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Homework Help: Logarithmic Antiderivative

  1. Apr 4, 2008 #1
    Find f. (x>0)
    f''(x) = x^(-2)
    x > 0
    f(1) = 0
    f(8) = 0

    Alright, everything was fine until one point....okay. First derivative:
    f''(x)=x[tex]^{-2}[/tex]
    f'(x)=-x[tex]^{-1}[/tex]+C

    Now here's where i'm not sure: in the next step, it will be x raised to zero, which is one. times 1/0, which is zero, but...my physics professor informed me that it is actually, the natural log of the absolute value of x.

    f(x)=-ln|x|+Cx+D

    Now, use the above input & output values of f(x) to divulge the constants.

    f(1)=C+D=0
    f(8)=-ln|8|+8c+D=0

    C+D=0
    8c+D=ln|8|
    8c-c=ln|8|
    7c=ln|8|

    C=[tex]\frac{ln|8|}{7}[/tex]


    Now, plug in:

    f(x)=-ln|x|+[tex]\frac{x*ln|8|}{7}[/tex]-[tex]\frac{ln|8|}{7}[/tex]

    Does this look right? I'm not quite sure about the x in the second term in that equation. I only have one more try on my online homework thing. :rofl: Thank you!
     
    Last edited: Apr 4, 2008
  2. jcsd
  3. Apr 4, 2008 #2

    Hootenanny

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    Looks good to me :approve:
     
  4. Apr 4, 2008 #3

    HallsofIvy

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    Since 8 is a positive number, |8|= 8. Your "online homework thing" may object to |8| rather than 8. (That's why I hate those things!)
     
    Last edited by a moderator: Apr 4, 2008
  5. Apr 4, 2008 #4
    Thank you! The answer was correct, and the checker decided to take my answer. What a surprise.
     
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