Question

b. $f(x)=2x^{2}-1$ from $x_1 = 1$ to $x_2 = 3$ net change: average rate of change:

Explanation:

Step1: Recall net - change formula

Net change of a function $y = f(x)$ from $x_1$ to $x_2$ is $\Delta y=f(x_2)-f(x_1)$.
Given $f(x)=2x^{2}-1$, $x_1 = 1$, $x_2 = 3$. First, find $f(x_1)$ and $f(x_2)$.
$f(x_1)=2(1)^{2}-1=2 - 1=1$.
$f(x_2)=2(3)^{2}-1=2\times9 - 1=18 - 1 = 17$.

Step2: Calculate net - change

$\Delta y=f(x_2)-f(x_1)=17 - 1=16$.

Step3: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ from $x_1$ to $x_2$ is $\frac{\Delta y}{\Delta x}=\frac{f(x_2)-f(x_1)}{x_2 - x_1}$.
Here, $\Delta x=x_2 - x_1=3 - 1 = 2$.
Since $\Delta y = 16$ and $\Delta x=2$, the average rate of change is $\frac{16}{2}=8$.

Answer:

Net change: 16
Average rate of change: 8