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Lecturer(s)
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Brzezina Miroslav, doc. RNDr. CSc., dr. h. c.
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Šimůnková Martina, RNDr. Ph.D.
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Course content
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The topic of the subject is to master the bases of number sets theory, fundamental terms and proper-ties of the differential calculus of real functions of one variable. Course program (syllabus): 1. Basic set and logic notation, number sets. Supremum and infimum of sets. The supremum and infimum theorem. Definitions of mapping, and real function of one real variable, types of mappings. 2. Basic properties of functions (even, odd, periodic, bounded functions, monotony). Arithmetic operations of functions. Composite functions, inverse functions. 3. Elementary functions and its transformations (identical function, power, polynomials, rational function, logarithmic and exponential functions. Trigonometric and cyclometric functions. Signum, absolute value, entire function. 4. Real sequences. Neighbourhood of a point. Definition of a limit of sequences. 5. Theorems about sequences, examples. Euler number. 6. Limit of a function, continuity. Limit and continuity of a composite function. 7. One-sided limits and one-sided continuity of functions. (In)finite limits at (in)finite points. Theorem about the grip function. Limit (sin x) / x for x 0. Asymptotes. 8. Derivative, geometric and physical applications, tangent line to a function. Derivative of a composite function. Derivative of an inverse function. 9. One-sided derivation. Derivatives of a higher order. Differential of a function, relationship with the derivative, applications. 10. Properties of continuous functions on the closed intervals. Mean value theorems. The l Hospital rule. 11. Applications of derivatives to studying of graphs of functions: monotony, local extremes; convexity, and concavity, point of inflexion. 12. Global extremes. Examples. 13. Taylor polynomial. 14. Reserve.
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Learning activities and teaching methods
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Monological explanation (lecture, presentation,briefing)
- Class attendance
- 84 hours per semester
- Preparation for exam
- 56 hours per semester
- Home preparation for classes
- 72 hours per semester
- Preparation for credit
- 28 hours per semester
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Learning outcomes
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The topic of the subject is to master the bases of number sets theory, fundamental terms of the differential calculus of real functions of one variable.
Functions of one variable. Differential calculus.
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Prerequisites
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Analytic thinking. Secondary school mathematics.
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Assessment methods and criteria
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Combined examination
Credit - see syllabus.
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Recommended literature
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Bittnerová, D. - Plačková, G.:. Louskáček 1 - Diferenciální počet funkcí jedné reálné proměnné (Sbírka úloh). Liberec, TUL 2006, 2007..
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Brabec, J. - Martan, F. - Rozenský, Z.:. Matematická analýza I. Praha, SNTL 1985..
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Černý, I.:. Matematická analýza, 1.část. [Skripta TU v Liberci.] Liberec 1995..
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Nekvinda, M.:. Matematika I. Liberec 1997..
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Nekvinda, M.- Vild, J.:. Matematické oříšky I. Liberec, TUL 1999, 2002, 2003..
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Veselý, J.:. Matematická analýza pro učitele, 1.díl. Praha, Matfyzpress 1997..
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