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 oflogic 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 localmaxima and minima, points ofinflexion. 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 localmaxima 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 partialtests:preliminary test (see a. above),first partial testduring the class period andsecond partial testat the end. More detailed information on the partialtests will be available on Blackboard.
NOTES
Basic knowledge will be included in the preliminary test and thus attendance at the pre-course classes is highly recommended. More detailed information on the pre-course will be available on Blackboard. An on line pre-course TEOREMAisalso accessible to the address .