Elements of Calculus

Lecturer:

Michal Kupsa

Tutor:

Michal Kupsa

Supervisor:

Department of Applied Mathematics

Synopsis:

Students acquire knowledge and understanding of the fundamentals of classical calculus so that they are able to apply mathematical way of thinking and reasoning and are able to use basic proof techniques. They get skills to practically handle functions of one variable in solving the problems in informatics. They understand the links between the integrals and sums of sequences. They are able to estimate lower or upper bounds of values of real functions and to handle simple asymptotic expressions.

Requirements:

The ability to think mathematically and knowledge of a high school mathematics.

Syllabus of lectures:

1. Introduction, real numbers, basic properties of functions.

2. Limits.

3. Continuity, introduction to derivatives.

4. Properties of derivatives, implicit differentiation, numerical and symbolic differentiation on a computer.

5. Classical theorems (Rolle, mean value, etc.), differentiation using limits, finding limits using derivatives (lHospitals rule).

6. Taylor polynomials and approximation, error estimation, root finding (bisection, regula falsi, Newtons method), monotony, extremes and optimization.

7. Convexity, function graph, primitive function, substitution.

8. Integration by parts, partial fractions.

9. Definite integral (properties, N-L formula).

10. Improper integral.

11. Uses of integrals, numerical methods for definite integrals.

12. Sequences and their limits.

13. Extended scales of infinity, small- and big-O notation, theta. Space and time complexity of algorithms.

Syllabus of tutorials:

1. Differentiating.

2. Domain of a function.

3. Basic properties of functions.

4. Limits of functions.

5. Tangents/normals, implicit differentiation, related rates.

6. Limits of functions.

7. Approximation, optimization.

8. Graphs of functions, primitive functions.

9. Indefinite integral.

10. Definite integral.

11. Improper integral.

12. Applications of integrals.

13. Sequences.

Study Objective:

Handling the elementary calculus is a necessary assumption to build mathematical skills and habits that are needed in both subsequent mathematical and theoretical modules. For purposes of analysis of algorithms, there is an overview of asymptotic estimation of the growth order of functions.

Study materials:

1. Strang, G. ''Calculus.'' Wellesley-Cambridge Press, 2009. ISBN 0961408820.

Note:

Time-table for winter semester 2011/2012:

	06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon
Tue
Fri
Thu	roomTH:A-1030 Kupsa M. 11:00–12:30 (lecture parallel1) DejviceroomTH:A-1030 Kupsa M. 12:45–14:15 (lecture parallel1 parallel nr.101) Dejvice
Fri	roomTH:A-1030 Kupsa M. 12:45–14:15 EVEN WEEK (lecture parallel1) Dejvice

Time-table for summer semester 2011/2012:

Time-table is not available yet

The course is a part of the following study plans:

• Information Systems and Management (Presented in English) (compulsory course in the program)
• Information Technologies (Presented in English) (compulsory course in the program)
• Computer Engineering (Presented in English) (compulsory course in the program)
• Software Engineering (Presented in English) (compulsory course in the program)
• Computer Science (Presented in English) (compulsory course in the program)
• Web and Multimedia (Presented in English) (compulsory course in the program)
• Informatics (Presented in English) (compulsory course in the program)