•GUIDA ECONOMIA 07-08 - Università degli studi di Udine

220 prospectus udine
Separate and contiguous classes of sets.
Intervals of R. Neighbourhood of a point,
neighboourhood of + ∞, - ∞, ∞. Internal
point, external point, frontier point, accumulation
point. Open and closed sets.
Elements of combinatorial calculus. Dispositions
with and without repetitions. Permutations.
Combinations.
Functions. Function and the graph of a
function. The domain of a function, the
range of a function, 1-1 functions. Some
elementary functions: constant, identical,
sign, absolute value. Sum and product of
functions. Inverse function. Composition
of functions. Bounded functions. The
maximum, minimum, supremum and
infimum of a function. Monotone functions,
symmetric functions, periodic
functions. Basic elementary functions:
powers and roots, exponentials and logarithmics,
trigonometrics, polinomials,
rational functions.
Continuity and limits. Continuity in a
given point. The finite limit in a given
point. The connection between limit and
continuity in a point. Right and left limits.
Functions approaching infinity. The
uniqueness of the limit. Theorem on the
sign of a function and the sign of its limit.
Comparison of limits. Limit of: an algebraic
sum of functions; of a product of
functions; a quotient of functions and
monotone functions. Indeterminate
forms. The continuity of: the sum of
functions; a product of functions; a quotient
of functions; the absolute value of a
function. The continuity of the inverse
function and of the composite function.
The continuity of basic elementary functions.
Theorems on continuous functions
defined on intervals: existence of a root,
intermediate value theorem, Weierstrass’
theorem on maximum and minimum.
Infinite and infinitesimals. Comparing
infinitesimals and infinites. Asymptotes.
Derivatives. Derivative in a point and
derivative function. Line tangent. Derivative
of a sum, a product, and a quotient of
functions. Derivative of the composite
and of inverse functions. Derivative of
basic functions. Locally increasing and
locally decreasing functions. Local maximum
and minimum of a function. Local
properties and derivatives. Rolle’s theorem,
Lagrange’s theorem, Corollaries of
Lagrange’s theorem. Limits of derivatives.
Graphs of functions. L’Hospital’s
theorem on indeterminate forms. Concave
and convex functions. Local concavity
and convexity. Local approximation of
functions. Taylor’s formulae: Peano and
Lagrange remainders. Applications of
Taylor’s formula.
Integrals. Indefinite integrals. Integration
by parts and by substitution. Integration
of rational fractions. The definite integral.
Basic properties of the definite integral.
Mean value, mean value theorem.
Fundamental theorem of definite integration.
Torricelli’s theorem. Integration
by parts and by substitution. Improper
integrals: unbounded function and
unbounded interval of integration.
Functions of two variables. Level sets. Quadratic
forms. Neighbourhood of a given
point of the plane. Limit and continuity.
Sign of continuous functions. Weierstrass’s
theorem on maximum and minimum.
Partial derivatives. Differentiability.
Tangent plane. First order conditions
for the maximum and the minimum.
Second order sufficient conditions for the
maximum and the minimum. Constrained
optimisation. Multipliers of
Lagrange’s method. Optimization on
bounded domains.
Pre-requisites
Algebraic structure of real numbers.
Inequalities. Elements of symbolic calculus.
Linear systems in 2 variables. Elements
of analytic geometry.