find the average rate of change of $f(x)=\frac{1}{x}$ over the interval $-5,3$. write your answer as an…

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, $a=-5$, $b = 3$, and $f(x)=\frac{1}{x}$.

Step2: Calculate $f(a)$ and $f(b)$

$f(-5)=\frac{1}{-5}=-\frac{1}{5}$, $f(3)=\frac{1}{3}$.

Step3: Substitute into the formula

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

Step4: Add the fractions in the numerator

$\frac{1}{3}+\frac{1}{5}=\frac{5 + 3}{15}=\frac{8}{15}$.

Step5: Divide by 8

$\frac{\frac{8}{15}}{8}=\frac{8}{15}\times\frac{1}{8}=\frac{1}{15}\approx0.1$.

Answer:

$\frac{1}{15}$ (or approximately $0.1$)