Syllabus
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1. Basic notions a) Sets, relations, mappings b) Axiomatics of real numbers, infimum and supremum *
2. Limits of sequences a) Limits and arithmetic operations, limits and inequalities, extension of reals b) Limits of monotone sequences, Cantor nested interval theorem, Bolzano-Cauchy condition c) Borel covering theorem. Cluster points of a sequence, lim sup *
3. Series of real numbers a) Convergent series, absolutely convergent series b) Cauchy's root and ratio tests, Leibniz's test. *
4. Limits and continuity of functions a) Theorems on limits, Heine's approach to limits of functions. Bolzano-Cauchy condition for the convergence of functions b) Limits and continuity, limit of a composition of functions, continuity of the inverse function c) Properties of continuous functions on a closed interval. Intermediate value property, extrems, uniform continuity *
5. Elementary transcendental functions a) Polynomials, rational functions, n-th root b) Exponential function, logarithm, power function c) Trigonometric and hyperbolic functions, cyclometric functions *
6. Derivative of function a) Definition, derivative as a function, applications b) Derivatives and arithmetic operations, derivative of composed and inverse function (chain rule) c) Higher derivatives, Leibniz's formula *
7. Properties of functions a) Theorems of Rolle, Lagrange and Cauchy (mean value theorems) b) Relation between derivative and monotonicity (convexity). c) Extreme values, points of inflection, asymptots
Annotation
The first part of a four-semester course in calculus for bachelor's program Financial Mathematics.