( english version )
Mathematical analisys AB ( 9 CFU )
Prof. Giuseppe Rosario Mingione
     Tel. 0521/032346 - Fax. 0521/032350           E-mail. giuseppe.mingione@unipr.it

The »eld of Real numbers R. Axioms for R: operations, order, completeness.
Maximum, minimum of a set. Relevant properties of sup and inf operators. The
triangle inequality.
The »eld of Complex numbers C. Complex numbers in the history of mathe-
matics: from the problem of solving algebraic equations to the abstract construc-
tion. Axioms for C. Algebraic and trigonometric form of complex numbers. Basic
properties and operations; miscellanea of manipulations. The triangle inequality in
C. The Gauss plane. Roots of zn = w and related problems. Geometric represen-
tation. The Fundamental Theorem of Algebra.
Limits. Elementary topology in R: intervals, neighborhoods and the fundamental
accumulation property. Functions and limits: basic properties. Operations with
limits: sum, products. Inde»nite forms. A bit of sequences.
The functions exp and log. Qualitative behavior of the functions exp and log;
growth of functions and limits. Elementary algebraic properties. New inde»nite
Continuous functions. Continuity of functions: basic properties. Right and
left limits. Zeros of a continuous function. Mean value theorem. Inversion of a
continuous function. Minima and maxima of a continuous function.
Derivatives. Di«erentiability of a function. Geometric interpretation, and tan-
gents to the graph. Di«erentiability and continuity. Derivatives and operations.
Basic use of derivatives: local maxima and minima. Checking increasing and de-
creasing functions via derivatives. Little Fermat's theorem. Classical theorems of
Rolle, Lagrange and Cauchy. How to use derivatives to compute limits: Taylor's
formula. Evaluation of error terms: at zero, at in»nity.
Integration. How to invert the di«erentiation procedure. Basic integration meth-
ods: by parts, substitutions, and elementary algebraic tricks. The classical problem
of area evaluation: a sketch of the classical Peano construction. Basic properties and
theorems from integration theory; the integral as a linear operator. Non-integrable
functions. Classes of integrable functions. Integral mean value theorem and the
Fundamental Theorem of Calculus; its variations and consequences.
Generalized integrals. The problem of computing the area of unbounded sets.
Basic examples, comparison theorems. Standard convergence criteria.
Series. How to de»ne the sum of in»nite numbers. Basic examples, comparison
theorems. Standard convergence criteria. Series and generalized integrals.
Ordinary di«erential equations (ode). De»nitions and physical motivations.
The Cauchy problem. Solvability of linear ode of »rst and second order.

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