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 The system of real numbers
01.01 Some elements of logic
01.02 Fundamental properties of the system of real numbers
01.03 Upper extremity and lower extremity
01.04 Natural, integer and rational numbers
01.05 Some notions of combinatorial type
01.06 The system of complex numbers

02 Continuity and limits
02.01 The continuous functions 
02.02 The neighborhoods 
02.03 Limit of a function
02.04 Upper limit and lower limit
02.05 Sequences
02.06 Special functions
02.07 Some properties of continue functions 
02.08 Series
02.09 Extensions to the complex case

03 Differential calculus
03.01 The derivative
03.02 Some properties of differentiable functions
03.03 L'Hopital theorems
03.04 Taylor's formula
03.05 Convex functions
03.06 Extensions to the complex case

04 Integral calculus
04.01 Lower integral and upper integral
04.02 Integrable functions
04.03 The fundamental theorem of integral calculus
04.04 Integration formulas
04.05 Improper integrals
04.06 Extensions to the complex case

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.

Attendance

The following elementary knowledges developed in the PreCourse of Mathematics are required:

Algebra:
First degree equations
First degree inequalities
First degree equations and inequalities with absolute value
Equations of higher degree than the first
Inequalities higher than the first
Equations and inequalities of higher degree than the first with absolute value
Irrational equations
Irrational inequalities

Analytic geometry:
Orthogonal systems 
The line
The conics

Exponentials and Logarithms:
Exponential equations
Exponential inequalities
Logarithmic equations
Logarithmic inequalities

Trigonometry:
Unitary circumference
Associated angles
Trigonometric expressions
Trigonometric formulas
Trigonometric equations
Trigonometric inequalities

We suggest the text: G. Malafarina, Mathematics for pre-courses, McGraw-Hill Education, ISBN: 8838665621, (2010) pp. 225.

Course books

Notes (Dispense) of "Analisi Matematica - Parte I" by Prof. M. Degiovanni - Università Cattolica del Sacro Cuore.

The didactic notes developed during the theoretical lectures as well as the proposal exercices will be available at the platform 
blended.uniurb.it

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 five exercises on the following topics:

Set theory: relations; functions between sets; combinatorics;
Real numerical sequences;
Complete study of a real function of one real variable;
On the properties of continuity and differentiability of real functions of one real variable;
Integrals of a real function of one real variable; calculation of areas of flat domains.

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.

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.

Attendance

The following elementary knowledges developed in the PreCourse of Mathematics are required:

Algebra:
First degree equations
First degree inequalities
First degree equations and inequalities with absolute value
Equations of higher degree than the first
Inequalities higher than the first
Equations and inequalities of higher degree than the first with absolute value
Irrational equations
Irrational inequalities

Analytic geometry:
Orthogonal systems 
The line
The conics

Exponentials and Logarithms:
Exponential equations
Exponential inequalities
Logarithmic equations
Logarithmic inequalities

Trigonometry:
Unitary circumference
Associated angles
Trigonometric expressions
Trigonometric formulas
Trigonometric equations
Trigonometric inequalities

We suggest the text: G. Malafarina, Mathematics for pre-courses, McGraw-Hill Education, ISBN: 8838665621, (2010) pp. 225.

Course books

Notes (Dispense) of "Analisi Matematica - Parte I" by Prof. M. Degiovanni - Università Cattolica del Sacro Cuore.

The didactic notes developed during the theoretical lectures as well as the proposal exercices will be available at the platform 
blended.uniurb.it

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 five exercises on the following topics:

Set theory: relations; functions between sets; combinatorics;
Real numerical sequences;
Complete study of a real function of one real variable;
On the properties of continuity and differentiability of real functions of one real variable;
Integrals of a real function of one real variable; calculation of areas of flat domains.

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.

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.

It is advisable to consult the following text which aims to show mathematics and the associated calculation tools how they could be used to consult and clarify a large number of biological phenomena:

Bodine Erin N. Suzanne Lenhart Gross Louis J. Caristi G. (cur.) Mozzanica M. (cur.) Tommei G. (cur.), Mathematics for the life sciences, UTET University, 2017.

Some simple models taken from the quoted text will be proposed as an exercise during the course.

We also suggest the following text which contains over 500 exercises carried out useful for understanding the main theoretical topics covered during the course:

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

Notes (Dispense) of "Analisi Matematica - Parte I" by Prof. M. Degiovanni - Università Cattolica del Sacro Cuore.