Università Roma Tre

REAL NUMBERS AND RELATIVE TOPOLOGY, REAL FUNCTIONS OF A REAL VARIABLE, LIMITS OF FUNCTIONS, DIFFERENTIABILITY, INTEGRALS, TAYLOR FORMULA, COMPLEX NUMBERS, NUMERICAL SERIES.

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

Number sets (N,Z,Q and R), axiomatic construction of R via supremum, Archimedean property, density of Q in R, construction of N and the inductive method, binomial formula and combinatorial calculus, real powers, Bernoulli inequality; topological concepts in R (accumulation and isolated points, open/closed sets and characterization, closure of a set) and Bolzano-Weierstrass theorem; complex numbers, polar representation and n-roots of unity; real functions with a real variable, domain, image and inverse functions; limits for functions and properties, limits of monotone functions; limits for sequences, special limits, the Napier number, the bridge theorem, limsup/liminf, sequences and topology, compact sets and characterization; continuous functions and their properties, continuity of the elementary functions, types of discontinuities and monotone functions, fundamental theorems on the continuous functions (zeroes, of intermediate values, Weierstrass); derivative of a function and properties, derivatives of elementary functions, fundamental theorems of the differential calculus (Fermat, Rolle, Cauchy, Lagrange, de l'Hopital, Taylor formula), monotonicity and sign of the derivative, degenerate local maxima/minima, convex/concave functions; graph of a function; Riemann integration and properties, integrability of continuous functions, primitives for elemntary functions, I and II fundamental theorems of the integral calculus, change of variables and integration by parts, rational functions, some special change of variables; series and convergence, geometric series, convergence criteria for positive series (comparison, asymptotic comparison, root, quotient, condensation) and for generic series (absolute convergence, Leibnitz); Taylor series, series of some elemtnary functions; improper integrals.

Analisi Matematica 1, C.D. Pagani, S. Salsa, editore Zanichelli
Analisi Matematica 1, E. Giusti, editore Bollati Boringhieri
Funzioni Algebriche e Trascendenti, B. Palumbo, M.C. Signorino, editore Accademica
Analisi Matematica, M. Bertsch, R. Dal Passo, L. Giacomelli, editore MCGraw-Hill

REAL NUMBERS AND RELATIVE TOPOLOGY, REAL FUNCTIONS OF A REAL VARIABLE, LIMITS OF FUNCTIONS, DIFFERENTIABILITY, INTEGRALS, TAYLOR FORMULA, COMPLEX NUMBERS, NUMERICAL SERIES.

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