find the average rate of change of the function over the given interval. y = \\sqrt{3x + 1}; between x = 1…

Explanation:

Step1: Recall the 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, $a = 1$, $b = 5$, and $y=f(x)=\sqrt{3x + 1}$.

Step2: Calculate $f(5)$

Substitute $x = 5$ into $y=\sqrt{3x + 1}$: $f(5)=\sqrt{3\times5+1}=\sqrt{15 + 1}=\sqrt{16}=4$.

Step3: Calculate $f(1)$

Substitute $x = 1$ into $y=\sqrt{3x + 1}$: $f(1)=\sqrt{3\times1+1}=\sqrt{3 + 1}=\sqrt{4}=2$.

Step4: Calculate the average rate of change

Using the formula $\frac{f(b)-f(a)}{b - a}$, we have $\frac{f(5)-f(1)}{5 - 1}=\frac{4 - 2}{4}=\frac{2}{4}=\frac{1}{2}$.

Answer:

$\frac{1}{2}$