EDUCATIONAL GOALS
The course presents the basic elements of Mathematical Analysis. The educational aims of the course foresee the acquisition of results and demonstration techniques, as well as the ability to use the related calculation tools.
Knowledge and understanding of the following aspects is expected:
1) First approaches to real numbers and functions.
2) Equations and inequalities.
3) Complex numbers.
4) Matrices.
5) Linear systems.
6) Domains and limits of real functions of one real variable.
7) Calculation of limits and continuous functions.
8) Derivatives of real functions of one real variable.
9) Study of function.
Knowledge and understanding of applicative type are expected on the following aspects:
1) Applications of theorems and rules studied to solve real problems within the context of Information and Communication Technologies.
2) Knowing how to identify the best solution strategies for equations in the complex field.
3) Ability to perform calculations with limits and derivatives.
4) Ability to study a real function of a real variable and draw a qualitative graph.

DELIVERY TEACHING
72 ONLINE VIDEO LESSONS (12 TEACHING UNITS - TWO HOURS DURATION FOR EACH CFU).

INTERACTIVE TEACHING
N. 2 INTERACTIVE LESSONS FOR CFU.
N. 5 THEMATIC DISCUSSIONS ON THE TEACHING FORUM (TOPIC) AND N. 2 POST FOR CFU AS DEFINED BY THE GUIDELINES ON THE EDUCATION OF THE PQA.
N. 2 E-TIVITY EACH 5 CFU.
N. 2 TESTS FOR EACH CFU WITH 8 MULTIPLE CHOICE QUESTIONS.

COURSE PROGRAM
Module 1. First approaches to numbers and real functions:
Axioms of real numbers, and related properties. Set theory: generality and representation; intersection, union, set of parts, difference; Cartesian product. Natural, integer, rational, real numbers. Upper and lower extremes, maximum and minimum of a numerical set. Natural, integer and rational numbers. Functions: domain, invertibility, increasing and decreasing functions. Elementary functions.

Module 2. Equations and inequalities:
First and second degree equations. First and second degree inequalities. Inequalities of higher than second degree, fractional inequalities and systems of inequalities. Irrational, exponential and logarithmic inequalities.

Module 3. Complex numbers:
Complex numbers in Cartesian form, trigonometric form, exponential form and polar form. Rationalization of complex numbers in Cartesian form. Properties of the product and of the ratio between complex numbers in trigonometric, exponential and polar form. Powers and roots of complex numbers. Solution of equations in the complex domain.

Module 4. Matrices:
Operations between matrices: addition, subtraction, multiplication between matrices and non-commutativity properties; multiplication between a scalar and a matrix. Determinants of square matrices: generalities; calculation rules; properties of the determinants. Rank of matrices: generality; calculation in the case of square and non-square matrices; relationship between rank of matrices and linear independence/dependence of row/column vectors.

Module 5. Linear systems:
Matrix formulation of a linear system. Characterization of the solutions of a linear system: uniqueness of solution, infinite solutions, incompatibility; geometric meaning of a linear system and relation with its solutions. Solving rules: inverse matrix method; Cramer's method. Homogeneous linear systems. Formalization of the solutions of a linear system via procedures that foresee elementary operations between rows and columns of complete and incomplete matrices.

Module 6. Domains and limits of real functions of one real variable:
Real functions of a real variable: techniques for the computation of the domain. Intuitive approach to the definition of limit: geometric meaning of a finite limit when the independent variable tends to a finite/infinite value; geometric meaning of an infinite limit when the independent variable tends to a finite/infinite value. Analytical formalization of finite/infinite limit when the independent variable tends to a finite/infinite value. Limits of compound functions. Uniqueness of the limit theorem. Comparison theorem. Indeterminate forms. Infinite and infinitesimal. Asymptotes of a function.

Module 7. Calculation of limits and continuous functions:
Calculation procedures for limits in an indeterminate form. Notable limits and their applications. Continuous functions in a point and in an interval. Points of discontinuity of a function. Weierstrass theorem, sign permanence theorem, zeros theorem, intermediate value theorem.

Module 8. Derivatives of real functions of a real variable:
Limit of the incremental ratio and definition of derivative. Geometric meaning of the derivative. Derivability and continuity. Derivatives of elementary functions. Derivatives of compound functions. Relative maxima and minima for real functions of a real variable. Fermat's theorem. Rolle's theorem. Lagrange's theorem. Concave, convex functions, and inflection points. De L’Hospital's theorem.

Module 9. Study of function:
Procedures for the study of a function and for the sketch of its qualitative graph. Insights about fractional rational functions.

METHOD OF VERIFYING PROFIT IN ITINERE

The degree of student learning is constantly monitored through verification tools and methodologies.
In particular, in order to make the verification and certification of the training results feasible, the teacher and the tutor will take into account the following aspects:
1) automatic tracking of training activities by the system - reporting;
2) didactic and technical monitoring (in terms of quantity and quality of interactions, compliance with didactic deadlines, delivery of the planned works, etc.);
3) ongoing training checks, also for self-assessment (e.g. multiple choice tests, true/false, sequence of questions with varying difficulty, simulations, mind maps, works, group projects, etc.);
4) final exam, during which it is considered and valued not only the performance during the exam, but also the work done online (remote activities, quantity and quality of online interactions, etc.) .
In this framework, the evaluation considers several aspects:
a) the result of a certain number of intermediate tests, if foreseen (in terms of online tests, development of papers, etc.);
b) the quality and quantity of participation in online activities (frequency and quality of interventions that can be monitored through the platform);
c) the results of the final exam.
It is specified that the final test is only of "written" type. The oral test is optional and is useful only and exclusively to refine the evaluation of the written test. The topics of the oral exam, if done, are agreed between the teacher and the student.
All the collected data will be, in general, evaluated by the teacher to give the student an evaluation that is objective and consistent with the university's aims, taking into account both the summative aspects and the formative aspects.

ASSESSMENT METHOD AND AIMS OF THE FINAL EXAM
The access to the exam is subordined to the recognition of attendance, which will be recognized, at the booking time for the exam, by an appropriate certificate, that will certify the development of the didactic verification activities in progress and at the level of the work made in the various exercises.
The exam will consist of a written test, with an optional oral one. The final evaluation will be expressed out of 30, with possible honors depending on the degree of maturity reached by the student.
The exam aims at the didactic aims described as follows. In particular, the student will have to:
a) be able to determine the domain of a real function of a real variable;
b) know how to find the solutions of an equation in the complex domain;
c) be able to recognize the best resolution strategy for limits of real functions of a real variable;
d) know how to study the qualitative graph of a real function of a real variable, identifying any asymptotes, any relative and absolute maximum/minimum, and any inflection points.
The written test foresees a score for all the previous aims, from point a) to point d). Each aim has a variable score. Honors will be awarded in those particular cases in which particular maturity is shown in solving the exam questions.

KNOWLEDGE AND UNDERSTANDING IN TERMS OF EXPECTED OUTCOMES (DUBLIN DESCRIPTOR NO. 1)
Knowledge of the theory of real functions of a real variable will allow the student to acquire awareness of the possible analytical modeling of phenomena that describe the technological world.

SKILLS TO APPLY KNOWLEDGE AND UNDERSTANDING IN TERMS OF EXPECTED OUTCOMES (DUBLIN DESCRIPTOR NO. 2)
The student will be able to use knowledge related to the theory of functions to solve analytical/numerical problems commonly seen in contexts related to Information and Communication Technology systems.

Paolo Marcellini, Carlo Sbordone, Elementi di Analisi Matematica 1, Versione semplificata per i nuovi corsi di laurea, Liguori Editore, 2016.
Ciro D’Apice, Rosanna Manzo, Verso l’esame di Matematica 1, Maggioli Editore, 2015.
Additional teaching material will be available in the section dedicated to teaching within the university platform.