find the average rate of change of f(x)=3x² - 8 on the interval 1,b. your answer will be an expression…

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, $a = 1$ and $f(x)=3x^{2}-8$.

Step2: Calculate $f(b)$ and $f(1)$

$f(b)=3b^{2}-8$, $f(1)=3\times1^{2}-8=3 - 8=-5$.

Step3: Substitute into the formula

$\frac{f(b)-f(1)}{b - 1}=\frac{(3b^{2}-8)-(-5)}{b - 1}=\frac{3b^{2}-8 + 5}{b - 1}=\frac{3b^{2}-3}{b - 1}$.

Step4: Simplify the expression

Factor the numerator: $3b^{2}-3 = 3(b^{2}-1)=3(b + 1)(b - 1)$. Then $\frac{3(b + 1)(b - 1)}{b - 1}=3(b + 1)$ for $b\neq1$.

Answer:

$3(b + 1)$