Mathematics

Mathematics
GR. A-B: PROF. ANNA TORRIERO; GR. C-DE: PROF. ROSA ALBANESE; GR. DI-LA: PROF.
ALESSANDRA CORNARO; GR. LE-O: PROF. ENRICO MIGLIERINA; GR. P-SA: PROF. MONICA
BIANCHI; GR. SB-Z: PROF. SALVATORE VASSALLO
COURSE AIMS
The course has two objectives: to present several fundamental mathematical tools
for dealing with economic-financial problems, and to help students to acquire a
precise and essential language. The course will emphasize how to develop a view
toward critically re-examining mathematical concepts which students will find in
their academic pursuits, and how to stimulate the capacity to use mathematical
methods, tools and models in a wide array of applications. The course will cover
basic topics in linear algebra, differential and integral calculus, and optimization;
taken together, these concepts will provide an effective tool for analysing
economic and business phenomena.
Basic knowledge: Natural, integer, rational and real numbers. Basic elements of
logic and set theory. Elementary algebra. Powers, logarithmic and exponential
functions. Equations and inequalities (polynomial, fractional, irrational,
logarithmic and exponential). Systems of equations and inequalities. Plane
analytical geometry. Basic notions of trigonometry.
These topics will be reviewed during the pre-course class.
COURSE CONTENTS
Elements of linear algebra. The linear space Rn. Subspaces, linear combinations,
linear independence. Matrices and corresponding operations. Determinant. Inverse
matrix. Matrix rank. Systems of linear equations. Rouchè-Capelli theorem,
Cramer’s rule.
Real functions of one variable
 Introductory concepts: Domain. Maximum, minimum, upper and lower
bounds. Bounded functions, monotonic functions, composition of functions,
inverse function. Convex functions.
 Limits and continuity: Limits and related theorems. Operations on limits and
indecision forms. Continuity of functions and related theorems. Asymptotes.
 Differential calculus: Incremental ratio and derivative. Differentiable functions.
Rules of differentiation. Derivative of composite and inverse functions.
Fundamental theorems of differential calculus. Taylor formula. Global and
local maxima and minima, points of inflexion. Necessary and/or sufficient
conditions for the existence of maxima and minima. Concavity, convexity.
 Integral Calculus: The indefinite integral. The Riemann (definite) integral and
related theorems. Some techniques of integration.
Real functions of two real variables
The euclidean space R2. Domain. Level sets. Global and local maxima and minima.
Saddle points. Continuity. Partial derivatives. Hessian. Concave, convex and
homogeneous functions. Taylor Formula. Unconstrained optimization: first and
second order conditions. Constrained optimization via the level set approach. The
Lagrange multiplier method, interpretation of the Lagrange multiplier.
READING LIST
1. A. TORRIERO-M. SCOVENNA-L. SCAGLIANTI, Manuale di Matematica, Metodi e applicazioni,
Cedam, 2013.
2. M. SCOVENNA-R.GRASSI, Esercizi di Matematica, Esercitazioni e temi d’esame, Cedam, 2011.
3. M. BIANCHI-L. SCAGLIANTI, Precorso di Matematica, Nozioni di base, Cedam, 2010.
4. F. BREGA-G. MESSINEO, Esercizi di Matematica Generale, Giappichelli, 2013 (5 volumes).
Online instructional material is available on Blackboard.
TEACHING METHOD
Lectures, assignments, pre-course classes. An on line pre-course TEOREMA is also
accessible to the address http://teorema.cilea.it .
ASSESSMENT METHOD
Grading will be based on written exams organized as follows:
a. an on-line preliminary test, concerning basic knowledge, essential to pass to the
written exam and given in the computer labs. Students who have correctly
answered 8 questions in the mathematical section of the admission Faculty test are
exempted. Preliminary test and exemption expire at the beginning of the new
academic year.
b. a written exam in which students will be required to answer open and multiple
choices questions, both of theoretical and practical content, concerning the whole
program.
For all students it is possible substitute the written exam set out in point b. with two partial
tests: the first during the class period and the second during the winter exam session. More
detailed information on the partial tests will be available on Blackboard.
NOTES
The preliminary test concerns basic knowledge, therefore attendance at the pre-course
classes is highly recommended.
Further information can be found on the lecturer's webpage at
http://docenti.unicatt.it/web/searchByName.do?language=ENG, or on the Faculty notice
board.