# MATHEMATICS

## LEARNING OUTCOMES OF THE COURSE UNIT

The course object is to supply the students with knowledge and understanding of limits, differential calculus and integral calculus for real functions of one real variable, EDO and elements of mathematical statistic. Give to them the competence to treat and apply these instruments.

Objects of the course are also : judgement independence, strong written and oral communication skills, learning ability, in accordance with the specific objects of the mathematical area.

## PREREQUISITES

Abilty to hand mathematical expressions and to resolve mathematical equalities and inequalities.

## COURSE CONTENTS SUMMARY

Basic notions of set theory and mathematical logic. Real numbers. Real functions of a real variable and their properties. Limit, continuity and Riemann integral. ODE. Brief introduction to Statistic theory.

## RECOMMENDED READINGS

M. Bramanti, C. D. Pagani, S. Salsa, "Analisi Matematica 1", Ed. Zanichelli.

A. Guerraggio, "Matematica per le scienze", Ed. Pearson.

P. Marcellini, C.Sbordone, "Elementi di Analisi Matematica Uno", Ed. Liguori.

## ASSESSMENT METHODS AND CRITERIA

The understanding check consists in a final written (test and, if it will be positive (his mark is greater or equal to 18/30) in an oral discussion. In both the tests the student has to demonstrate knowledge, comprehension and to be able to connect knowledge and comprehension about limits, differential calculus and integral calculus for real functions of one real variable, ODE and elements of mathematical statistic. The written test consists in 5 or 6 open questions regarding the above arguments.

The marks will be attributed considering : the accuracy of the exposition and the operating methods.

The mark of every test has to be greater or equal to 18/30.

The final exam will be considered sufficient if the average of the total amount of the written and oral tests is greater or equal to 18/30.

## TEACHING METHODS

The teaching consists in frontal lessons where both theoretical and applicable aspects are expounded. The exercises are programmed in order to they can solve independently the problems arising from the theoretical lessons.

## FURTHER INFORMATIONS

None

## COURSE CONTENTS

1.Basic notions of mathematical logic. Basic notions of set theory.

2.Integers, rational numbers, irrational numbers. Upper bound, maximum, least upper bound (supremum). The completeness axiom. Functions and terminology concerning functions. Composite functions. One-to-one functions and inverses. Elementary functions and their diagrams : absolute value, rational, exponential, logarithmic, power with real exponent, trigonometric.

3.Limit of a function. One-side limit of a function. Properties of the limits of functions. Continuous functions. Theorems concerning continuous functions on an interval.

4.Definition of derivative. Derivatives and continuity. Algebra of derivatives. The chain rule. One-sided derivatives and infinite derivatives. Zero derivatives and local extrema. Rolle's theorem. The mean-value theorem for derivatives. Higher order derivatives. Taylor's formula with remainder. Convexity of a function. Diagram of a function.

5.Definition of the Riemann integral. Linear properties. Integration by parts. Change of variable in a Riemann integral. Mean value for the Riemann integral. The integral as a function of the interval. Fundamental theorems of integral calculus. Generalized integrals and comparison theorems.

6.Differential equations. Cauchy problems. Linear first order equations. Separable variables equations. Second oreder linear equations with constant coefficients. 7. Basic notions of probability and statistic.

7. Complements : (a) complex numbers (b) matrices and systems (c) ellipse, parabola, hyperbole.