Mathematical Analysis I

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Course Unit Code470-2102/01
Number of ECTS Credits Allocated4 ECTS credits
Type of Course Unit *Compulsory
Level of Course Unit *First Cycle
Year of Study *First Year
Semester when the Course Unit is deliveredWinter Semester
Mode of DeliveryFace-to-face
Language of InstructionCzech, English
Prerequisites and Co-Requisites There are no prerequisites or co-requisites for this course unit
Name of Lecturer(s)Personal IDName
JAH02RNDr. Pavel Jahoda, Ph.D.
VOD03Mgr. Petr Vodstrčil, Ph.D.
Summary
In the first part of this subject, there are fundamental properties of the set of real numbers mentioned. Further, basic properties of elementary functions are recalled. Then limit of sequence, limit of function, and continuity of function are defined and their basic properties are studied. Differential and integral calculus of one-variable real functions is essence of this course.
Learning Outcomes of the Course Unit
Students will get basic practical skills for work with fundamental concepts, methods and applications of differential and integral calculus of one-variable real functions.
Course Contents
Lectures:

Real numbers. Supremum and infimum. Principle of mathematical induction.
Real one-variable functions and their basic properties.
Elementary functions.
Sequences of real numbers. Limit of sequence.
Theorems on limit of sequences, calculation of limits.
Limit of a function. Theorems on limits.
Continuity of a function. Theorems on limits and continuity of composite function.
Derivative and differential of a function. Calculation of derivatives.
Basic theorems of differential calculus. L'Hospital rule.
Intervals of monotony of a function. Local extremes of a function.
Convexity and concavity. Asymptotes of graphs. Course of a function.
Global extremes of a function. Weierstrass-theorem.
Taylor's theorem.
Fundamental principles of integral calculus.

Exercises:
Application of principle of mathematical induction. Supremum and infimum of various sets.
Functions and their properties. Graph of a function. Functions with absolute value.
Elementary functions. Calculation of inverse function.
Finding domain of definition of a function. Arithmetic and geometric sequence.
Calculation of limits of sequences.
Calculation of limits of functions.
Limits of functions. Continuity of a function.
Calculation of derivatives.
Tangent and normal line. L'Hospital rule.
Monotony of a function. Local extremes.
Convexity and concavity, asymptotes. Course of a function.
Global extremes of a function.
Taylor's polynom and error estimation.
Calculation of antiderivatives and Riemann integrals.
Recommended or Required Reading
Required Reading:
J. Bouchala, M. Sadowská: Mathematical Analysis I, VŠB-TUO.
J. Bouchala: Matematická analýza 1, skripta VŠB-TUO.
J. Bouchala: Matematická analýza ve Vesmíru, http://www.am.vsb.cz/bouchala
P. Šarmanová, J. Kuben, Š. Hošková, P. Račková: Diferenciální a integrální počet funkcí jedné proměnné, http://www.am.vsb.cz/sarmanova/cd
Recommended Reading:
L. Gillman, R. H. McDowell: Calculus, New York, W.W. Norton & Comp. Inc. 1973.
J. Brabec, F. Martan, Z. Rozenský: Matematická analýza I. Praha, SNTL 1985.
B. Budinský a J. Charvát: Matematika I. Praha, SNTL 1987.
K. Rektorys a kol.: Přehled užité matematiky I a II. Praha, Prometheus 1995.


M. Demlová, J. Hamhalter: Calculus I, skripta ČVUT Praha 1996 (anglicky).
J. Stewart: Calculus, Belmont, California, Brooks/Cole Pub. Comp. 1987 (anglicky).
Planned learning activities and teaching methods
Lectures, Tutorials
Assesment methods and criteria
Task TitleTask TypeMaximum Number of Points
(Act. for Subtasks)
Minimum Number of Points for Task Passing
Exercises evaluation and ExaminationCredit and Examination100 (100)51
        Exercises evaluationCredit30 (30)10
                Tests 1Written test15 0
                Tests 2Written test15 0
        ExaminationExamination70 21