Chapter 9.6, Problem 52E

Energy use Energy use per dollar of GDP indexed to 1980 means that energy use for any year is viewed as a percent of the use per dollar of GDP in 1980. The following data show the energy use per dollar of GDP, as a percent, for selected years from 1985 and projected to 2035.

Energy Use per Dollar of GDP

Year	Percent	Year	Percent
1985	83	2015	51
1990	79	2020	45
1995	75	2025	41
2000	67	2030	37
2005	60	2035	34
2010	56

Source: U.S. Department of Energy

These data can be modeled with the function

$E\left(t\right)=0.0039{\left(0.4t+2\right)}^3-0.13{\left(0.4t+2\right)}^2-1.4\left(0.4t+2\right)+91$ where E(t) is the energy use per dollar of GDP (indexed to 1980) and t is the number of years past 1980.

(a) Find E'(t).

(b) Use this model to find and interpret the instantaneous rates of change of energy use per dollar of GDP in 2000 and 2025.

(c) Use the data in the table to find an average rate of change that approximates the 2025 instantaneous rate.