Question

given (f(x)=\frac{1}{x + 5}), find the average rate of change of (f(x)) on the interval (3,3 + h). your answer will be an expression involving (h).

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ on the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a = 3$, $b=3 + h$, and $f(x)=\frac{1}{x + 5}$.

Step2: Find $f(3 + h)$ and $f(3)$

$f(3 + h)=\frac{1}{(3 + h)+5}=\frac{1}{h + 8}$, and $f(3)=\frac{1}{3+5}=\frac{1}{8}$.

Step3: Calculate the average rate of change

The average rate of change is $\frac{f(3 + h)-f(3)}{(3 + h)-3}=\frac{\frac{1}{h + 8}-\frac{1}{8}}{h}$.
First, find a common denominator for the numerator: $\frac{1}{h + 8}-\frac{1}{8}=\frac{8-(h + 8)}{8(h + 8)}=\frac{8 - h - 8}{8(h + 8)}=\frac{-h}{8(h + 8)}$.
Then, $\frac{\frac{-h}{8(h + 8)}}{h}=\frac{-h}{8h(h + 8)}$.
Cancel out the $h$ terms (assuming $h
eq0$), we get $\frac{-1}{8(h + 8)}$.

Answer:

$\frac{-1}{8(h + 8)}$