Suppose that a population y grows according to the logistic model given by formula: y=(L)/1+Ae^-kt . a. At what rate is y increasing at time t=0 ? b. In words, describe how the rate of growth of y varies with time. c. At what time is the population growing most rapidly?
"y=\\frac{l}{1+Ae^{-kt}}"
a) "\\frac{dy}{dt}=\\frac{-lAe^{-kt}(-k)}{(1+Ae^{-kt})^2}"
"=\\frac{kAle^{-kt}}{(1+Ae^{-kt})^2}"
"y'(t=0)=\\frac{kAl}{1+A)^2}"
At time t y is increasing at a rate "=\\frac{kAl}{1+A)^2}"
b) the rate of growth of y increases as t increases this is because it is always positive.
c) "y''=\\frac{(1+Ae^{-kt})^2(-k^2lAe^{-kt})-kAle^{-kt}\\cdot2(1+Ae^{-kt})(-kA)}{(1+Ae^{-kt})^4}=0"
"\\implies (1+Ae^{-kt})(-1)+2=0"
"\\implies 1+Ae^{-kt}=2"
"e^{-kt}=\\frac{1}{A}"
"t=\\frac{ln\\frac{1}{A}}{-k}"
The population is growing most rapidly at time, "t=\\frac{ln\\frac{1}{A}}{-k}"
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