Question

find the average rate of change of f(x)=-2x^3 + 5x^2 from x=-2 to x=3. simplify your answer as much as possible.

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 given by $\frac{f(b)-f(a)}{b - a}$. Here, $a=-2$, $b = 3$, and $f(x)=-2x^{3}+5x^{2}$.

Step2: Calculate $f(-2)$

Substitute $x=-2$ into $f(x)$:
\[

$$\begin{align*} f(-2)&=-2(-2)^{3}+5(-2)^{2}\\ &=-2(-8)+5\times4\\ &=16 + 20\\ &=36 \end{align*}$$

\]

Step3: Calculate $f(3)$

Substitute $x = 3$ into $f(x)$:
\[

$$\begin{align*} f(3)&=-2(3)^{3}+5(3)^{2}\\ &=-2\times27+5\times9\\ &=-54 + 45\\ &=-9 \end{align*}$$

\]

Step4: Calculate the average rate of change

\[

$$\begin{align*} \frac{f(3)-f(-2)}{3-(-2)}&=\frac{-9 - 36}{3 + 2}\\ &=\frac{-45}{5}\\ &=-9 \end{align*}$$

\]

Answer:

$-9$