Logistic Model Derivative Analysis
A research team collected population data of a species at different time intervals. The data collected is as follows: Time (years) Population 0 500 2 800 4 1200 6 1500 8 1700 Using this dataset: (a) Estimate the instantaneous rate of change of the population at t = 4 years using a central difference method. […]
Part a
Using the provided data, estimate the instantaneous rate of change of the population at t = 4 years using the central difference method with h = 2 years.
Part b
Assuming the logistic model \(P(t)=\frac{2000}{1+Ae^{-rt}}\) applies, use your derivative estimate from part (a) to refine the estimate for the growth rate \(r\) at t = 4 years. Hint: First, use the initial condition P(0) = 500 to find the value of A. Then, differentiate the logistic model to relate \(P'(t)\) with \(r\).
Graphical Slope Approximation vs. Derivative Confirmation
A graph of a function f is given. At x = 2, approximate the slope of the tangent by using two points on the curve. The graph suggests that f(1.8) = 3.6 and f(2.2) = 4.4. \n\nProceed as follows:\n(a) Use the two points to estimate the slope of f at x = 2.\n(b) If f(x) […]