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27210 - Mathematical Analysis 1

Academic Year 2011/2012

Teacher Alberto Venni
Number of credits 11
SSD MAT/05
Teaching Mode Traditional lectures
Lingua Italian
Versione italiana

Learning outcomes

At the end of the course the students acquire basic notions on differential and integral calculus for functions of one real variable and on related topics. Moreover they are able to use classical tools of Mathematical Analysis that have useful applications in different fields.

Course contents

Sets - Ordered couples and relations - Functions - Equivalences - Orderings - Basic definitions on groups and fields - Real numbers - The extended real line - Natural, integer and rational numbers - Powers and roots - Finite and infinite sets - Combinatoric calculus - Complex numbers - Polynomials.
Metric spaces - Topological properties - Limits - Operations on limits - Some remarkable limits - Continuous and uniformly continuous functions - Compact spaces - Bolzano-Weierstraß theorem - Complete metric spaces.
Series - Convergence criteria - Absolutely convergent series - p-adic representation of the real numbers - Sequences and series of functions - Power series - Exponentials, logarithms, trigonometric and hyperbolic functions - The fundamental theorem of algebra.
Derivatives - Rules of computation - Derivatives of the elementary functions - Differentiable functions on an interval - Higher order derivatives - Taylor's formula - Riemann integral - Sufficient conditions for integrability - Integrals of vector-valued functions - Primitives and integrals - Higher order primitives - Improper integrals.

Readings/Bibliography

Lecture notes.

Teaching methods

Lectures

Assessment methods

Written and oral exam.

Language of instruction

Italian

Office hours

See the Faculty member's website