Question

question 6
given $f(x)=\frac{1}{x + 2}$, find the average rate of change of $f(x)$ on the interval $5,5 + h$.
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, $a = 5$, $b=5 + h$, and $f(x)=\frac{1}{x + 2}$.

Step2: Calculate $f(5 + h)$ and $f(5)$

$f(5 + h)=\frac{1}{(5 + h)+2}=\frac{1}{h + 7}$, and $f(5)=\frac{1}{5+2}=\frac{1}{7}$.

Step3: Substitute into the formula

The average rate of change is $\frac{f(5 + h)-f(5)}{(5 + h)-5}=\frac{\frac{1}{h + 7}-\frac{1}{7}}{h}$.

Step4: Simplify the numerator

$\frac{1}{h + 7}-\frac{1}{7}=\frac{7-(h + 7)}{7(h + 7)}=\frac{7 - h - 7}{7(h + 7)}=\frac{-h}{7(h + 7)}$.

Step5: Divide by $h$

$\frac{\frac{-h}{7(h + 7)}}{h}=\frac{-h}{7(h + 7)}\cdot\frac{1}{h}=-\frac{1}{7(h + 7)}$.

Answer:

$-\frac{1}{7(h + 7)}$