Question

given the function $f(x)=x^{2}+3x - 2$, determine the average rate of change of the function over the interval $-2leq xleq2$.

Explanation:

Step1: Recall average - rate - of - change formula

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

Step2: Calculate $f(a)$ and $f(b)$

First, find $f(-2)$:
\[

$$\begin{align*} f(-2)&=(-2)^{2}+3\times(-2)-2\\ &=4-6 - 2\\ &=-4 \end{align*}$$

\]
Then, find $f(2)$:
\[

$$\begin{align*} f(2)&=2^{2}+3\times2-2\\ &=4 + 6-2\\ &=8 \end{align*}$$

\]

Step3: Calculate the average rate of change

Substitute $f(-2)=-4$, $f(2)=8$, $a=-2$, and $b = 2$ into the average - rate - of change formula:
\[

$$\begin{align*} \frac{f(2)-f(-2)}{2-(-2)}&=\frac{8-(-4)}{2 + 2}\\ &=\frac{8 + 4}{4}\\ &=\frac{12}{4}\\ &=3 \end{align*}$$

\]

Answer:

$3$