what is the average rate of change of the function over the interval x = 0 to x = 8? f(x) = (3x - 1)/(3x +…

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, $a = 0$, $b = 8$, and $f(x)=\frac{3x - 1}{3x+3}$.

Step2: Calculate $f(0)$

Substitute $x = 0$ into $f(x)$: $f(0)=\frac{3\times0 - 1}{3\times0+3}=\frac{-1}{3}=-\frac{1}{3}$.

Step3: Calculate $f(8)$

Substitute $x = 8$ into $f(x)$: $f(8)=\frac{3\times8 - 1}{3\times8+3}=\frac{24 - 1}{24 + 3}=\frac{23}{27}$.

Step4: Calculate the average rate of change

$\frac{f(8)-f(0)}{8 - 0}=\frac{\frac{23}{27}-(-\frac{1}{3})}{8}=\frac{\frac{23}{27}+\frac{9}{27}}{8}=\frac{\frac{23 + 9}{27}}{8}=\frac{\frac{32}{27}}{8}=\frac{32}{27}\times\frac{1}{8}=\frac{4}{27}$.

Answer:

$\frac{4}{27}$