increasing and concave down, decreasing and concave up, decreasing and concave down.-We can solve for P, if we wish.dP/dt=0.......
P = (500 +/- sqrt(130000)) / 6
P = (500 +/- 100 * sqrt(13) / 6
P = (100/6) * (5 +/- sqrt(13))
P = (50/3) * (5 +/- sqrt(13))
Since the leading coefficient of the second derivative is negative, then the second derivative will be going from Q3 to Q4
-inf < P < (50/3) * (5 - sqrt(13))
(50/3) * (5 - sqrt(13)) < P < (50/3) * (5 + sqrt(13))
(50/3) * (5 + sqrt(13)) < P < inf
The first and third domains, the population graph is concave down
The second domain, the population is concave up. Match that up with the domains of the first derivative of the equation and see when the population is increasing and concave up, increasing and concave down, decreasing and concave up, decreasing and concave down.
We can solve for P, if we wish.
dP/dt=0.3(1-(P/200))((P/50)-…
dP/dt = (3/10) * (1/200) * (1/50) * (200 - P) * (P - 50) * P
dP / ((200 - P) * (P - 50) * P) = (3 / 100000) * dt
To integrate the left-hand side, we'll need to decompose the fraction.
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1 / ((200 - P) * (P - 50) * P) =>
A / (200 - P) + B / (P - 50) + C / P
A * (P - 50) * P + B * (200 - P) * P + C * (200 - P) * (P - 50) = 0P^2 + 0P + 1
A * (P^2 - 50P) + B * (200P - P^2) + C * (200P - P^2 + 50P - 10000) = 0P^2 + 0P + 1
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