Question

find the average rate of change of the function f(x) = 1/(x - 7) as x changes from x = - 1 to x = 2. the average rate of change is . (type an integer or a simplified fraction.)

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ from $x = a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a=-1$, $b = 2$, and $f(x)=\frac{1}{x - 7}$.

Step2: Calculate $f(-1)$ and $f(2)$

$f(-1)=\frac{1}{-1 - 7}=-\frac{1}{8}$, $f(2)=\frac{1}{2 - 7}=-\frac{1}{5}$.

Step3: Compute the average rate of change

$\frac{f(2)-f(-1)}{2-(-1)}=\frac{-\frac{1}{5}-(-\frac{1}{8})}{3}=\frac{-\frac{1}{5}+\frac{1}{8}}{3}=\frac{-\frac{8 - 5}{40}}{3}=\frac{-\frac{3}{40}}{3}=-\frac{1}{40}$.

Answer:

$-\frac{1}{40}$