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find the average rate of change of f(x)=12x^3 between x = 1 and x = 3.
average rate of change =
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Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a = 1$, $b = 3$, and $f(x)=12x^{3}$.

Step2: Calculate $f(3)$ and $f(1)$

First, find $f(3)$: $f(3)=12\times3^{3}=12\times27 = 324$. Then, find $f(1)$: $f(1)=12\times1^{3}=12$.

Step3: Compute the average rate of change

Substitute into the formula: $\frac{f(3)-f(1)}{3 - 1}=\frac{324 - 12}{2}=\frac{312}{2}=156$.

Answer:

156