# MATHEMATICS

## Learning outcomes of the course unit

The aim of the course is to teach the students the fundamentals of differential and integral calculus and to help them master the techniques of calculus.

## Course contents summary

Introduction: The ordered field of the real numbers. Maximum, minimum, least upper bound and greatest lower bound. Inequalities. Basic concepts of real variable real functions and principal properties of elementary functions. Elements of combinatorial analysis. Overview of the field of complex numbers.

Limits. Limits of a sequence and of a function. Operations with limits and indeterminate forms. Comparison theorems, significant limits. Continuous functions and the main theorems.

Differential calculus and applications. The concept of derivative. Rules of derivation and derivatives of elementary functions. Study of function: relative extrema and theorems of Fermat and Lagrange; test of increase (decrease), convexity, concavity, asymptotes and graph. Taylor’s formula. Expansions of the elementary functions and applications.

Integral calculus and applications. Definition of the integral; properties and geometric meaning. The mean value theorem and basic formula of integral calculus (Torricelli’s theorem). Indefinite integral and techniques of integration. Areas of plane domains. Improper integrals. Differential equations: various models and study of some types of first order differential equations.

Elements of linear algebra. Vector spaces and matrices. Determinant, characteristic, inverse matrix. Linear systems. Self values and self vectors.

Elements of probability and statistics. Definitions of probability and axioms; conditioned probability and law of product. Binomial distribution.

Data representation; means, variance, standard deviation. The normal distribution. The regression line and the calculation of regression coefficients; covariance and correlation coefficient. Linear-related models.

## Recommended readings

M. BRAMANTI , C.D. PAGANI , S. SALSA: “Mathematica”. Zanichelli.

P. MARCELLINI, C. SBORDONE: "Calcolo", Liguori Editore, Napoli.

V. VILLANI: "Matematica per discipline biomediche", McGraw Hill, Libri Italia.

For exercises:

P. MARCELLINI, C. SBORDONE:"Esercitazioni di Matematica", I volume, parte prima e seconda, Liguori Editore.

## Teaching methods

The teaching method is based on classroom lessons in which theoretical treatment of the material covered is supplemented by numerous examples and exercises. In addition, enrolment in the IDEA project also makes it possible for the student to utilize supplementary exercises.

The final exam involves a written test and an oral interview: The student must demonstrate to have acquired the methods and techniques of differential and integral calculus as well as adequate comprehension of the basic concepts of the various topics covered.