1. Sequences.

The limit of a real sequence and the elementary properties of the limit. Monotone, strictly monotone, bounded sequences and their properties. Subsequences. Comparison test for sequences. Sandwich theorem. Elementary limits (n^q, , q^n, n^(1/n), (1+1/n)^n). Definition and properties of limes inferior and superior. Concept of Cauchy sequences. Cauchy criterion (WP).

 

2. Real functions.

Extension of elementary functions (exp, sin and cos) to real numbers and the properties of these functions. Definition of inverses of elementary functions (log, arcsin, arccos and arctg). Definition and properties of hyperbolic functions and their inverses. Concepts of even, odd, (strictly) monotonically increasing/decreasing, concave/convex and periodic functions. Jensen inequality. The limit of a function and the elementary properties of the limit. Left and right limit and their relationship with the limit. Continuity of a function. The sum, product and composition of continuous functions are continuous functions. Weierstrass' maximum-minimum principle (WP). Bolzano's theorem (WP). Concept of uniform continuity and Heine's theorem (WP).

 

3. Basics of differential calculus.

The concept of differentiation and the derivative of a function. The relation between continuity and differentiability. Derivatives of elementary functions and differentiation rules. Intermediate value theorems: Rolle, Lagrange, Cauchy. L'Hospital rule. Concept of n times and infinitely differentiable function. Continuously differentiable functions. Connection between monotonicity and the derivative. Connection between convexity/concavity and the second derivative. Inequality between arithmetic mean and geometric mean. Hölder's inequality and Minkowski's inequality. Taylor series analysis. Binomial series. The concept of local extreme value and its connection with higher-order derivatives. Graphing of functions using first and second order derivatives.

 

4. Indefinite integral.

Concept of indefinite integral and elementary indefinite integrals. Substitution and partial integration. Partial fractions. Integration of rational functions.

 

5. Elementary integral calculus.

Concept of partition. Lower and upper Riemann sums and the oscillatory sum of a function for a given partition. Lower and upper integrals of a bounded function. Definition of Riemann integrability. Newton-Leibniz theorem (WP). Application of integral: arc length and centroid of a curve; centroid of plane figure; volume, surface area and centroid of surface and solid of revolution. Pappus–Guldinus theorem. Improper integral. Comparison test for improper integrals.