General Mathematics

Academic Year 2022/2023 - Teacher: SALVATORE CORRENTE

Expected Learning Outcomes

KNOWLEDGE AND UNDERSTANDING: The student will receive the basic tools allowing to deal with the modern formal approaches to economics and business management. The focus will be on the basic principles of mathematics applied to economics rather than on a sterile technicality. The course will give also an idea of the possible applications of the introduced tools. More generally, the course will try to educate students to a rigorous approach to the analysis of economic and management business phenomena. The accuracy of the mathematical treatment will allow the student to acquire a mindset that will be beneficial for all the other subjects of the university courses and, more generally, for all the professional issues that have to be handled.
APPLYING KNOWLEDGE AND UNDERSTANDING: The student will be given the opportunity to analyze rigorously a mathematical problem and to use the basic concepts in order to draw appropriate conclusions. The student will be able to solve simple but not trivial mathematical problems. The student will be able to conduct a mathematical reasoning through the introduction of rigorous definitions and the proofs of some theorems particularly significant. The student will also be able to apply the knowledge, learned during the course, to the formalization of some basic economic and management business issues such as profit maximization and utility maximization.
MAKING JUDGMENTS: Students will be educated to elaborate by themselves the most appropriate approach to the proposed problems avoiding sterile applications of repetitive patterns. The student will be educated to judge the considered mathematical formalization by different points of view such as the elegance of the model, the power of the mathematical tools, the computational burden and so on.
COMMUNICATION SKILLS: Students will be able to use the technical terms and they will learn how properly express the mathematical formalization of the problem and the results obtained with it. Students will be able to give a formal presentation of economic phenomena and management business. The student will critically discuss quantitative models related to economic and management business phenomena.
LEARNING SKILLS: Taking into account the ever increasing use of mathematical formalization in economics and management business, the course will enable to access the more qualified literature in these areas, giving an essential basis for future learning in both educational and professional level.

Course Structure

Classrooom-taught lessons including also exercising sessions during which it will be shown how to apply the theoretical concepts introduced during the course.

Required Prerequisites

Even if there is not any formal requisite, the knowledge of the following concepts is however very useful: the four operations e their properties; prime numbers, factors integer decomposition, greatest common divisor and least common multiple; fractions and their operations; powers, roots and logarithms; monomials, polynomials and polynomial decomposition; first and second order equations and inequalities; fractional equations and inequalities; equations and inequalities in absolute value; Euclidean geometry (circumferences, polygons, Pythagorean theorem).

Detailed Course Content

1^st PART

ELEMENTS OF MATHEMATICAL LOGIC: languages and propositions; connectives; quantifiers.

SET THEORY: properties, subsets, operations. Functions. Binary relations. Real numbers and inequalities. Basics of trigonometry.

COMBINATORICS: dispositions, combinations and permutations. Binomial theorem, binomial coefficients.

MATRICES AND DETERMINANTS: definitions and classifications. Sum and product between matrices. Inverse matrix. Determinant and its property. Rank of matrix.

LINEAR SYSTEMS: linear forms. Definitions and properties. Normal linear systems: Cramer’s rule. Rouché-Capelli Theorem. Solution of parameterized systems.

2^nd PART

ANALYTICAL GEOMETRY: Cartesian coordinate system. Straight line equation in the plane.

REAL FUNCTIONS OF REAL VARIABLE: definitions, classifications, geometrical representation. Composite functions and inverse functions. Limits: definitions and theorems. Continuous functions. Infinitesimals and infinities.

DERIVATIVES AND DIFFERENTIALS: definitions, properties and their geometric interpretation. Derivatives of elementary functions. Derivatives and differentials of sum, product and quotient of functions. Derivatives of composite and inverse functions. Derivatives and differentials of n-th order. Main theorems on differentiable functions.

APPLICATIONS OF DIFFERENTIAL CALCULUS: Taylor’s and Mac Laurin’s formulas. Indeterminate forms. Monotonic functions, convex functions, local and global extrema, inflection points, asymptotes. Study of function. Elasticity of a function.

3^rd PART

INTEGRALS: indefinite integral and primitives. Definite integral and its geometric interpretation. Main methods of integration.

Textbook Information

S. Corrente, S. Greco, B. Matarazzo, S. Milici, "Matematica Generale", Giappichelli Editore, Torino, 2021.
B. Matarazzo, M. Gionfriddo, S. Milici, “Esercitazioni di Matematica” ed. Tringale, Catania,1990.
A. Giarlotta, S. Angilella, “Matematica generale. Teoria e pratica con quesiti a scelta multipla. VOLUME 1. Logica – Insiemistica – Combinatorica - Insiemi numerici”. Giappichelli Editore, 2013.

Course Planning

	Subjects	Text References
1	1. Teoria degli insiemi: unione, intersezione, prodotto cartesiano, differenza di insiemi, insieme potenza, cardinalità di un insieme e teorema dei quattro cardinali, partizione di un insieme.	Corrente, Greco, Matarazzo e Milici: Capitolo Primo
2	2. Teoria degli insiemi. Applicazioni: definizione, applicazioni iniettive, suriettive e corrispondenze biunivoche, funzione inversa. Relazioni binarie: definizioni, proprietà, relazioni di equivalenza, relazioni d’ordine, teoremi che legano una relazione di equivalenza e una partizione di un insieme.	Corrente, Greco, Matarazzo e Milici: Capitolo Primo
3	3. Numeri: numeri naturali, numeri interi, numeri razionali, numeri reali.	Corrente, Greco, Matarazzo e Milici: Capitolo Primo e Capitolo Secondo
4	4. Calcolo combinatorio: disposizioni semplici, permutazioni semplici in linea aperta e in linea chiusa, inversioni di una permutazione, combinazioni semplici, proprietà di simmetria, legge di Stifel, triangolo di Tartaglia.	Corrente, Greco, Matarazzo e Milici: Capitolo Quinto
5	5. Calcolo combinatorio: disposizioni con ripetizione, combinazioni con ripetizione, permutazione con ripetizione, binomio di Newton.	Corrente, Greco, Matarazzo e Milici: Capitolo Quinto
6	6. Matrici: definizione, vari tipi di matrici (quadrate, rettangolari, vettori, nulle, opposte, trasposte, diagonali, scalari, simmetriche, estratte, complementari). Operazioni tra matrici: somma, prodotto scalare.	Corrente, Greco, Matarazzo e Milici: Capitolo Sesto
7	7. Matrici. Operazioni su matrici: prodotto righe per colonne, matrice inversa, teoremi sulla matrice inversa. Determinante di una matrice: definizione.	Corrente, Greco, Matarazzo e Milici: Capitolo Sesto
8	8. Determinante di una matrice: proprietà, I e II teorema di Laplace, Teorema di Binet, Matrice aggiunta, Rango di una matrice, proprietà del rango, Teorema di Pascal.	Corrente, Greco, Matarazzo e Milici: Capitolo Sesto
9	9. Sistemi lineari: forme lineari, principi di equivalenza dei sistemi lineari.	Corrente, Greco, Matarazzo e Milici: Capitolo Settimo
10	10. Sistemi lineari: teorema di Cramer, metodo di Cramer, teorema di Rouché-Capelli, sistemi lineari omogenei e proprietà.	Corrente, Greco, Matarazzo e Milici: Capitolo Settimo
11	11. Elementi di metrica. Piano cartesiano, distanza tra punti nel piano.	Corrente, Greco, Matarazzo e Milici: Capitolo Ottavo
12	12. Cenni di trigonometria: circonferenza trigonometrica, seno, coseno, tangente e cotangente, relazioni fondamentali.	Corrente, Greco, Matarazzo e Milici: Capitolo Nono
13	13. Insiemi numerici: maggioranti e minoranti, estremo inferiore ed estremo superiore, insiemi separati e contigui. Cenni di topoligia: intorni, punti interni, di frontiera, insiemi aperti e chiusi, punti di accumulazione, teorema di Bolzano-Weierstrass	Corrente, Greco, Matarazzo e Milici: Capitolo Decimo
14	14. Equazione generale di una retta. Casi particolari e varie forme dell’equazione di una retta. Intersezione di due rette. Condizione di parallelismo e condizione di perpendicolarità. Distanza di un punto da una retta.	Corrente, Greco, Matarazzo e Milici: Capitolo Undicesimo
15	15. Funzioni reali di variabile reale: dominio, codominio, restrizione, prolungamento, grafico di una funzione, insieme di esistenza.	Corrente, Greco, Matarazzo e Milici: Capitolo Dodicesimo
16	16. Limite di una funzione: funzione convergente, divergente, teoremi fondamentali sui limiti.	Corrente, Greco, Matarazzo e Milici: Capitolo Dodicesimo
17	17. Limite di una funzione: operazioni sui limiti, forme indeterminate, limiti notevoli.	Corrente, Greco, Matarazzo e Milici: Capitolo Dodicesimo
18	18. Funzioni continue, teoremi sulle funzioni continue con particolare riferimento al teorema di esistenza degli zeri e teorema di Weierstrass. Punti di discontinuità di prima, seconda e terza specie.	Corrente, Greco, Matarazzo e Milici: Capitolo Dodicesimo
19	19. Funzioni monotone, funzioni inverse, funzioni composte, funzioni pari e dispari, funzioni periodiche, infinitesimi e infiniti, confronto tra infinitesimi e infiniti.	Corrente, Greco, Matarazzo e Milici: Capitolo Dodicesimo
20	20. Derivata di una funzione: rapporto incrementale e sua interpretazione geometrica; definizione di derivata, sua interpretazione geometrica, calcolo delle derivate di alcune funzioni, punti di non derivabilità: punti angolosi, punti cuspidali, punti a tangente verticale.	Corrente, Greco, Matarazzo e Milici: Capitolo Tredicesimo
21	21. Derivabilità e continuità. Differenziale. Regole di derivazione. Derivate di funzioni composte e inverse. Derivate di ordine successivo al primo.	Corrente, Greco, Matarazzo e Milici: Capitolo Tredicesimo
22	22. Teoremi di Rolle, Cauchy e Lagrange.	Corrente, Greco, Matarazzo e Milici: Capitolo Quattordicesimo
23	23. Teorema di de l’Hopital. Formula di Taylor e formula di Mac-Laurin.	Corrente, Greco, Matarazzo e Milici: Capitolo Quattordicesimo
24	24. Funzioni crescenti e decrescenti in un punto. Massimi e minimi relativi. Teorema di Fermat. Condizioni sufficienti per l’esistenza di massimi e minimi relativi. Ricerca degli estremi di una funzione.	Corrente, Greco, Matarazzo e Milici: Capitolo Quattordicesimo
25	25. Funzioni convesse e funzioni concave. Punti di flesso. Asintoti. Elasticità di una funzione.	Corrente, Greco, Matarazzo e Milici: Capitolo Quattordicesimo
26	26. Integrale indefinito: primitive, definizione dell’integrale indefinito, proprietà dell’integrale indefinito, integrali immediati.	Corrente, Greco, Matarazzo e Milici: Capitolo Quindicesimo
27	27. Integrale indefinito: metodo di integrazione per decomposizione in somma, metodo di integrazione per parti, integrale di funzioni razionali fratte, metodo di integrazione per sostituzione.	Corrente, Greco, Matarazzo e Milici: Capitolo Quindicesimo
28	28. Integrale definito: definizione dell’integrale secondo Riemann, condizioni integrabilità, interpretazione geometrica.	Corrente, Greco, Matarazzo e Milici: Capitolo Quindicesimo
29	29. Integrale definito: proprietà dell’integrale definito, teorema della media, teorema di Torricelli-Barrow e sue applicazioni.	Corrente, Greco, Matarazzo e Milici: Capitolo Quindicesimo

Learning Assessment

Learning Assessment Procedures

The final exam is composed of a written and an oral part. The oral exam can be done only if the written one has been passed.

Examples of frequently asked questions and / or exercises

What are the main set operations (union, intersection, Cartesian product, set difference, complement) and which are their properties?
What is the power set of a set and which is its cardinality?
Which equation relates the cardinalities of unions and intersections of sets?
What is a function?
What is a bijective function?
What is the inverse function and when can it be computed?
What is a relation? When is it called binary?
What is a set partition?
What is an equivalence relation? Can you enunciate and prove the two theorems linking a set partition with an equivalence relation?
What is a partial ordering relation and when is it said total?
What are simple and with repetition dispositions, combinations and permutations?
What is the binomial coefficient and which are its properties? (Symmetry property and Stifel formula)
What is the Pascal's triangle? How is it linked to the Stifel formula?
What is the Newton binomial theorem?
What is a matrix and which are the main operations can be done on them?
What is submatrix of a given matrix?
What is the inverse matrix of a given matrix? Which are its properties?
What is the determinant of a matrix and which are its properties?
Can you enunciate the two Laplace theorems?
Can you enunciate the Binet theorem?
What is a minor of order r of a given matrix? What is the complement of a minor of a given matrix? What is the cofactor of the elements of a given matrix?
What is the adjoint matrix and which are its properties?
What is the rank of a matrix and which are its properties?
Can you enunciate the Kronecker-Pascal theorem?
What is a linear form?
When a linear system is possible, impossible, determined or indetermined?
Which are the equivalence principles for linear systems?
Can you enunciate and prove the Cramer theorem?
Can you enunciate the Rouchè-Capelli theorem?
Can you write the straight line equation and can you prove the parallelism and perpendicularity conditions between two straight lines?
What are the supremum and the infimum of a given numbers set? Which are their properties? What are the minimum and the maximum of a given numbers set?
When two numbers sets are separated and when are they contiguous?
What are the interior, boundary and accumulation points of a given numbers set?
What is the closure of a given numbers set?
When is a set said closed or open?
Can you enunciate the Bolzano-Weierstrass theorem?
What are the domain, codomain, supremum, infimum, maximum, minimum and diagram of a function?
What is the limit of a function? When is a function said convergent for x tending towards x0?
When is a function said divergent?
Can you enunciate the main theorems about limits?
What is a continuous function? What are the point of discontinuity of a function? How many types of them do exist?
Can you enunciate the main theorems about continuous functions?
When is a function said monotonic?
What are odd, even and periodic functions?
What is the comparison between infinitesimals? What is the comparison between infinities?
What is the derivative of a function? Which is its meaning from the geometrical point of view?
What are corners, cusps and vertical tangents?
Which is the relationship between differentiability and continuity of a function?
Can you enunciate the theorems related to the derivative of the sum of two functions, of the product of two functions, of the ratio of two functions and of the composed functions?
What is the differential of a function and which is its mathematical meaning?
What is the elasticity of a function?
Can you enunciate and prove the Rolle, Lagrange and Cauchy theorems?
Which is the geometrical interpretation of the Rolle and Lagrange theorems?
Can you enunciate the de l'Hopital theorems?
What are the Taylor and MacLaurin formula? What are they used for?
When is a function said increasing in a point?
Can you enunciate and prove the main theorems related to increasing functions in a point?
What is a relative maximum point?
Can you enunciate and prove the Fermat theorem? Which is its geometrical interpretation? Can the theorem be inverted?
Can you enunciate the theorems giving sufficient conditions for the existence of relative maximum and minimum points?
When is a function said convex? When is a function said concave?
Can you enunciate the main theorems related to convex and concave functions?
What is an inflection point?
What is an asymptote? How many types of asymptotes do exist?
What is a primitive of a function?
Can you prove that the difference between two primitive of a function is a constant?
What is the indefinite integral and which are its main properties?
What is the integration by parts method?
What is the definite integral and which is its geometrical interpretation? Which are the integrable functions?
Which are the properties of the definite integral?
Can you enunciate and prove the mean value theorem? Which is its geometrical interpretation?
Can you enunciate and prove the Torricelli-Barrow theorem?
Why is the Torricelli-Barrow theorem said "fundamental theorem of the integral calculus"? Which is the link between indefinite integral and definite integral?

VERSIONE IN ITALIANO

Corso di laurea in

Economia