27991 - Mathematical Analysis T-1

Learning outcomes

Fornire gli strumenti matematici di base (limiti, derivate, integrali) per la analisi qualitativa delle funzioni e la risoluzione di problemi applicativi.

Course contents

PROPERTIES OF REAL NUMBERS, LIMITS AND CONTINUOUS FUNCTIONS. Generalities about functions: composition of functions, invertible functions and inverse functions. Pecularities of real-valued functions of one real variable. Definition of convergent and of divergent sequences of real numbers. Theorems about limits of sequences: uniqueness of the limit, comparison theorems. The algebra of limits. Monotone sequences and their limits. The number e. Decimal representation of real numbers. Definition of a continuous function of one real variable. The Weierstrass theorem and the intermediate value theorem. Definition of limit of a real function of one real variable; generalization of results established for sequences. Continuity of the composition of two continuous functions and the theorem on the change of variable in a limit. One-sided limits. Monotone functions and their limits. Asymptotes. The inverse circular functions. The hyperbolic functions and their inverse functions.
DIFFERENTIAL CALCULUS. Definition of a differentiable function and of derivative of a function. The algebra of derivatives. The chain rule. The mean value theorem and its application to study the monotonicity of a function. Higher order derivatives. Taylor's formula with Peano and Lagrange forms of the remainder. Relative maxima and minima of a function: definitions, necessary conditions, sufficient conditions. Convex functions.
INTEGRAL CALCULUS. Definition of the Riemann integral. Properties of the integral: linearity, additivity, monotonicity, the mean value theorem. Sufficient conditions of integrability. The fundamental theorems of the integral calculus. The theorems of integration by substitution and of integration by parts. Piecewise continuous functions and propeties of their integrals. Improper integrals: definitions, absolute convergence, comparison theorem.
COMPLEX NUMBERS. Definition and operations on complex numbers. Algebraic form of a complex number, modulus and argument of a complex number, exponential form of a complex number. De Moivre formula, roots of a complex number, algebraic equations in C, the complex exponential function.
LINEAR DIFFERENTIAL EQUATIONS. Linear differential equations of first order: general integral for homogeneous and non homogeneous equations, the Cauchy problem. Linear differential equations of second order with constant coefficients: general integral for homogeneous and non homogeneous equations, the Cauchy problem. Generalization to variable coefficients and arbitrary order equations.

Readings/Bibliography

Marco Bramanti, Carlo Domenico Pagani, Sandro Salsa, Analisi matematica 1. Ed. Zanichelli.

oppure

Marcellini P.-Sbordone C.: Analisi Matematica 1 - Liguori Editore

oppure

M. Bertsch, R. Dal Passo, L. Giacomelli - Analisi Matematica, ed. McGraw Hill. (seconda edizione)

oppure

G.C. Barozzi, G. Dore, E. Obrecht: Elementi di Analisi Matematica, vol. 1, ed. Zanichelli.

In general, the student may use any good textbook of Mathematical Analysis which contains the arguments of the program. The student will check with the professors the validity of the chosen alternative textbook depending on the program.

Exercices book:

M. Bramanti - Esercitazioni di Analisi 1, Ed. Esculapio, Bologna, 2011

M. Amar, A.M. Bersani - Esercizi di Analisi Matematica 1, Ed. Esculapio, Bologna, 2011

S. Abenda - Esercizi di Analisi Matematica Vol 1, Ed. Progetto Leonardo -Bologna
oppure
S. Salsa & A. Squellati: Esercizi di Matematica, Vol. I, Ed. Zanichelli

Teaching methods

The course consists of lessons describing the fundamental concepts of differential and integral calculus real for real functions of one real variable. Lessons are completed with examples and counterexamples illuminating the theoretical content. Futhermore a lot of exercises are solved in the classroom.

Assessment methods

The assessment consists in a written part, lasting three hours, containing both the resolution of various exercises and theoretical questions (definitions and theorems, possibly with proofs).

The written test is passed if one gets a grade greater or equal than 18/30.

Students, that pass the written test of Mathematical Analysis T1 have the possibility to face a further theoretical test by registering on a suitable Almaesami list. In any case the obtained result can not be modified more than two points (positively or negatively). Otherwise we proceed to verbalize the result of the written test by tacit assent, a week later from the publication on Almaesami of the results of the written test.

The theoretical part of the exam dwells upon the comprehension of the relevant concepts and on the knowledge of definitions and the statements of fundamental theorems. Proofs of some theorems, clearly detailed, may be required.

Teaching tools

Textbook and exercices book, online material available on http://www.dm.unibo.it/~baldi [http://www.dm.unibo.it/%7Ebaldi] and on INSEGNAMENTI ONLINE [https://iol.unibo.it/].

Tutorship (if appointed).

Office hours

See the website of Annalisa Baldi

See the website of Enrico Smargiassi