Question

How do you find the average rate of change of `f(x)=11x^3+11` over [1,3]?

Answer 1

Average rate of change `= 132`

Explanation:

Average rate of change `= (Delta f'(x))/(Delta x)`

Given
`color(white)("XXX")f(x) = 11x^3+11`

`f'(x) = 33x^2`

Over the interval `[1,3]`
`Delta f'(x) = f'(3) - f'(1)`
`color(white)("XXXX") = 33*9 - 33*1`
`color(white)("XXXX") = 33*8`
`color(white)("XXXX") =264`
and
`Delta x = 3-1 = 2`

So
the average rate of change `= 264/2 = 132`

Answer 2

The average rate of change of `f` over `[a,b]` is defined to be `(f(b)-f(a))/(b-a)`

Explanation:

So find `(f(3) - f(1))/(3-1)`

`f(3) = 11(3)^3+11`

`f(x) = 11(1)^3+11`

Now do the arithmetic.