# MATHEMATICAL ANALYSIS

## Learning outcomes of the course unit

Basic notions of calculus (limits, derivatives, integrals). The student should be able to plot the graph of a function, and to handle standard integration methods

## Prerequisites

None

## Course contents summary

Limits for sequences. Differential and integral calculus for real functions of one real variable

## Course contents

Sets and numbers. Basic set theory, operations between sets. Number systems: N, Z, Q, R, C. Representation of real numbers on a line; maximum, minimum, supremum, infimum of a set of real numbers; integer part and absolute value of real numbers; powers and roots. Complex numbers in various forms.

Functions: injective, surjective and bijective functions. Composition of functions; inverse function. Graphs. Real functions of one real variable. Monotonic functions. Powers with real exponent. Exponential and logarithmic functions. Angles; trigonometrical functions. Cardinality.

Sequences and series. Limits of sequences. Numeric series and convergence.

Limits and continuity. Limits of real functions of one real variable; properties. Continuity of real functions of one real variable; properties of continuous functions.

Differential calculus. Derivative and its geometric interpretation. Derivation rules (sum, product, ratio, inverse); chain rule; derivatives of the elementary functions. Relative and absolute maxima and minima; stationary points; monotony and the sign of the derivative. Main theorems (Fermat, Rolle, Lagrange aka mean-value, De l'Hopital); higher order derivatives; Taylor series development. Graphs.

Integral calculus. Primitive of a function defined in an interval; indefinite integrals. Geometric interpretation. Main properties. Fundamental theorem of the integral calculus. Integration techniques: by parts, by substitution; integration of rational functions.

## Recommended readings

M. Bertsch, R. Dal Passo, L. Giacomelli, Analisi Matematica, Mc Graw-Hill

## Teaching methods

Traditional lecture. Detailed exercises

## Assessment methods and criteria

Written and oral tests, that will confirm that the studenit is really able to deal with the basic notions of calculus (limits, derivatives and integrals of real functions of one real variable)