions; Weierstrass theorem.

The concept of derivative, its geometric and physical interpretation; one-variable differential calculus (mean value theorem, local and global extremes, concavity and convexity of functions, Taylor's formula, indeterminate expressions, investigation of functions).

The elements of a geometry and topology in the R^n space. Functions of several variables and their exemplary application.

W. Rudin: Principles of mathematical analysis

A. Browder: Mathematical analysis: an introduction

Student finishing the course:

1) knows the most important elementary functions (some algebraic functions, trigonometric, exponential and logarithmic functions),

2) efficiently uses the concepts of the limit of a sequence and the limit of a function,

3) can show the convergence of basic series,

4) knows the concept of continuity and differentiability of functions, is able to determine derivatives of elementary functions, is able to investigate of a function given by the formula,

5) knows and is able to practically use Taylor's formula,

6) understands a notion of distance in multivariate space;

7) knows basic application of multivariate functions,

8) is prepared to continue learning mathematical subjects covered by the program in the further course of study,

9) understands the importance and usefulness of mathematical modeling of natural phenomena and the precision of mathematical methods, and is aware of the limited scope of applicability of specific models.

FINAL SCORE WILL BE GIVEN ON THE BASIS OF:

short tests on current topics – 60 points

homework — 20 points

activity during classes – 20 points

written exam – 100 points.

For a positive grade, it is necessary to obtain at least 50% of the points.

Zero exam: students who obtain min. 85% from tests and homwork.

Re-take exam: the grade will be given only on the basis of the exam.