Download Calculs

Calculs
GR. A-CL: PROF. ANNA TORRIERO; GR. CO-LA: PROF. GUIDO CECCAROSSI; GR. LE-PO: PROF.
SALVATORE VASSALLO; GR. PR-Z: PROF. PAOLA BIFFI
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 prove an effective tool for analysing economic
and business phenomena.
COURSE CONTENT
BASIC KNOWLEDGE (pre-course class)
Natural, integer, rational and real numbers. Fundamentals of logic and basic set
theory. Elementary algebra, arithmetic, analytical geometry, trigonometry.
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. 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.
ELEMENTS OF LINEAR ALGEBRA
Vectors and matrices and corresponding operations. Determinant. Inverse matrix.
Matrix rank. Systems of linear equations. Rouchè-Capelli theorem, Cramer’s rule.
REAL FUNCTION OF TWO REAL VARIABLES
The euclidean space R2. Domain. Level sets. Global and local maxima and minima.
Saddle points. Continuity. Partial derivatives. Unconstrained optimization: first and
second order conditions. Constrained optimization via the level set approach. The
Lagrange multiplier method.
READING LIST
A. TORRIERO-M. SCOVENNA-L. SCAGLIANTI, Manuale di matematica, metodi e applicazioni,
CEDAM, 2009.
M. SCOVENNA-R. GRASSI, Esercizi di matematica, esercitazioni e temi d’esame, CEDAM,
2011.
M. BIANCHI-L. SCAGLIANTI, Precorso di matematica, CEDAM, 2010 (2ª ed.).
F. BREGA-G. MESSINEO, Esercizi di matematica generale, Giappichelli, 2006 vol. I, e 2008
vol. II.
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, 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
and some simple proofs can be asked in addition to the proofs of the following theorems:
Uniqueness of limits, Sign preservation, Continuity of a differentiable function, Fermat’s
theorem, Rolle’s Theorem, Mean value theorem, Mean value theorem for integral calculus,
Fundamental theorem of integral calculus.
For all students it is possible to take partial tests (preliminary test and 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 TEOREMA is also accessible to the address
http://teorema.cilea.it.
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.