## Calculus (3rd Edition)

See proof Average rate of change in $x=2$: 12
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 $x=2$: $2^2+2(2)+4=12$