Calculus

Lecturer:

Petr Habala (gar.)

Tutor:

Petr Habala (gar.)

Supervisor:

Department of Mathematics

Synopsis:

This course covers the standard basics of continuous mathematics. First, for functions of one variable we cover limits, derivatives and integration, which is followed by sequences and series of real numbers. The acquired skills are then applied to functions of more variables, where we use partial derivatives to find extrema. The focus is on practical computational skills and on understanding the meaning of notions and calculations. The course is concluded by a survey of power series and a brief introduction to ordinary differential equations, whose main purpose is to show students that continuous mathematics is a powerful

tool.

Requirements:

Syllabus of lectures:

1. Introduction. Limit of a function.

2. Continuity. Introduction to derivatives.

3. Differentiation and basic theorems, l'Hospital's rule.

4. Monotonicity and extrema. Applications of derivative (Taylor polynomial).

5. Graph sketching. Introduction to indefinite integral.

6. Properties of integral, methods of evaluation.

7. Definite integral.

8. Improper integral. Applications of integral.

9. Sequences. Introduction to series.

10. Series. Introduction to functions of more variables.

11. Functions of more variables (including extrema without and with constraints).

12. Series of functions (region of convergence, expanding a function in a power series).

13. Brief introduction to differential equations.

14. Back-up class.

Syllabus of tutorials:

1. Review, domains of functions.

2. Limit of a function.

3. Differentiation, tangent and normal lines.

4. Limit using l'Hospital's rule.

5. Monotonicity and extrema.

6. Taylor polynomial. Graph sketching.

7. Basic methods of integration.

8. Definite integral.

9. Improper integral. Applications of integral.

10. Limit of a sequence, intuitive evaluation. Scale of powers.

11. Testing series convergence.

12. Partial derivative, local extrema.

13. Constrained extrema. Power series.

14. Solving differential equations by separation.

Study Objective:

Study materials:

1. M. Demlová, J. Hamhalter: Calculus I. ČVUT Praha, 1994.

2. P. Pták: Calculus II. ČVUT Praha, 1997.

3. Habala, P.: Math Tutor, http://math.feld.cvut.cz/mt/

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:

• Otevřená informatika - Počítačové systémy (compulsory course in the program)
• Otevřená informatika - Informatika a počítačové vědy (compulsory course in the program)
• Otevřená informatika - Softwarové systémy (compulsory course in the program)
• Otevřená informatika, před rozřazením do oborů (compulsory course in the program)