Question

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

Answer 1

41

Explanation:

The `color(blue)"average rate of change"` of 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 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)|)))`

here a = -2 and b = 4

`f(4)=4+(4)^4=260`

and `f(-2)=-2+(-2)^4=14`

The average rate of change between (-2 ,14) and (4 ,260) is

`(260-14)/(4-(-2))=41`

This means that the average of all the slopes of lines tangent to the graph of f(x) in the interval [-2 ,4] is 41.