Elements of Calculus
Code  Completion  Credits  Range  Language 

BIEZMA  Z,ZK  6  3+2 
 Lecturer:
 Antonella Marchesiello
 Tutor:
 Antonella Marchesiello
 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 (l'Hospital's rule).
6. Taylor polynomials and approximation, error estimation, root finding (bisection, regula falsi, Newton's method), monotony, extremes and optimization.
7. Convexity, function graph, primitive function, substitution.
8. Integration by parts, partial fractions.
9. Definite integral (properties, NL 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 bigO 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.'' WellesleyCambridge Press, 2009. ISBN 0961408820.
 Note:
 Further information:
 https://courses.fit.cvut.cz/BIEZMA/
 Timetable for winter semester 2018/2019:

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 Fri  Timetable for summer semester 2018/2019:
 Timetable is not available yet
 The course is a part of the following study plans:

 Bc Branch Security and Information Technology, Presented in English, Version 2015, 16, 17 and 18 (compulsory course in the program)
 Bc. Branch WSI, Specialization Software Engineering, Presented in English, Version 2015, 16, 17, 18 (compulsory course in the program)
 Bc. Branch Computer Science, Presented in English, Version 2015, 2016, 2017 and 2018 (compulsory course in the program)