Question: A projection of population trends suggests that t years from now, the population of a certain community will be \(P\left( t \right)=-2{{t}^{3}}+18{{t}^{2}}+96t+74\) thousand people.

(a) Find the rate of change of the population \(R\left( t \right)\) with respect to time

(b) At what rate does the population growth rate \(R\left( t \right)\) change with respect to time?

(c) Use increments to estimate how much the growth rate of the population changes during the first month of the fourth year.

(d) What is the actual change in the growth rate during this period?