Math and science::Analysis::Tao::10: Differentiation of functions
Newton's approximation
Informally, if is differentiable at , then one has the
approximation , and conversely.
The formal version:
Newton's approximation
Let be a subset of , let be a limit
point of , let be a function, and let
be a real number. Then the following statements are logically equivalent:
- is differentiable at with derivative .
- For any , there exists a such that
for all , is
[...]-close to [...].
In other words, we have:
2. can be prased as: fix . Given a challenge of , it's possible to find a neighbourhood around for which along with and can be used to approximate for all of the neighbourhood, with an approximation error at most .