I have:

http://img12.imageshack.us/img12/6121/capturerhf.png

Is the function differentiable in (0,2)? If so, find its Tangent Plane.

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- Dec 13th 2011, 01:10 PMGIPCIs the following function differentiable?
I have:

http://img12.imageshack.us/img12/6121/capturerhf.png

Is the function differentiable in (0,2)? If so, find its Tangent Plane. - Dec 13th 2011, 02:43 PMFernandoRevillaRe: Is the following function differentiable?
We have $\displaystyle (\nabla f)(0,2)=(f_x(0,2).f_y(0,2))=\ldots=(0,1)$ , so

**if**$\displaystyle f$ is differentiable at $\displaystyle (0,2)$ the only possible differential is $\displaystyle \lambda (h,k)=(\nabla f)(0,2)(h,k)^t=k$ . Now, analyze if $\displaystyle \displaystyle\lim_{(h,k) \to (0,0)} \frac{|f(0+h,2+k)-f(0,2)-\lambda (h,k)|}{ \left\|{(h,k)}\right\|}=0$ . - Dec 14th 2011, 03:08 AMGIPCRe: Is the following function differentiable?
I came to that lim term myself and that's where I kind of get stuck with all the epsilons. I can't draw a concrete conclusion from there if it is indeed 0 or just a small epsilon bigger than 0. That's what making this exercise difficult.

- Dec 14th 2011, 03:49 AMFernandoRevillaRe: Is the following function differentiable?
Use $\displaystyle \sin h(2+k)\approx h(2+k)$ in a neighborhood of $\displaystyle (0,0)$ in the precise terms of the Taylor's formula.

- Dec 14th 2011, 06:35 AMGIPCRe: Is the following function differentiable?
I came to a conclusion that the function is indeed differentiable, but I have to show a strict rigorous proof.

How do I expand f to its Taylor Series?

And from there, how do I find the tangent plane? - Dec 14th 2011, 10:29 AMFernandoRevillaRe: Is the following function differentiable?
That is a correct conclusion.

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but I have to show a strict rigorous proof.

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How do I expand f to its Taylor Series?

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And from there, how do I find the tangent plane?

- Dec 14th 2011, 10:38 AMGIPCRe: Is the following function differentiable?
Sorry for the follow up questions. Even with the Taylor expansion, how do I use it in the lim term you prescribed earlier?

And also, after i prove differentiability, how do I find the appropriate P0 to plug into the formula for the tangent plane?

I'm sorry for asking pathetic questions :(