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Course, academic year 2017/2018
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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 2017 to 2017
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: Mgr. Tereza Klimošová, Ph.D.
Mgr. Petr Honzík, Ph.D.
doc. Hans Raj Tiwary, M.Sc., Ph.D.
Class: Informatika Bc.
Classification: Mathematics > Real and Complex Analysis
Is co-requisite for: NMAI056
Annotation -
Last update: doc. RNDr. Pavel Töpfer, CSc. (26.01.2018)

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.
Course completion requirements -
Last update: doc. RNDr. Pavel Töpfer, CSc. (26.01.2018)

There will be two small tests for the credit for the tutorial.

It is required to obtain 50% of the points to pass the tutorial.

The material for the tests will be the one covered during the lectures.

Obtaining the credit is necessary before the final exam.

There is provision for repeated attempts for the credit.

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.

Requirements to the exam -
Last update: doc. Hans Raj Tiwary, M.Sc., Ph.D. (12.10.2017)

There will be one written exam for the main course, with oral part where the students explain

their solutions at the end of the exam. The material for the exam will be the same as taught in the lecture.

Syllabus -
Last update: doc. RNDr. Pavel Töpfer, CSc. (26.01.2018)

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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