Elements of Calculus
Code  Completion  Credits  Range  Language 

BIKZMA  Z,ZK  6  20KP+4KC  Czech 
 Garant předmětu:
 Tomáš Kalvoda
 Lecturer:
 Ivo Petr
 Tutor:
 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:

Knowledge of highschool mathematics.
 Syllabus of lectures:

1. Introduction, real numbers, basic properties of functions.
2. Sequences and their limits.
3. Extended scales of infinity, small and bigO 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/BIZMA/
 No timetable has been prepared for this course
 The course is a part of the following study plans:

 Bachelor program Informatics, unspecified branch, in Czech, parttime, 2015 – 2021 (compulsory course in the program)
 Bachelor branch Security and Information Technology, in Czech, parttime, 2015  2019 (compulsory course in the program)
 Bachelor branch Web and Software Engineering, spec. Software Engin., in Czech, parttime, 2015–2020 (compulsory course in the program)
 Bachelor branch Security and Information Technology, parttime, in Czech, 2020 (compulsory course in the program)
 Bachelor branch Security and Information Technology, in Czech, parttime, 2015 commuters (compulsory course in the program)
During a review of study plans, the course BIKMA1.21 can be substituted for the course BIKZMA.