General mathematics
PROF. FAUSTO MIGNANEGO; PROF. ALESSANDRO SBUELZ
COURSE AIMS
To provide students with the mathematical instruments they need to understand the
quantitative aspects of the banking, finance and insurance businesses.
COURSE CONTENT
DETAILED TEACHING AIMS
At the end of the course, participants will have learnt technical skills, complete
with a stimulating overview of some practical applications. In particular, they will
be able to
– study a function and its derivatives analytically (with possible applications to
the value of bonds, their duration and their convexity);
– maximize a function subject to equality constraints (with possible applications
to firm profit maximization and consumer choice);
– solve a linear system (with possible applications to no-arbitrage asset pricing);
– calculate integrals (with possible applications to the valuation of bonds in the
presence of a time-varying interest rate);
– calculate the present value of assets providing a finite/infinite stream of cash
flows;
– draw up a mortgage redemption plan.
COURSE CONTENT
– Real numbers: operations with real numbers and their rules, real sets and
intervals, operations with sets, definition of upper and lower bound of sets of
real numbers, cluster point and neighbourhood of a point.
– Real functions of a real variable: domain and codomain of a function,
asymptotes, summary of elementary functions, definition of a composite
function, definition of an inverse function, graphs of inverse functions, injective
and surjective functions, relationships between injective, monotone and inverse
functions.
– Limits: definitions of limit of functions, infinitesimal and infinite order, limit
uniqueness theorem, operations with limits, and sign permanence theorem.
– Continuous functions: definition of a continuous function, elementary functions
as continuous functions, limits of continuous functions, and discontinuities;
theorems on continuous functions: Weirstrass' theorem, zero existence theorem,
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Darboux's theorem, and theorems on the continuity of composite and inverse
functions.
Differential calculus: difference quotients of continuous functions, definition of
first derivative, geometrical meaning of the first derivative, definition and
geometrical meaning of the differential, equation of a tangent to a continuous
graph, differentiability of real intervals, differentiability as a sufficient
condition for continuity, indeterminate forms, limits and l'Hôpital's theorem,
definition of relative and absolute extreme values, necessary and sufficient
conditions for a relative extreme value, Fermat's theorem, definition of a point
of inflection, concavity and convexity, Rolle's theorem and Lagrange's theorem,
their meanings and consequences, relationships between first derivative and
monotony and between second derivative and concavity, study of real functions
of a real variable and their graphs, and overview of the Taylor-McLaurin
formulas.
Real functions of two real variables: representation of domains on the R2 plane,
definition of partial derivatives and their calculation, and determination of
unconstrained and constrained extreme values (Lagrangian multipliers method).
Sequences and series: limits of sequences, overview of numerical series,
definitions of converging and diverging series, geometrical series, harmonic
series, alternating-sign series, and asymptotic comparison principle.
Integral calculus: Riemann integral, the fundamental theorem of integral
calculus, primitives of elementary functions, and integration by parts and by
substitution.
Linear algebra: Rn space, vectors and matrices, determinant of a matrix and
linear systems.
Rudiments of financial mathematics: capitalization and earnings systems
READING LIST
The study material chosen (exercises, exam essays, etc.) will be made available on the
course Blackboard site.
PECCATI-SALSA-SQUELLATI, Matematica per l’Economia e l’Azienda, EGEA, Milan, 1999.
SCAGLIANTI-TORRIERO, Matematica metodi e applicazioni, Cedam, Padua, 2002.
CASTELLANI-GOZZI, Matematica di base per l’economia e l’azienda-esercizi e testi d’esame
svolti, Esculapio, Bologna, 2001.
TEACHING METHOD
Lectures and class exercises.
ASSESSMENT METHOD
Written exam, which may be supplemented with an oral examination on the student's
request; one or two tests take place during the course.
NOTES
Further information can be found on the lecturer's webpage
http://www2.unicatt.it/unicattolica/docenti/index.html or on the Faculty notice board.
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