Chapter 1.4, Problem 40E

a.

To determine

To Find: The average rate of change in spending from 2012 to 2017.

Answer to Problem 40E

The average rate of change in spending from 2012 to 2017 is:

  $-10.6\text{ billion/year}$ .

Explanation of Solution

Given information:

Table for National Defense Spending:

  $\begin{array}{cc}\text{Year}& \text{National Defense Spending }\left(\text{\$billion}\right)\\ 2012& 678\\ 2013& 633\\ 2014& 622\\ 2015& 598\\ 2016& 624\\ 2017& 656\end{array}$

Concept used:

The average rate of change of a function $f\left(x\right)$ over the interval $\left[a,b\right]$ is calculated as:

  $\frac{f\left(b\right)-f\left(a\right)}{b-a}$ .

Calculation:

The average rate of change in spending from 2012 to 2017 is:

  $\begin{array}{c}\frac{f\left(b\right)-f\left(a\right)}{b-a}=\frac{f\left(2017\right)-f\left(2012\right)}{2017-2012}\\ =\frac{625-678}{2017-2012}\\ =-\frac{53}{5}\\ =-10.6\end{array}$

Conclusion:

Hence the average rate of change in spending from 2012 to 2017 is: $-10.6\text{ billion/year}$ .

b.

To determine

To Find: The average rate of change in spending from 2012 to 2015.

Answer to Problem 40E

The average rate of change in spending from 2012 to 2015 is: $-26.667\text{ billion/year}$ .

Explanation of Solution

Given information:

Table for National Defense Spending:

  $\begin{array}{cc}\text{Year}& \text{National Defense Spending }\left(\text{\$billion}\right)\\ 2012& 678\\ 2013& 633\\ 2014& 622\\ 2015& 598\\ 2016& 624\\ 2017& 656\end{array}$

Concept used:

The average rate of change of a function $f\left(x\right)$ over the interval $\left[a,b\right]$ is calculated as:

  $\frac{f\left(b\right)-f\left(a\right)}{b-a}$ .

Calculation:

The average rate of change in spending from 2012 to 2015 is:

  $\begin{array}{c}\frac{f\left(b\right)-f\left(a\right)}{b-a}=\frac{f\left(2015\right)-f\left(2012\right)}{2015-2012}\\ =\frac{598-678}{2015-2012}\\ =-\frac{80}{3}\\ =-26.667\end{array}$

Conclusion:

Hence the average rate of change in spending from 2012 to 2017 is: $-26.667\text{ billion/year}$ .

c.

To determine

To Find: The average rate of change in spending from 2015 to 2017.

Answer to Problem 40E

The average rate of change in spending from 2015 to 2017 is: $-13.5\text{ billion/year}$ .

Explanation of Solution

Given information:

Table for National Defense Spending:

  $\begin{array}{cc}\text{Year}& \text{National Defense Spending }\left(\text{\$billion}\right)\\ 2012& 678\\ 2013& 633\\ 2014& 622\\ 2015& 598\\ 2016& 624\\ 2017& 656\end{array}$

Concept used:

The average rate of change of a function $f\left(x\right)$ over the interval $\left[a,b\right]$ is calculated as:

  $\frac{f\left(b\right)-f\left(a\right)}{b-a}$ .

Calculation:

The average rate of change in spending from 2015 to 2017 is:

  $\begin{array}{c}\frac{f\left(b\right)-f\left(a\right)}{b-a}=\frac{f\left(2017\right)-f\left(2015\right)}{2017-2015}\\ =\frac{625-598}{2017-2015}\\ =-\frac{27}{2}\\ =-13.5\end{array}$

Conclusion:

Hence the average rate of change in spending from 2015 to 2017 is: $-13.5\text{ billion/year}$ .

d.

To determine

To Explain: why it is difficult to predict about the rate of change of national defense spending on the basis of the data given in the table/

Answer to Problem 40E

There is a huge difference in the average rate of change on the basis of the table given. Hence it is difficult to predict the rate of change of national defense using the table provided.

Explanation of Solution

Given information:

Table for National Defense Spending:

  $\begin{array}{cc}\text{Year}& \text{National Defense Spending }\left(\text{\$billion}\right)\\ 2012& 678\\ 2013& 633\\ 2014& 622\\ 2015& 598\\ 2016& 624\\ 2017& 656\end{array}$

Concept used:

The average rate of change of a function $f\left(x\right)$ over the interval $\left[a,b\right]$ is calculated as:

  $\frac{f\left(b\right)-f\left(a\right)}{b-a}$ .

Calculation:

From part (a), part (b) and part (c) it can be concluded that the rate of change of national defense spending differs in every situation.

The average rate of change in spending from 2012 to 2017 is: $-10.6\text{ billion/year}$ .

The average rate of change in spending from 2012 to 2015 is: $-26.667\text{ billion/year}$ .

The average rate of change in spending from 2015 to 2017 is: $-13.5\text{ billion/year}$ .

From above three cases there is a huge difference in the average rate of change. Hence it is difficult to predict about the rate of change of national defense on the basis of the given table.