# CALCULUS 1 PAR. 1

## Learning outcomes of the course unit

The aim of the course consists in providing students with the basic of Mathematical Analysis

## Course contents summary

1. Real numbers.

Axiomatic definition of real numbers, maximum, minimum,

least upper and greatest lower bound; integer part and modulus of real numbers;

powers, roots, nth roots of non-negative numbers;

rational and irrational numbers; intervals, distance; neighbourhoods,

cluster points, isolated points, interior points; closed sets, open sets, frontier.

injective, surjective, and bijective functions, inverse function; graphs;

real functions of a real variable, monotone functions, exponential

and logarithmic functions; trigonometric functions.

2. Functions.

One to one, surjective and bijective functions; inverse functions; graphs;

monotone functions; exponential and logarithmic functions; trigonometric functions.

3. Limits.

Limits of functions with real values, uniqueness of the limit, limits of the restrictions;

limit of the sum, product, quotient of two functions; permanence of the sign,

comparison theorems; right and left limit; limits of monotone functions;

orders of infinitesimals and infinities, asymptotics.

4. Continuous functions.

Continuity of real functions of a real variable, restrictions of continuous functions,

composition of continuous functions; sum, product, quotient of continuous functions;

examples of continuous functions; discontinuity, examples of discontinuous functions;

zero theorem; continuity and intervals; continuity and monotony;

continuity of inverse functions; Weierstrass theorem.

5. Differential calculus.

Incremental ratio, derivatives, right and left derivatives;

geometrical significance of the derivative; derivation rules:

derivatives of the sum, product, quotient of two functions;

derivatives of compound functions and inverse functions;

derivatives of elementary functions; relative maximums and minimums;

stationary points; relationship between monotony and sign of the derivative;

Rolle's theorem and Lagrange's theorem and their geometrical interpretation,

Cauchy's theorem and de l'Hopital's theorem; convex functions,

derivatives of convex functions,

relationship between convexity and sign of the second derivative;

Taylor's formula with Peano, Lagrange remainder;

study of local maxima and minima via the study of higher order derivatives.

6. Integrals.

Partitions of an interval; upper and lower integral, integrable functions in an interval,

integrability of continuous functions and monotone functions;

geometrical interpretation of the integral;

properties of integrals; mean of an integrable function; integrals on directed intervals;

fundamental theorem of integral calculus; primitives, indefinite integrals;

integration by parts and by substitution; integrals of rational functions.

Taylor expansion with integral remainder.

## Recommended readings

1. E. Acerbi, G. Buttazzo: Primo corso di Analisi Matematica, Ed. Pitagora, 1997.

2. E. Acerbi, G. Buttazzo: Analisi matematica ABC, Ed. Pitagora, 2000.

3. M. Bramanti, C.D. Pagani, S. Salsa: Analisi Matematica 1, Ed. Zanichelli, 2008.

4. M. Giaquinta, L. Modica: Analisi Matematica 1, vol. 1 & 2, Ed. Pitagora, 1998.

## Teaching methods

Lectures.

## Assessment methods and criteria

The exam consists in a written part and an oral part in a different date.

Two evaluations in itinere are fixed:

one at the end of the first term and the second at the end of the second term.

If both of them are positively marked, the student can skip the written part of the exam.