find the average rate of change of the function on the given interval. $f(x)=-x^{3}-4x^{2}+1$, interval: $0,2$

Explanation:

Step1: Recall the average - rate - of - change formula

The average rate of change of a function $y = f(x)$ on the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a = 0$, $b = 2$, and $f(x)=-x^{3}-4x^{2}+1$.

Step2: Calculate $f(2)$

Substitute $x = 2$ into $f(x)$: $f(2)=-(2)^{3}-4\times(2)^{2}+1=-8 - 16+1=-23$.

Step3: Calculate $f(0)$

Substitute $x = 0$ into $f(x)$: $f(0)=-(0)^{3}-4\times(0)^{2}+1 = 1$.

Step4: Calculate the average rate of change

Using the formula $\frac{f(b)-f(a)}{b - a}$, we have $\frac{f(2)-f(0)}{2 - 0}=\frac{-23 - 1}{2}=\frac{-24}{2}=-12$.

Answer:

$-12$