To find P'(0), we need to take the derivative of the population formula with respect to time t and evaluate it at t=0:
P'(t) = 0.04P(1−P/20000)
P'(0) = 0.04P(1−P/20000) |t=0
P'(0) = 0.04(1000)(1−1000/20000)
P'(0) = -0.4
This means that at the beginning of 2015, the population was decreasing at a rate of 0.4 individuals per year. More specifically, for every 1 individual in the population, the population was decreasing by 0.4/1000 (or 0.04%) per year.
The rate of change of a population P of an environment is determined by the logistic derivative formula dP/dt= 0.04P(1−P/20000) where t is in years since the beginning of 2015. So P(1) is the population at the beginning of 2016. Suppose P(0) = 1000. (a) Calculate derivative P'(0). Explain what this number means.
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