what is the average rate of change of the function over the interval x = 0 to x = 8?\n$f(x)=\frac{2x - 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 $f(x)=\frac{2x - 1}{3x+5}$.

Step2: Calculate $f(0)$

Substitute $x = 0$ into $f(x)$: $f(0)=\frac{2\times0 - 1}{3\times0+5}=\frac{- 1}{5}=-\frac{1}{5}$

Step3: Calculate $f(8)$

Substitute $x = 8$ into $f(x)$: $f(8)=\frac{2\times8 - 1}{3\times8+5}=\frac{16 - 1}{24 + 5}=\frac{15}{29}$

Step4: Calculate the average rate of change

$\frac{f(8)-f(0)}{8 - 0}=\frac{\frac{15}{29}-\left(-\frac{1}{5}\right)}{8}=\frac{\frac{15}{29}+\frac{1}{5}}{8}=\frac{\frac{15\times5+29}{29\times5}}{8}=\frac{\frac{75 + 29}{145}}{8}=\frac{\frac{104}{145}}{8}=\frac{104}{145\times8}=\frac{13}{145}$

Answer:

$\frac{13}{145}$