Question

How do you find the average rate of change of y with respect to x on the interval [1,4], where `y=x^2+x+1`?

Answer 1

6

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(4)=4^2+4+1=21`

and `f(1)=1^2+1+1=3`

The average rate of change between (1 ,3) and (4 ,21) is

`(21-3)/(4-1)=18/3=6`

This means that the average of all the slopes of lines tangent to the graph of y between (1 ,3) and (4 ,21) is 6.