Question

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

Step2: Calculate $g(3)$

Substitute $x = 3$ into $g(x)$:
$g(3)=3^{3}-2\times3^{2}+3\times3$
$=27-2\times9 + 9$
$=27-18 + 9$
$=18$.

Step3: Calculate $g(1)$

Substitute $x = 1$ into $g(x)$:
$g(1)=1^{3}-2\times1^{2}+3\times1$
$=1-2 + 3$
$=2$.

Step4: Calculate the average rate of change

$\frac{g(3)-g(1)}{3 - 1}=\frac{18 - 2}{2}$
$=\frac{16}{2}=8$.

Answer:

$8$