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Is the following function differentiable?

  1. Dec 14, 2011 #1
    I have:
    http://img12.imageshack.us/img12/6121/capturerhf.png [Broken]


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

    So far I have
    We have [itex](\nabla f)(0,2)=(f_x(0,2).f_y(0,2))=\ldots=(0,1)[/itex] , so if [itex]f[/itex] is differentiable at [itex](0,2)[/itex] the only possible differential is [itex]\lambda (h,k)=(\nabla f)(0,2)(h,k)^t=k[/itex] .So I have to analyze [itex]\displaystyle\lim_{(h,k) \to (0,0)} \frac{|f(0+h,2+k)-f(0,2)-\lambda (h,k)|}{ \left\|{(h,k)}\right\|}=0[/itex] but I can't seem to solve it.

    And I also don't know how to find a tangent plane
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Dec 14, 2011 #2
    I came to a conclusion that f is indeed differentiable.

    Can someone please help me understand how to prove it? Someone suggested that i use the Taylor expansion but i don't know how to use it. So i'm hoping someone could show me :)
     
  4. Dec 14, 2011 #3

    D H

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    As a prerequisite, the function needs to be continuous at the point in question for a derivative to exist at that point. Is this function continuous at (x,y)=(0,2)?

    Assuming it is continuous, this is a function of two variables, so you need to investigate both the partial with respect to x and with respect to y. For x not equal to zero you will get one pair of partial derivatives, for x equal to zero you will get another. Is the first set equal to the second set in the limit x→0?
     
  5. Dec 14, 2011 #4
    when i try to take the partial derivatives i can't really take them to x->0 and i get stuck with the epsilon business.

    The general steps for showing differentiability doesn't really apply here.
     
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