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Parametric equations from partial derivatives

  1. Oct 18, 2012 #1
    1. The problem statement, all variables and given/known data
    The surface z=f(x,y)=√(9-2x2-y2) and the plane y=1 intersect in a curve. Find parametric equations for the tangent line at (√(2),1,2).


    2. Relevant equations
    Partial derivatives


    3. The attempt at a solution
    Okay, so I'm just trying to work through an example in my textbook, so technically I have the answer, I just want the in between steps that I can't figure out. I know that you take the partial derivative with respect to x and get

    fx(x,y)=0.5(9-2x2-y2)-0.5×(-4x)

    and then you plug in the given point (√(2),1,2) and come out with -√(2). But then it goes to say, 'It follows that this line has direction vector <1,0,-√(2)>'.

    Where did the 1 and 0 come from? And how do I get the parametric equations from the 'direction vector'?
     
  2. jcsd
  3. Oct 18, 2012 #2

    SammyS

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    Hello nindelic. Welcome to PF !
    Find an equation which describes the curve of intersection. This will give z as a function of x along this curve.
     
  4. Oct 18, 2012 #3
    Thank you, SammyS! And okay, I don't know what that means. A big part of my problem I'd say is understanding math speak. If all I have is numbers then I can do about anything, but throw some words in there and I am completely lost. So what exactly does that mean? Like using different coordinates?

    I tried finding the partial derivative with respect to y but that didn't help at all. I tried solving the equation and the partial WRT x for the other variables but that didn't help. I just don't know where the 1 or 0 came from the in the direction vector...
     
  5. Oct 18, 2012 #4

    SammyS

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    The initial statement in your Original Post, is:
    "The surface z=f(x,y)=√(9-2x2-y2) and the plane y=1 intersect in a curve."​
    Can you give the equation describing this curve?
     
  6. Oct 18, 2012 #5
    No, I'm not sure how to find it.
     
  7. Oct 18, 2012 #6

    SammyS

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    Plug y = 1 into the equation of the surface.
     
  8. Oct 19, 2012 #7
    Okay so then the equation of the intersecting curve is:

    z=f(x,y)=√(9-2x^2-1)?

    But how does that help me?
     
  9. Oct 19, 2012 #8

    SammyS

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    Now you can find dz/dx along the curve of intersection?

    What can that tell you?
     
  10. Oct 19, 2012 #9
    It tells you the rate of change of the intersection with respect to x.

    dz/dx= (0.5(8-2x^2)^-0.5)*(-4x)
     
  11. Oct 19, 2012 #10

    SammyS

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    What is this rate when x = 1 ?
     
  12. Oct 20, 2012 #11
    -2/sqrt(6)
     
  13. Oct 20, 2012 #12

    SammyS

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    DUH! ...

    I should have asked, "What is this rate when x = √2 ?".
     
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