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Lecturer(s)
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Course content
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Week 1: Mathematical reasoning - open statements and quantifiers, negating quantified statements; 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; Bolzano-Weierstrass theorem; convergent and divergent sequences; Cauchy 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: Functions of one real variable; graphical representation; inverse functions; composition of functions; polynomial, trigonometric, exponential, and hyperbolic functions; Week 6: Local and global behaviour of a function; limits; one-sided limits; algebra of limits; Week 7: Continuity of a function at a point; points of discontinuity; continuity in a closed interval; Week 8: Derivative and differential of a function - definition and both the geometrical and the physical meaning; differentiability and continuity of a function; Week 9: 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 10: Indefinite integral; fundamental theorem of calculus; integration by parts and integration by substitution; Week 11: Definite integral and its applications; improper integrals; inequalities for integrals; Week 12: Higher order derivatives and differentials; Taylor's theorem; Week 13: Applications of differential and integral calculus in solving optimization and physical problems. Further information and the lecture notes can be found on the web page http://analyza.kma.zcu.cz.
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Learning activities and teaching methods
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Lecture supplemented with a discussion
- 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
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| prerequisite |
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| Knowledge |
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| There is no prerequisite for this course. Students should be familiar with a high school algebra and trigonometry. |
| learning outcomes |
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| By the end of the course, a successful student should be able to: 1. Understand logical constructions and to be able to read mathematical text; 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 the elementary properties of 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 |
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| Lecture supplemented with a discussion |
| assessment methods |
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| Combined exam |
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Recommended literature
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Děmidovič, Boris Pavlovič. Sbírka úloh a cvičení z matematické analýzy. Havlíčkův Brod : Fragment, 2003. ISBN 80-7200-587-1.
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Drábek, Pavel; Míka, Stanislav. Matematická analýza I. Plzeň : Západočeská univerzita, 1999. ISBN 80-7082-558-8.
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Míková, Marta; Kubr, Milan; Čížek, Jiří. Sbírka příkladů z matematické analýzy I. Plzeň : Západočeská univerzita, 1999. ISBN 80-7082-568-5.
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Polák, J. Přehled středoškolské matematiky.. Praha : Prometheus, 2008. ISBN 978-80-7196-356-1.
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Pultr, Aleš. Matematická analýza I. Praha : Matfyzpress, 1995. ISBN 80-8586-3-09-X.
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