Question

find the average rate of change of f(x)=x^3 - 2x^2 + 3x 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 $\frac{f(b)-f(a)}{b - a}$. Here, $a = 2$, $b=3$, and $f(x)=x^{3}-2x^{2}+3x$.

Step2: Calculate $f(2)$

Substitute $x = 2$ into $f(x)$:
$f(2)=2^{3}-2\times2^{2}+3\times2=8-2\times4 + 6=8 - 8+6=6$.

Step3: Calculate $f(3)$

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

Step4: Calculate the average rate of change

$\frac{f(3)-f(2)}{3 - 2}=\frac{18 - 6}{1}=12$.

Answer:

$12$