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?
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
(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.
To find the average rate of change (AROC) of a function, we need to calculate the change in the function value over a given interval divided by the change in the input variable over the same interval.
In this case, we want to find the AROC of the function f(x) = x^3 - 3 as x changes from 1 to 1 + h.
Step 1: Calculate the value of f(x) at x = 1+h:
f(1 + h) = (1 + h)^3 - 3
Step 2: Calculate the value of f(x) at x = 1:
f(1) = (1)^3 - 3
Step 3: Calculate the change in f(x) over the interval 1 to 1 + h:
Δf(x) = f(1 + h) - f(1)
Step 4: Calculate the change in x over the interval 1 to 1 + h:
Δx = (1 + h) - 1
Step 5: Calculate the AROC:
AROC = Δf(x) / Δx
Substituting the values we found, we have:
AROC = (f(1 + h) - f(1)) / ((1 + h) - 1)
Now, simplify the expression to find the AROC.