Università Roma Tre

Numerical sets (N, Z, Q and R), axiomatic construction of R, construction of N and principle of induction, complex numbers; elements of topology in R and Bolzano-Weierstrass theorem; real functions of real variable, limits of functions and their properties, limits of sequences, notable limits, the Napier number; continuous functions and their properties; derivative of functions and their properties, the fundamental theorems of differential calculus (Fermat, Rolle, Cauchy, Lagrange, de l'Hopital, Taylor's formula), convex / concave functions; function graph; Riemann integration and properties, integrability of continuous functions, fundamental theorem of integral calculus, integration by substitution and by parts, integration rules; numerical series, simple and absolute convergence, convergence criteria for series with positive terms and for series with any terms; Taylor series; improper integrals.

Analisi matematica I Marcellini-Sbordone
Analisi matematica I Pagani-Salsa
Esercitazioni di Matematica Marcellini-Sbordone

N and induction principle, Newton binomial; Z, integers modulo n; Q, axiomatic construction of R, Archimedean property, density of Q in R, powers with real exponent; complex numbers, polar representation and n-th roots of the unit; rudiments of topology of R (isolated and accumulation points, open / closed sets); real functions of a real variable, domain, co-domain, and inverse functions; Limits of functions and their properties, limits of monotone functions ; limits of sequences, significant limits, Nepero number, bridge theorem infinite series and convergence, geometric series, convergence tests for series with positive terms (comparison, asymptotic comparison, root, ratio, condensation) and for generic series (absolute convergence, Leibniz); continuous functions and their properties, continuity of elementary functions, types of discontinuities and monotone functions, fundamental theorems on continuous functions (zeros, intermediate values, Weierstrass); derivatives of a functions, derivatives of elementary functions, fundamental theorems of differential calculus (Fermat, Rolle, Cauchy, Lagrange, de l' Hospital, Taylor's formula), monotony and sign of the derivative , local maxima /minima, convex / concave functions; graph of a function; Riemann integral and its properties, integrability of continuous functions, primitive of elementary functions, I and II fundamental theorem of (integral) calculus; integration by substitution and by parts, rational functions, some special substitutions; improper integrals; Taylor series expansions, expansions of some elementary functions.

A. Laforgia, Calcolo differenziale e integrale, Ed. Accademica;
P. Marcellini e C. Sbordone, Esercizi di Matematica, Vol. 1, tomi 1--4, Ed. Liguori;