# 00013 - Mathematical Analysis

## Learning outcomes

The aim of the course is to provide the capability of the student to face both theoretical and practical problems in Mathematical Analysis, referring to the analysis of the behavior of a real function of a real variable, computation of definite integrals, development of a function in power series.

## Course contents

The set R of real numbers.  The main subsets of R: Natural, Integers and Rational numbers. Completeness axiom. Archimedean property. Mathematical Induction. Factorial and binomial coefficients, Newton formula. Bernoulli inequality. Arithmetic Geometric inequality.

Sequences. Limit of a sequence. Monotonic sequences and the number e.

Real functions. Limits and elementary functions. Asymptotics and Landau symbols. Continuous functions. Bolzano theorem on intermediate value and Weierstrass theorem on maxima and minima.

Derivatives. Theorems of Rolle, Lagrange, Cauchy and De l'Hopital. Graph of a function, extrema. Convex and concave functions. Inflexion points. Asymptotes. Taylor polynomials and series.

Riemann integral. Fundamental theorems of Calculus. Integration methods.

Series.  Cesaro Stolz Theorems. Geometric series. Series with positive terms and convergence tests. Series with alternating terms.

Improper integrals General convergence criterion. Connection with series theory. The probability integral

Ordinary Differential Equation. Introduction to elementary differential equation of first order: separable, homogeneos, linear and Bernoulli.

Daniele Ritelli. Lezioni di Analisi Matematica. Esculapio

2015 ISBN: 9788874888870

Marco Bramanti. Esercitazioni di Analisi Matematica 1. Esculapio  ISBN: 9788874884445

Marco Bramanti. Precalculus. Esculapio

ISBN: 9788874880201

Robert Carlson. A Concrete Introduction to Real Analysis, second edition. 2018 CRC Press ISBN 9781498778138

## Teaching methods

Lessons ex cathedra using also video beamer. Homework. Computer algebra will also be employed to support thoretical arguments.

## Assessment methods

Written examination of 2 hours, where is possible to use calculators and books. The exam is completed by an oral examination if the written examination is satisfactory. The aim of the exam is to detect the capability of the student to face both theoretical and practical problems in Mathematical Analysis. The written examination can be divided, for the first call, in to two partial examinantion and is composed by multiple choice questions and solution of exercises.

## Teaching tools

Video beamer and blackboard.  Computer algebra to illustrate important topics.

The teaching material presented in class will be made available to the student in electronic format through the university's institutional portal. Username and password are reserved for students enrolled at the University of Bologna.

The teacher responds to e-mail messages, duly signed by the student with Name, Surname and matriculation number, and which concern appointment requests or topics that are not covered by the course information presented here.