# 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.

## Recommended readings

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 methods

Teaching method: classroom lectures and classroom exercises

Assessment method: written and oral examination