00013 - Mathematical Analysis

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 R 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.

 

 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 Stieltjes integral. Fundamental theorems of Calculus. Integration methods. 

Series.  Cesaro Stolz Theorems. 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

https://www.editrice-esculapio.com/ritelli-lezioni-di-analisi-matematica/

Further readings

Marco Bramanti. Esercitazioni di Analisi Matematica 1. Esculapio  ISBN: 9788874884445

Manfred Stoll. Introduction to real analysis 3rd edition. Taylor and Francis 2021 ISBN 9780-367-48688-4

Charles H.C. Little, Kee L. Teo and Bruce van Brunt: Real Analysis via Sequences and Series. Springer 2015 ISBN 978-1-4939-2651-0

Peter R. Mercer: More Calculus of a Single Variable. Springer 2014 ISBN 978-1-4939-4685-5

Robert Carlson. A Concrete Introduction to Real Analysis, second edition. 2018 CRC Press ISBN 9781498778138

James R. Kirkwood An Introduction to Analysis, 3ed Taylor and Francis ISBN 9780367702359 

 

John D. Ross, Kendall C. Richards

Introductory Analysis: An Inquiry Approach, CRC Press ISBN 13:978-0-815-37144-1

Suggested preliminary reading

Marco Bramanti. Precalculus. Esculapio

ISBN: 9788874880201

 

 

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 on the Unibo institutional portal.



Lectures will be recorded on Teams

The lectures will be recorded on the Teams platform.

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.
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

Download link

Video beamer and 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/h4occ96pmiae1ws/CV_dr.pdf?dl=0

Office hours

See the website of Daniele Ritelli