Elements of Calculus

Relations:

It is not possible to register for the course BIK-ZMA if the student is concurrently registered for or has already completed the course BIK-MA1.21 (mutually exclusive courses).

During a review of study plans, the course BIK-MA1.21 can be substituted for the course BIK-ZMA.

It is not possible to register for the course BIK-ZMA if the student is concurrently registered for or has previously completed the course BIK-MA1.21 (mutually exclusive courses).

Course guarantor:

Lecturer:

Tutor:

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:

Knowledge of high-school mathematics.

Syllabus of lectures:

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

2. Sequences and their limits.

3. Extended scales of infinity, small- and big-O notation, theta.

4. Limits.

5. Continuity, introduction to derivatives.

6. Derivatives and properties of derivatives.

7. Classical theorems (Rolle, mean value, etc.), l'Hospital's rule.

8. Taylor polynomials and approximation, error estimation, root finding (bisection, Newton's method), monotony, extremes and optimization.

9. Convexity, function graph, primitive function, substitution.

10. Integration by parts, partial fractions.

11. Definite integral (properties, Newton's formula).

12. Improper integral.

13. Uses of integrals.

14. Space and time complexity of algorithms.

Syllabus of tutorials:

1. Domain of a function.

2. Basic properties of functions.

3. Sequences.

4. Limits of functions.

5. Differentiating.

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

7. Limits of functions.

8. Approximation, optimization.

9. Graphs of functions, primitive functions.

10. Indefinite integral.

11. Definite integral.

12. Improper integral.

13. Applications of integrals.

Study Objective:

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.

Study materials:

1. J. Tkadlec: Diferenciální a integrální počet funkcí jedné proměnné. ČVUT Praha, 2004.

Note:

Further information:

https://courses.fit.cvut.cz/BI-ZMA/

No time-table has been prepared for this course

The course is a part of the following study plans:

• Bachelor program Informatics, unspecified branch, in Czech, part-time, 2015 – 2021 (compulsory course in the program)
• Bachelor branch Security and Information Technology, in Czech, part-time, 2015 - 2019 (compulsory course in the program)
• Bachelor branch Web and Software Engineering, spec. Software Engin., in Czech, part-time, 2015–2020 (compulsory course in the program)
• Bachelor branch Security and Information Technology, part-time, in Czech, 2020 (compulsory course in the program)