Mathematics
PROF. ANNA TORRIERO; PROF. ROSA ALBANESE; PROF. ALESSANDRA CORNARO, PROF.
GABRIELE BOLAMPERTI; PROF. ENRICO MIGLIERINA; PROF. MONICA BIANCHI; 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.
COURSE CONTENTS
Basic knowledge (pre-course class). Natural, integer, rational and real numbers.
Fundamentals of logic and basic set theory. Elementary algebra, arithmetic,
analytical geometry, trigonometry.
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 function 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 (course and pre-course classes), assignments.
ASSESSMENT METHOD
Grading will be based on
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,
c. an oral exam for students having achieved a grade on the written test of 15/30, 16/30
or 17/30 and also in other cases as specified in Blackboard. The oral exam concerns
all the programme of the course (definitions, theorems, properties and their
applications).
For all students it is possible to take partial tests: preliminary test (see a. above), first
partial test during the class period and second partial test at the end. More detailed
information on the partial tests will be available on Blackboard.
NOTES
Basic knowledge will be included in the preliminary test and thus attendance at the precourse classes is highly recommended. More detailed information on the pre-course will be
available on Blackboard. An on line pre-course TEORE MA is also accessible to the address
http://teorema.cilea.it.