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Course, academic year 2018/2019
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Mathematical Analysis I - NMAI054
Title in English: Matematická analýza I
Guaranteed by: Department of Applied Mathematics (32-KAM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2018 to 2018
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.
doc. Mgr. Robert Šámal, Ph.D.
RNDr. Dušan Pokorný, 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)

Exam will be oral with time for written preparation. A student must obtain credit from the tutorial to take the exam.

To obtain credit from the tutorial, a student needs to satisfy all of the following conditions:

  • at most three absences (with possible exceptions for serious medical and personal reasons),
  • score at least 50% in the test at the end of the term,
  • score average 60% of credit from homework, three lowest marks/undelivered homework will not be considered into this average.
Literature -
Last update: doc. RNDr. Pavel Töpfer, CSc. (26.01.2018)

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. RNDr. Pavel Töpfer, CSc. (26.01.2018)

Exam will be oral with time for written preparation. A student must obtain credit from the tutorial to take the exam. The material for the exam corresponds to the syllabus to the extent to which topics were covered during lectures and tutorials. Ability to generalize and apply theoretical knowledge to solving problems will be required.

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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