Calculus

The rate of growth of the profit is approximated by P'(x)=xe^(-x^2) where x represents time measured in years. The total profit in the third year that the new technology is in operation is $10,000.

1.Find the total profit function

2. what happens to the total amount of profit in the long run?

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  1. dp/dt = t e^-t^2

    if p = c - .5e^-t^2
    p' = t e^-t^2 =

    when t = 3
    10,000 = c - .5e^-9
    c = 10,000 + .5 e^-9
    c = 10,000 + .0000617
    so
    p = 10,000 - .5/ e^(t^2) essentially

    It becomes 10,000 as t---> oo

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