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Math Help - Improper Integral (have answer, need clarification)

  1. #1
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    Improper Integral (have answer, need clarification)

    This is from a an AP Calculus BC 2006 exam.



    The answer is


    My question is, why start at t = 2? Why add the initial value?
    I wrote my improper integral from 0 to infinity.
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  2. #2
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    Quote Originally Posted by DiggOlive View Post
    This is from a an AP Calculus BC 2006 exam.



    The answer is


    My question is, why start at t = 2? Why add the initial value?
    I wrote my improper integral from 0 to infinity.
    y(t)-y(2)=\int_2^t \frac{dy}{dt}dt and y(2)=-3
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  3. #3
    Senior Member Pinkk's Avatar
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    Well, you are only given the derivatives of x(t) and y(t), so there are an infinite number of possible functions with these derivatives. So you need an initial value condition to determine which function you are dealing with. Usually a problem gives you the initial function of y(0)=k, but this case you are given the initial condition is y(2)=-3. So to find the horizontal asymptote, you have to find the limit of the function (which will be the integral of the derivative) from t=2, because we know at y(2)=-3. If you weren't given an initial condition, there'd be an infinite number of functions with these derivatives, and therefore there'd be an infinite amount of respective horizontal asymptotes.
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