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Math Help - Is the following function differentiable?

  1. #1
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    Is the following function differentiable?

    I have:


    Is the function differentiable in (0,2)? If so, find its Tangent Plane.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    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 .
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  3. #3
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    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.
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    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.
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    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?
    Last edited by GIPC; December 14th 2011 at 08:19 AM.
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    Re: Is the following function differentiable?

    Quote Originally Posted by GIPC View Post
    I came to a conclusion that the function is indeed differentiable,
    That is a correct conclusion.

    but I have to show a strict rigorous proof.
    There are many ways to show a rigorous proof.

    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.

    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.
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  7. #7
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    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
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