27991 - Mathematical Analysis T-1

Course contents

PROPERTIES OF REAL NUMBERS. 
LIMITS AND CONTINUOUS FUNCTIONS. 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. Generalities about functions: composition of functions, invertible functions and inverse functions. Pecularities of real-valued functions of one real variable. 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

G.C. Barozzi, G. Dore, E. Obrecht: Elementi di Analisi Matematica, vol. 1, Zanichelli (2009)
An exercise book on functions of one  real  variable, such as, for example: M. Bramanti: Esercitazioni di Analisi Matematica 1, Progetto Leonardo - Esculapio (2011)

Teaching methods

The course consists of lessons describing the fundamental concepts of real numbers, sequences and numerical series, and, especially, of 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 and an oral exam. In the written part, lasting three hours, the solution various exercises is required. Access to the oral part is allowed only to the students passing the written part. The oral 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. The oral part of the exam must be passed in the same session of the written part. Only in the period january - february the oral part may be passed in the subsequent session. The written part may be replaced by two partial examinations, lasting two hours each, showing the student studies regularly.

Teaching tools

Tutorship (if appointed)

Office hours

See the website of Enrico Obrecht