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Find an equation for the tangent plane to a surface (using gradient)

  1. Oct 8, 2015 #1
    1. The problem statement, all variables and given/known data
    Find an equation for the tangent plane to a surface [itex]xz^2 +x^2y-z=-1[/itex] at the point (1,-3,2).

    2. Relevant equations
    [itex]
    (\vec{r}-\vec{r_p}) \cdot \nabla f(\vec{r_p}) = 0
    [/itex]

    3. The attempt at a solution

    First I found the gradient of the function
    [itex]
    \nabla f = (z^2+2xy)\hat{i} + x^2 \hat{j} + (2xz-1)\hat{k}
    [/itex]
    and then evaluated at (1,-3,2)
    [itex]
    \nabla f(\vec{r_p}) = (2^2+2(1)(-3))\hat{i} + 1^2 \hat{j} + (2(1)(2)-1)\hat{k} = (-2,1,3)
    [/itex]

    And then used the relation in the relevant equations section:
    [itex]
    (\vec{r}-\vec{r_p}) \cdot \nabla f(\vec{r_p}) = 0 \\
    (x-1,y+3,z-2) \cdot (-2,1,3) = 0 \\
    (-2x+2)+(y+3)+(3z-6) = 0 \\
    -2x+y+3z=1
    [/itex]

    This is the first time I've ever done this and I am just going off of a similar example that the solution was given for. It is not exactly the same question so not sure my process is even correct. Would appreciate any help :)
     
    Last edited: Oct 8, 2015
  2. jcsd
  3. Oct 8, 2015 #2

    Mark44

    Staff: Mentor

    Looks good to me.

    As a check you can do the following:
    • Confirm that the normal to the plane is parallel to the gradient at the given point.
    • Confirm that the plane contains the given point.
     
  4. Oct 8, 2015 #3
    Thanks for taking a look, and for the advice on how to check if correct, appreciate it :)
     
  5. Oct 9, 2015 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Mark44 told you that!
    "As a check you can do the following:
    • Confirm that the normal to the plane is parallel to the gradient at the given point.
    • Confirm that the plane contains the given point."
     
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