# 27210 - Mathematical Analysis 1

## Learning outcomes

At the end of the course, the student acquires basic knowledge of infinitesimal and integral calculus, developing both the habit of scientific reasoning and sensitivity to the analysis of mathematical models, especially by studying the asymptotic development of functions. He also knows how to perform a detailed study of functions in one variable, of sequences, of numerical series and of series of functions.

## Course contents

Sets, relations, functions. Real and complex numbers, R^n. infimum and supremum, completeness. The induction principle. Limits of sequences, monotone sequences. n-th roots, exponential, logarithm, circular functions. Upper limit and the Bolzano-Weierstrass theorem. Topology of R^n. Limits of functions. Continuity and uniform continuity. Compactness. Differential and integral calculus for one-real-variable functions, Taylor formulas. Convessity, local maxima and minima. Generalized integrals. Series. Sequences and series of functions, uniform convergence. Power series. Taylor series. Ordinary differential equations.

Ermanno Lanconelli, Analisi Matematica 1 e 2, Ed. Pitagora.
Enrico Giusti, Analisi Matematica 1 e 2, Ed. Boringhieri.
Pagani, C.D.-Salsa, S., Analisi Matematica 1 e 2, Ed. Zanichelli.
Eserciziari Lanconelli-Obrecht, Esercizi di Analisi 1 e 2, Ed. Pitagora.

## Teaching methods

Frontal classroom lectures on the blackboard.

## Assessment methods

The final examination involves a written test and an oral part. Both parts in the same day. To sign up you should use the system AlmaEsami [https://almaesami.unibo.it/] . For a schedule of examinations, please refer always to AlmaEsami. In the written test, the student has to solve some exercises and to illustrate some topics of "theory", to demonstrate that he has acquired and know how to use the tools provided during the course. Overcoming the written test allows to the oral examination, which consists of a discussion on the written test and in questions that tend to establish the theoretical knowledge of the course contents, the acquisition of the methodological rigor and the ability to reason about topics related to the course. The oral examination aims in particular to verify the achievement of the expected knowledge and skills, that is of the fundamentals of the infinitesimal and integral calculus, of the habit to scientific reasoning and of sensitivity to analysis of mathematical models, especially through the study of the asymptotic expansion of functions. Both the written and oral test have the additional purpose of verifying the learning of the general methods of mathematical analysis and the acquisition of critical judgment in relation to the solution of mathematical problems. The final score, out of thirty, takes into account both tests.

## Office hours

See the website of Francesco Uguzzoni