Chapter 3.9, Problem 99E

 World population (part 2) The relative growth rate r of a function f measures the rate of change of the function compared to its value at a particular point. It is computed as r(t) = f′(t)/f(t).

  a.    Confirm that the relative growth rate in 1999 (t = 0) for the logistic model in Exercise 98 is r(0) = P′(0)/P(0) = 0.015. This means the world’s population was growing at 1.5% per year in 1999.

  b.    Compute the relative growth rate of the world’s population in 2010 and 2020. What appears to be happening to the relative growth rate as time increases?

  c.    Evaluate $\underset{t\to \infty }{\lim }\left(t\right)=\underset{t\to \infty }{\lim }\frac{P^{\prime }\left(t\right)}{P\left(t\right)}$ where P(t) is the logistic growth function from Exercise 98. What does your answer say about populations that follow a logistic growth pattern?