Question

3.3 rates of change and behavior of graphs
score: 4/22 answered: 4/22
question 5
find the average rate of change on the interval specified for real numbers h.
h(x)=2x³ on 3,3 + h
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Explanation:

Step1: Recall average - rate - of - change formula

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

Step2: Calculate $h(3 + h)$ and $h(3)$

$h(3 + h)=2(3 + h)^{3}=2(27 + 27h+9h^{2}+h^{3})=54 + 54h + 18h^{2}+2h^{3}$; $h(3)=2\times3^{3}=2\times27 = 54$.

Step3: Substitute into the formula

$\frac{h(3 + h)-h(3)}{(3 + h)-3}=\frac{(54 + 54h + 18h^{2}+2h^{3})-54}{h}=\frac{54h + 18h^{2}+2h^{3}}{h}$.

Step4: Simplify the expression

Since $h
eq0$, we can cancel out the $h$ in the numerator and denominator: $\frac{54h + 18h^{2}+2h^{3}}{h}=54 + 18h+2h^{2}$.

Answer:

$2h^{2}+18h + 54$