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. Elements of (naive) Set Theory.
01.01 Sets: generalities, definitions and operations.
01.02 Relations between sets. Operations between relations.
01.03 Equivalence relations.
01.04 Quotient set.
01.05 Functions between sets: generalities.
01.06 Image and inverse image of a function.
01.07 Injective, surjective and bijective functions.
01.08 Order relations.
01.09 Algebraic structures.
01.10 Compatibility of the order with respect to the operations.
01.11 Operations and orders on the quotient set.
01.12 Cardinality of a set.
01.13 Elements of combinatorics: permutations, dispositions,
combinations.

02. Numeric sets.
02.01 The set of natural numbers.
02.02 The principle of mathematical induction.
02.03  Significative examples.
02.04 The ring of integers.
02.05 The field of rational numbers.
02.06 Irrational numbers: existence.
02.07 The field of real numbers.
02.08 The Dedekind completeness property.
02.09 Real subsets: boundedness and related definitions.
02.10 Archimedean property and concept of density.
02.11 Bernoulli's inequality; Newton's binomial; Mean values.

03. Outline of Topology on the real line.
03.01 Intervals.
03.02 Distance function.
03.03 Open (closed) balls.
03.05 Internal (external), closure, derivative of a set.
03.06 Open (closed) sets.
03.07 Neighborhoods.
03.08 Characterization properties of the open (closed) topology.
03.09 Metric spaces. Metric subspaces. The topology of the plane.

04. Real numerical sequences.
04.01 Uniqueness of the limit
04.02 Subsequences of a sequence.
04.03 Theorem (permanence of the sign).
04.04 Comparison theorems.
04.05 Bounded sequences.
04.06 Carabinieri theorem.
04.07 Infinitesimals.
04.08 Operations with finite limits.
04.09 Operations with infinite limits: sum.
04.10 Operations with infinite limits: product.
04.11 Operations with infinite limits: reciprocal and quotient.
04.12 Indeterminate forms.
04.13 Monotone sequences.
04.14 Arithmetic and geometric succession.
04.15 Monotonous subsequences.
04.16 Appendix: compactness and completeness.

05. Limits of the real functions of a real variable.
05.01 Special cases.
05.02 Convergent and divergent functions.
05.03 Limits of restrictions; left and right limits.
05.04 Limits of monotone functions.
05.05 Theorems on limits.
05.06 Indeterminate forms.
05.07 Some special limits.
05.08 Exercises.

06. Continuity for real functions of a real variable.
06.01 Operations with continuous functions.
06.02 Limits and continuity of compound functions.
06.03 Continuity of restrictions and extensions.
06.04 Continuity throughout the domain.
06.05 Extension for continuity.
06.06 Continuous images of intervals are intervals.
06.07 Continuity of monotone functions.
06.08 Homeomorphisms.
06.09 Points of discontinuity.
06.10 Absolute extremes.
06.11 Local extremes.
06.12 Continuous functions on compacts.
06.13 Exercises.

07. Elementary functions.
07.01 Review of functions between sets.
07.02 Powers: natural, roots, radicals, real.
07.03 Exponential function.
07.04 Logarithmic function.
07.05 The natural exponential (logarithm).
07.06 Formal properties.
07.07 Circular functions.
07.08 Inverse functions of circular functions.
07.09 Hyperbolic functions.
07.10 Inverse hyperbolic functions.
07.11 Exercises.

08. Derivatives for real functions of a real variable.
08.0l Variation of a function relative to the variable
independent.
08.02 Derivative of a real function of a real variable.
08.03 Left and right derivatives.
08.04 Differentiability implies continuity.
08.05 Derived function; derivatives of elementary functions.
08.06 Linearity of the derivation.
08.07 Derivation of products.
08.08 Derivation of the reciprocal and of the quotient.
08.09 Derivation of compositions: rule of the chain.
08.10 Derivative of the modulo of a function.
08.11 Derivation of inverse functions.
08.12 Diffeomorphisms.
08.13 A never differentiable continuous function.
08.14 Exercises.

09. Fundamental theorems of differential calculus.
09.01 Derivatives and local extremes.
09.02 Rolle's theorem, classic version.
09.03 Mean value theorem.
09.04 Corollaries of the mean value theorem.
09.05 The theorem of finite increments.
09.06 The de l'Hopital rule.
09.07 Corollaries.
09.08 Derivatives with higher order.
09.09 Other considerations.
09.10 Points of internal local extremes and successive derivatives
09.11 Convex and concave functions.
09.12 Asymptotes.
09.13 Asymptotic developments and successive derivatives: Taylor's formula.
09.14 Applications and Exercises: qualitative study of the graph of a
real function of real variable.
09.15 The Gauss function: on the theory of errors.

10. The Riemann integral.
10.1 Step functions.
10.2 Integral of step functions with compact support.
10.3 Area of a plane set.
10.4 Integrable Riemann functions.
10.5 Properties of the integral.
10.6 Integral and area of the trapezoid.
10.7 An often useful observation.
10.8 Integral extended to an interval.
10.9 Local integrability of continuous functions.
10.10 Integral extended to an oriented interval.
10.11 Indefinite integral.
10.12 Primitive, or anti-derivative.
10.13 The fundamental theorem of integral calculus.
10.14 Indefinite integrals.
10.15 Integration of rational functions (hints).
10.16 Integration by parts.
10.17 Integration by replacement.
10.18 Integration defined by parts and by replacement.
10.19 The logarithm as a quadrature of the hyperbola.
10.20 Average of an integrable function.
10.21 A generalization of the mean theorem.
10.22 Exercises.

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

Elementary knowledge relating to the following topics developed in the PreCourse of Mathematics is 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.

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.

Disabilità e DSA

Le studentesse e gli studenti che hanno registrato la certificazione di disabilità o la certificazione di DSA presso l'Ufficio Inclusione e diritto allo studio, possono chiedere di utilizzare le mappe concettuali (per parole chiave) durante la prova di esame.

A tal fine, è necessario inviare le mappe, due settimane prima dell’appello di esame, alla o al docente del corso, che ne verificherà la coerenza con le indicazioni delle linee guida di ateneo e potrà chiederne la modifica.

Teaching

Theoretical lessons and exercises.

Attendance

Elementary knowledge relating to the following topics developed in the PreCourse of Mathematics is 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.

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.

Disabilità e DSA

Le studentesse e gli studenti che hanno registrato la certificazione di disabilità o la certificazione di DSA presso l'Ufficio Inclusione e diritto allo studio, possono chiedere di utilizzare le mappe concettuali (per parole chiave) durante la prova di esame.

A tal fine, è necessario inviare le mappe, due settimane prima dell’appello di esame, alla o al docente del corso, che ne verificherà la coerenza con le indicazioni delle linee guida di ateneo e potrà chiederne la modifica.

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.