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Course, academic year 2016/2017
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Mathematical Analysis I - NMAI054
Czech title: Matematická analýza I
Guaranteed by: Department of Applied Mathematics (32-KAM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2016
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/2 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Guarantor: doc. RNDr. Martin Klazar, Dr.
prof. RNDr. Stanislav Hencl, Ph.D.
prof. RNDr. Jan Rataj, CSc.
Class: Informatika Bc.
Classification: Mathematics > Real and Complex Analysis
Is co-requisite for: NMAI056, NMAI055
Annotation -
Last update: KLAZAR/MFF.CUNI.CZ (05.02.2009)

Introductory course for students of informatics covering basic elements of differential calculus of functions of one variable (limits, continuity, derivative, the Taylor polynomials), sequences and series of real numbers.
Literature -
Last update: doc. RNDr. Martin Klazar, Dr. (26.11.2012)

T. M. Apostol, Mathematical Analysis, Addison-Wesley, 1974 (2nd edition).

Ch. Ch. Pugh, Real Mathematical Analysis, Undergraduate Text in Mathematics, Springer, 2002.

T. Tao, Analysis I, Hindustan Book Agency, 2006.

T. Tao, Analysis II, Hindustan Book Agency, 2006.

V. A. Zorich, Mathematical Analysis I, Universitext, Springer, 2004.

V. A. Zorich, Mathematical Analysis II, Universitext, Springer, 2004.

Syllabus -
Last update: KLAZAR/MFF.CUNI.CZ (05.02.2009)

Number sets, real numbers, the least upper bound property and its corollaries.

Sequences and series of real numbers. Limits of sequences and their basic properties. Bolzano-Weierstrass theorem and Bolzano-Cauchy theorem.

Series. Criteria of convergence. Absolute and non-absolute convergence. Alternating series.

Limits and continuity of real functions. Basic theorems on functions continuous on an interval (Darboux property, image of interval, boundedness and existence of extrema of a function continuous on a closed interval, continuity of inverse function).

Basic elementary functions and their properties.

Derivatives. Definition of a derivative and calculus of derivatives. Derivatives of higher orders. Applications of derivatives (necessary condition for a local extremum, derivative and monotonicity, l'Hospital rule, convex and concave functions, Taylor polynom, forms of remainder, Taylor series).

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