# MATHEMATICAL ANALYSIS 1

## Learning outcomes of the course unit

Knowledge and understanding:

At the end of this course the student should know the essential definitions

and results in analysis in one variable, and he should be able to grasp how

these enter in the solution of problems.

Applying knowledge and understanding:

The student should be able to apply the forementioned notionsto solve medium

level problems, and to understand how they will be used in a more applied

context.

Making judgements:

The student should be able to evaluate coherence and correctness of the

results obtained by him or presented him.

Communication skills:

The student should be able to communicate in a clear and precise way, also

in a context broader than mere calculus.

## Prerequisites

Elementary algebra, elementary equation and inequality, elementary logic

## Course contents summary

Integral and differential calculus

## Course contents

PREREQUISITES: elementary algebra, trigonometry, analytic geometry, rational powers, exponentials and logarithms; elementary functions.

PROGRAM

LOGIC: propositions and predicates, sets, functions, order relations and equivalence.

NUMERICAL SETS: natural numbers and the principle of induction, combinatorics and elementary probability, integers, rational, real numbers, complex numbers and n-th roots.

REAL FUNCTIONS: extrema of real functions, monotone functions, even and odd functions, powers, absolute value, trigonometric functions, hyperbolic functions, graphs of real functions.

SEQUENCES: overview of topology, sequences and their limits; comparison theorems and algebraic theorems, continuity, monotone sequences, theorems of Bolzano-Weierstrass and Cauchy, key examples, the number of Napier; recursively defined sequences.

CONTINUOUS FUNCTIONS: limits of functions, continuity, first properties of continuous functions, continuous functions on an interval (zeros, intermediate values); Weierstrass theorem, uniformly continuous functions, theorem of Heine-Cantor; lipschitz functions; infinitesimals.

DERIVATIVES: definition of the derivative, the first properties; algebraic operations on derivatives, derivatives and local properties of functions; theorems of Rolle, Lagrange, Cauchy; indeterminate forms, de l'Hôpital theorem, Taylor's formula and various remains, asymptotic developments; functions convex qualitative study of functions.

INTEGRATION: construction and first properties of the integral, primitive, fundamental theorem of integral calculus, methods of integration, improper integrals, integration of rational functions.

SERIES: standard definition and first properties; convergence criteria set in terms of non-negative; series in terms of alternating sign.

## Recommended readings

• Enrico GIUSTI “Analisi matematica vol.1” Boringhieri

• Emilio ACERBI, Giuseppe BUTTAZZO “Analisi matematica ABC.

1-Funzioni di una variabile” Pitagora

Testi di esercizi

• V. DEMIDOVICH “Esercizi e problemi di Analisi Matematica”

Editori Riuniti.

• Enrico GIUSTI “Esercizi e complementi di analisi matematica vol.1”

Boringhieri

• Domenico MUCCI “Analisi matematica - Esercizi 1. Funzioni di

una variabile” Pitagora

## Teaching methods

Frontal lesson, exercise to little groups, use of tablet PC

## Assessment methods and criteria

The examination is both written and oral.

In the written part, the student will show his basis knowledge and his ability in solving some particular paroblem. In the oral part, the student will show his knowledge of the foundamental theorems of Mathematical Analisys 1. The oral exposition must be done using a proper mathematical formalism.

## Other informations

The lessons, in pdf format, can be downloaded from my webpage.