Content of the Final Exam

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

Integral calculus

Primitive functions, indefinite integral, “tabular integrals”, methods of integration by parts (per partes) and substitutions for indefinite integrals. Integration of rational functions (with simple imaginary roots in denominators at most, real roots of arbitrary multiplicity). Integration by special substitutions.
Definite integral, Newton-Leibniz's formula, per partes and substitutions for definite integrals, comparing of integrals without their computation. Improper integrals, convergence and divergence, calculation.
Applications to integral calculus: area of plane figures (plane sheets), volume of solids of revolution, length of graphs of functions, static moments and coordinates of the centre of gravity of plane figures.

Functions of several variables

Domains of definition. For functions of two variables level curves and graphs. Partial derivatives of the 1^st order and their geometric meaning. Partial derivatives of higher orders. Directional derivatives and their geometric meaning. Gradient. Total differential and its applications.
Implicit functions, existence, derivatives of implicit functions of one variable, partial derivatives of implicit functions of two variables. Equations of tangent lines and normal lines of plane curves, equations of tangent planes and normal lines of surfaces. Extremes, local, constrained, global.

Differential equations

Identification of differential equations of the 1^st order, general solutions, particular solutions with initial conditions (Cauchy problems). Separable differential equations (including Cauchy problems). Homogeneous differential equations of the 1st order (including Cauchy problems). Linear differential equations of the 1st order (including Cauchy problems). Exact differential equations (including Cauchy problems).

Please note: Details of the concepts will be presented during the lectures.