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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 $$\mathbb{R}$$, let $$x_0 \in X$$ be a limit point of $$X$$, let $$f : X \to \mathbb{R}$$ be a function, and let $$L$$ be a real number. Then the following statements are logically equivalent:

1. $$f$$ is differentiable at $$x_0$$ with derivative $$f'$$.
2. For any $$\varepsilon > 0$$, there exists a $$\delta > 0$$ such that for all $$x \in \{i \in X : |i - x_0| < \delta \}$$, $$f(x)$$ is $$\varepsilon |x - x_0|$$-close to $$f(x_0) + f'(x_0)(x-x_0)$$.

In other words, we have:
$$|f(x) - (f(x_0) + f'(x_0)(x - x_0))| \le \varepsilon|x - x_0|$$

2. can be prased as: fix $$x_0$$. Given a challenge of $$\varepsilon$$, 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 $$\varepsilon$$.

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 $$\varepsilon > 0$$ and I can can give you a $$\delta > 0$$ such that any point that is within $$\delta$$ from $$x_0$$, an estimate of $$f(x_0) + f'(x_0)(x - x_0)$$ will be at most $$\delta \varepsilon$$ off. And in fact, I can do slightly better, the error will be within $$|x - x_0| \varepsilon$$.

p255