Real Functions of a Real Variable:

1. Sequences of real numbers, convergence, divergence, limits of sequences and methods for their calculating.
2. Basic concepts of functions, fundamental elementary and elementary functions, graphs, polynomials, rational functions, composite and inverse functions, inverse trigonometric functions, limits (proper and improper) and methods for their calculating.
3. Continuity. Bolzano’s and Weierstrass’s theorems and their applications.
4. Derivatives and their geometric and physical meaning, applications of derivatives.
5. Differentials and applications, higher derivatives and differentials and their applications.
6. Lagrange’s theorem and its consequences, l’Hospital’s rule.
7. Properties of functions based on the 1^st and 2^nd derivatives, intervals of monotony, local extremes, stationary (critical) points, necessary and sufficient conditions, convexity, concavity, points of inflection, asymptotes.
8. Global (absolute, total) extremes on compact intervals, word problems.
9. Taylor’s theorem, Taylor’s polynomial and its applications.

Linear Algebra:

1. Vector (linear) spaces, the vector space of ordered n-tuples, R², R³, Rⁿ, linear combinations, linear independence and dependence, bases, the dimension, subspaces.
2. Linear hull, matrices, the rank of a matrix, Gauss’s algorithm.
3. Systems of linear algebraic equations, basic methods for solving, Gaussian elimination, Frobenius theorem.
4. Matrix multiplication, inverse matrices and their applications, matrix equations.
5. Determinants of the 2^nd and 3^rd orders, Sarrus’s rule, inverse matrices by means of determinants, Cramer’s rule.

Analytic Geometry in Space:

1. Fundamental properties of geometric vectors.
2. General form and parametric representation of a plane.
3. Parametric equations of straight lines.
4. A straight line as the intersection of two planes.
5. Relationship problems on straight lines and planes.
6. Deviations of planes and straight lines.
7. Distances between various geometric bodies (points, straight lines, planes).
8. Application of analytic methods for solving geometric problems in the space.

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

References:

1. Bubeník F.: Mathematics for Engineers, Prague.
2. Bubeník F.: Problems to Mathematics for Engineers, Prague.
3. Rektorys K.: Survey of Applicable Mathematics, Vol. I, II.

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