Question

Suppose
f(x)=x^(3)−3

What is the average rate of change of f(x) with respect to x as x changes from 1 to 1+h?

The AROC is?

Answer 1

f(1) = 1^3 - 3 = -2
f(1+h) = (1+h)^3 - 3
= 1 + 3h + 3h^2 + h^3 - 3
= h^3 + 3h^2 + 3h - 2

avg rate of change = (h^3 + 3h^2 + 3h - 2 + 2)/(1+h - 1)
= (h^3 + 3h^2 + 3h)/h
= h^2 + 3h + 3 , h ≠ 0

Answer 2

(x+h)^3 = (x+h)(x^2 + 2 hx + h^2)
= x^3
+ 2 h x^2 + h^2 x
+ 1 h x^2 +2h^2 x + h^3
--------------------------
= x^3 + 3 h x^2 + 3 h^2 x + h^3
now subtract x^3

f(x+h) - f(x)
= 3 h x^2 + 3 h^2 x + h^3
NOW let x = 1 :)
= 3 h + 3 h^2 + h^3
==========================

Now I did that for x in general rather than just for x = 1 for a reason
Look at what happens to
[ f(x+h) - f(x) ] / h
which is the slope
as h ---> 0
(3 h x^2 + 3 h^2 x + h^3)/ h
= 3 x^2 + 3 h x + h^2
as h ---> 0
that is just 3 x^2

Next year you will hear that this is called the DERIVATIVE of the function, and that is where this problem is leading you.