Question

How do you find the average rate of change of `f(z) = 4 − 3z^2` from [-2,0]?

Answer 1

6

Explanation:

The `color(blue)"average rate of change"` of f(z) over an interval between 2 points (a ,f(a)) and (b ,f(b)) is the slope of the `color(blue)"secant line"` connecting the 2 points.

To calculate the average rate of change between the 2 points use.

`color(red)(|bar(ul(color(white)(a/a)color(black)((f(b)-f(a))/(b-a))color(white)(a/a)|)))`

`f(0)=4-3(0)^2=4`

`f(-2)=4-3(-2)^2=4-12=-8`

The average rate of change between (-2 ,-8) and (0 ,4) is

`(4-(-8))/(0-(-2))=12/2=6`

This means that the average of all the slopes of lines tangent to the graph of f(z) is 6.