Learning objectives
Provide the basic tools of Mathematical Analysis
Course unit content
<div align="justify">Real numbers. Axiomatic definition of real numbers, maximum, minimum, least upper and greatest lower bound; whole part and modulus of real numbers; powers, roots, nth roots of non-negative numbers; rational and irrational numbers; intervals, distance; neighbourhoods, cluster points, isolated points, interior points; closed sets, open sets, frontier. injective, surjective, and bijective functions, inverse function; graphs; real functions of a real variable, monotone functions, exponential and logarithmic functions; trigonometric functions. <br />
<br />
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. <br />
<br />
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<br />
of discontinuous functions; zero theorem; continuity and intervals; continuity and monotony; continuity of inverse functions; Weierstrass theorem. <br />
<br />
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. <br />
<br />
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. </div>
Bibliography
E. Acerbi, G. Buttazzo, Primo corso di Analisi Matematica, Pitagora editrice, Bologna (1997).<br />
E. Acerbi, G. Buttazzo, Analisi Matematica ABC, Pitagora editrice, Bologna (2000).<br />
J. Cecconi, G. Stampacchia: Analisi Matematica 1, Ed. Liguori, 1974.<br />
M. Giaquinta, G. Modica: Analisi Matematica 1: Funzioni di una variabile, Ed. Pitagora, 1998.<br />
E. Giusti: Analisi Matematica 1, Ed. Boringhieri, 1983.
Teaching methods
Teaching method: classroom lectures and classroom exercises<br />
Assessment method: written and oral examination
