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

Newton's approximation

Informally, if f is differentiable at x0, then one has the approximation f(x)f(x0)+f(x0)(xx0), and conversely.

The formal version:

Newton's approximation

Let X be a subset of R, let x0X be a limit point of X, let f:XR be a function, and let L be a real number. Then the following statements are logically equivalent:

  1. f is differentiable at x0 with derivative f.
  2. For any ε>0, there exists a δ>0 such that for all x{iX:|ix0|<δ}, f(x) is [...]-close to [...].

    In other words, we have:
    |f(x)(f(x0)+f(x0)(xx0))|ε|xx0|

2. can be prased as: fix x0. Given a challenge of ε, it's possible to find a neighbourhood around x0 for which x0 along with f(x0) and f(x0) can be used to approximate f for all of the neighbourhood, with an approximation error at most ε.