Is the following function differentiable?

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December 13th 2011, 01:10 PM
GIPC

Is 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.
December 13th 2011, 02:43 PM
FernandoRevilla

Re: Is the following function differentiable?

We have $(\nabla f)(0,2)=(f_x(0,2).f_y(0,2))=\ldots=(0,1)$ , so if $f$ is differentiable at $(0,2)$ the only possible differential is $\lambda (h,k)=(\nabla f)(0,2)(h,k)^t=k$ . Now, analyze if $\displaystyle\lim_{(h,k) \to (0,0)} \frac{|f(0+h,2+k)-f(0,2)-\lambda (h,k)|}{ \left\|{(h,k)}\right\|}=0$ .
December 14th 2011, 03:08 AM
GIPC

Re: 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.
December 14th 2011, 03:49 AM
FernandoRevilla

Re: Is the following function differentiable?

Use $\sin h(2+k)\approx h(2+k)$ in a neighborhood of $(0,0)$ in the precise terms of the Taylor's formula.
December 14th 2011, 06:35 AM
GIPC

Re: 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?
December 14th 2011, 10:29 AM
FernandoRevilla

Re: Is the following function differentiable?

Quote:

Originally Posted by GIPC

I came to a conclusion that the function is indeed differentiable,

That is a correct conclusion.

Quote:

but I have to show a strict rigorous proof.

There are many ways to show a rigorous proof.

Quote:

How do I expand f to its Taylor Series?

You only need $\sin h(2+k)=h(2+k)-\frac{h^3(2+k)^3}{3!}+\ldots$ for finding in a comfortable way the limit in my answer #2.

Quote:

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

Use the well known formula $f_x(P_0)(x-x_0)+f_y(P_0)(y-y_0)+f_z(P_0)(z-z_0)=0$ where $P_0(x_0,y_0,z_0)$ belongs to the surface.
December 14th 2011, 10:38 AM
GIPC

Re: 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 :(