Question

Find the average rate of change of f ( x ) = 3 x 2 − 9 on the interval [ 2 , b ] . Your answer will be an expression involving b

Answer 1

`m = 3b+6`

Explanation:

Given

`f(x)=3x^2 - 9`

Required

The average rate over
`[2,b]`

Average rate (m) is calculated using:

`m = (f(b) - f(a))/(b - a)`

Where

`[a,b] = [2,b]`

So, we have:

`m = (f(b) - f(2))/(b - 2)`

Calculate f(b) and f(2)

`f(x)=3x^2 - 9`

`f(b)=3b^2 - 9`

`f(2)=3*2^2 - 9 = 12 - 9 = 3`

So, we have:

`m = (f(b) - f(2))/(b - 2)`

`m = (3b^2 - 9 - 3)/(b - 2)`

`m = (3b^2 - 12)/(b - 2)`

Expand the numerator

`m = (3b^2 + 6b-6b-12)/(b - 2)`

Factorize

`m = (b(3b + 6)-2(3b+6))/(b - 2)`

Factor out 3b + 6

`m = ((b -2)(3b+6))/(b - 2)`

Cancel out b - 2

`m = 3b+6`