I need help to Find the total profit function P(x).

Question

The rate of growth of profit (in millions) from an invention is approximated by P'(x)=xe^-x2, where x represents time in years. The total profit in year 1 that the invention is in operation is $20,000. Find the total profit function P(x).

Answer 1

We need to integrate P'(x). The derivative of e^-x^2 is -2xe^-x^2, so the integral of P' is P(x)=-(e^-x^2)/2+k, where k is constant of integration. The total profit in the first year (x=1) is $20000 which is 20000/1000000=$M0.02, where $M1 is one million dollars. P(1)=-(e^-1)/2+k. P(1)=0.02=-0.18394+k. So k=0.20394 and P(x)=0.2039-(e^(-x^2))/2.