M1100B Mathematical Analysis I

Faculty of Science
Autumn 2025
Extent and Intensity
4/2/0. 6 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
In-person direct teaching
Teacher(s)
prof. RNDr. Roman Šimon Hilscher, DSc. (lecturer)
Guaranteed by
prof. RNDr. Roman Šimon Hilscher, DSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Prerequisites
!OBOR(AMV) && !OBOR(FINPOJ) && !OBOR(UM) && !OBOR(OM) && !OBOR(STAT) && !OBOR(MOD)
High school mathematics
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The first course of the mathematical analysis. The content is the differential and integral calculus of functions of one real variable. Attention is paid to the fact that students come from middle schools with various level of mathematical knowledge. Students will understand theoretical and practical methods of differential and integral calculus of functions of one variable and will able to apply these methods to concrete problems.
Learning outcomes
At the end of the course students will be able to:
define and interpret the basic notions from the calculus of functions of one real variable;
formulate relevant mathematical theorems and to explain methods of their proofs;
analyse problems from the topics of the course;
understand to theoretical and practical methods of calculus of functions of one variable;
apply the methods of calculus to concrete problems.
Syllabus
  • Introduction: Real numbers and their basic properties, general properties of real functions, elementary functions. Axioms of real numbers and their properties.
  • Functions and sequences: Sequences of real numbers, limit and continuity of functions, properties of continuous functions.
  • Differential calculus in one variable: Basic rules of derivative and its properties, geometric interpretation, Taylor formula, behaviour of functions, planar curves.
  • Integral calculus in one variable: Primitive function and its properties, basic methods of integration, special methods of integrations (integrals of goniometric, irrational, and other types of elementary functions).
  • Riemann integral and its properties: Construction of Riemann integral and its calculation (Newton-Leibniz formula), applications of integrals (area of planar objects, length of curves, volume and surface of solids of revolution).
Literature
    required literature
  • DOŠLÁ, Zuzana and Jaromír KUBEN. Diferenciální počet funkcí jedné proměnné (Differential Calculus of Functions of One Variable). 2. vyd. Brno: Masarykova univerzita, 2012, vi, 209. ISBN 9788021058149. info
  • DOŠLÝ, Ondřej and Petr ZEMÁNEK. Integrální počet v R (Integral Calculus in R). 1. dotisk 1. Brno: Masarykova univerzita, 2024, 214 pp. ISBN 978-80-210-5635-0. info
    recommended literature
  • ZEMÁNEK, Petr and Petr HASIL. Sbírka řešených příkladů z matematické analýzy I. 3., aktual. vyd. Brno: Masarykova univerzita, 2012. Elportál. ISBN 978-80-210-5882-8. url PURL info
    not specified
  • DEMIDOVIČ, Boris Pavlovič. Sbírka úloh a cvičení z matematické analýzy. 1. vyd. Havlíčkův Brod: Fragment, 2003, 460 s. ISBN 8072005871. info
  • BABULA, Kamil. Protipříklady v matematické analýze (Counterexamples in calculus). Brno: Masarykova univerzita, 2008, 44 pp. Bakalářská práce. info
  • NOVÁK, Vítězslav. Diferenciální počet v R. 2. přeprac. vyd. Brno: Masarykova univerzita, 1997, 231 s. ISBN 8021015616. info
  • NOVÁK, Vítězslav. Integrální počet v R. 2. vyd. Brno: Masarykova univerzita, 1994, 148 s. ISBN 8021009918. info
  • Diferenciální počet. Edited by Vojtěch Jarník. Vyd. 6. nezměn. Praha: Academia, 1974, 391 s. URL info
  • Integrální počet. Edited by Vojtěch Jarník. 5. vyd. Praha: Academia, 1974, 243 s. URL info
  • BRAND, Louis. Advanced calculus : an introduction to classical analysis. New York: John Wiley & Sons, 1955, x, 574. info
  • ADAMS, R. A. and Christopher ESSEX. Calculus : a complete course. 7th ed. Toronto: Pearson, 2010, xvi, 973. ISBN 9780321549280. info
Teaching methods
Lectures about the theory with illustrative solved problems. Seminar group devoted to solving theoretical and practical numerical problems.
Assessment methods
Lectures: 4 hours/week. Seminars (compulsory): 2 hours/week.
Written intrasemestral tests in seminars (25% of the overall evaluations).
Final exam: Written test (50%) including computational problems and oral exam (25%) including theoretical questions. In case of problematic epidemic situation only written exam containing also theoretic questions (75%), or even a complete online exam.
To pass: at least 1/2 points from intrasemestral tests, then 50% in total.
Results of the intrasemestral tests are included in the overall evaluation. All percentages are given relative to the overall total for the whole semester.
The conditions for final evaluation may be specified later depending on the pandemic situation and legal regulations.
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught annually.
The course is taught every week.
Listed among pre-requisites of other courses
  • Enrolment Statistics (recent)
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