How do you find the average rate of change for the function f(x)=x3 on the indicated intervals [2,h]?

Question

umkhululiueyj · Accepted Answer

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

\frac{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:

\frac{8 - h^{3}}{2 - h}

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

\frac{h^{3} - 8}{h - 2}

The top in this case is a difference of cubes, and can be simplified to this:

\frac{(h - 2) (h^{2} + 2 h + 4)}{h - 2}

We can then cancel the (h-2) value on the top and bottom.

\frac{h - 2 (h^{2} + 2 h + 4)}{h - 2}

This leaves us with our answer

h^{2} + 2 h + 4

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How do you find the average rate of