find the average rate of change of $f(x)=2x^{3}-2x^{2}-3$ from $x = - 1$ to $x = 2$. simplify your answer as…

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ from $x = a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a=-1$, $b = 2$, and $f(x)=2x^{3}-2x^{2}-3$.

Step2: Calculate $f(a)$

Substitute $x=-1$ into $f(x)$: $f(-1)=2(-1)^{3}-2(-1)^{2}-3=2\times(-1)-2\times1 - 3=-2 - 2-3=-7$.

Step3: Calculate $f(b)$

Substitute $x = 2$ into $f(x)$: $f(2)=2(2)^{3}-2(2)^{2}-3=2\times8-2\times4 - 3=16 - 8-3=5$.

Step4: Calculate the average rate of change

Use the formula $\frac{f(b)-f(a)}{b - a}=\frac{5-(-7)}{2-(-1)}=\frac{5 + 7}{2 + 1}=\frac{12}{3}=4$.

Answer:

$4$