Mathematics 1

Lecturer:

Radek Fučík (gar.)

Tutor:

Radek Fučík (gar.), Zuzana Farová, Jan Mach

Supervisor:

Department of Mathematics

Synopsis:

The course is devoted to the study of the basics of calculus of one variable. It includes an introduction to differential and integral calculus, with particular emphasis on applications in practical problems.

Requirements:

Syllabus of lectures:

1. Functions and their properties.

2. Limits of functions.

3. Continuity.

4. The derivative, tangent to a curve, some differentiation formulas, derivatives of higher order.

5. Rolle's theorem, the mean value theorem (Lagrange). Extreme values, asymptotes, concavity and point of inflections, curve sketching.

6. The definite integral. The antiderivate function, indefinite integral, substitution, integration by parts. Newton's theorem, the area calculation. Primitive functions to trigonometric functions, mean integral.

7. The transcendental functions: logarithm function, e number, exponential function, hyperbolic functions.

8. Applications of the definite integral: the length of a curve, the volume and the area of a revolved curve.

Syllabus of tutorials:

1. Functions and their properties: domain of definition, range, inverse, absolute value, inequalities, quadratic inequalities, graphs, composition of functions, polynomials, division of polynomials.

2. Limits of functions: the limits of basic functions, the limits of trigonometric functions.

3. Continuity: The investigation of continuity of functions from the definition, identification of types of discontinuities.

4. Derivatives: derivative computation by definition, rules for derivatives of basic functions, tangents, higher order derivatives.

5. Rolle's theorem, the mean value theorem (Lagrange). Extreme values, asymptotes, concavity and point of inflections, curve sketching.

6. Integral calculus: the antiderivate functions, the method of substitution, the method of integration by parts, advanced techniques of integration of trigonometric functions, definite integrals, Newton's formula.

7. Transcendental functions: logarithm definition, characteristics, exponential, hyperbolic and trigonometric functions and their derivatives.

8. Applications of the definite integral: area under the graph of the function, length of a graph, volume and surface the area of a revolved curve.

Study Objective:

Knowledge:

Elementary notions of mathematical analysis of the differential and integral calculus of functions of one real variable.

Abilities:

Understanding the basics of mathematical logic and mathematical analysis.

Study materials:

Key references:

[1] Calculus, One Variable, S.L.Salas, Einar Hille, John Wiley and Sons, New York, Chichester, Brisbane, Toronto, Singapore, 1990 (6th edition), ISBN 0-471-51749-6

Note:

Time-table for winter semester 2011/2012:

Time-table is not available yet

Time-table for summer semester 2011/2012:

Time-table is not available yet

The course is a part of the following study plans: