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

1 Answer
Oct 10, 2015

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!