Mathematical Analysis I

Lectures:

Real numbers. Supremum and infimum. Principle of mathematical induction.
Real one-variable functions and their basic properties.
Elementary functions.
Sequences of real numbers. Limit of sequence.
Theorems on limit of sequences, calculation of limits.
Limit of a function. Theorems on limits.
Continuity of a function. Theorems on limits and continuity of composite function.
Derivative and differential of a function. Calculation of derivatives.
Basic theorems of differential calculus. L'Hospital rule.
Intervals of monotony of a function. Local extremes of a function.
Convexity and concavity. Asymptotes of graphs. Course of a function.
Global extremes of a function. Weierstrass-theorem.
Taylor's theorem.
Fundamental principles of integral calculus.

Exercises:
Application of principle of mathematical induction. Supremum and infimum of various sets.
Functions and their properties. Graph of a function. Functions with absolute value.
Elementary functions. Calculation of inverse function.
Finding domain of definition of a function. Arithmetic and geometric sequence.
Calculation of limits of sequences.
Calculation of limits of functions.
Limits of functions. Continuity of a function.
Calculation of derivatives.
Tangent and normal line. L'Hospital rule.
Monotony of a function. Local extremes.
Convexity and concavity, asymptotes. Course of a function.
Global extremes of a function.
Taylor's polynom and error estimation.
Calculation of antiderivatives and Riemann integrals.