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Thread: Find the equation for the tangent plane to the surface. Answer doesn't seem right.

  1. #1
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    Find the equation for the tangent plane to the surface. Answer doesn't seem right.

    $\displaystyle z=ln(3x^{2}+6y^{2}+1)$, $\displaystyle P(0,0,0)$

    Ok, so I found $\displaystyle \frac{\partial f}{\partial x}$ and $\displaystyle \frac{\partial f}{\partial y}$

    $\displaystyle \frac{\partial f}{\partial x}=\frac{6x}{3x^{2}+6y^{2}+1}$

    $\displaystyle \frac{\partial f}{\partial y}=\frac{12y}{3x^{2}+6y^{2}+1}$

    So then,

    $\displaystyle f_{x}(0,0)=\frac{0}{1}=0$

    $\displaystyle f_{y}(0,0)=\frac{0}{1}=0$

    And I have a normal vector $\displaystyle n=(0,0,-1)$ at point $\displaystyle (0,0,0)$

    So then,

    $\displaystyle 0(x-0)+0(y-0)-(z-0)=0$

    $\displaystyle =>z=0$

    Did I do this correctly? Thanks for looking.
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  2. #2
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    Quote Originally Posted by downthesun01 View Post
    $\displaystyle z=ln(3x^{2}+6y^{2}+1)$, $\displaystyle P(0,0,0)$
    You should have looked at $\displaystyle F(x,y,z)=\ln(3x^{2}+6y^{2}+1)-z$.

    Where is your $\displaystyle F_z?$
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    $\displaystyle f_{z}=-1$

    $\displaystyle f_{z}(0,0,0)=-1$

    $\displaystyle n=0i+0j-k$ at point $\displaystyle (0,0,0)$

    $\displaystyle 0(x-0)+0(y-0)-(z-0)=0$

    $\displaystyle z=0$

    Still getting z=0

    And it still seems like an odd answer for some reason.
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  4. #4
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    Quote Originally Posted by downthesun01 View Post
    $\displaystyle f_{z}=-1$

    $\displaystyle f_{z}(0,0,0)=-1$

    $\displaystyle n=0i+0j-k$ at point $\displaystyle (0,0,0)$

    $\displaystyle 0(x-0)+0(y-0)-(z-0)=0$

    $\displaystyle z=0$

    Still getting z=0

    And it still seems like an odd answer for some reason.
    1. Which reasons?

    2. I've attached the graph of z and the tangent plane. And it looks as if your calcualtions are OK. So again: which reasons?
    Attached Thumbnails Attached Thumbnails Find the equation for the tangent plane to the surface. Answer doesn't seem right.-tangplane.png  
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  5. #5
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    Honestly, z=0 just seems kind of boring. I know, not a very good reason. Thanks for the graph though. It makes it really easy to see that the answer is correct. Much appreciated.
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