Is the following function differentiable?

• Dec 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.
• Dec 13th 2011, 02:43 PM
FernandoRevilla
Re: 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 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.
• Dec 14th 2011, 03:49 AM
FernandoRevilla
Re: 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 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?
• Dec 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 $\displaystyle \sin h(2+k)=h(2+k)-\frac{h^3(2+k)^3}{3!}+\ldots$ 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 $\displaystyle f_x(P_0)(x-x_0)+f_y(P_0)(y-y_0)+f_z(P_0)(z-z_0)=0$ where $\displaystyle P_0(x_0,y_0,z_0)$ belongs to the surface.
• Dec 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 :(