# MATHEMATICAL ANALYSIS 1

## Learning outcomes of the course unit

Knowledge and understanding.

Students must demonstrate knowledge and understanding of the basic results of differential and integral calculus for functions of one real variable.

Applying knowledge and understanding.

Students must be able to apply the forementioned notions to solve medium level problems related to the field of study and to understand how they can be used for solving problems in a more applied context.

Making judgements.

Students must be able to evaluate coherence and correctness of results obtained by themselves or by others.

Communication skills.

Students must be able to communicate in a clear and precise way mathematical statements in the field of study, also in a context broader than mere calculus.

## Prerequisites

Students are supposed to be familiar with the basic mathematical tools taught in High School: basic set theory, elementary algebra, logarithms and exponential, trigonometry, equations and inequalities, basic plane analytical geometry.

## Course contents summary

Differential and integral calculus for functions of one real variable.

## Course contents

1) Number systems.

Rational and real numbers. Dedekind's completeness axiom. Least upper bound and greatest lower bound of sets. Archimedean property of integers. Density of rational numbers.

Induction principle, Bernoulli's inequality and Newton's binomial formula.

Combinatorics: permutations, dispositions and simple combinations.

Complex numbers. Algebraic and geometric representation of complex numbers. De Moivre's formula. Roots of complex numbers. The fundamental theorem of algebra.

2) Numerical sequences and series.

Convergent and divergent sequences. Theorems about limits of sequences. Monotone sequences. Nepero's number and some fundamental limits of sequences. Bolzano-Weierstrass theorem. Cauchy sequences and completeness of real numbers.

Convergent and divergent series. Necessary condition for convergence. Series with nonnegative terms: comparison, root and ratio tests. Leibnitz's test for alternating series. Absolute convergence.

Complex sequences and series.

3) Limits and continuity for functions of one real variable.

Finite and infinite limits, limits at infinity. Theorems about limits of functions. Limits of monotone functions. Some fundamental limits of functions.

Continuous functions. Composition of continuous functions. Continuity of inverse functions. Continuity of elementary functions. The intermediate value theorem and its consequences. Weierstrass theorem. Uniform continuity and Heine-Borel theorem

4) Differentiation of functions of one real variable.

The derivative of a real function of one variable and its geometrical meaning. The algebra of derivatives. The chain rule and the inverse function theorem. Derivatives of elementary functions. Theorems by Fermat, Rolle and Lagrange and their consequences. Higher order derivatives. Taylor's formula. Lagrange's remainder. Maxima and minima of differentiable functions. Convex functions and their properties. Antiderivatives. Integration by parts and by substitution. Antiderivatives of rational functions.

5) The Riemann integral.

The Riemann integral of a bounded real function of one variable and its geometrical meaning. Properties of the integral. Integrals of monotone and continuous functions. The mean value theorem for integrals. The fundamental theorem of calculus and its consequences. Generalized integrals. Integral test for series.

## Recommended readings

E. GIUSTI "Analisi Matematica 1", II ed., Bollati Boringhieri, Torino 1988

D. MUCCI "Analisi Matematica esercizi vol.1", Pitagora, Bologna 2004

## Teaching methods

Lectures in classroom. Laboratory activities in smaller groups.

## Assessment methods and criteria

The final exam consists of a written text (two parts) followed by a colloquium.