Average Rate of Change

Date: 10/30/2002 at 18:23:27
From: Brant Langer Gurganus
Subject: Average rate of change (using average of 2 derivatives vs. 
slope formula)

Recently, a math quiz had the following problem:

Find the average rate of change of y with respect to x on the interval
[2, 3], where y = x^3 + 2.

Our teacher has taught us that average rate of change can be figured 
by averaging the derivative of the end points or by using slope 
formula. I and some others chose the derivative version.

y = x^3 + 2
y' = 3x^2
y'(2) = 3(2)^2 = 12
y'(3) = 3(3)^2 = 27
(12 + 27)/2 = 39/2 = 19.5

Others in class chose the slope version.

y(2) = 2^3 + 2 = 10
y(3) = 3^3 + 2 = 29
(29 - 10)/(3-2) = 19

As you can see, the two methods yielded different results. Our
teacher could not explain why and accepted both results. However, both 
methods yielded the same answer with each method. I tried to determine 
why with a few other math teachers and we could not figure out why.

Here are the hypotheses we tested, and they were proved wrong:
 * odd-sized intervals yield equivalent answers
 * odd-degree functions yield different answers

So far, the best explanation is that beyond 2nd-degree functions, some
sort of error is introduced. Since the amount of error seems to
increase with the degree of the function, it appears that there may be
some way to correct for the error.

Any comments would be highly appreciated.

Date: 10/31/2002 at 09:45:54
From: Doctor Rick
Subject: Re: Average rate of change (using average of 2 derivatives 
vs. slope formula)

Hi, Brant.

It is simply not correct that average rate of change can be figured 
by averaging the derivative of the end points. To follow this 
approach, you need to average the derivative at ALL points in the 
interval [2,3] - not just the endpoints.

Let's say you have a sequence: 10, 13.391, 17.625, 22.797, 29. What is 
the average of these numbers? Is it the average of the first and last 
numbers?

  (10 + 29)/2 = 19.5

No, you must average all four numbers:

  (10 + 13.391 + 17.625 + 22.797 + 29)/5 = 18.563

The rate of change of your function is dy/dx = 3x^2 + 2. How do we 
average ALL values of this function in the interval [2,3]? We add them 
up and divide by the number of points, as always - but in the case of 
a continuous function, this translates into integrating the function 
over the interval, and dividing by the width of the interval:

              3
  1/(3-2) Integral 3x^2 dx
             x=2

What is this integral? It's x^3 evaluated at x=3, minus x^3 evaluated 
at x=2.

  1/(3-2)(3^3 - 2^3) = 19

We get the same answer as the slope of the line joining the endpoints. 
But do you see what we've done? We took the derivative of a function, 
and then we integrated that derivative - getting back the original 
function, apart from a constant of integration, which cancels out 
anyway when we take the difference of the endpoints. We've gone 
through a lot of extra work in order to take the difference of the 
value of the function at the endpoints, divided by the difference of 
the x coordinates of the endpoints - in other words, the slope of the 
line.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/