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Math Help - Gradients and directional derivatives

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    Gradients and directional derivatives

    1. The problem statement, all variables and given/known data[/b]
    Let z=3x2-y2. Find all points at which ||∇Z||=6


    First things first


    I took the partials dz/dx and dz/dy


    dz/dx=6x


    dx/dy=-2y


    I know that √(36x2+4y2)=6 or (36x2+4y2)=36


    Then using the above relation I solved for each variable getting
    1.y=√(9-x2)


    2.x=√(1-1/9y2)


    We already have the relation from the partials


    A.-2y=0


    B.6x=0
    So I plugged in 1 into A and 2 into B to get the following


    x=+-1 and y=+-3


    However the answers don't check out


    Where did I go wrong?
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    Re: Gradients and directional derivatives

    Quote Originally Posted by mdhiggenz View Post
    1. The problem statement, all variables and given/known data[/b]
    Let z=3x2-y2. Find all points at which ||∇Z||=6


    First things first


    I took the partials dz/dx and dz/dy


    dz/dx=6x


    dx/dy=-2y


    I know that √(36x2+4y2)=6 or (36x2+4y2)=36
    Have you considered that 36x^2 + 4y^2 = 36 is an ellipse?

    -Dan
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    Re: Gradients and directional derivatives

    Hey Dan thanks for the response.
    No,I didnt consider an ellipse but it isn't obvious to me how that would help. Can you elaborate, and explain why what I did was incorrect?
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    Forum Admin topsquark's Avatar
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    Re: Gradients and directional derivatives

    Quote Originally Posted by mdhiggenz View Post
    Hey Dan thanks for the response.
    No,I didnt consider an ellipse but it isn't obvious to me how that would help. Can you elaborate, and explain why what I did was incorrect?
    Honestly I didn't spend any time on it that far down. I think the simplest way to state your answer is that the solution set is all points on that ellipse.

    -Dan
    Last edited by topsquark; November 6th 2012 at 05:15 PM.
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    Re: Gradients and directional derivatives

    Quote Originally Posted by mdhiggenz View Post
    A.-2y=0


    B.6x=0
    Why did you set these to 0? That would imply | \nabla z |  = 0 which is in the solution set, but is not the whole solution set.

    -Dan
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