Università Roma Tre

Numerical sets (N, Z, Q and R), R assiomatic construction of R by upper extremity, Archimede properties, Q in R density, N construction and induction principle, Newton binomial and combinatorial calculus, Inequality of Bernoulli; Topology elements in R (isolated and accumulated points, open / closed sets and characterization, closure of a set) and Bolzano-Weierstrass theorem; Complex numbers, polar representation and n-th roots of the unit; Real functions of real variable, domain, co-domain, and inverse functions; Function and property limits, monotone function limits; Succession limits, significant limits, Nepero number, bridge theorem, limsup / liminf, sequences and topology, compact sets, and characterization; Continuous functions and their properties, continuity of elementary functions, types of discontinuities and monotone functions, fundamental theorems on continuous functions (zeros, intermediate values, Weierstrass); Function and function derivative, elementary function derivatives, fundamental theorems of differential calculus (Fermat, Rolle, Cauchy, Lagrange, de Hopital, Taylor's formula), monotony and derivative sign, local degradation maxima / minima, convex functions / concave; Function graph; Integration according to Riemann and property, integrability of continuous functions, primitive elementary functions, I and II fundamental theorem of integral calculation, integration for substitution and parts, rational functions, some special replacements; Numerical series and convergence, geometric series, convergence criteria for positive terms series (comparison, asymptotic comparison, root, ratio, condensation) and for series at any time (absolute convergence, Leibnitz); Taylor series developments, developments of some elementary functions; Improper integrals.

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