Question

How do you find the average rate of change of `f(x) = 5x^2` from [4,4+h]?

Answer 1

Average rate of change is `40+5h`

Explanation:

Given
`color(white)("XXX")f(x)=5x^2`

Then
`color(white)("XXX")f(4)=5xx4^2`
`color(white)("XXXXXX")=80`
and
`color(white)("XXX")f(4+h) = 5xx(4+h)^2`
`color(white)("XXXXXXXX")=5xx(16+8h+h^2)`
`color(white)("XXXXXXXX")=80+40h+5h^2`

Change between `f(4)` and `f(4+h)`
`color(white)("XXX")f(4+h) - f(4)`
`color(white)("XXXXXXXXXXXX")=40h+5h^2`

This change occurs over an interval of `(4+h) - (4) = h`

So the average rate of change is
`color(white)("XXX")(40h+5h^2)/h = 40+5h`