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Integrals and Inverse Functions

  1. Feb 25, 2009 #1
    1. The problem statement, all variables and given/known data
    Set f(x)=[tex]\int^{2x}_{1}[/tex][tex]\sqrt{16 + t^{4}}[/tex]dt.
    A. Show that f has an inverse.
    B. Find ([tex]f^{-1}[/tex])'(0).


    2. Relevant equations
    ([tex]f^{-1}[/tex])'(x)=1/(f'([tex]f^{-1}[/tex](x)))


    3. The attempt at a solution
    A. f'(x)=[tex]\sqrt{16 + t^{4}}[/tex] >0, so f is always increasing, hence one-to-one. By definition of inverse functions, f would have an inverse.

    B. ([tex]f^{-1}[/tex])'(0)=1/(f'([tex]f^{-1}[/tex](0)))

    I'm not sure where to go from here for part B. Help?
     
  2. jcsd
  3. Feb 25, 2009 #2
    I think maybe you can find the inverse of the function firstly,then subtitute x by 0 ? I am also focusing on part B
     
  4. Feb 25, 2009 #3

    Hurkyl

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    (Solving f(x) = 0 should be easy....)

    You did forget something, though: to have an inverse, a function must be one-to-one and onto. So, for example, you would need to show that [tex]\lim_{x \rightarrow +\infty} f(x) = +\infty[/itex], and similarly for [itex]-\infty[/itex].
     
  5. Feb 25, 2009 #4
    Thanks, I think I've got it. I was overlooking some details.
     
  6. Feb 26, 2009 #5

    HallsofIvy

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    It is not necessary to actually determine f or f-1. Since f is defined by an integral, the "fundamental theorem of calculus" gives you its derivative.
     
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