Question

find the average rate of change of $g(x)=-x^{4}+16$ over the interval $1,3$. write your answer as an integer, fraction, or decimal rounded to the nearest tenth. simplify any fractions.

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 $g(x)=-x^{4}+16$.

Step2: Calculate $g(3)$

Substitute $x = 3$ into $g(x)$:
$g(3)=-(3)^{4}+16=-81 + 16=-65$.

Step3: Calculate $g(1)$

Substitute $x = 1$ into $g(x)$:
$g(1)=-(1)^{4}+16=-1 + 16 = 15$.

Step4: Calculate the average rate of change

$\frac{g(3)-g(1)}{3 - 1}=\frac{-65-15}{2}=\frac{-80}{2}=-40$.

Answer:

$-40$