# 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 of the analysis in one variable, and he should be able to grasp how these enter in the solution to problems.

Applying knowledge and understanding:

The student should be able to apply the forementioned notions to 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 himself or offered 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

Prerequisites: elementary algebra; trigonometry; analytic geometry; powers and rational numbers; exponentials and logarithms; elementary functions.

## Course contents summary

Functions depending on one variable.

## Course contents

Elementary algebraic properties of the real numbers (standard

types of equations and inequations); logic and set theory.

Numerical sets: natural numbers and induction principle;

combinatoric calculus; rational numbers; real numbers and supremum

of a set; complex numbers and n-roots.

Real functions: maximum and supremum; monotonicity; 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.

Properties of continuous functions (including mean value,

existence of a maximum, Lipschitz continuity); limits of functions

and of sequences of real numbers; 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.

All statements are rigorously proved.

## Recommended readings

Theory and basic examples:

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

Exercises for cross-examination:

E. ACERBI: "Esami di Analisi Matematica 1", Pitagora Editore, Bologna, 2012

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

## Teaching methods

Teaching methods:

Lectures in classrom. Laboratory activities in smaller groups of students.

Examination methods:

Written text (divided in two parts) and oral test.

## Assessment methods and criteria

Written and oral text at the end of the class.

Knowledges and ability in understanding are verified the first one by the management of the oral examination and the second one by the decoding of the text of the problems.

Skills are checked by means of the resolution of the proposed problems.

Judgement independence is verified if the student is able to select the credible answers and descard the impossible ones.

Communication skill is checked by evaluating the way and correctedness of expression, both in written and oral way, of the mathematical concepts.