B020629 - REAL ANALYSIS I

Italian

Real and complex numbers.
Functions of one real variable.
Limits, derivatives.
Taylor formula. Rieman Integrals. Generalized integrals.

Analisi Matematica I, G. Anichini G. Conti, Pearson Ed.

In tthis part of the course we provides for the acquisition and further development of basic knowledge of the concepts of differential calculus and integral calculus in one dimension.
The training objective is to develop a clear understanding of the theoretical approach to mathematics, improving, at the same time, the familiarity with the computation.

Fundamental concepts and techniques learned in mathematics courses of secondary school. In particular: formal calculation, polynomials, algebraic equations and inequalities, analytical geometry elements (the Cartesian plan, the straight lines, circle, parabola, etc.), Trigonometric functions.

For detailed informations, check on the web site
http://www.dimai.unifi.it/users/zecca

Multiple choice test

Numbers:
Sets (union, intersection, difference, empty set,
complementary). natural numbers, relative, rational. Real numbers:
algebraic axioms, ordering. logical quantifiers.
Inequalities. Absolute value. Powers and roots. Logarithms.
Intervals. Maximum, minimum, upper bounds, lower bounds, and lower bound
top of a set. Property 'of the real completeness. Property of
Archimedes. Density 'of the rational. Applications between sets, applications
injective, surjective, bijective. Domain, codomain, image and graphic
of an application.

Real functions of one variable (limits and continuity):
Real functions of real variable. limited functions. monotone functions.
Inverse functions. Polynomial and rational functions. main
transcendental functions (exponential and logarithmic functions, trigonometric functions
and their inverses, hyperbolic functions). The function integer part. elements of
topology of the real line: surroundings of a point, accumulation points, isolated points.
Maximum and absolute and relative minimum. Function limits (finite and
infinite). uniqueness' of the limit theorem. Theorem of sign permanence.
Theorem of the police. Theorem on the operations for the calculation of the limits. forms
indeterminate. left and right limit. Function limit composed.
change of variable limits. theorem
existence of limits for monotonous functions. fundamental limits and consequences. Continuity'. Theorem continuity 'functions
combined (sum, product, quotient and composition). classification of
discontinuity'. Value theorem. Intermediate value theorem and
applications. Theorem
of continuity 'of an inverse function. theorem
Weierstrass.

Real functions of one variable (derivative):
Definition of derivative. Derived right and left. angular points. Interpretation
geometric derivative. Differential. derivation rules (sum, product, quotient,
composition and inverse function). Derivative of the main functions. Fermat's theorem.
Theorems of Rolle and Lagrange. Consequences of the Theorem. Theorems of de L'Hopital. Higher order derivatives. Asymptotes of a
function. convex functions in a range. sufficient conditions for the existence of
maximum and relative minimum. inflection points. function studies.
Infinitesimal and the infinite.
The symbol-or small. Taylor's formula with the rest in the form of Peano. of formula
Taylor with the rest in the form of Lagrange. of MacLaurin formula. Applications of the formula
Taylor the calculation of the limits and to some approximation problems.

Simple integrals:
Primitive. indefinite integrals. integration by parts formula for indefinite integrals.
integration formula for replacement for indefinite integrals.
Integration of elementary or derived from elementary functions.
Integration of rational functions. Some integration methods.
Definition of definite integral. negligible sets and necessary and
sufficient for the integrability '. Property 'of definite integrals (linearity',
monotony, additivity '). integration by parts for definite integrals.
integration formula for substitution for definite integrals. Theorem of the average for
integrals. fundamental theorem of calculus. fundamental formula of integral calculus. Definition of the logarithm by the integral. Application
of the definite integral to the calculation of areas of plane figures and to the calculation of volumes of
solid of rotation. improper integrals. the convergence criteria (comparison, comparison
asymptotic, absolute convergence). The error function.