Elements of Calculus

Lecturer:

Martin Holeňa (gar.), Ivo Petr

Tutor:

Martin Holeňa (gar.), Ivo Petr

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:

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.

14. Space and time complexity of algorithms.

Syllabus of tutorials:

1. Domain of a function. Basic properties of functions. Limits of functions. Differentiating. Tangents/normals, implicit differentiation, related rates.

2. Limits of functions. Approximation, optimization. Graphs of functions, primitive functions. Indefinite integral. Definite integral.

3. Improper integral. Applications of integrals. Sequences. Assessment.

Study Objective:

Study materials:

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:

• Informtion Systems and Management (Presented in Czech) (compulsory course in the program)
• Information Technology (Presented in Czech) (compulsory course in the program)
• Computer engineering (Presented in Czech) (compulsory course in the program)
• Software engineering (Presented in Czech) (compulsory course in the program)
• Computer Science (Presented in Czech) (compulsory course in the program)
• Web and Multimedia (Presented in Czech) (compulsory course in the program)
• Informatics - Plan for Period Before Assignement of Specialization (Presented in Czech) (compulsory course in the program)