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

Question

Isabella Ackerman · Accepted Answer

#2/3#.

Let #y=f(x)=-1/(x+2), x in [-1,-1/2]#. By Defn., the Average Rate of Change of #y" over "[-1,-1/2]# #=(f(-1/2)-f(-1))/(-1/2-(-1))=(f(-1/2)-f(-1))/(1/2)=2{f(-1/2)-f(-1)}#. #=2{(-1/(-1/2+2))-(-1/(-1+2))}# #=2{-1/(3/2)+(1/1)}# #=2(-2/3+1)# #=2(1/3)# #=2/3#.

Emilia Aguilar · Answer

undefined To find the average rate of change of $ y = -\frac{1}{x+2} $ over the interval $[-1, -\frac{1}{2}]$, you use the formula:

$$ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} $$

Where:
$ f(a) $ represents the value of the function at the lower bound of the interval,
$ f(b) $ represents the value of the function at the upper bound of the interval,
$ a $ is the lower bound of the interval,
and $ b $ is the upper bound of the interval.

Plugging in the values:

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

$$ \text{Average Rate of Change} = \frac{-\frac{2}{5} - (-1)}{-\frac{1}{2} - (-1)} = \frac{-\frac{2}{5} + 1}{\frac{1}{2}} = \frac{\frac{3}{5}}{\frac{1}{2}} = \frac{3}{5} \times 2 = \frac{6}{5} $$

So, the average rate of change of $ y = -\frac{1}{x+2} $ over $[-1, -\frac{1}{2}]$ is $ \frac{6}{5} $.