The course is aimed at the acquisition of the theoretical and applicative principles of basic mathematics. The aim of the course is to present the main notions of elementary algebra and mathematical analysis. In particular, the tools necessary for the qualitative study of the real functions of a real variable will be introduced. For this purpose, the fundamental notions of differential and integral calculus for real functions of one real variable will be presented.

01. Introduction
01.01 Overview of Set Theory
01.02 Some elements of logic
01.03 Fundamental properties of the real number system
01.04 Upper bound and lower bound
01.05 Natural, integer and rational numbers
01.06 Some notions of combinatorial type
01.07 Elements of Analytical Geometry

02. Continuity and limits
02.01 Continuous Functions
02.02 The neighborhoods
02.03 Limit of a function
02.04 Upper limit and lower limit
02.05 Sequences
02.06 Elementary functions
02.07 Some properties of the continuous functions

03. Differential calculus
03.01 The derivative
03.02 Some properties of differentiable functions
03.03 The fundamental theorems of Differential Calculus
03.04 Examples and applications

04. Study of a real function of a real variable
04.01 General
04.02 Classic examples
04.03 Maximum (minimum) problems

05. Integral calculus
05.01 Lower integral and upper integral
05.02 Integrable functions
05.03 The fundamental theorem of integral calculus
05.04 Integration formula

06. Elements of Analytic Geometry
06.01 The line
06.02 The Conics
06.03 The resolution of the Geometric Problem

No additional information.

Knowledge and understanding. At the end of the course the student must have acquired a good knowledge of the mathematical topics covered in the course. He must be able to argue correctly and with language properties on the topics covered in the program. Examples and working methods are shown in the classroom during the lessons and proposed in the exercises.

Applied knowledge and understanding. At the end of the course the student must have acquired a good ability to use the main tools of basic mathematics. He must be able to correctly apply the formulation studied and must be able to solve general mathematical problems similar to those studied. In particular, he must be able to apply the acquired knowledge even in contexts slightly different from those studied, and have the ability to use the acquired knowledge to independently solve problems that may appear new. Examples of such applications are shown in the classroom during the lessons and proposed in the exercises.

Autonomy of judgment. At the end of the course the student must have acquired a good ability to analyze topics and problems in general mathematics, the ability to critically evaluate any proposed solutions, and a correct interpretation of similar topics.

Communication skills. At the end of the course the student must have acquired a good ability to clearly communicate his / her statements and considerations concerning general mathematics problems. The working method is shown in the classroom during the lessons and proposed in the exercises.

Ability to learn. At the end of the course the student must have acquired a good capacity for autonomy in the study of the discipline, in the reading and interpretation of a qualitative phenomenon, in the search for useful information to deepen the knowledge of the topics covered.

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

There are no supporting activities.

Teaching

Theoretical lessons and exercises.

Attendance

Although strongly recommended, course attendance is not mandatory.

Course books

G. De Marco, Analisi Uno. Teoria ed Esercizi. Zanichelli, Bologna, 1986.

G. Devillanova - G. Molica Bisci, Elements of Set Theory and Recursive Arguments, Atti della Accademia Peloritana dei Pericolanti
Classe di Scienze Fisiche, Matematiche e Naturali, ISSN 1825-1242 - Vol. 99, No. S1, A? (2021).

G. Devillanova - G. Molica Bisci, The Faboulous Destiny of Richard Dedekind, Atti della Accademia Peloritana dei Pericolanti
Classe di Scienze Fisiche, Matematiche e Naturali, ISSN 1825-1242 - Vol. 98, No. S1, A1 (2021).

G. Malafarina, Matematica per i precorsi, McGraw-Hill Education, ISBN: 8838665621, (2010) pp.225.

C. Marcelli, Analisi matematica 1. Esercizi con richiami di teoria. Ediz. MyLab. Con aggiornamento online, Pearson, 2019.

C. Pagani - S. Salsa, Analisi Matematica, Vol. 1 Zanichelli, Bologna, 2015.

G. Prodi, Analisi Matematica, Bollati Boringhieri, Torino, 1970.

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

Assessment

The expected objectives are verified through the following two tests, both mandatory:

1. A formative assessment test: consisting of a written paper - lasting 2 hours - divided into three exercises on the following topics:

A complete study of a single-valued real function;

Resolution of a geometrical problem;

Integrals of single-valued real function (and applications).

2. An oral interview: consisting of the discussion of the written paper and three open questions on the theoretical topics covered in the course.

For both tests, the evaluation criteria are as follows:
- relevance and effectiveness of the responses in relation to the contents of the program;
- the level of articulation of the response;
- adequacy of the disciplinary language used.

The overall evaluation is expressed with a mark out of thirty taking into account both tests.

Precisely: the written exam consists of exercises related to the topics of the course. The written exam is passed if the mark is, at least, 15/30.  

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.

The final mark of Fundamentals of Mathematics 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.

During the course, several exercises will be proposed to be carried out on the theoretical topics covered in class.

These exercises will be functional to the overcoming of the expected written paper.

Students are strongly advised to do them.