given $f(x)=\frac{1}{x + 10}$, find the average rate of change of $f(x)$ on the interval $4,4 + h$. your…

Explanation:

Step1: Recall the formula for average rate of change

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 = 4) and (b=4 + h), and (f(x)=\frac{1}{x + 10}).

Step2: Calculate (f(4 + h)) and (f(4))

• (f(4 + h)=\frac{1}{(4 + h)+ 10}=\frac{1}{h + 14})
• (f(4)=\frac{1}{4 + 10}=\frac{1}{14})

Step3: Substitute into the average - rate - of - change formula

[ \begin{align*} \frac{f(4 + h)-f(4)}{(4 + h)-4}&=\frac{\frac{1}{h + 14}-\frac{1}{14}}{h}\ &=\frac{\frac{14-(h + 14)}{14(h + 14)}}{h}\ &=\frac{\frac{14 - h-14}{14(h + 14)}}{h}\ &=\frac{\frac{-h}{14(h + 14)}}{h}\ &=\frac{-h}{14(h + 14)}\times\frac{1}{h}\ &=-\frac{1}{14(h + 14)} \end{align*} ]

Answer:

(-\frac{1}{14(h + 14)})