Course: Mathematical Analysis 1

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Course title Mathematical Analysis 1
Course code KMA/MA1
Organizational form of instruction Lecture + Tutorial
Level of course Bachelor
Year of study 1
Semester Winter and summer
Number of ECTS credits 6
Language of instruction Czech
Status of course Compulsory, Compulsory-optional
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
Lecturer(s)
  • Stehlík Petr, Doc. RNDr. Ph.D.
  • Volek Jonáš, RNDr. Ph.D.
  • Brandner Marek, Doc. Ing. Ph.D.
  • Nečesal Petr, Ing. Ph.D.
  • Tomiczková Světlana, RNDr. Ph.D.
  • Kotsu Matas Aleš, Ing. Ph.D.
Course content
Week 1: Sets and elementary operations; subsets of real numbers; absolute value; maximum, minimum, least upper bound, and greatest lower bound of a subset of real numbers; Week 2: Sequences of real numbers; subsequences; bounded and monotone sequences; recursively defined sequences; convergent and divergent sequences; Week 3: Algebra of limits and fundamental theorems concerning the properties of a limit; Week 4: Conditions ensuring the convergence of infinite sequences and series; Week 5: Absolute and relative convergence, alternating series; Week 6: Functions of one real variable; graphical representation; inverse functions; composition of functions; Week 7: Local and global behaviour of a function; limits; one-sided limits; algebra of limits; Week 8: Continuity of a function at a point; points of discontinuity; continuity in a closed interval; Week 9: Derivative and differential of a function - definition and both the geometrical and the physical meaning; differentiability and continuity of a function; Week 10: Differentiation from first principles, product rule and chain rule, Rolle's theorem, Langrange's and Cauchy's mean value theorems; stationary points of a function; l'Hospital's rule; Week 11: Indefinite integral; fundamental theorem of calculus; integration by parts and integration by substitution; Week 12: Definite integral and its applications; mean value theorem inequalities for integrals; Week 12: Improper integrals; higher order derivatives and differentials; Taylor's theorem;

Learning activities and teaching methods
Interactive lecture, Lecture supplemented with a discussion, Task-based study method
  • Contact hours - 78 hours per semester
  • Preparation for comprehensive test (10-40) - 24 hours per semester
  • Preparation for an examination (30-60) - 56 hours per semester
prerequisite
professional knowledge
There is no prerequisite for this course. Students should be familiar with a high school algebra and trigonometry.
learning outcomes
By the end of the course, a successful student should be able to: 1. Read mathematical text and to use logical constructions; 2. Use rigorous arguments in calculus and ability to apply them in solving problems on the topics in the syllabus; 3. Demonstrate knowledge of the definitions and fundamental theorems concerning sequences, series and continuous and differentiable functions of one real variable; 4. Use both the definition of derivative as a limit and the rules of differentiation to differentiate functions; 5. Sketch the graph of a function using asymptotes, critical points, and the derivative test for increasing/decreasing and concavity properties; 6. Set up max/min problems and use differentiation to solve them; 7. Use l'Hospital's rule; 8. Evaluate integrals using techniques of integration, such as substitution, inverse substitution, partial fractions and integration by parts; 9. Apply integration to compute areas and volumes by slicing; 10.Find the Taylor series expansion of a function near a point, with emphasis on the several starting terms; 11.Use developed theory in solving problems on physical systems.
teaching methods
Lecture supplemented with a discussion
Interactive lecture
Task-based study method
assessment methods
Combined exam
Skills demonstration during practicum
Recommended literature
  • Děmidovič, Boris Pavlovič. Sbírka úloh a cvičení z matematické analýzy. Havlíčkův Brod : Fragment, 2003. ISBN 80-7200-587-1.
  • Drábek, Pavel; Míka, Stanislav. Matematická analýza I. Plzeň : Západočeská univerzita, 1999. ISBN 80-7082-558-8.
  • Míková, Marta; Kubr, Milan; Čížek, Jiří. Sbírka příkladů z matematické analýzy I. Plzeň : Západočeská univerzita, 1999. ISBN 80-7082-568-5.
  • Polák, J. Přehled středoškolské matematiky.. Praha : Prometheus, 2008. ISBN 978-80-7196-356-1.
  • Pultr, Aleš. Matematická analýza I. Praha : Matfyzpress, 1995. ISBN 80-8586-3-09-X.


Study plans that include the course
Faculty Study plan (Version) Branch of study Category Recommended year of study Recommended semester
Faculty of Education Physics in Education (16) Physics courses 2 Winter
Faculty of Education Chemistry in Education (15) Chemistry courses 2 Winter
Faculty of Education Geography in Education (15) Pedagogy, teacher training and social care 2 Winter
Faculty of Education Geography in Education (13) Pedagogy, teacher training and social care 2 Winter
Faculty of Applied Sciences Financial Informatics and Statistics (1) Economy 1 Winter
Faculty of Applied Sciences Computations and Design (2016) Special and interdisciplinary fields 1 Winter
Faculty of Applied Sciences Geomatics (2012) Construction industry, geodesy and cartography 1 Winter
Faculty of Applied Sciences Mathematics for Business Studies (2017) Mathematics courses 1 Winter
Faculty of Applied Sciences Mathematics for Business Studies (2016) Mathematics courses 1 Winter
Faculty of Applied Sciences Financial Informatics and Statistics (2017) Economy 1 Winter
Faculty of Applied Sciences General Mathematics (2012) Mathematics courses 1 Winter
Faculty of Education Chemistry in Education (16) Chemistry courses 2 Winter
Faculty of Education Biology in Education (16) Biology courses 2 Winter
Faculty of Education Physics in Education (15) Physics courses 2 Winter
Faculty of Education Geography in Education (1) Pedagogy, teacher training and social care 2 Winter
Faculty of Education Physical Education in Education (1) Physical education and sport 2 Winter
Faculty of Education Physics in Education (17) Physics courses 2 Winter
Faculty of Applied Sciences Computer Modelling (2014) Special and interdisciplinary fields 1 Winter
Faculty of Applied Sciences Mathematics and its Applications (2015) Mathematics courses 1 Winter
Faculty of Applied Sciences Mathematics and its Applications (2017) Mathematics courses 1 Winter
Faculty of Applied Sciences Mathematics and its Applications (2015) Mathematics courses 1 Winter
Faculty of Education - (16) Pedagogy, teacher training and social care 2 Winter
Faculty of Education Information Technologies in Education (14) Informatics courses 2 Winter
Faculty of Education Studies in Mathematics (16) Mathematics courses 2 Winter
Faculty of Education Biology in Education (17) Biology courses 2 Winter
Faculty of Applied Sciences General Mathematics (2012) Mathematics courses 1 Winter
Faculty of Applied Sciences Mathematics and Management (2011) Mathematics courses 1 Winter
Faculty of Applied Sciences Mathematics and its Applications (2017) Mathematics courses 1 Winter
Faculty of Philosophy and Arts Analytic Philosophy and Philosophy of Sciences (11-1,2,3) Philosophy, theology 2 Winter
Faculty of Applied Sciences Computations and Design (1) Special and interdisciplinary fields 1 Winter
Faculty of Education Chemistry in Education (14) Chemistry courses 2 Winter
Faculty of Education Information Technologies in Education (15) Informatics courses 2 Winter
Faculty of Education Physics in Education (13) Physics courses 2 Winter
Faculty of Education Studies in Mathematics (15) Mathematics courses 2 Winter
Faculty of Education Studies in Mathematics (13) Mathematics courses 2 Winter
Faculty of Applied Sciences Financial Informatics and Statistics (2016) Economy 1 Winter
Faculty of Education - (17) Pedagogy, teacher training and social care 2 Winter
Faculty of Education Studies in Mathematics (17) Mathematics courses 2 Winter
Faculty of Applied Sciences Financial Informatics and Statistics (2007) Economy 1 Winter
Faculty of Applied Sciences Mathematics for Science (2012) Mathematics courses 1 Winter
Faculty of Applied Sciences Mathematics for Business Studies (2015) Mathematics courses 1 Winter
Faculty of Applied Sciences Scientific Computing and Modelling (2011) Mathematics courses 1 Winter
Faculty of Applied Sciences Mathematics for Business Studies (2017) Mathematics courses 1 Winter
Faculty of Education Geography in Education (16) Pedagogy, teacher training and social care 2 Winter
Faculty of Education Biology in Education (15) Biology courses 2 Winter
Faculty of Education Studies in Mathematics (1) Mathematics courses 2 Winter
Faculty of Applied Sciences Computer Modelling (2016) Special and interdisciplinary fields 1 Winter
Faculty of Applied Sciences Computations and Design (2017) Special and interdisciplinary fields 1 Winter
Faculty of Education Information Technologies in Education (16) Informatics courses 2 Winter
Faculty of Education Biology in Education (14) Biology courses 2 Winter
Faculty of Education Information Technologies in Education (13) Informatics courses 2 Winter
Faculty of Education Information Technologies in Education (17) Informatics courses 2 Winter
Faculty of Applied Sciences Mathematics for Business Studies (2016) Mathematics courses 1 Winter
Faculty of Applied Sciences Financial Informatics and Statistics (2016) Economy 1 Winter
Faculty of Applied Sciences Mathematics for Business Studies (2011) Mathematics courses 1 Winter
Faculty of Education Physics in Education (14) Physics courses 2 Winter
Faculty of Education Chemistry in Education (13) Chemistry courses 2 Winter
Faculty of Education Biology in Education (13) Biology courses 2 Winter
Faculty of Applied Sciences Computations and Design (2014) Special and interdisciplinary fields 1 Winter
Faculty of Education Chemistry in Education (17) Chemistry courses 2 Winter
Faculty of Education Geography in Education (17) Pedagogy, teacher training and social care 2 Winter
Faculty of Applied Sciences Mathematics and its Applications (2016) Mathematics courses 1 Winter
Faculty of Applied Sciences Financial Informatics and Statistics (2017) Economy 1 Winter
Faculty of Education Geography in Education (14) Pedagogy, teacher training and social care 2 Winter
Faculty of Education Studies in Mathematics (14) Mathematics courses 2 Winter
Faculty of Education Physical Education in Education (17) Physical education and sport 2 Winter
Faculty of Applied Sciences Mathematics and its Applications (2016) Mathematics courses 1 Winter
Faculty of Applied Sciences Computer Modelling (2017) Special and interdisciplinary fields 1 Winter