0
$\begingroup$

Suppose that the functions $f: \mathbb R^2 \to \mathbb R$ and $h: \mathbb R^2 \to \mathbb R$ are continuously differentiable. Find necessary and sufficient conditions for these functions to be first-order approximations of each other at the point (0,0).

$\endgroup$
  • $\begingroup$ Do you know the definition of "first-order approximation"? $\endgroup$ – Martín-Blas Pérez Pinilla Apr 8 at 16:47
  • $\begingroup$ Yes, from the theorem: let O be an open subset of R^n and suppose that the function f: O -> R is cont diff. Let x be a point in O. Then lim as h -> 0 (f(x+h)-[f(x)+<grad*f(x),h>])/ ||h|| = 0 $\endgroup$ – Amanda Lococo Apr 8 at 16:54
  • $\begingroup$ Then, start calculating the gradients of the functions. $\endgroup$ – Martín-Blas Pérez Pinilla Apr 8 at 16:58
  • $\begingroup$ Ok, so for f, we get that f' = df/dx(0,0)+df/dy(0,0) and similarly for h' = dh/dx(0,0)+dh/dy(0,0). Should I put these in the equation and simply solve to see when they are equivalent? $\endgroup$ – Amanda Lococo Apr 8 at 17:03
  • $\begingroup$ Gradient is a vector... $\endgroup$ – Martín-Blas Pérez Pinilla Apr 8 at 18:35
0
$\begingroup$

See https://en.wikipedia.org/wiki/Order_of_approximation#First-order.

First-order approximation means coincidence of functions and first derivatives in the point, so $$f(0,0) = h(0,0)$$ $$\cdots = Df(0,0) = Dh(0,0) = \cdots$$ where you can write $Df(0,0)$, $Dh(0,0)$ using partial derivatives...

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.