Integral calculus

1. Indefinite integral, primitive functions, tabular integrals. Fundamental methods for calculating indefinite integrals: per partes, substitutions.
2. Integration of rational functions (with simple imaginary roots in denominators at most one).
3. Integration by special substitutions.
4. Definite integral, fundamental methods for calculating definite integrals: Newton-Leibniz's formula, per partes, substitutions.
5. Improper integrals, convergence and divergence of improper integrals, methods of computation.
6. Geometrical and physical applications of integral calculus: area of a plane figure (plane sheet), volume of a solid of revolution, length of the graph of a function, static moments and the centre of gravity of a plane figure.

Functions of several variables

7. Domains of definitions, in case of two variables also level curves and graphs. Partial derivatives, partial derivatives of higher orders.
8. Directional derivatives. Gradient. Total differential. Derivatives and partial derivatives of functions defined implicitly.
9. Equations of tangent and normal lines of a plane curve and tangent planes and normal lines of a surface.
10. Local extremes and local extremes with respect to a set (constrained extremes).
11. Global extremes on a set.

Differential equations

12. Differential equations of the 1^st order, separation of variables, homogeneous equations. Cauchy problems.
13. Linear differential equations of the 1^st order, variation of a constant. Exact equations. Cauchy problems.

Please note: In general the exercises at each tutorial shall be based on the subject matter of the previous lecture.

The final exam and tutorial tests are primarily based on the textbook Bubeník F.: Mathematics for Engineers, Prague (see References).