MATHEMATICAL ANALYSIS

Sets and operations on them. Sets of numbers: N, Z, Q and R. The representation of the real numbers via the real line and the order relationship. Absolute value.

Intervals. Bounded sets. Supremum and infimum of a set. Neighborhood of a point: circular, open and closed neighborhood.

Functions: domain, graph, image, preimage, boundedness, injectivity, surjectivity, bijectivity, monotonicity. Some examples. Inverse function. Theorem on inverse function.

Composition. Translation, rescaling. Even, odd and periodic functions.

Elementary functions: powers, polynomial and rational functions. Exponential and logarithmic functions. Trigonometric functions and their inverse. Hyperbolic goniometric functions.

Sequences.

Limits, limits from the left and from the right.Theorems: uniqueness, sum, product, sign (proof), comparison I and II (proof). Algebra of limits.

Continuity, continuity from the left and from the right, discontinuity. Basic theorems on continuous functions: theorems coming from those on limits, continuity of the composition and of the inverse function.

Indeterminate forms: sum, product and exponential type. lim x?0sinx/x , lim x?+?(1+1/x)^x.

Local comparison, infinitesimal and infinite.

Asymptotes.

Theorems: zeros, Weierstrass, intermediate values.

Derivatives: definitions, rules, relationship between derivability and continuity (proof).

De l’Hopital theorem.Extrema and critical points of a function. Theorems: Fermat, Rolle, Lagrange (and its consequences), Cauchy. Monotonicity test. Convexity.

Taylor’s formula.

Series: basic definition. Geometric series. Leibniz and quotient criterions.

Last update: 09-09-2024

The students may adopt the book

Claudio Canuto, Anita Tabacco, Mathematical Analysis I, EDIZ. MYLAB. CON AGGIORNAMENTO ONLINE, Pearson 2022

or any edition of

Claudio Canuto, Anita Tabacco, Mathematical Analysis I, Springer.

Last update: 12-09-2024

This module aims to present the basic notions of mathematical analysis. We introduce real functions of a real variable, the definitions of limit, continuity and differentiability, together with their main properties and some basic results. We apply the concepts above to the study of the graph of real functions.

Last update: 10-09-2024

Basic knowledge, generally provided in secondary schools.

Last update: 10-09-2024

Lectures. Upon request, teaching material (handouts, tests, exercises) will be provided to students.

Last update: 10-09-2024

The exam consists of a final written test and an oral test, which can be accessed if at least a predetermined minimum score (15/30) is achieved in the final written test. Passing any ongoing written and/or oral tests exempts the Student from discussing, in the final exam, the part on which he has already been assessed.

The written test includes five multiple choice questions. The student will have to choose among the answer’s options the one he believes is correct and justify his choice.

Passing the written test gives the right to take the oral exam, which will take place immediately after the written test. The oral exam is an interview on the topics of the course. During the exam we evaluate the student's ability to communicate, by an appropriate scientific language, its knowledge, as well to explain the techniques used in the written test.

The grade of the exam will be assigned according to the following evaluation criterion:

30 - 30 cum laude: complete, in-depth and critical knowledge of the topics, excellent language skills, complete and original interpretative ability, full ability to autonomously apply knowledge to solve the proposed problems;

26 - 29: complete and in-depth knowledge of the topics, full ownership of language, complete and effective interpretative ability, able to autonomously apply knowledge to solve the proposed problems;

24 - 25: knowledge of the topics with a good level of learning, good language skills, correct and confident interpretative ability, ability to correctly apply most of the knowledge to solve the proposed problems;

21 - 23: knowledge of the topics, but lack of mastery of them, satisfactory language skills, correct interpretative ability, limited ability to autonomously apply knowledge to solve the proposed problems;

18 - 20: basic knowledge of the main topics, basic knowledge of technical language, sufficient interpretative ability, ability to apply the basic knowledge acquired in elementary contexts;

Insufficient: does not have acceptable knowledge of the topics covered during the course.

Last update: 12-09-2024

In the first part of this course we provide the student with the basic notions of differential calculus for real functions of several real variables and we focus on maxima and minima for those functions. Next, we develop the main topics regarding integral calculus: we start with functions of a real variable and then we pass to multiple integrals. Subsequently and gradually, we will move on to targeted in-depth studies that will allow the deal with complex problems inherent to curves, ordinary differential equations, curvilinear and surface integrals.

Last update: 09-09-2024

C. Canuto, A.Tabacco, Mathematical analysis 2, Ediz. MyLab. Pearson 2023

P. Marcellini, C. Sbordone, N. Fusco, Analisi Matematica Due, Zanichelli 2020.

Last update: 09-09-2024

In the first part of this course we provide the student with the basic notions of differential calculus for real functions of several real variables and we focus on maxima and minima for those functions. Next, we develop the main topics regarding integral calculus: we start with functions of a real variable and then we pass to multiple integrals. Subsequently and gradually, we will move on to targeted in-depth studies that will allow the deal with complex problems inherent to curves, ordinary differential equations, curvilinear and surface integrals.

Last update: 09-09-2024

Mathematical Analysis I. The basic notions of functions of one real variable, the definitions of limit, continuity and differentiability, together with their main properties. 

Last update: 09-09-2024

The course, in order to achieve the expected objectives, mainly takes place through lectures. There are also practical based lessons, guided exercises with teacher support, and exam simulations with the aim of stimulating a critical thinking together with an autonomous the approach to problem solving.

Last update: 09-09-2024

No further information

Last update: 09-09-2024

The topics and the level of the exercises correspond to the program delivered and to the reference texts indicated.

The exam consists of a written test followed by an oral test. During the written test, students are asked to perform the complete development of some exercises. The time assigned for the written test is two hours. The evaluation of the written test is scored out of thirty. The student pass the written test if the overall evaluation is not less than 14/30. Once passed the written test, the student is entitled to participate to the oral examination only in the same session in which he passed the written examination.

The written test evaluates the critical skills achieved by the students in the arguments treated during the course and the methodological rigor of the resolutions proposed in response to the questions.

The oral exam consists of an interview on the topics listed on the course program and it assesses the student's ability to communicate, by an appropriate scientific language, the notions learned, as well as his ability to present the theoretical aspects that underlie the various types of exercises contained in written test. The score of the oral exam will be assigned according to the following evaluation criterions:

30-30 cum laude: Comprehensive, deep and critical knowledge of the course topics, excellent skills in understanding and applying independently the acquired knowledge to solve the proposed questions;

29-27: Comprehensive and deep knowledge of the course topics, very good skills in understanding and applying independently the acquired knowledge to solve the proposed questions;

26-25: Comprehensive knowledge of the course topics, good skills in understanding and applying independently the acquired knowledge to solve the proposed questions;

24-22: Proper knowledge of the course topics, skills in understanding and applying independently the acquired knowledge to solve the proposed questions;

21-18: Basic knowledge of the course topics, basic skills in understanding and applying the acquired knowledge to solve the proposed questions.

Last update: 09-09-2024

Goal 4: Ensure inclusive and equitable quality education

Last update: 09-09-2024