Mathematical Analisys AB ( 9 CFU )

Finalità

The course provide the basic notions of calculus for functions of one variable.

Programma

Number systems.

Natural numbers and Induction principle; combinatorics; elements of calculus of probability.

Integer and Rational numbers; Dedekind's completness axiom; least upper bound and greatest lower bound; the function absolute value; interval of real numbers.

Complex numbers; algebraic and trigonometric form; De Moivre's formula; n-th roots of a complex number; the foundamental theorem of algebra.

Continuous functions, limits, numerical sequences.

An introduction to continuity; the definition of a continuous function; Lipschitz functions; theorems about continuous functions (the theorem of existence of zeroes, the intermediate values theorem, the Weierstrass theorem).

Some elements of topology; an introduction to the definition of limit for a function of one variable; the property of limits; some foundamental limits of functions.

Numerical sequences; properties of limits of numerical sequences and comparison with the limits of functions; foundamental limits; Nepero's number.

The algebra of the Landau "little-o" symbol.

Differential calculus for functions of one variable.

Differentiable functions; the derivative of a function and its geometric meaning; the derivative of a function and its local property; Rolle and Lagrange's theorem and their consequences; antiderivatives; primitives of a function.

Taylor's formula with Peano's form of the remainder; Taylor's formula with Lagrange's form of the remainder; the qualitative study of the graph of a function of one variable.

Integrals and Series.

The Area's problem and an introduction to the integral of a continuous function; the foundamental theorem of calculus and its consequences; the Torricelli's theorem; integration by parts and by sostitution; generalized integrals.

Introduction to numerical series; convergence criterion; comparison in between generalized integrals and numerical series.

Elements of Ordinary Differential Equations.

Introduction; linear equations of first order; separable differential equations; linear equations of second oder with constant coefficients homogeneous and not homogeneous.

Natural numbers and Induction principle; combinatorics; elements of calculus of probability.

Integer and Rational numbers; Dedekind's completness axiom; least upper bound and greatest lower bound; the function absolute value; interval of real numbers.

Complex numbers; algebraic and trigonometric form; De Moivre's formula; n-th roots of a complex number; the foundamental theorem of algebra.

Continuous functions, limits, numerical sequences.

An introduction to continuity; the definition of a continuous function; Lipschitz functions; theorems about continuous functions (the theorem of existence of zeroes, the intermediate values theorem, the Weierstrass theorem).

Some elements of topology; an introduction to the definition of limit for a function of one variable; the property of limits; some foundamental limits of functions.

Numerical sequences; properties of limits of numerical sequences and comparison with the limits of functions; foundamental limits; Nepero's number.

The algebra of the Landau "little-o" symbol.

Differential calculus for functions of one variable.

Differentiable functions; the derivative of a function and its geometric meaning; the derivative of a function and its local property; Rolle and Lagrange's theorem and their consequences; antiderivatives; primitives of a function.

Taylor's formula with Peano's form of the remainder; Taylor's formula with Lagrange's form of the remainder; the qualitative study of the graph of a function of one variable.

Integrals and Series.

The Area's problem and an introduction to the integral of a continuous function; the foundamental theorem of calculus and its consequences; the Torricelli's theorem; integration by parts and by sostitution; generalized integrals.

Introduction to numerical series; convergence criterion; comparison in between generalized integrals and numerical series.

Elements of Ordinary Differential Equations.

Introduction; linear equations of first order; separable differential equations; linear equations of second oder with constant coefficients homogeneous and not homogeneous.

Attività d'esercitazione

A 40 hours exercise course will take place. Students are supposed to follow in small groups.

Modalità d'esame

A final written exam will take place

Propedeuticità

Are required basic notion of set theory, logic, functions, trigonometry and analitic geometry. All those notions are provided during the preliminary of the course.

Testi consigliati

E. Acerbi - G. Buttazzo: Analisi matematica ABC vol.1, Pitagora, Bologna, 2003

D. Mucci: Analisi matematica. Esercizi/1. Funzioni di una variabile, Pitagora, Bologna, 2004

D. Mucci: Analisi matematica. Esercizi/1. Funzioni di una variabile, Pitagora, Bologna, 2004