Course: Calculus II

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Course title Calculus II
Course code UMB/565
Organizational form of instruction Lecture + Lesson
Level of course Bachelor
Year of study not specified
Frequency of the course In each academic year, in the summer semester.
Semester Summer
Number of ECTS credits 8
Language of instruction Czech
Status of course Compulsory
Form of instruction unspecified
Work placements unspecified
Recommended optional programme components None
Lecturer(s)
  • Eisner Jan, Mgr. Dr.
  • Kalová Jana, doc. RNDr. Ing. Ph.D.
Course content
Content of lectures: 1. Sequences: definition, examples, basic concepts, convergence, absolute and relative convergence, theorems on sequences, definition of limsup and liminf of sequences, arithmetic and geometric sequences 2. Infinite series: sum of a series, Bolzano-Cauchy condition on series convergence, properties of series, criteria for convergence of series, absolute convergence 3. Functional sequences and series: point and uniform convergence, example of functional sequence with point convergence but not with uniform convergence, Weierstrass criterion of uniform convergence, properties of uniformly convergent sequences and series 4. Power series: range of convergence of power series and its determination, basic properties of power series, Taylor series 5. Indefinite integral: definition, basic properties, substitution method, integration by parts, methods of integration of rational, trignometric and irrational functions 6. Riemann (definite) integral: definition, basic concepts, existence, substitution method and integration by parts for Riemann integral, relationship between indefinite and Riemann integral (Newton formula) 7. Improper integral: definition, convergence, integral criterion for convergence of infinite series 8. Geometric application of Riemann integral: mean value of function, derivation of formulae for area between two curves, surfaces and volumes of solids of revolution, and length of curves Content of practicals: Practicing theoretical concepts on specific examples.

Learning activities and teaching methods
Monologic (reading, lecture, briefing), Dialogic (discussion, interview, brainstorming), Demonstration, Projection
  • Class attendance - 70 hours per semester
  • Preparation for classes - 28 hours per semester
  • Preparation for credit - 28 hours per semester
  • Preparation for exam - 56 hours per semester
Learning outcomes
Introduction to series and single variable integration.
Student should be able to explain what are sequences of numbers and infinite series and also why a sum of infinite amount of numbers can be a finite number. Student should also know how to calculate limits of sequences and whether a given infinite series converges or not. Last but not least, student should understand meaning of integration of functions, why is it needed and which practical applications integration has. (S)he should also know how to define Riemann integral and how to work with it.
Prerequisites
Students are assumed to have knowledge of calculus covered by the courses Matematická analýza I (UMB564).
UMB/CV010
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UMB/CV551
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UMB/CV564
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UMB/010
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UMB/551
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UMB/564

Assessment methods and criteria
Combined exam, Test, Interim evaluation

100% attendance of practicals (excused absence permitted), performance of all written tests on practicals and at least 50% success in these tests
Recommended literature
  • S.I. Grossman: Calculus. John Wiley & Sons, Inc. 2005.
  • V. Jarník: Diferenciální počet I, II, Integrální počet I, Academia Praha, 1984.
  • M. Giaquinta, G. Modica. Mathematical Analysis. Function of One Variable.. Birkhäuser, Boston, 2003.
  • M. Giaquinta, G. Modica. Mathematical Analysis. An Introduction to Functions of Several Variables. Birkhauser Boston, 2009. ISBN 978-08176-4507-6.
  • Š. Hošková, J. Kuben, P. Račková. Integrální počet funkcí jedné proměnné. VŠB-TU Ostrava, 2006. ISBN 80-248-1191-X.


Study plans that include the course
Faculty Study plan (Version) Category of Branch/Specialization Recommended year of study Recommended semester
Faculty: Faculty of Science Study plan (Version): Mathematics for future teachers (1) Category: Mathematics courses 1 Recommended year of study:1, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Secondary Schools Teacher Training in Physics (2012) Category: Pedagogy, teacher training and social care - Recommended year of study:-, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Secondary Schools Teacher Training in Mathematics (1) Category: Pedagogy, teacher training and social care - Recommended year of study:-, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Secondary Schools Teacher Training in Mathematics (1) Category: Pedagogy, teacher training and social care - Recommended year of study:-, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Secondary Schools Teacher Training in Mathematics (1) Category: Pedagogy, teacher training and social care - Recommended year of study:-, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Physics for future teachers (1) Category: Physics courses 1 Recommended year of study:1, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Physics (1) Category: Physics courses 1 Recommended year of study:1, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Applied Mathematics (2010) Category: Mathematics courses 1 Recommended year of study:1, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Mathematics for future teachers (1) Category: Mathematics courses 1 Recommended year of study:1, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Mechatronics (1) Category: Special and interdisciplinary fields 1 Recommended year of study:1, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Mathematics for future teachers (1) Category: Mathematics courses 1 Recommended year of study:1, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Chemistry (1) Category: Chemistry courses 1 Recommended year of study:1, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Measuring and Computer Technology (1) Category: Electrical engineering, telecommunication and IT 1 Recommended year of study:1, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Secondary Schools Teacher Training in Mathematics (2012) Category: Pedagogy, teacher training and social care - Recommended year of study:-, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Biophysics (1) Category: Physics courses 1 Recommended year of study:1, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Mathematics for future teachers (1) Category: Mathematics courses 1 Recommended year of study:1, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Biophysics (1) Category: Physics courses - Recommended year of study:-, Recommended semester: Summer