Question

How do you find the average rate of change of `y = 9x^3 - 2x^2 + 6` between x = -4 and x = 2?

Answer 1

`-80`

Explanation:

The `color(blue)"average rate of change"` of y = f(x) 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.

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

here a = -4 and b = 2

`f(-4)=9(-4)^3-2(-4)^2+6=550`

`f(2)=9(2)^3-2(2)^2+6=70`

The average rate of change between (-4 ,550) and (2 ,70) is

`(70-550)/(2-(-4))=(-480)/6=-80`

This means that the average of all the slopes of lines tangent to the graph of f(x) between (-4 ,550) and (2 ,70) is - 80.