# 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

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

## Course contents summary

Logic and set theory.

Numerical sets.

Combinatorial calculus and elementary probability.

Real functions.

Sequences.

Continuous functions.

Properties of differentiable functions.

Indefinite and definite integral.

Numerical series.

Full proof is provided for every statement.

## 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:

E. ACERBI: Esami di Analisi matematica 1, Pitagora editore, Bologna, 2012

## Teaching methods

Teaching method:

Oral lessons, practical lessons in small groups.

Exams:

Written test divided into two parts followed by a colloquium; bi-weekly written exercises during the semester.

## Assessment methods and criteria

Written and oral examination at end.

Knowledge and decoding ability are checked, the first by answering theoretical questions during the oral examination, the second by understanding and decoding the text of complex problems.

The ability to apply knowledge is checked through the solution of proposed problems. The first part of the written test contains both exercises where only basic knowledge has to be applied (if z=5+7i then the suqre of z is...) and exercises requiring understanding of the subject (if z=1+i then the 3257th power of z is...)

The ability to make judgements is checked through the selection of reasonable and unreasonable answers.

Communication skills are checked through an appropriate use of technical language during the oral and written parts.