This course represents a bridge between high school and college ways of teaching and learning. The first task of this course is to bring students with different backgrounds up to date with the pre-requisite material. Adequate time (about two weeks) will be devoted to this introductory part. Preliminaries that should be known from previous studies (in particular inequalities, elementary graphs, trigonometry) will be revisited and put in a more general setting.

The main goal of the course is to provide the students with tools to understand and elaborate logical arguments, from simple to more elaborate ones, and to introduce the elements of differential and integral calculus for functions of one variable with applications to ordinary differential equations of the first order and of the second order linear case. Complex numbers will also be introduced and applied to the solutions of linear equations of the second order with constant coefficients.

Ability to follow a chain of logical arguments. Essentials of differential and integral calculus for functions of one variable. Computational ability.

Preliminaries:
Numbers, equations and inequalities, analytical geometry and trigonometry. First properties of functions.

Calcolo differenziale:
limits, continuity,infinitesimals and infinite functions. Differential calculus for functions of one variable and applications. Taylor formula.

Calcolo integrale:
Integral calculus for functions of one variable. Improper integrals.

Funzioni di piu' variabili:
continuity, partial derivatives, gradient. Line integral of a force.

Equazioni differenziali:
preliminaries on complex numbers. Ordinary differential equations of the first order. Linear ordinary differential equations of the second order with constant coefficients.

A. Bacciotti, F. Ricci, Analisi matematica 1, Levrotto e Bella
A. Bacciotti, Il calcolo differenziale e integrale. Prima parte:
funzioni reali di una variabile reale, Celid.
C. Canuto - A. Tabacco, Analisi Matematica I, Springer-Verlag
F. Flandoli, introduzione all'analisi matematica, McGraw-Hill libri Italia.
Marcellini Sbordone, Analisi Matematica I, Ed. Liguori
L. Pandolfi, Analisi Matematica 1, Bollati-Boringhieri
V. Demidovich, Esercizi e problemi di Analisi Matematica, Editori Riuniti
S. Lancelotti, Esercizi di Analisi Matematica, Celid
P. Marcellini, C. Sbordone, Esercizi di matematica, Vol. 1-2 Liguori

The exam consists of two parts: a written one and an oral par. The written part ascertains the familiarity of the student with the arguments of the course while the oral part is mostly concerned with the theory.
In order to be admitted to the oral part of the exam, students should score 15/30 or higher in the written part.