ECTS - University of Pisa
| Modules | Area | Type | Hours | Teacher(s) | |
| PRINCIPLES OF MATHEMATICS | SECS-S/06 | LEZIONI | 63 |
|
Il corso si propone di fornire le conoscenze e le tecniche matematche necessarie per una comprensione approfondita delle materie afferenti alle aree disciplinari di riferimento del Dipartimento di Economia e Management.
The course aims at providing the necessary knowledge and suitable techniques to adequately deal with the
disciplines in the areas of the courses belonging to the Department of Economics and Management.
La conoscienza dello studente sarà verificate tramite esame scritto seguito da esame orale.
The student's knowledge will be verified by carrying out written and an oral tests.
Alla fine del corso lo studente avrà acquisito padronanza degli strumenti e concetti matematici presentati. Tale competenza sarà utile alla comprensione dei temi caratterizzanti il corso di laurea, con particolare riferimento a quelli che rientrano nell'area economica e finanziaria. In particolare, lo studente saprà
At the end of the course the student have to master the mathematical concepts presented during teh lessions. This competence will be useful in understanding the contents of subsequent courses, with particular reference to those in the economic and financial areas. In detail, the student must be able to
Durante la perte scritta dell'esame, lo studente dovrà risolvere correttamente gli esercizi assegnati. La capacità di applicare i risultati teorici alla risoluzione di eserci sarà oggetto di specifica valutazione. Inoltre, durante l'esame orale, lo studente dovrà essere capace di discutere i principali risultati teorici utilizzando lnguaggio e terminologia appropriati.
During the written test, the student will have to appropriately solve the given exercises. The ability to apply theoretical results to solve exercises will be subject to specific evaluation. In addition, within the oral examination, the student must state and discuss the main results using an appropriate mathematical terminology.
Alla fine del corso lo studente amplierà le proprie capacità di comprensione, formalizzazione e risoluzione di problemi attraverso il linguaggio matematico.
At the end of the course, the student will broaden his skills in understanding, formalizing and solving problems by means of the mathematical language.
Durante l'esame, lo studente dovrà dimostrare le sue capacità di applicare i concetti matematici appresi durante il corso.
During the exam, the student will demonstrate his skills to apply the mathematical concepts he has learned in the course.
Equazioni e disequazioni di primo e secondo grado. Generalità sui polinomi, frazioni algebriche tra polinomi. Equazioni e disequazioni logaritmiche ed esponenziali. Basi di geometria analitica (rette e coniche)
Integers, prime factorization, fractions, decimals, percent, radicals.
Algebraic expressions, polynomial factorization, algebraic equations and inequalities, irrational equations
and inequalities.
Logarithms and their properties, exponential and logarithmic equations and inequalities,
Cartesian Plane, Lines, Circles, and Parabolas
Metodo d'insegnamento: lezioni frontali (frequenza fortemente consigliata). Attività di apprendimento: frequentazione delle lezioni con esercizi, studio individuale.
Teaching method: lectures (attendance is strongly recommended).
Learning activities: attendance at lectures and exercises, individual study
Parte I - Funzioni di una variabile reale
Concetto di funzione. Funzioni elementari di uso comune in Economia. Funzioni inverse.
Concetto di limite di una funzione. Comportamento del limite rispetto alle operazioni algebriche. Calcolo di semplici limiti. Unicità del limite. Teorema della permanenza del segno
Continuità di una funzione e proprietà delle funzioni continue. Teorema degli zeri.
Derivata di una funzione. Significato economico della derivata. Relazione tra derivabilità e continuità. Regole di derivazione. Differenziale di una funzione.
Massimi e minimi relativi e assoluti di una funzione. Condizioni di ottimalità del I ordine. Teoremi di Rolle e di Lagrange. Funzioni crescenti e decrescenti. Condizioni di ottimalità del II ordine. Funzioni convesse e concave. Interpretazione geometrica ed economica delle funzioni concave e convesse. Studio di funzioni polinomiali, razionali fratte, logaritmiche ed esponenziali.
Parte II - Elementi di algebra lineare
Matrici, vettori e loro operazioni. Determinante di una matrice quadrata e relative proprietà. Inversa di una matrice. Sistemi lineari: Teorema di Rouché Capelli, metodi risolutivi.
Rette e piani nello spazio.
Parte III - Funzioni di più variabili
Curve di livello di una funzione. Lettura delle curve di livello in termini di crescenza o decrescenza dei livelli.
Derivate parziali prime e loro significato economico. Derivazione di funzioni composte. Il differenziale totale e applicazioni economiche. Derivate parziali seconde.Funzioni concave e convesse. Condizioni di ottimalità per problemi di massimo e minimo liberi. Problemi di ottimo vincolato: metodo della sostituzione.
Program:
Parte I – Functions in one variable
Definition of a function. Concept of domain of existence of a function and exercises. Operations between
functions. Elementary functions. Inverse function. Euclidian Metric of R.
Limit of a function, Theorem on the uniqueness of the limit, Sign permanence theorem, asymptotes,
calculus of limits, Indeterminate forms
Continuous functions: introduction and first definitions. Theorems on continuous functions. Classification
of discontinuities, Weierstrass Theorem and its conseguences.
Derivative of a function. Tangent Line to the graph of function. Derivatives of elementary functions.
Connection between differentiability and continuity of a function. Derivation rules. De l'Hospital Theorem
and application to computation of limits. Introduction to local/global maxima and minima. Definitions on
maxima and minima points. Critical points and extremum points. The relation between critical points and
extremum points: the Fermat Theorem. The Rolle Theorem, The Lagrange theorem and its application to
study the monotonicity properties of functions. Two sufficient conditions for a critical point to be an
extremum point. Definition of convexity of twice differentiable functions on intervals. Two sufficient
condition for convexity expressed in terms of the first and the second derivative respectively. Study of
functions (polynomial functions, rationals functions, irrationals functions, absolute value functions,
exponential functions, logarithmic functions, piece wise defined functions)
Parte II – Linear Algebra
Generalities on matrices and vectors. Special matrices. Examples of operations among matrices.
Determinant of square matrices and its properties. Binet theorem. Computation of the determinant: the the
Laplace rule. Rank of matrices. Inverse of a matrix. Linear systems. The Cramer's rule. General linear
systems and the Rouchè-Capelli theorem. Lines and Planes in R 3 .
Parte III – Functions in two variables
Topology in R 2 : circular neighborhoods. Domain. Continuity. Level curves. Definition of partial
derivatives. Gradient. Directional derivative. Differentiability. Second order partial derivatives. Hessian
matrix. Sufficient optimality conditions of the second order for critical points.
Introduction to constrained optimization problems (equality constraint): use of the substitution method for
solving constrained optimization problems on a compact region
Mathematics for economics, third edition. M.Hoy, J.Livernois, C.McKenna, R.Rees, T.Stengos. The MIT Press.
Mathematics for economics, third edition. M.Hoy, J.Livernois, C.McKenna, R.Rees, T.Stengos. The MIT Press.
L'esame consiste in un test scritto (che sarà preceduto da un test sui prerequiti di 30 minuti, il cui superamento sarà vincolante per poter procedere con lo svolgimento dell'esame) seguito da colloquio. Il test scritto, della durata di 2 ore, consiste nella risoluzione di esercizi sui contenuti del corso (funzioni reali di variabile reale, sistemi lineari ed linee e piani nello spazio, funzioni reali di due variabili reali). Il test scritto è sufficiente se lo studente raggiunge il minimo punteggio di 18, necessario per essere ammessi all'esame orale. L'esame orale deve essere sostenuto nello stesso appello dello scritto o in quello successivo (ma solo una volta).
The exam consists of a written and an oral test.
The exams of Principles of Mathematics will be carried out as follows:
1. A written pre-selection test, lasting 30 minutes, in which you will be asked to perform 10 basic
exercises on prerequisites. You pass the test if you answer at least 6 questions correctly (
correct questions≥6 ). This part does not give any score in the final rating
2. After passing the pre-selection (you will immediately have the score of your test), a written test,
lasting 2 hours, in which you will be asked to perform exercises on the topics of the course. The
written test will take place immediately after the pre-selection test.
3. After passing the written test (the results of which will be published as usual on the elearning
website), an oral test where you will be asked both questions of Theory (definitions, theorems
without demonstrations) and exercises.
The written test is sufficient if the student reaches a total score of 18, which is necessary to be admitted to the oral examination. The oral exam must be held in the same or in the next round as the written test (but only once).
The written test is sufficient if the student reaches a total score of 18, which is necessary to be admitted to the oral examination. The oral exam must be held in the same or in the next round as the written test (but only once).
The platforms that will be used for the tests will be https://elearning.ec.unipi.it/course/view.php?id=1632 for the pre-selection and Google Meet (Join using the classroom of the course https://classroom.google.com/c/MTc2NTAzOTQwNTIx?cjc=exsdqve) for the other two tests. In this regard, the entry codes and further details will be specified close to the date of the exam. We recommend that you access Google Meet exclusively with your institutional account.
The dates of the exams are those specified in the exam site (https://esami.unipi.it/).
The text of the second written exam will be shown to you by sharing my screen. You will be required:
- to take the test on paper sheets;
- at the end of it, to photograph all the sheets you have written on and send them to the email address a010639@gsuite.unipi.it using only your institutional email;
- to remain connected to the platform until I have informed you of the receipt of your attachments.
Then, what you will need to have to take the exam is:
a) a connection instrument with webcam and microphone (laptop, desktop computer, tablet), through which you must be connected to the platform for the duration of the test (the webcam must remain on); be sure that the camera and the microphone of your computer work properly. YOU ARE NOT ALLOWED TO DO THE TEST WITH EARPHONES; a mobile phone to photograph all the sheets you have written on
b) your student book or identity document, that can be shown on webcam at the beginning of the exam;
c) the usual stationery and paper sheets.
Given the need to organize the exam in the best possible way, you are invited to sign to the exam on the first days of opening for the registration. Students who are not registered for the exam cannot be admitted.
If you pass the written part (preselection test + written test), you can decide to do the oral part in the same or in the next session.
In the case you don't pass the oral part (so if you don't pass the exam), then in the new session you have to do the complete exam (preselection+written+oral).
