FUNCTIONS WITH ONE VARIABLE A
Learning outcomes of the course unit
Provide the basic tools of Mathematical Analysis
Course contents summary
Limits. limits of functions with real values, uniqueness of the limit, limits of the restrictions; limit of the sum, product, quotient of two functions; permanence of the sign, comparison theorems; right and left limit; limits of monotone functions; orders of infinitesimals and infinities, asymptotics.
Continuous functions. continuity of real functions of a real variable, restrictions of continuous functions, composition of continuous functions; sum, product, quotient of continuous functions; examples of continuous functions; discontinuity, examples
of discontinuous functions; zero theorem; continuity and intervals; continuity and monotony; continuity of inverse functions; Weierstrass theorem.
Differential calculus. difference quotients, derivatives, right and left derivatives; geometrical significance of the derivative; derivation rules: derivatives of the sum, product, quotient of two functions; derivatives of compound functions and inverse functions; derivatives of elementary functions; relative maximums and minimums; stationary points; relationship between monotony and sign of the derivative; Rolle's theorem and Lagrange's theorem and their geometrical interpretation, cauchy's theorem and l'Hopital's theorem; convex functions, derivatives of convex functions, relationship between convexity and sign of the second derivative; Taylor's formula with Peano, Lagrange and integral remainder; study of local maximums and minimums with calculation of successive derivatives.
Integrals. partitions of an interval; upper and lower integral, integrable functions in an interval, integrability of continuous functions and monotone functions; geometrical interpretation of the integral; properties of integrals; mean of an integrable function; integrals on directed intervals; fundamental theorem of integral calculus; primitives, indefinite integrals; integration by parts and by substitution; integrals of rational functions.
E. Acerbi, G. Buttazzo, Primo corso di Analisi Matematica, Pitagora editrice, Bologna (1997).
E. Acerbi, G. Buttazzo, Analisi Matematica ABC, Pitagora editrice, Bologna (2000).
J. Cecconi, G. Stampacchia: Analisi Matematica 1, Ed. Liguori, 1974.
M. Giaquinta, G. Modica: Analisi Matematica 1: Funzioni di una variabile, Ed. Pitagora, 1998.
E. Giusti: Analisi Matematica 1, Ed. Boringhieri, 1983.
Teaching method: classroom lectures and classroom exercises
Assessment method: written and oral examination