Question

what is the average rate of change for the function f(x)=3x^2 - 5 on the interval -3≤x≤1? write the average rate of change in the box.

Explanation:

Step1: Recall 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, $a=-3$, $b = 1$, and $f(x)=3x^{2}-5$.

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

First, find $f(-3)$:
\[

$$\begin{align*} f(-3)&=3\times(-3)^{2}-5\\ &=3\times9 - 5\\ &=27-5\\ &=22 \end{align*}$$

\]
Then, find $f(1)$:
\[

$$\begin{align*} f(1)&=3\times1^{2}-5\\ &=3 - 5\\ &=-2 \end{align*}$$

\]

Step3: Calculate the average rate of change

\[

$$\begin{align*} \frac{f(1)-f(-3)}{1-(-3)}&=\frac{-2 - 22}{1 + 3}\\ &=\frac{-24}{4}\\ &=-6 \end{align*}$$

\]

Answer:

$-6$