Mathematics 2

Lecturer:

Radek Fučík (gar.)

Tutor:

Radek Fučík (gar.)

Supervisor:

Department of Mathematics

Synopsis:

The course, which is the continuation of Mathematics 1, is devoted to the integration techniques, improper Riemann integral, introduction to parametric curves (especially in polar coordinates), the basics of sequences and infinite series, and finally to the Taylor and power series and their applications.

Requirements:

Mathematics 1.

Syllabus of lectures:

1. Integration techniques.

2. The improper integral and the convergence criteria.

3. Conic sections: ellipse, hyperbole, parable.

4. Polar coordinates.

5. Parametric curves: length of a curve, tangent to a curve, surfaces, volumes and surfaces of revolution.

6. Sequences: limits of sequences, important limits, the convergence criteria.

7. Series: the convergence criteria, absolute and non-absolute convergence, alternating series.

8. Power series. Differentiation and integration of power series.

9. Taylor polynomial and Taylor series.

Syllabus of tutorials:

1. Advanced integration techniques: integrals of rational functions, partial fractions, integration of trigonometric functions.

2. Improper Riemann integral: calculating improper integrals, convergence criteria.

3. Conic sections: circle, ellipse, hyperbole, parable, conic sections identification, description of conics through the distance between points and between a point and a line.

4. Polar coordinates: the transformation of points and equations between the cartesian and polar coordinates.

5. Parametric curves: length of a curve, tangent to the curve, surfaces, volumes and surfaces of revolution.

6. Properties of sets: finding suprema and infima of sets.

7. Sequences: limits of sequences, important limits, convergence criteria.

8. Infinite series: convergence criteria, absolute and relative convergence, alternating series.

9. Power series: convergence criteria, differentiation and integration of power series, sum of infinite series.

10. Taylor polynomials and Taylor series: the expansion of important functions in power series.

Study Objective:

Knowledge:

Advanced integration techniques, improper Riemann integral, numerical sequences, and infinite power series.

Abilities:

Understanding the basics of mathematical logic and mathematical analysis. Taylor series expansion.

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: