#### Answer

See proof
Average rate of change in $x=2$: 12

#### Work Step by Step

We are given the function:
$f(x)=x^3$ over $[2,x]$.
Determine the average rate of change of $f$ over $[2,x]$:
$\dfrac{\Delta f}{\Delta x}=\dfrac{f(x)-f(2)}{x-2}=\dfrac{x^3-2^3}{x-2}=\dfrac{(x-2)(x^2+2x+4)}{x-2}=x^2+2x+4$
So the average rate of change of $f$ over $[2,x]$ is $x^2+2x+4$.
Determine the average rate of change of $f$ at $x=2$:
$2^2+2(2)+4=12$