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Integration with inverse functions

  1. Aug 12, 2011 #1
    1. The problem statement, all variables and given/known data
    let f(x)=(4t^3+4t)dt(between 2 and x)
    if g(x) = f^(-1)(x), then g'(0)=?


    2. Relevant equations



    3. The attempt at a solution
    f'(x) = 4x^3+4x
    annd i already don't know where to go from here.. help?
     
  2. jcsd
  3. Aug 12, 2011 #2
    There's a formula for the derivative of inverses
    http://en.wikipedia.org/wiki/Inverse_functions_and_differentiation" [Broken]

    If you start with f(f-1(x)) = x, differentiate both sides and rearrange and you'll get something like
    b80fffc5e854e0c44c3e6accbfadf7a2.png
     
    Last edited by a moderator: May 5, 2017
  4. Aug 12, 2011 #3
    so
    [f^(-1)(0)]' = 1/[f'(f^(-1)(0))]
    but where do i go from here?
    because i don't know what f^(-1)(0) is...
     
    Last edited by a moderator: May 5, 2017
  5. Aug 12, 2011 #4
    If f(b) = 0, then taking the inverse of both sides gives you f-1(0) = b. Then you apply this to the original function you were given to find f-1(0)
     
  6. Aug 12, 2011 #5
    got it :)
    one other thing, at the beginning you started with f(f^(-1)(x))=x
    ... where did that come from?
     
  7. Aug 12, 2011 #6
    That's the purpose of the inverse functions: the compositions of inverse functions return the input x, f(f-1(x)) = f-1(f(x)) = x. http://en.wikipedia.org/wiki/Inverse_function" [Broken] has a pretty good article with examples.
     
    Last edited by a moderator: May 5, 2017
  8. Aug 12, 2011 #7
    thanks!
     
    Last edited by a moderator: May 5, 2017
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