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Lecturer(s)
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Finěk Václav, doc. RNDr. Ph.D.
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Šimůnková Martina, RNDr. Ph.D.
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Course content
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This course theoretically extends and complements the concurrently taught course KMA/PAN1. Emphasis is placed on precise mathematical statements, proofs of selected theorems, counterexamples, and connections between concepts. 1 - The language of mathematics. Sets, relations, mappings, logical connectives, and quantifiers. Direct and indirect proofs, proof by contradiction, and mathematical induction. 2 - Real numbers. Supremum, infimum, and the completeness axiom. The Archimedean property and the nested interval principle. 3 - Functions of one real variable. Composition and inverse functions, boundedness, monotonicity, elementary functions, and plane curves. 4 - Sequences of real numbers. Finite and infinite limits, uniqueness of the limit, algebra of limits, and the squeeze theorem. 5 - Monotone sequences, subsequences, and the Bolzano-Weierstrass theorem. 6 - Cauchy sequences and the Cauchy convergence criterion in R. Relationship with the completeness of the real numbers. 7 - Limits of functions. Definitions using neighbourhoods and the epsilon-delta condition, one-sided limits, and infinite limits. 8 - Continuity, algebra of continuous functions, and continuity of composite and inverse functions. 9 - Properties of continuous functions on a closed interval: boundedness, existence of extrema, and the intermediate value property. 10 - Derivative and differential of a function of one variable. Geometrical interpretation and linear approximation. Relationship between differentiability and continuity. 11 - Differentiation rules, derivatives of composite and inverse functions, and higher-order derivatives. 12 - Rolle's theorem, Lagrange's mean value theorem, Cauchy's mean value theorem, and their consequences. 13 - L'Hôpital's rule. Monotonicity and local and global extrema. 14 - Convexity and concavity, inflection points, asymptotes, curve sketching, and parametrically defined curves.
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Learning activities and teaching methods
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Monological explanation (lecture, presentation,briefing), Self-study (text study, reading, problematic tasks, practical tasks, experiments, research, written assignments)
- Class attendance
- 28 hours per semester
- Preparation for credit
- 14 hours per semester
- Preparation for exam
- 28 hours per semester
- Home preparation for classes
- 20 hours per semester
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Learning outcomes
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Elementary theory of a real function of one real variable and the differential calculus of a real fuction of one real variable.
After completing the course, the student is able to: 1 - State and explain fundamental definitions and theorems concerning sequences, limits, continuity, and derivatives. 2 - Present selected proofs, distinguish between the assumptions and conclusions of theorems, and use appropriate counterexamples. 3 - Apply theoretical results when analysing the properties and graphs of functions.
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Prerequisites
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Upper-secondary-school mathematics and the ability to think analytically. Concurrent enrolment in KMA/PAN1 is required.
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Assessment methods and criteria
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Combined examination, Test
Course credit is awarded for regular work on assigned theoretical problems and successful completion of a credit test. The examination consists of written and oral parts. It assesses knowledge of definitions and theorems, understanding of their assumptions, the ability to present selected proofs, and the ability to apply theory when solving problems.
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Recommended literature
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Černý, I. Matematická analýza, 1. část. [Skripta TU v Liberci.]. Liberec: TUL, 1995.
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Černý, I. Matematická analýza, 2. část. [Skripta TU v Liberci.]. Liberec: TUL, 1996.
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Jarník, V. Diferenciální počet I. Praha 1963..
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Jirásek, F., Kriegelstein, E., Tichý, Z. Sbírka řešených příkladů z matematiky. Praha, 1982.
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