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.
To calculate the derivative P'(0), we need to differentiate the given population function with respect to t.
Given:
dP/dt = 0.04P(1−P/20000)
Now, we need to find dP/dt at t = 0, which is represented as P'(0).
To find this, we substitute t = 0 into the derivative expression:
dP/dt = 0.04P(1−P/20000)
Therefore, substituting t = 0,
P'(0) = 0.04(1000)(1 - 1000/20000)
Simplifying the equation:
P'(0) = 0.04(1000)(1 - 0.05)
P'(0) = 0.04(1000)(0.95)
P'(0) = 38
Hence, P'(0) = 38.
Explanation:
The derivative P'(0) in this context represents the rate of change of the population at t = 0, which is the beginning of the year 2015. In this case, since P'(0) = 38, it means that at the beginning of 2015, the population was increasing at a rate of 38 individuals per year.
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.