Maths in english

Maths In English

FundamentalsLimits

ContinuityDerivates

Classe V G:Marcelli, Pacini, Pallini

FundamentalsProperties of the set R of real numbers

1.The sum defines an associative and commutative operation on R for which 0 is the identity element. Moreover, for every element x in R, there is an element, -x, which is the inverse of x with respect to addition.2.The product defines an associative and commutative operation on R for which 1 is the identity element. Moreover, for every elemnt x in R differente from 0, there is an element, 1/x, which is the inverse of x with respect to multiplication.3.Given three elements of R: a,b, and c and the order relation “≤”: if a≤b then a+c≤b+c: moreover, if a≤b and c0, then ac≤bc.4.For every couple of subsets A and B of R so that a≤b, ∨ a A and b B, then there ∈ ∈exists a real number z such that a≤z≤b, ∨a A and b B∈ ∈

Given the fulfilment of the properties 1-4:

THE SET OF REAL NUMBERS R IS A COMPLETE ORDERED FIELD

Assume now that A is a nonempty subset of R. if, given m R, we have m≥a, ∈ ∨a A, then ∈we will say that m is an UPPER BOUND of A and A is UPPER BOUNDED. Similarly, when m≤a, ∨a A, we will say taht m is a LOWER BOUND of A and that A is LOWER BOUNDED.∈

Maximum and Minimum of a set

Let A be a nonempty subset of R. An element m belonging to A is said to be the maximum of A when m is also an upper bound of A.We write m=max A

Similarly, an element m belonging to A is said to be the minimum of A (m=min A) when m is also a lower bound of A

Supremum and Infimum of a set

Let A be an upper bounded subset of R, the supremum of A (SupA) is defined as the minimum of the upper bounds of A.Let A be an lower bounded subset of R, the infimum of A (InfA) is defined as the maximum of the lower bounds of A.

If a subset of R is unbounded from above we set: supA=+∞If a sub of R is unbounded from below we set: infA=-∞

R*= R U (+∞, -∞) EXTENDED SET OF REAL NUMBERS

Closed interval ab [a,b]={x R/a≤x≤b}∈Open interval ab (a,b)={x R/a<x<b}∈Interval ab closed on the left and pen on the right [a,b)={x R/a≤x<b}∈Interval ab open on the left and closed on the right (a,b]={x R/a<x≤b}∈

A function f of domani D and codomain C associates an argument x, which is an element of D, to the value of the function y=f(x) that is a unique element of C

The set of the elements y of C for which y=f(x) for some x in D is the range of f.

Neighbourhood of a point

A neighbourhood of a point x0 is an open interval (a,b) containing x0

The interval (x0 –r, x0 +r) is a particular neighbourhood of x0 called open ball of radius r

Accumulation points

Given a subset A of R, x0 belonging to A is an accumulation point if any meighbourhood of x0 contains element of A different from x0 .If x0 belonging to A is not an accumulation point, it is called isolated point

LimitsIt’s useful to extend the concept of neighbourhood by defining the neighbourhoods of +∞ and -∞

You should notice that in the definition of a limit the value of the function f(x) in x0 is irrelevant. As a matter of fact, f(x) is not even required to be defined in x0.

Theorems of limits

When l is infinitive, the following result can be used:

In the above formulae it is always assumed that the right-hand-side does no give rise to an intermediate form. The presence of an indeterminate form, does not mean necessarily that the integral does not exist.

The following is a list of some useful indeterminate forms that have a well defined limit.

ContinuityFrom the algebraic prperties of limits and the definition of a continuous function it follows that sums, products, quotients of continuous funciots are stille continuous functions. Moreover it can be shown that also the composite of two continuous functions and the inverse of a continuous.

The accumulation points fro the domain of a function where the function is not defined or where it is defined, but it is not continuous are called discontinuities.

Theorems of Continuity

DerivativesLet us consider a function f(x); given h>0, assume that both of point x0 and x0+h lie in the domani of f(x). The secant to the graph of the function is the unique line passinge through the two points (x0,f(x0)) and (x0 +h, f(x0 +h)) on the Cartesian plan. This line is described by the function

S(x)=mx0,x0+h .(x-x0) + f(x0 )Where mx0,x0+h , the slope of the line is given by

Mx0,x0+h =f(x0 +h) – f(x0 )h

Which is also called difference quotient of f(x) at x0. when tha point x0 +h is approaching x0 the secant will be an increasingly good approximation of the tangent at x0

We will now review the main properties of the derivative at a point. Similarly to what we have done when studying limits and continuous functions, it is useful to analyse the algebra of derivates.

For the derivative of the composite functions we have:

Also the differentiability of the inverse function can be deduced from the differentiability of f(x), in fact:

List of the most common functions

Just like in the study of continuity, it is useful to classify the points of non-differentiability, i.e. the points at which the derivative of a function does not exist.