College Algebra, Chapter 3, Review Exercises, Section Review Exercises, Problem 62

Suppose that $f(x) = 8 - 3x$. (a) Determine the average rate of change of $f$ between $x = 0$ and $x = 2$ and the average rate of change of $f$ between $x = 15$ and $x = 50$ (b) Were the two average rates of change that you found in part(a) the same? Explain why or why not.

Recall that the formula for averate rate is...
$\displaystyle \frac{f(b) - f(a)}{b - a}$
Thus,

$\begin{equation} \begin{aligned} \text{a.) } \frac{f(2) - f(0)}{2 - 0} &= \frac{8-3(2) - [8-3(0)]}{2}\\ \\ &= \frac{8-6-8}{2}\\ \\ &= -3 \\ \\ \\ \frac{f(50) - f(15)}{50-15} &= \frac{8-3(50) - [ 8 - 3 (15) ] }{35}\\ \\ &= \frac{8-150-8+45}{35}\\ \\ &= \frac{-105}{35}\\ \\ &= -3 \end{aligned} \end{equation}$

b.) The average rates are the same in part(a). This shows that the distance covered between the two pair of points have equal slopes.