81854 - Mathematical Analysis 1B

Learning outcomes

At the end of the course, the student will deepen the basic knowledge of Mathematical Analysis, as a unique and creative science of fundamental importance. He/she will gain the knowledge of the concepts of integral and of generalized integral of real function of one real variable (including the notion of convexity), and of a numerical series and sequences of functions. He/she will be able to study real functions of one real variable. In particular, the student will know how to apply this knowledge to the solution of simple practical problems, appearing in the pure and applied sciences.

Course contents

-Taylor's Formula (Peano and Lagrange remainders).

-Convex functions: main definitions and criteria of convexity.

-Study of the qualitative graph of a function of one real variable (further enriched by the knowledge of the related topics of the course of Mathematical Analysis 1A).

-Riemann integral of a real function of one real variable. The Fundamental Theorems of Integral Calculus. Primitive functions. Generalized integrals.

-Numeric series: definitions, main theorems and convergence criteria.

(If time remains: sequences and series of functions: pointwise and uniform convergence; power series; Taylor series.)

-Metric spaces (a general outline). The n-dimensional real Euclidean space.

Readings/Bibliography

In order to deepen the study of the topics of the course, students can consult the following textbooks.

--Theory:

E. Lanconelli, Lezioni di Analisi Matematica 1, ed. Pitagora

P. Marcellini - C. Sbordone: Analisi Matematica 1, ed. Liguori

E. Giusti, Analisi Matematica 1, ed. Boringhieri

--Exercises:

M. Bramanti, Esercitazioni di Analisi Matematica 1, ed. Esculapio

P. Marcellini - C. Sbordone: Esercitazioni di Matematica, volume 1, parte seconda, ed. Liguori

E. Giusti, Esercizi e complementi di Analisi Matematica, volume 1, ed. Boringhieri

**VERY IMPORTANT:

During the course, pdf sheets of exercises will be available on the UniBo website "VIRTUALE" (Virtual Learning Environment) https://virtuale.unibo.it/

For the preparation to the written examination, it is of a fundamental importance that the student carried out all the exercises uploaded by the teacher on the VIRTUALE dedicated website.

Teaching methods

Lectures and exercises in the classroom (and, if need be, according to the dispositions of UniBo, via mixed learning or distance-learning).

Uploads on the VIRTUALE website of several pdf's of exercises.

After the experience imposed by the Covid19 pandemic in the Academic Year 2019/20, the teacher will activate two Telegram chats (which will be made available to students during the first semester) in order to be able to communicate in a constant, constructive and useful way with the class.

Assessment methods

The examination consists of a preliminary written test and of an oral examination.

The written test consists of some exercises related to the arguments of the course. In order to take the written/oral examinations, student must register at least five days before the exam through the website AlmaEsami https://almaesami.unibo.it/ .

The written test remains valid for the oral exam in the same examination period.

The oral test follows the written test; it mainly concerns the theoretical aspects of the course. The student must exhibit his/her knowledge of the concepts given during the course (in particular definitions, theorems and their proofs), and he/she must show how to connect these concepts with each other.

Teaching tools

During the course, pdf sheets of exercises will be made available, uploaded on the UniBo "VIRTUAL" website

https://virtuale.unibo.it/

These pdf's are very important for the written-exam preparation (and were generally considered by previous students to be comprehensive for the written-exam preparation).

Office hours

See the website of Andrea Bonfiglioli

See the website of Annalisa Baldi