# MATHEMATICAL ANALYSIS

## Learning outcomes of the course unit

The course provides the basic mathematical instruments for a solid comprehension of the other courses.

## Prerequisites

Preliminary requirements: elementary algebra; trigonometry; analytic geometry; rational powers; exponential and logarithm; elementary functions.

## Course contents summary

Logic and set theory; equivalence and ordering.

Numerical sets: natural numbers and induction principle; combinatorial calculus and elementary probability; integers and rationals; real numbers and supremum; complex numbers and their n-th roots.

Real functions: maximum and supremum; monotone, odd and even functions; powers; irrational functions; absolute value; trigonometric, exponential and hyperbolic functions; graphs of the elementary functions and geometric transformations of the same.

Sequences: topology; limits and related theorems; monotonic sequences; Bolzano-Weierstrass and Cauchy theorems; basic examples; the Neper number "e"; recursive sequences; complex sequences.

Continuous functions: limits of functions; continuity and properties of continuous functions (including intermediate values, Weierstrass theorem); uniform continuity and Heine-Cantor theorem; Lipschitz continuity; infinitesimals.

Properties of differentiable functions (including Rolle, Lagrange, Hopital theorems); Taylor expansion (with Peano and Lagrange remainder); graphing a function.

Indefinite and definite integral: definition and computation (straightforward, by parts, by change of variables); integral mean and fundamental theorems; Torricelli theorem; generalised integrals: definition and comparison principles.

Numerical series: definition, convergence criteria, Leibniz and integral criteria.

## Course contents

Preliminary requirements: elementary algebra; trigonometry; analytic geometry; rational powers; exponential and logarithm; elementary functions.

Logic and set theory; equivalence and ordering.

Numerical sets: natural numbers and induction principle; combinatorial calculus and elementary probability; integers and rationals; real numbers and supremum; complex numbers and their n-th roots.

Real functions: maximum and supremum; monotone, odd and even functions; powers; irrational functions; absolute value; trigonometric, exponential and hyperbolic functions; graphs of the elementary functions and geometric transformations of the same.

Sequences: topology; limits and related theorems; monotonic sequences; Bolzano-Weierstrass and Cauchy theorems; basic examples; the Neper number "e"; recursive sequences; complex sequences.

Continuous functions: limits of functions; continuity and properties of continuous functions (including intermediate values, Weierstrass theorem); uniform continuity and Heine-Cantor theorem; Lipschitz continuity; infinitesimals.

Properties of differentiable functions (including Rolle, Lagrange, Hopital theorems); Taylor expansion (with Peano and Lagrange remainder); graphing a function.

Indefinite and definite integral: definition and computation (straightforward, by parts, by change of variables); integral mean and fundamental theorems; Torricelli theorem; generalised integrals: definition and comparison principles.

Numerical series: definition, convergence criteria, Leibniz and integral criteria.

## Recommended readings

Theory and basic exercises:

E. ACERBI e G. BUTTAZZO: "Primo corso di Analisi matematica", Pitagora editore, Bologna, 1997

D. MUCCI: "Analisi matematica esercizi vol.1", Pitagora editore, Bologna, 2004

esamination exercises:

A. COSCIA e A. DEFRANCESCHI: "Primo esame di Analisi matematica", Pitagora editore, Bologna, 1997

## Teaching methods

Teaching method:

Oral lessons, practical lessons in small groups.

Exams:

Written test divided into two parts followed by a colloquium.

## Assessment methods and criteria

Written and oral examination at end