Question

How do you find the average rate of change of `y=2x^2-2x+1` over [-1,-1/2]?

Answer 1

`-5`

Explanation:

The `color(blue)"average rate of change"` of y 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(b)=f(-1/2)=2(-1/2)^2-2(-1/2)+1=2 1/2`

`f(a)=f(-1)=2(-1)^2-2(-1)+1=5`

The average rate of change between `(-1,5)" and " (-1/2,2 1/2)` is

`(2 1/2-5)/(-1/2-(-1))=(-2 1/2)/(1/2)=-5`

This means that the average of all the slopes of lines tangent to the graph of y between `(-1,5)"and " (-1/2,2 1/2)` is - 5