Heres my problem,
A biologist determines that, t hours after a bacterial colony was established, the population of bacteria in the colony is changing at a rate given by
$P′(t)=2e^t/(1+e^t)$
million bacteria per hour, $0\le t\le7.$
If the bacterial colony started with a population of 1 million, how many bacteria are present in the colony after the 7-hour experiment? Round your answer to 2 decimal places.
I tried to solve the problem as 1000000+integration from 0 to 7 of the given equation but it didnt work. any advice or help?
Best Answer
First solution $$P(t)=P(t_0)+P'(t_0)(t-t_0)+R_2(x)$$ where $$R_2(x)=\frac{1}{2}\int_{x_0}^{x}f''(t)(x-t)^2dt$$ Second solution $$\int \frac{2e^t}{1+e^t}dt=2\ln(1+e^t)+c$$ in the other words $$P(7)=P(0)+2\ln(1+e^7)-2\ln(1+e^0)$$