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

Real numbers.   Completeness axiom. Archimedean property. Mathematical Induction. Factorial and binomial coefficients, Newton formula. Bernoulli inequality. Arithmetic Geometric inequality.

 

Complex numbers. Algebraic and trigonometric form. Powers and roots (De Moivre formulas).

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

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

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

Differential calculus. Theorems of Rolle, Lagrange, Cauchy and De l'Hopital. 

Riemann  integral. Fundamental theorems of Calculus. Integration methods.  

Complements of differential and integral calculus


De l'Hopital's theorems. Search for relative and absolute extremes. Concavity and convexity, Flexions, Asymptotes. Taylor polynomials. Power series. Real exponential and complex exponential. A short look at the the Stieltjes integral.

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

Ordinary Differential Equation. Introduction to elementary differential equation of first order: separable, 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.



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


The lessons will be recorded to facilitate the organisation of notes but will only be available once the lesson is over.


The lecturer is available to address your queries. Please ensure your e-mail messages are signed with your full name, surname, and matriculation number. The lecturer will respond to 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

 

Links to further information

https://www.dropbox.com/scl/fi/4k8tqlg4rrsevdw9d3jla/Materiali2024.pdf?rlkey=cioq2yhys1ya0gwy1pwid7s6c&dl=0

Office hours

See the website of Daniele Ritelli