27991 - Mathematical Analysis T-1

Learning outcomes

At the end of the course, after the final examination, the student should possess basic knowledge relating basic elements of Mathematical Analysis, and in particular relating the real functions of a real variable: interpretation of graphs, limits, derivatives, integration and their meaning.

Course contents

SET THEORY AND NUMBERS. Operations with sets. Numerical sets (natural, integer, rational and real 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.

FUNCTIONS. Generalities about functions: composition of functions, invertible functions and inverse functions.

SEQUENCES. 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.

CONTINUOUS FUNCTIONS. 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. Monotone functions and their limits.

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.

ORDINARY 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. Separable ODEs.

Readings/Bibliography

THEORY. 

1. M. Bertsch, R. Dal Passo, L. Giacomelli: Analisi Matematica (seconda edizione), McGraw-Hill (2011)
2. G.C. Barozzi, G. Dore, E. Obrecht: Elementi di Analisi Matematica, vol. 1, Zanichelli (2009)

EXERCISES. M. Bramanti: Esercitazioni di Analisi Matematica 1, Progetto Leonardo - Esculapio (2011)

Teaching methods

Lectures and weekly student reception.

Assessment methods

At the end of the course there will be a written and an oral test, both of which are compulsory.

The written test will consist of exercises. It will last 2 hours and 30 minutes. It is forbidden to use notes, books nor pocket calculators.

Access to the subsequent oral test is allowed only to those who have passed the written test with a score greater than or equal to 18/30.

The oral test will focus on verifying understanding of the fundamental concepts and knowledge of the definitions and statements of the main results, examples and counterexamples.

There will be 6 rounds (each of which will have both written and oral test): 3 in the winter session (January-February) and 3 in the summer session (2 between June-July + 1 in September).

The oral test must be taken within the same session in which you passed the written test.

Teaching tools

Sheets of exercises will be made available, uploaded on the UniBo "VIRTUAL" website

Office hours

See the website of Eugenio Vecchi