Answer
See proof
Average rate of change at $x=1$: 3
Work Step by Step
We are given the function:
$f(x)=x^3$ over $[1,x]$.
Determine the average rate of change of $f$:
$\dfrac{\Delta f}{\Delta x}=\dfrac{f(x)-f(1)}{x-1}=\dfrac{x^3-1^3}{x-1}=\dfrac{(x-1)(x^2+x+1)}{x-1}=x^2+x+1$
So the average rate of change of $f$ on $[1,x]$ is $x^2+x+1$.
Determine the average rate of change of $f$ at $x=1$:
$1^2+1+1=3$
