Question

what is the average rate of change of $g(x)=\frac{2}{x + 3}$ over the interval $1leq xleq3$?

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = g(x)$ over the interval $[a,b]$ is $\frac{g(b)-g(a)}{b - a}$. Here, $a = 1$, $b = 3$, and $g(x)=\frac{2}{x + 3}$.

Step2: Calculate $g(3)$

Substitute $x = 3$ into $g(x)$: $g(3)=\frac{2}{3+3}=\frac{2}{6}=\frac{1}{3}$.

Step3: Calculate $g(1)$

Substitute $x = 1$ into $g(x)$: $g(1)=\frac{2}{1 + 3}=\frac{2}{4}=\frac{1}{2}$.

Step4: Calculate the average rate of change

$\frac{g(3)-g(1)}{3 - 1}=\frac{\frac{1}{3}-\frac{1}{2}}{2}$. First, find a common - denominator for the numerator: $\frac{1}{3}-\frac{1}{2}=\frac{2-3}{6}=-\frac{1}{6}$. Then, $\frac{-\frac{1}{6}}{2}=-\frac{1}{12}$.

Answer:

$-\frac{1}{12}$