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calculus

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assume that during the first 3 minutes after a foreign substance is introduced into the blood,the rate R'(t) at which new antibodies are produced(in thousands of antibodies per minute) is given by R'(t)=t/(t^2+1) where t is in minutes.find the total quantity of new antibodies in the blood at the end of 3 minutes.

  • calculus - ,

    dR/dt = t/(t^2+1)
    Assume R = 0 at t = 0
    R = integral of t dt/(t^2 + 1)
    = (1/2)ln(t^2 + 1)
    At t = 3 minutes, R = (1/2) ln(10) = 1.15
    Since R is measured in thousands, the answer is 1150.

  • calculus - ,

    i don't understand this part:dt/(t^2 + 1)
    = (1/2)ln(t^2 + 1)

  • calculus - ,

    It's not dt/(t^2 + 1)
    It is t dt/(t^2 + 1)

    That is just (1/2) du/u if u=t^2+1. That's where the log comes from

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