f, h cont diff. Find necessary and sufficient conditions for these functions to be first-order approximations of each other at the point (0,0).

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).

• Do you know the definition of "first-order approximation"? – Martín-Blas Pérez Pinilla Apr 8 at 16:47
• 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 – Amanda Lococo Apr 8 at 16:54
• Then, start calculating the gradients of the functions. – Martín-Blas Pérez Pinilla Apr 8 at 16:58
• 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? – Amanda Lococo Apr 8 at 17:03
• Gradient is a vector... – Martín-Blas Pérez Pinilla Apr 8 at 18:35

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...