## Calculus: Early Transcendentals 8th Edition

(a) The average rate of change from 1990 to 2005 is $1169.6th/da.yr$ (b) The instantaneous rate of change in 2000 is $1397.8th/da.yr$
(a) The amount of oil in unit of thousands of barrels per day in 1990 is $66,533$ and in 2005 is $84,077$. So, the average rate of change from 1990 to 2005 is $$r=\frac{84,077-66,533}{2005-1990}=1,169.6\hspace{0.1cm}(th/da.yr)$$ The unit of the numerator is $th/da$ (thousands of barrels per day) and that of the denominator is $yr$ (year). Therefore, the unit of the average rate of change is $th/da.yr$ (b) The amount of oil in unit of thousands of barrels per day in 1995 is $70,099$, in 2000 is $76,784$ and in 2005 is $84,077$. So, the average rate of change (i) from 1995 to 2000: $\frac{76,784-70,099}{2000-1995}=1337(th/da.yr)$ (ii) from 2000 to 2005: $\frac{84,077-76,784}{2005-2000}=1458.6(th/da.yr)$ The instantaneous rate of change in 2000 would be the average of the average rates of change from 1995 to 2000 and from 2000 to 2005: $$\frac{1337+1458.6}{2}=1397.8(th/da.yr)$$ Taking the average does not change the unit. That means the unit here is still $th/da.yr$