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Quadratic function

  1. Feb 23, 2005 #1
    Let [tex]f(x)[/tex] be a quadratic function such that [tex]f(0) = -4[/tex] and

    [tex]\frac{f(x)}{x^2(x-5)^8}dx[/tex]
    is a ration function.

    Determine the value of [tex]f'(0)[/tex].

    [tex]f'(0)=______[/tex]

    i dont really have a clue on how to do this. I can only think of integrating the function then find the derivative of it and plug in 0 for the x's, which doesnt seem to be correct.
     
  2. jcsd
  3. Feb 23, 2005 #2

    xanthym

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    Because the integrand's numerator is a polynomial of lesser order than the denominator, we can use partial fraction decomposition to obtain:
    f(x)/{(x^2)(x-5)^8} =
    = a_1/x + a_2/x^2 + b_1/(x - 5) + b_2/(x - 5)^2 + b_3/(x - 5)^3 + ... + b_8/(x - 5)^8

    Since the integral is given to be purely rational, {a_1=0} and {b_1=0} since otherwise LOG terms would result. Thus, adding remaining terms on the right with common denominator of {(x^2)(x-5)^8}, we can equate f(x) with the numerator:
    f(x) = {a_2*(x-5)^8} + {b_2*(x^2)*(x-5)^6} + {b_3*(x^2)*(x-5)^5} + ... + {b_8*(x^2)}

    Because it's given that f(0)=(-4), we have:
    f(0) = (-4) = {a_2*((0) - 5)^8} + 0 + 0 + ... + 0
    (-4) = a_2*(5^8)
    a_2 = (-4)/(5^8)

    Furthermore, f'(x) will have the form:
    f'(x) = (8)*(a_2)*(x-5)^7 + {terms involving either (x) or (x^2)}
    so that substituting x=(0) and a_2={(-4)/(5^8)} from above:
    f'(0) = (8)*{(-4)/(5^8)}*{(0) - 5)^7} + 0 + 0 + .... + 0
    f'(0) = (8)*(4)*(5^7)/(5^8)
    f'(0) = (32/5)


    ~~
     
    Last edited: Feb 23, 2005
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