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Homework Help: Fundamental Theorem

  1. Jul 13, 2008 #1
    The problem statement, all variables and given/known data
    Question One:
    Find a continuous function f and a number a such that

    [tex]2 + \int_{a}^{x} \frac {f(t)} {t^{6}} \,dt = 6 x^{-1}[/tex]

    Question Two:
    At what value of x does the local max of f(x) occur?
    [tex]f(x) = \int_0^x \frac{ t^2 - 25 }{ 1+\cos^2(t)} dt[/tex]

    The attempt at a solution
    I just need some pointers of where to get started.
    Question One:

    So I used FTC1 on both sides,

    [tex]2 + f(x) / x^{6} = 6x^{-1}[/tex]

    [tex]f(x)= 6x^{5} - 2[/tex]

    I'm not sure how to find a, evaluation theorem?

    Question Two:
    Last edited: Jul 13, 2008
  2. jcsd
  3. Jul 13, 2008 #2
    I don't know about an analytic solution, but the second part of the problem is very feasible numerically. You can solve in Mathematica in only a few lines by turning it into a minimization problem.
  4. Jul 13, 2008 #3
    Well, for Question one:

    Can anyone confirm that [tex]f(x) = 6x^{5}[/tex] and a = 2.

    I'm pretty sure that a = 2 since,

    F(x) - F(a) = [ [tex]6x^{5} / x^{6}[/tex] ] - [ 2 ] = [tex]6x^{-1} - 2[/tex]
  5. Jul 13, 2008 #4

    matt grime

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    Homework Helper

    How does the FTC just let you drop an integral sign out like that? (In 1.)
  6. Jul 14, 2008 #5


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    No, that is not correct. You have differentiated the left side of the equation but not the right.

    Once you have found the function, put it into the integeral.
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