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Math Help - Partial derivative question

  1. #1
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    Partial derivative question

    Let f(x,y) = \frac{x^{2}(x - y)}{x^{2} + y{2}} if (x,y) doesnt equal (0,0) and

    0 if x = y = 0

    a) Find f_x(0,0) and f_y(0,0)

    b) Is f continuous at (0,0)?

    c) Is f differentiable at (0,0)? Explain.

    Attempt:

    a) F_x(0,0) and f_y(0,0) both = 0

    b) No since f(x,y) = 0 (if x = y = 0) and \frac{x^{2}(x - y)}{x^{2} + y{2}} (if (x,y) doesnt equal (0,0)

    c) It isnt differentiable at (0,0) similar reasoning as b
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  2. #2
    MHF Contributor chisigma's Avatar
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    a)

    \displaystyle f_{x} (0,0) = \lim_{\delta \rightarrow 0} \frac{1}{\delta}\ \frac{\delta^{3}}{\delta^{2}} = 1

    \displaystyle f_{y} (0,0) = \lim_{\delta \rightarrow 0} \frac{1}{\delta}\ (-\frac{0}{\delta^{2}}) = 0

    Kind regards

    \chi \sigma
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  3. #3
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    Quote Originally Posted by SyNtHeSiS View Post
    Let f(x,y) = \frac{x^{2}(x - y)}{x^{2} + y{2}} if (x,y) doesnt equal (0,0) and

    0 if x = y = 0

    a) Find f_x(0,0) and f_y(0,0)

    b) Is f continuous at (0,0)?

    c) Is f differentiable at (0,0)? Explain.

    Attempt:

    a) F_x(0,0) and f_y(0,0) both = 0
    It would be better if you showed how you got these- then we could comment. f_x is certainly NOT 0. Since f(0, 0)= 0, the partial derivative with respect to x is give by
    \lim_{h\to 0}\frac{f(h, 0)- 0}{h}= \lim_{h\to 0}\frac{\frac{h^2(h)}{h^2+ 0}}{h}= \lim_{h\to 0}\frac{h^3}{h^3}

    b) No since f(x,y) = 0 (if x = y = 0) and \frac{x^{2}(x - y)}{x^{2} + y{2}} (if (x,y) doesnt equal (0,0)
    What? You appear to be saying that any non-constant function is not continuous!
    What is the limit of f as (x, y) goes to (0, 0)?

    c) It isnt differentiable at (0,0) similar reasoning as b
    What reasoning? All you did was repeat the definition of the function!
    what is the definition of "differentiable" for a function of two variables?
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