# 27991 - Mathematical Analysis T-1

## Learning outcomes

At the end of the course, after the final examination, the student should possess basic knowledge relating basic elements of Mathematical Analysis, and in particular relating the real functions of a real variable: interpretation of graphs, limits, derivatives, integration and their meaning. The student will be able to solve the typical problems of Mathematical Analysis concerning the above topics (limits; the Induction Principle; continuity; monotonicity and convexity; Taylor expansions; Riemann definite and indefinite integration; complex numbers; real series).

## Course contents

• Introduction: Properties of the real numbers and of N, Z, Q, R. Real-valued functions of one real variable; injectivity, surjectivity, invertibility, inverse function, composition of function, monotone functions. Elementary functions: basic facts.
• Limits: Accumulation point, definitions of limit; one/two-sided limits. Elementary properties of limits: unicity, locality. Algebraic properties of the limit, comparison theorems, limits of monotone functions. Indeterminate forms. Landau symbols.
• Continuity: Definition of a continuous function of one real variable. The Weierstrass Theorem, the Bolzano Theorem and the Intermediate Value Theorem. Continuity of the composition of two continuous functions.
• Differential calculus and applications: Definition of a differentiable function and of the derivative of a function. The algebra of derivatives. The mean value theorems and their application in the study of the monotonicity of a function. Higher order derivatives. Hopital's Rule. Taylor's formula. Local maxima and minima of a function: definitions, necessary conditions, sufficient conditions. Convex functions.
• Integration: Definition of the Riemann integral. Properties of the integral: linearity, additivity, monotonicity, the mean value theorem. The fundamental theorems of the integral calculus. The theorems of integration by substitution and of integration by parts.
• Improper integrals: basic definitions and the comparison theorem.
• Complex numbers: Definiton of the field of the complex numbers. Algebraic form. Modulus and argument of a complex number. Exponential form of a complex number. De Moivre's formula. Complex roots of a complex number. Algebraic equations in C.
• Real series: Basic definitions; necessary condition for the convergence of a series; series of nonnegative real numbers: comparison theorems; the root and ratio criteria. Leibniz theorem.

G.C. Barozzi, G. Dore, E. Obrecht: Elementi di Analisi Matematica, Volume 1, Zanichelli Editore (Bologna), 2009

Online pdf files will regularly be uploaded on the site "VIRTUALE" (Virtual Learning Environment)

https://virtuale.unibo.it/

S. Salsa, A. Squellati: Esercizi di Analisi Matematica 1, Zanichelli Editore (Bologna), 2011

Provided that the student carefully solves the exercises presented by the teacher in the online pdf's, he/she will not need any further texts for his/her study. For the theory part: it is sufficient that the student regularly attends ALL the frontal lessons given in the classroom, and that he/she carefully studies the theory on the notes taken in the classroom.

An exercise pdf sheet of paper will provide the student with the COMPLETE list of theoretical questions for the oral examination.

The adopted textbook is an optional tool. Non-attending students are strongly advised to obtain lecture notes taken by some regularly attending students. This will allow the non-attending student to save time and effort in preparing the written and oral exams. Obviously, non-attending students have the right to prepare for the exam also through the use of the recommended texts.

## 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 2020/21, the teacher will activate a Telegram chat (which will be made available to students from the very beginning of the semester) in order to be able to communicate in a constant, constructive and useful way with the class (and to upload useful didactic material).

## Assessment methods

Written and oral examinations. A detailed program for the oral part will be published in the institutional web-site VIRTUALE. The written part of the examination will check the knowledge of ALL the topics presented in the exercises, regularly published online. During the oral examination, the student will be asked at least three theorems/proofs/examples/definitions, presented during the lectures.

Dates:

3 exams in January/February

1 in June, 1 in July, 1 in September

In order to take the written/oral examinations, students 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.

## Teaching tools

Regularly, exercises will be published on-line on the VIRTUALE website. As for the preparation for the oral examination, at the end of the semester, a very-detailed list of the questions for the oral part will be published online.

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 Eugenio Vecchi