# 27991 - Mathematical Analysis T-1

## Learning outcomes

Basic knowledge of Integral and Differential Calculus for real funcions of a real variable.

## Course contents

PROPERTIES OF REAL NUMBERS.
LIMITS AND CONTINUOUS FUNCTIONS. Definition of convergent and of divergent sequences of real numbers. Theorems about limits of sequences: uniqueness of the limit, comparison theorems. The algebra of limits. Monotone sequences and their limits. The number e. Decimal representation of real numbers. Generalities about functions: composition of functions, invertible functions and inverse functions. Pecularities of real-valued functions of one real variable. Definition of a continuous function of one real variable. The Weierstrass theorem and the intermediate value theorem. Definition of limit of a real function of one real variable; generalization of results established for sequences. Continuity of the composition of two continuous functions and the theorem on the change of variable in a limit. One-sided limits. Monotone functions and their limits. Asymptotes. The inverse circular functions. The hyperbolic functions and their inverse functions.
DIFFERENTIAL CALCULUS. Definition of a differentiable function and of derivative of a function. The algebra of derivatives. The chain rule. The mean value theorem and its application to study the monotonicity of a function. Higher order derivatives. Taylor's formula with Peano and Lagrange forms of the remainder. Relative maxima and minima of a function: definitions, necessary conditions, sufficient conditions. Convex functions.
INTEGRAL CALCULUS. Definition of the Riemann integral. Properties of the integral: linearity, additivity, monotonicity, the mean value theorem. Sufficient conditions of integrability. The fundamental theorems of the integral calculus. The theorems of integration by substitution and of integration by parts. Piecewise continuous functions and propeties of their integrals. Improper integrals: definitions, absolute convergence, comparison theorem.
COMPLEX NUMBERS. Definition and operations on complex numbers. Algebraic form of a complex number, modulus and argument of a complex number, exponential form of a complex number. De Moivre formula, roots of a complex number, algebraic equations in C, the complex exponential function.
LINEAR DIFFERENTIAL EQUATIONS. Linear differential equations of first order: general integral for homogeneous and non homogeneous equations, the Cauchy problem. Linear differential equations of second order with constant coefficients: general integral for homogeneous and non homogeneous equations, the Cauchy problem. Generalization to variable coefficients and arbitrary order equations.

M. Bertsch, R. Dal Passo, L. Giacomelli - Analisi Matematica, ed. McGraw Hill. (seconda edizione)

or

G.C. Barozzi, G. Dore, E. Obrecht: Elementi di Analisi Matematica, vol. 1, ed. Zanichelli.

or

Marco Bramanti, Carlo Domenico Pagani, Sandro Salsa, Analisi matematica 1. Ed. Zanichelli.

There are some useful texts with exercises:
M. Bramanti - Esercitazioni di Analisi 1, Ed. Esculapio, Bologna, 2011

M. Amar, A.M. Bersani - Esercizi di Analisi Matematica 1, Ed. Esculapio, Bologna, 2011

S. Abenda - Esercizi di Analisi Matematica Vol 1, Ed. Progetto Leonardo -Bologna

or

S. Salsa & A. Squellati: Esercizi di Matematica, Vol. I, Ed. Zanichelli

## Teaching methods

The course consists of lessons describing the fundamental concepts of real numbers, sequences and numerical series, and, especially, of real functions of one real variable. Lessons are completed with examples and counterexamples illuminating the theoretical content. Futhermore a lot of exercises are solved in the classroom.

## Assessment methods

The assessment consists in a written and an oral exam. In the written part, lasting three hours, the solution various exercises is required. Access to the oral part is allowed only to the students passing the written part. The oral part of the exam dwells upon the comprehension of the relevant concepts and on the knowledge of definitions and the statements of fundamental theorems. Proofs of some theorems, clearly detailed, may be required. The oral part of the exam must be passed in the same session of the written part. Only in the period january - february the oral part may be passed in the subsequent session.

## Teaching tools

The reading book suggested and, in addition, some exercices and other material will be made available online.

## Office hours

See the website of Nicola Arcozzi

See the website of Eleonora Cinti

See the website of Eleonora Cinti