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  1. #11
    Glowy Goopy Goodness The_Liquid_Laser's Avatar
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    It's been a while since I've seen this material, but the notation doesn't look quite right to me. (Perhaps I am just not familiar with it.) From the context of the problem f is a function with multiple independent variables, e.g. f(x,y). Therefore f(x/y) doesn't make sense to me. Is this the same as f(x/y, 0) ?
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  2. #12
    Courage is immortality Valiant's Avatar
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    Good LORD! This is why we entj's need the backup of specialists

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  3. #13

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    Quote Originally Posted by The_Liquid_Laser View Post
    It's been a while since I've seen this material, but the notation doesn't look quite right to me. (Perhaps I am just not familiar with it.) From the context of the problem f is a function with multiple independent variables, e.g. f(x,y). Therefore f(x/y) doesn't make sense to me. Is this the same as f(x/y, 0) ?
    Aha! That interpretation works. (Clever, LL)

    Note the "d" are actually denoting partials.

    f(x,y)=x(x/y,0)
    df(x,y)/dx=f(x/y,0)+x[df(x/y,0)/d(x/y)](d(x/y)/dx)=f(x/y,0)+(x/y)[df(x/y,0)/d(x/y)]
    df(x,y)/dy=x[df(x/y,0)/d(x/y)](d(x/y)/dy)=-(x/y)^2[df(x/y,0)/d(x/y)]

    Now note for all x,y:
    (df(x,y)/dx)x+(df(x,y)/dx)y=xf(x/y,0)+(x^2/y)[df(x/y,0)/d(x/y)]-(x^2/y)[df(x/y,0)/d(x/y)]=xf(x/y,0)=f(x,y).

    Note that this is exactly the condition we need.
    f(x,y)=(df(x,y)/dx)x+(df(x,y)/dx)y

    So, the equation for a tangent plane at point (x0, y0, f(x0,y0)) is:
    z-f(x0,y0)=(df(x,y)/dx|x=x0)(x-x0)+(df(x,y)/dx)(y-y0)

    The plane goes throught (0,0,0) if and only if:
    0-f(x0,y0)=(df(x,y)/dx|x=x0)(0-x0)+(df(x,y)/dx)(0-y0)

    Which is the same equation as:
    f(x0,y0)=(df(x,y)/dx|x=x0)x0+(df(x,y)/dx)y0

    And we know the above equation is simply:
    f(x,y)=(df(x,y)/dx)x+(df(x,y)/dx)y with (x,y)=(x0,y0)

    So all the tangent planes intercept (0,0,0).

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  4. #14
    Senior Member
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    Quote Originally Posted by YourLocalJesus View Post
    Good LORD! This is why we entj's need the backup of specialists
    This is why we ENTJs know enough that we can hold our own with the specialists, but our strength lies in knowing how to best utilize the resources that the specialists present us with.
    I am an ENTJ. I hate political correctness but love smart people ^_^

  5. #15
    pathwise dependent FDG's Avatar
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    Quote Originally Posted by IlyaK1986 View Post
    This is why we ENTJs know enough that we can hold our own with the specialists, but our strength lies in knowing how to best utilize the resources that the specialists present us with.
    Yeah allright, but my problem with this problem in the end was about the notation of the function, i wasn't sure on how to take the derivative...I've already thnaked ygolo in private for having clarified it to me : )
    ENTj 7-3-8 sx/sp

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