Question

find the average rate of change of $f(x)=\frac{-7}{x}$ over the interval $-14, -7$. write your answer as an integer, fraction, or decimal rounded to the nearest tenth. simplify any fractions.

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a=-14$, $b = - 7$, and $f(x)=\frac{-7}{x}$.

Step2: Calculate $f(-7)$ and $f(-14)$

$f(-7)=\frac{-7}{-7}=1$; $f(-14)=\frac{-7}{-14}=\frac{1}{2}$.

Step3: Substitute into the formula

$\frac{f(-7)-f(-14)}{-7-(-14)}=\frac{1-\frac{1}{2}}{-7 + 14}$.
First, simplify the numerator: $1-\frac{1}{2}=\frac{2 - 1}{2}=\frac{1}{2}$.
The denominator is $-7 + 14=7$.
So, $\frac{\frac{1}{2}}{7}=\frac{1}{2}\times\frac{1}{7}=\frac{1}{14}\approx0.1$.

Answer:

$\frac{1}{14}$ (or approximately $0.1$)