28622 - Mathematical Analysis T-A

Learning outcomes

Theoretical and computational aspects of differential and integral calculus of one real variable.

Course contents

LIMITS AND CONTINUOUS FUNCTIONS. Definition of convergent and of divergent sequences of real numbers. Theorems on limits of sequences: uniqueness of the limit, comparison theorems. The algebra of limits. Monotone sequences and their limits. Generalities about functions: composition of functions, invertible functions and inverse functions. Monotone 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.

DIFFERENTIAL CALCULUS. Definition of a differentiable function of derivative and differential 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. 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.

Readings/Bibliography

Theoretical aspects: 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 two hours, the solution various exercises is required. Access to the oral part is allowed only to the students who passed both the part of analysis and geometry of the course.

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.

The oral part of the exam must be passed before the starting of nexr semester of lessons.

Teaching tools

tutoring: the students will be supported by a tutor, who will help them to overcome any difficulty which can arise

Links to further information

http://www.dm.unibo.it/~citti/

Office hours

See the website of Giovanna Citti