How do you find the average rate of change of f(x)= -2/(3x+5) over [-1,3]?

Apr 30, 2017

The answer is $= \frac{3}{14}$

Explanation:

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

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

Here, we have

$f \left(x\right) = - \frac{2}{3 x + 5}$

and the interval is $\left[- 1 , 3\right]$

so,

$f \left(3\right) = - \frac{2}{3 \cdot 3 + 5} = - \frac{2}{14} = - \frac{1}{7}$

$f \left(- 1\right) = - \frac{2}{3 \cdot - 1 + 5} = - \frac{2}{2} = - 1$

So,

The average rate of change is

$= \frac{f \left(3\right) - f \left(- 1\right)}{3 - \left(- 1\right)} = \frac{- \frac{1}{7} + 1}{3 + 1} = \frac{6}{7} \cdot \frac{1}{4} = \frac{3}{14}$