Aim of the course is to give to the students some basic tools and topics in mathematical analysis, for functions of one variable.

01. Preliminaries:

01.01  Natural numbers, integers, rational numbers and real numbers.

01.02  Absolute value and distance in the real line and in the plane.

01.03  Intervals in the real line.

01.04  Bounded and unbounded sets in the real line.

01.05  Maximum and minimum of a subset of the real line.

01.06  Supremum and infimum of a subset of the real line.

01.07  Proof by induction and proof by contradiction.

02.  Functions of one variable:

02.01  What is a function.

02.02  Real functions of one real variable.

02.03  Elementary functions.

02.04  Operations with functions.

02.05  Limits of functions.

02.06  Theorems on limits (unicity of the limit*, sign-stability theorem*, squeeze theorem*).

02.07  Sequences.

02.08  Limits of sequences.

02.09  Continuous functions.

02.10  Theorems on continuous functions (algebra of continuous functions, sign-stability theorem).

02.11  Points of discontinuity.

02.12  Continuous functions on an interval: Bolzano Theorem (bisection method)*, Weierstrass Theorem and Intermediate-Value Theorem*.

03. Differential calculus for functions of one variable:

03.01  Derivative of a function and its geometric interpretation.

03.02  Derivatives of elementary functions.

03.03  Relations between differentiability and continuity*.

03.04  Differentiation rules.

03.05  Non-differentiability points.

03.06  Critical points, local and global maxima and minima.

03.07  Fermat Theorem*.

03.08  Mean-Value Theorem* and its applications (monotonicity test and characterization of functions with zero derivative in an interval).

03.09  De L’Hospital Rule.

03.10  Second-order derivatives of a function.

03.11  Concavity and convexity of a function.

03.12  Inflection points.

03.13  Sketching the graph of a function.

04. Integral calculus for functions of one variable:

04.01  Antiderivative and indefinite integral of a function.

04.02  Integrable functions.

04.03  Definite integral and its geometrical interpretation.

04.04  Properties of definite integral.

04.05  The Mean-Value Theorem for integrals*.

04.06  The Fundament Theorem of integral calculus*.

04.07  The Torricelli-Barrow Theorem*.

04.08  Techniques of integration: scomposition, substitution and by parts.

* : this means that the proof is required.

There are no mandatory prerequisites.

Knowledge and understanding:

The student will learn the basic notions of mathematical analysis for the study of functions of one variable and the calculation of derivatives and integrals.

Applying knowledge and understanding:

The student will learn the methodologies of mathematical analysis and will be able to apply them to the study of various problems.

Making judgements:

The student will be able to apply the techniques of mathematical analysis in order to solve new problems, also coming from real-world applications.

Communications skills:

The student will have the ability to express the fundamental notions of mathematical analysis using a rigorous terminology.

Learning skills:

The student will learn the ability to study the notions of mathematical analysis, also in order to use it in solving different kind of problems.

The teaching material prepared by the lecturer in addition to recommended textbooks (such as for instance slides, lecture notes, exercises, bibliography) and communications from the lecturer specific to the course can be found inside the Moodle platform › blended.uniurb.it

Teaching

Theorical and practical lessons.

Attendance

Although strongly recommended, course attendance is not mandatory.

Course books

Adams, Calcolo Differenziale 1, Casa Editrice Ambrosiana

Adams - Essex, Calculus: a complete course, Pearson Canada

Bertsch - Dall’Aglio - Giacomelli, Epsilon 1, Mc Graw Hill

Bramanti - Pagani - Salsa, Analisi matematica 1, Zanichelli

Bramanti - Pagani - Salsa, Analisi matematica 1 con elementi di geometria e algebra lineare, Zanichelli

Conti - Ferrario - Terracini - Verzini, Analisi matematica, Vol.1, Apogeo

Salsa - Squellati, Esercizi di Analisi matematica 1, Zanichelli

Rudin, Principles of Mathematical Analysis. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill, New York-Auckland-Dusseldorf, 1976.

Assessment

The exam of Principles of Mathematical Analysis consists of a written exam and an oral one, both of them mandatory.

The written exam, to carry out in two hours, consists of exercises related to the topics of the course. The written exam is passed if the mark is, at least, 15/30. During the written exam it is not allowed to use textbooks, workbooks or notes. Moreover, it is not allowed to use scientific calculators and mobile phones, under penalty of disqualification.

The oral exam consists of a discussion related to the topics of the course.  The oral exam can be taken only if the written one has been passed. If so, the oral exam can be taken only in the same call in which the written exam has been passed or in the other calls of the same session.

The final mark of Principles of Mathematical Analysis is the average of the marks of the written exam and the oral one.

Disability and Specific Learning Disorders (SLD)

Students who have registered their disability certification or SLD certification with the Inclusion and Right to Study Office can request to use conceptual maps (for keywords) during exams.

To this end, it is necessary to send the maps, two weeks before the exam date, to the course instructor, who will verify their compliance with the university guidelines and may request modifications.

Teaching

As for attending students.

Attendance

As for attending students.

Course books

As for attending students.

Assessment

As for attending students.

Disability and Specific Learning Disorders (SLD)

Students who have registered their disability certification or SLD certification with the Inclusion and Right to Study Office can request to use conceptual maps (for keywords) during exams.

To this end, it is necessary to send the maps, two weeks before the exam date, to the course instructor, who will verify their compliance with the university guidelines and may request modifications.