Question

How do you find the average rate of change of `f(x)=x^3-3x+5` over [-1,5]?

Answer 1

Find the difference in the value of the rate of change, `f'(x)`, between the two end points of the interval, then divide that difference by the interval width to get the average rate of change.

Explanation:

Given `f(x)=x^3-3x+5`
`color(white)("XXX")`Rate of change `= f'(x) = 3x^2-3`

`f'(-1) = 3(1)-3 = 0`
`f(5) = 3(25)-3 = 72`
Difference in `f'(x)` between the two end points: `Delta f_(5:-1)(x) = 72-0 = 72`

Width of interval: `Delta x= 5-(-1) = 6`

Average rate of change: `=72/6 = 12`