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Trouble computing ∂^2 f/∂x^2 (1,1)

  1. Feb 24, 2012 #1

    s3a

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    1. The problem statement, all variables and given/known data
    Question:
    "Given that the surface x^4 * y^7 + y^6 * z^8 + z^7 * x^9 + 4xyz = 7
    has the equation z = f(x,y) in a neighborhood of the point (1,1,1) with f(x,y) differentiable, find the derivatives.

    Find:
    a) ∂f/∂x(1,1)
    b) ∂f/∂y(1,1)
    c) ∂^2 f/∂x^2"

    Answers:
    ∂f/∂x(1,1) = -17/19 = -0.894736842105263
    ∂f/∂y(1,1) = -17/19 = -0.894736842105263
    ∂^2 f/∂x^2 (1,1) = -2.2399766729844


    2. Relevant equations
    Just taking the derivative of
    4x^3 * y^7 + 9x^8 * z^7 + 4yz + ∂f/∂x(8z^7 * y^6 + 7z^6 + 4yz).
    I also know that ∂f/∂x(1,1) = -17/19.


    3. The attempt at a solution
    I successfully get every single part of this question except the ∂^2 f/∂x^2 part.

    Instead of isolating for ∂f/∂x and trying to differentiate that again with respect to x (which seems very difficult, if not impossible, to do by hand), I just implicitly differentiate for the second time treating ∂f/∂x as a function. I then just plug in -17/19 for ∂f/∂x(1,1) and plug in the point (1,1,1) and get the wrong answer. I tried computing ∂^2 f/∂x^2 several times and keep getting it wrong so any help would be greatly appreciated.

    Thanks in advance!
     

    Attached Files:

  2. jcsd
  3. Feb 24, 2012 #2

    SammyS

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    Here's the Latex of the equation of the surface: [itex]x^4 \, y^7 + y^6 \, z^8 + z^7 \, x^9 + 4xyz = 7[/itex]

    Here's Latex for your result for the first partial derivative:
    [itex]\displaystyle
    4x^3 \, y^7 + 9x^8 \, z^7 + 4yz + \frac{\partial f}{\partial x}(8z^7 \, y^6 + 7z^6 + 4yz)=0[/itex]​

    I got something different:
    [itex]\displaystyle
    4x^3 \, y^7 + 9x^8 \, z^7 + 4yz + \frac{\partial f}{\partial x}(8z^7 \, y^6 + 7z^6\,x^9 + 4yz)=0[/itex]

    Here's the image of your work, so it's easier to check out.
    attachment.php?attachmentid=44347&d=1330124781.jpg

    That's a correct method.
     
  4. Feb 25, 2012 #3

    s3a

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    I kept getting it wrong so many times but I finally got it right. What I had shown here is wrong for the first differentiation with respect to x and so was yours.

    Here is my work if you can read the handwriting and care.

    Thank you!
     

    Attached Files:

  5. Feb 25, 2012 #4

    SammyS

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    It took me a long time to see what I had wrong.

    Should be 4xy, not 4yz in the parentheses.

    Here's the corrected first derivative.

    [itex]\displaystyle
    4x^3 \, y^7 + 9x^8 \, z^7 + 4yz + \frac{\partial f}{\partial x}(8z^7 \, y^6 + 7z^6\,x^9 + 4xy)=0[/itex]
     
  6. Feb 25, 2012 #5

    s3a

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    Yeah, I saw it. :smile: Also, sorry, I should have told you and spared your wasteful search.
     
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