Elements of Calculus

Přednášející:

Michal Kupsa

Cvičící:

Michal Kupsa

Předmět zajišťuje:

katedra aplikované matematiky

Anotace:

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.

Požadavky:

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

Osnova přednášek:

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.

Osnova cvičení:

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.

Cíle studia:

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.

Studijní materiály:

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

Poznámka:

Rozsah=prednasky+proseminare+cviceni4p+2c, Prednasejici: doc. RNDr. Alena Šolcová Ph.D.

Rozvrh na zimní semestr 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
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Čt	místnost TH:A-1030 Kupsa M. 11:00–12:30 (přednášková par. 1) Dejvicemístnost TH:A-1030 Kupsa M. 12:45–14:15 (přednášková par. 1 paralelka 101) Dejvice
Pá	místnost TH:A-1030 Kupsa M. 12:45–14:15 SUDÝ TÝDEN (přednášková par. 1) Dejvice

Rozvrh na letní semestr 2011/2012:

Rozvrh není připraven

Předmět je součástí následujících studijních plánů:

• Information Systems and Management (povinný předmět programu)
• Information Technologies (povinný předmět programu)
• Computer Engineering (povinný předmět programu)
• Software Engineering (povinný předmět programu)
• Computer Science (povinný předmět programu)
• Web and Multimedia (povinný předmět programu)
• Informatics (povinný předmět programu)