# How do you find the average rate of change of f(x)=x^3-3x+5 over [-1,5]?

Oct 17, 2015

Find the difference in the value of the rate of change, $f ' \left(x\right)$, between the two end points of the interval, then divide that difference by the interval width to get the average rate of change.

#### Explanation:

Given $f \left(x\right) = {x}^{3} - 3 x + 5$
$\textcolor{w h i t e}{\text{XXX}}$Rate of change $= f ' \left(x\right) = 3 {x}^{2} - 3$

$f ' \left(- 1\right) = 3 \left(1\right) - 3 = 0$
$f \left(5\right) = 3 \left(25\right) - 3 = 72$
Difference in $f ' \left(x\right)$ between the two end points: $\Delta {f}_{5 : - 1} \left(x\right) = 72 - 0 = 72$

Width of interval: $\Delta x = 5 - \left(- 1\right) = 6$

Average rate of change: $= \frac{72}{6} = 12$