Question

find the average rate of change of $f(x)=8x^{2}-4$ on the interval $3,b$. your answer will be an expression involving $b$.

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 $f(x)=8x^{2}-4$.

Step2: Find $f(b)$ and $f(3)$

First, find $f(b)$: $f(b)=8b^{2}-4$. Then find $f(3)$: $f(3)=8\times3^{2}-4=8\times9 - 4=72 - 4 = 68$.

Step3: Calculate the average rate of change

Substitute $f(b)$ and $f(3)$ into the average - rate - of - change formula: $\frac{f(b)-f(3)}{b - 3}=\frac{(8b^{2}-4)-68}{b - 3}=\frac{8b^{2}-72}{b - 3}$. Factor the numerator: $8b^{2}-72 = 8(b^{2}-9)=8(b + 3)(b - 3)$. Then $\frac{8(b + 3)(b - 3)}{b - 3}=8(b + 3)$.

Answer:

$8(b + 3)$