Question

3.3 rates of change and behavior of graphs
score: 6/22 answered: 6/22
question 7
find the average rate of change on the interval specified for real numbers b (where b ≠ -3). simplify your answer.
f(x)=3x^{2}-2 on -3,b
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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, $a=-3$ and $f(x)=3x^{2}-2$. So we need to find $f(b)$ and $f(-3)$.

Step2: Calculate $f(b)$ and $f(-3)$

$f(b)=3b^{2}-2$ and $f(-3)=3\times(-3)^{2}-2=3\times9 - 2=27 - 2=25$.

Step3: Substitute into the formula

The average rate of change on the interval $[-3,b]$ is $\frac{f(b)-f(-3)}{b-(-3)}=\frac{(3b^{2}-2)-25}{b + 3}=\frac{3b^{2}-27}{b + 3}$.

Step4: Simplify the expression

Factor the numerator: $3b^{2}-27 = 3(b^{2}-9)=3(b + 3)(b - 3)$. Then $\frac{3(b + 3)(b - 3)}{b + 3}=3(b - 3)$.

Answer:

$3(b - 3)$