93744 - ANALISI MATEMATICA

Learning outcomes

The aim of the course is to provide the capability of the student to face both theoretical and practical problems in Mathematical Analysis, referring to the analysis of the behavior of a real function of a real variable, computation of definite integrals, development of a function in power series.

Course contents

The set of real numbers.  The main subsets of R: Natural, Integers and Rational numbers. Completeness axiom. Archimedean property. Mathematical Induction. Factorial and binomial coefficients, Newton formula. Bernoulli inequality. Arithmetic Geometric inequality.

Sequences. Limit of a sequence. Monotonic sequences and the number e. Bolzano-Weierstrass Theorem. Cauchy sequences. Cesaro Stolz Theorems.

 

 Real functions. Limits and elementary functions. Asymptotics and Landau symbols. Continuous functions. Bolzano theorem on intermediate value and Weierstrass theorem on maxima and minima.

Derivatives. Theorems of Rolle, Lagrange, Cauchy and De l'Hopital. Graph of a function, extrema. Convex and concave functions. Inflexion points. Asymptotes. Taylor polynomials and series.

Riemann  integral. Fundamental theorems of Calculus. Integration methods.  Stieltjes integral.

Series.   Geometric series. Series with positive terms and convergence tests. Series with alternating terms.

Improper integrals General convergence criterion. Connection with series theory. The probability integral

Ordinary Differential Equation. Introduction to elementary differential equation of first order: separable, homogeneos, linear and Bernoulli. 

Readings/Bibliography

Main text

Daniele Ritelli. Lezioni di Analisi Matematica IV edizione. Esculapio 2021 ISBN:978-88-9385-257-9

 

Further readings

Teaching methods

Video projection. Blackboard. Computer Algebra will be used to illustrate relevant topics.

Assignment of exercises to be carried out and commented on with the teacher and tutors

The teaching material presented in class will be made available to students in electronic format



Lectures will be recorded.

The lecturer will reply to e-mail messages signed by the students, with name, surname and matriculation number, concerning requests for appointments, explanations and information not already present on the website.

Assessment methods

Written examination of 2 hours, where is possible to use calculators and books. The exam is completed by an oral examination if the written examination is satisfactory. The aim of the exam is to detect the capability of the student to face both theoretical and practical problems in Mathematical Analysis. The written examination can be divided, for the first call, in to two partial examinantion and is composed by multiple choice questions and solution of exercises. 

Evaluation scheme
18-19-E: minimal preparation and ability to analyze, relating to only instrumental mastery of problems presented in the course, correct use of methods at elementary level only after instructor's directions;
20-23-D: sufficient preparation and ability to analyze, but related to an only instrumental mastery of the problems presented in the course, correct and autonomous use of the methods at an elementary level;
24-27-C: technically adequate preparation but with some limits with respect to the topics dealt with, good analytical skills, even if not particularly articulated, correct use of the methods at an intermediate level;
28-29-B: technically adequate preparation with respect to the topics dealt with, good analytical skills in complex problems, correct use of methods at standard level
30-A: excellent and very thorough and exhaustive knowledge of the topics covered in the course, full mastery of methods.
30L-A+: excellent and very deep and exhaustive knowledge of the topics dealt with in the course, full mastery of the methods also from a theoretical point of view.

If the written test is insufficient in a non-serious way, it will be possible, within one week of the publication of the results, to integrate it through targeted exercises.
In the case of an online exam, the teacher will provide a Dropbox link for downloading the text and one for uploading, where the student will have to upload a pdf file with the grid of the answers given and the solution of the proposed problems.

Teaching tools

Video beamer and traditional blackboard.  Computer algebra to illustrate important topics.

The teaching material presented in class will be made available to the student in electronic format through the university's institutional portal. Username and password are reserved for students enrolled at the University of Bologna.

Office hours can be delivered using Teams

The teacher responds to e-mail messages, duly signed by the student with Name, Surname and matriculation number, and which concern appointment requests or topics that are not covered by the course information presented here.

Links to further information

https://www.dropbox.com/s/sdqomqlhx0da9h7/Materiali2022.pdf?dl=0

Office hours

See the website of Daniele Ritelli