27991 - Mathematical Analysis T-1

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's 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

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

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

Other suggestions for textbooks:

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

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

Other suggestions for exercises books:

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

P. Marcellini-C. Sbordone: Esercitazioni di Matematica, volume 1, ed. Liguori

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 and a number of exercises are solved.

Assessment methods

The examination consists of a preliminary written test (exercises) and a test about the theoretical part.
The preliminary written test (exercises) will be structured according to the restrictions and rules prescribedby the University for the Covid's infection. It consists in solving exercises related to the topics  of the course. In order to sustain it the student must register at the test through AlmaEsami [https://almaesami.unibo.it/] . If this written test is passed (at least 16/30), the student can sit for the test concerning the theoretical aspects of the course. There, the student must show to know the concepts explained during the course (in particular definitions and theorems) and how to connect them. The theoretical part of the exam must be passed in the same "sessione" of the preliminary written test (exercises).

Teaching tools

Textbook and exercices book, online material available on  INSEGNAMENTI ONLINE [https://iol.unibo.it/]. Tutorship (if appointed).

Office hours

See the website of Giovanni Cupini