# How do you find the average rate of change of f(x)=11x^3+11 over [1,3]?

Oct 23, 2015

Average rate of change $= 132$

#### Explanation:

Average rate of change $= \frac{\Delta f ' \left(x\right)}{\Delta x}$

Given
$\textcolor{w h i t e}{\text{XXX}} f \left(x\right) = 11 {x}^{3} + 11$

$f ' \left(x\right) = 33 {x}^{2}$

Over the interval $\left[1 , 3\right]$
$\Delta f ' \left(x\right) = f ' \left(3\right) - f ' \left(1\right)$
$\textcolor{w h i t e}{\text{XXXX}} = 33 \cdot 9 - 33 \cdot 1$
$\textcolor{w h i t e}{\text{XXXX}} = 33 \cdot 8$
$\textcolor{w h i t e}{\text{XXXX}} = 264$
and
$\Delta x = 3 - 1 = 2$

So
the average rate of change $= \frac{264}{2} = 132$

Oct 23, 2015

The average rate of change of $f$ over $\left[a , b\right]$ is defined to be $\frac{f \left(b\right) - f \left(a\right)}{b - a}$

#### Explanation:

So find $\frac{f \left(3\right) - f \left(1\right)}{3 - 1}$

$f \left(3\right) = 11 {\left(3\right)}^{3} + 11$

$f \left(x\right) = 11 {\left(1\right)}^{3} + 11$

Now do the arithmetic.