Question

How do you find the average rate of change for the function `f(x) =x^3` on the indicated intervals [2,h]?

Answer 1

`h^2 + 2h + 4` would be the average rate of change over the interval [2,h]

Explanation:

The average rate of change is slope, which can be formulated by:

`(y_1 - y_2)/(x_1 - x_2)`

We have both x values, 2 and h, and therefore we can get the y values by plugging the x values back into the function:

`f(2) = 2^3 = 8`
`f(h) = h^3`

Now we can plug everything into the original equation:

`(8-h^3)/(2-h)`

We can take out a negative to make it look like this:

`(h^3 - 8)/(h-2)`
The top in this case is a difference of cubes, and can be simplified to this:
`((h-2)(h^2 + 2h + 4))/(h-2)`
We can then cancel the `(h-2)` value on the top and bottom.
`(cancel(h-2)(h^2 + 2h + 4))/(cancel(h-2)`

This leaves us with our answer `h^2 + 2h + 4`.
Hope this helped!