Math and science::Analysis::Tao::10: Differentiation of functions

Newton's approximation

Informally, if $f$ is differentiable at $x_{0}$ , then one has the approximation $f (x) \approx f (x_{0}) + f^{'} (x_{0}) (x - x_{0})$ , and conversely.

The formal version:

Newton's approximation

Let $X$ be a subset of $R$ , let $x_{0} \in X$ be a limit point of $X$ , let $f : X \to R$ be a function, and let $L$ be a real number. Then the following statements are logically equivalent:

$f$ is differentiable at $x_{0}$ with derivative $f^{'}$ .
For any $ε > 0$ , there exists a $δ > 0$ such that for all $x \in {i \in X : | i - x_{0} | < δ}$ , $f (x)$ is $ε | x - x_{0} |$ -close to $f (x_{0}) + f^{'} (x_{0}) (x - x_{0})$ .

In other words, we have:
$| f (x) - (f (x_{0}) + f^{'} (x_{0}) (x - x_{0})) | \leq ε | x - x_{0} |$

2. can be prased as: fix $x_{0}$ . Given a challenge of $ε$ , it's possible to find a neighbourhood around $x_{0}$ for which $x_{0}$ along with $f (x_{0})$ and $f^{'} (x_{0})$ can be used to approximate $f$ for all of the neighbourhood, with an approximation error at most $ε$ .

This seems like such a confusing way to formulate the idea.

An attempt at the formal version in words as a challenge: for any

x_{0}

at which

f

is differentiable you may choose any

ε > 0

and I can can give you a

δ > 0

such that any point that is within

δ

from

x_{0}

, an estimate of

f (x_{0}) + f^{'} (x_{0}) (x - x_{0})

will be at most

δ ε

off. And in fact, I can do slightly better, the error will be within

| x - x_{0} | ε

Source

p255