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Lecturer(s)
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Šimůnková Martina, RNDr. Ph.D.
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Course content
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Course Description The content of the course is an introduction to the most important parts of the differential calculus of functions of one real variable and their basic applications. Fundamentals of Mathematics and Mapping Language of mathematics, naive set theory, real numbers, and the significance of the completeness axiom (the difference between rational and real numbers). The concept of mapping, basic properties, and operations with mappings (injective mapping, composite mapping, inverse mapping). Basic types of proofs: direct, indirect, by contradiction, and by mathematical induction. Functions and Sequences Functions of a real variable and the basic properties of functions (functions being bounded, periodic, monotonic, odd, and even). Application of function monotonicity to solving inequalities. Sequences of real numbers and their properties. Limit of a sequence, convergent sequences, and convergence of bounded monotonic sequences. Theorems on limits and algebraic operations. Cauchy sequences, and the theorem on Cauchy and convergent sequences. Types of Functions Real functions of one real variable, polynomials, number of roots of a polynomial, and factorization into root factors. Rational functions, partial fractions, and the decomposition of a rational function into a linear combination of a polynomial and partial fractions. Limits and Continuity Continuity of a function at a point and on an interval. Operations with continuous functions. Limit of a function, limits and algebraic operations, and their relationship to continuity. Methods of calculating limits (modification on a punctured neighborhood, limit of a continuous function, theorem on the arithmetic of limits). Heine's theorem on continuity and sequences. Properties of continuous functions on a closed interval: Weierstrass theorem on the extrema of a continuous function, and Bolzano's theorem on the root of a continuous function. Application of Bolzano's theorem to solving inequalities. Intermediate value theorem, and the theorem on the image of an interval under a continuous function. Inverse functions and their properties (monotonicity, continuity). Differential Calculus Derivative of a function at a point, geometric meaning of the derivative, approximation by a linear function, and the equation of a tangent line. Relationship between derivative and continuity, and the derivative as a function. Lagrange's and Rolle's mean value theorems. Relationship between derivative and monotonicity of a function. Derivative of an inverse function, and differentiation and algebraic operations. Derivative of a composite function. Local and global extrema of a function of a real variable, necessary condition for a local extremum, and stationary points of a function. Derivatives of higher orders, convex and concave functions, and sufficient condition for a local extremum. Taylor polynomial, and the remainder in Lagrange form.
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Learning activities and teaching methods
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Monological explanation (lecture, presentation,briefing)
- Class attendance
- 14 hours per semester
- Preparation for credit
- 28 hours per semester
- Preparation for exam
- 28 hours per semester
- Home preparation for classes
- 38 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.
Functions of one variables, differential calculus.
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Prerequisites
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Secondary school mathematics.
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Assessment methods and criteria
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Oral exam, Written exam
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Recommended literature
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Bittnerová, D. - Plačková, G. Louskáček 1 - diferenciální počet funkcí jedné reálné proměnné (Sbírka úloh). [Skripta TU v Liberci.] Liberec 2005.. Liberec, 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: 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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Nekvinda, M. - Vild, J. Matematické oříšky I. Liberec: TUL, 2003.
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Nekvinda, M. - Vild, J. Náměty pro samostatné referáty. Liberec 1995..
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Nekvinda, M. Matematika I. Liberec 1997 a další..
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Veselý, J. Matematická analýza pro učitele, 1.díl. Praha, Matfyzpress 1997..
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