# MATHEMATICS

## Learning outcomes of the course unit

At the level of knowledge and skills, the course aims to provide students with a good theoretical understanding of the topics, as well as a good ability to perform the calculations and to apply the methods required to solve exercises and problems. This ability is necessary for the effective application of mathematics in other scientific courses.

Another important goal of the course is to provide the student with an adequate mastery of the logical-mathematical language as a model of communication of rigorous scientific contents.

In summary, at the end of the course the student is able to:

1) Know the meaning and interpretation of the mathematical tools that are the subject of the course. For example, the meaning of derivative as the growth rate or as the slope of the tangent line to the graph.

2) Perform the calculations according to the rules presented in the course. For example, being able to apply the rules of calculation of integration.

3) Apply the methods for the study of graphs of functions and their interpretation. For example, apply differentiation to determine maximum and minimum of a function.

4) Communicate the statement of a theorem in a language formally correct.

## Prerequisites

Integers and rational numbers. Basic algebra including simple algebraic equations/inequations. Basic notions of analytic geometry (Cartesian plane, linear equations). Basic notions of trigonometry.

## Course contents summary

The course covers basic concepts and methods of differential and integral calculus for functions of one real variable: number sets, sequences, limits, graphs of functions, derivatives and integrals. Although the presentation of the arguments privileges understanding of the concepts and techniques of calculation with respect to formal rigor, some selected theorems with proof are presented.

## Course contents

1) Number sets and real functions.

Natural numbers, integers, rational numbers, real numbers.

Set theory.

Functions and Cartesian representation.

Monotone functions.

Linear functions.

Power, exponential, logarithm.

Trigonometric functions.

Mathematical induction.

2) Sequences.

Limits.

Operations with limits.

Indeterminate forms.

Napier's number e.

3) Continuous functions.

Limits of functions.

Definition of continuous function: examples and properties.

Weierstrass theorem.

Zeros of continuous functions.

4) Differentiation.

Geometrical meaning of the derivative.

Differentiation rules.

Higher order derivatives.

5) The fundamental theorems of differential calculus.

Rolle, Lagrange, Cauchy theorems and applications.

Points of growth, of degression, maximum and minimum of a

function.

Convex functions.

Taylor's formula.

6) Theory of Riemann integration.

Integral of a continuous function.

Definite Integrals: geometric interpretation.

The fundamental theorem of calculus. Fundamental formula of calculus. Indefinite integrals. Integration sum decomposition.

Integration by parts. Integration by substitution.

## Recommended readings

P. Marcellini, C. Sbordone: Calcolo, Liguori Editore.

P. Marcellini, C. Sbordone: Esercitazioni di matematica, Volume 1, parte prima e seconda, Liguori Editore.

Vinicio Villani: Matematica per discipline bio-mediche, quarta edizione, Mc Graw-Hill

## Teaching methods

1) Lectures where concepts, calculation rules and methods are explained.

2) Recitations during which exercises and problems are solved with the methods discussed in the lectures.

3) Homework. The exercises will be afterwards discussed in the classroom with the active participation of students.

## Assessment methods and criteria

The final exam consists of a written test and an oral test. As an alternative to the final written test, there will be three written tests on specific topics during the semester.

The written test is designed to measure the ability to solve basic mathematical problems based on the course material.

The oral test is aimed at evaluating the theoretical skills and exposure capability of the student.

To this end, a topic selected by the student from the material course is discussed, afterwards the student must show an adequate knowledge of another subject chosen by the examiner.