Question

How do you find the average rate of change of `f(x) = x^2+x+3` from [1,x]?

Answer 1

The average rate of change of a function `f` over an interval `[a,b]` is `(f(b)-f(a))/(b-a)`.

Explanation:

The average rate of change of `f(x)=x^2+x+3` over `[1,x]` is

`(f(x)-f(1))/(x-1) = ([x^2+x+3]-[(1)^2+(1)+3])/(x-1)`

`= (x^2+x-2)/(x-1)`

`= (cancel((x-1))(x+2))/cancel((x-1))`

`= x+2`