Module Topic (from proposal)
 
I want to develop a module that any of our calculus teachers can use to define the derivative of a polynomial function using the limit of a difference quotient.
 
CALCULUS MODULE
by Charles Cope, Morehouse College
 
Title: Using a difference quotient to find the derivative of a polynomial function
 N.B.      The derivative of a polynomial function is the limit of the difference quotient as the denominator tends toward zero.
 
        The procedure follows these steps:
1) Choose a polynomial function.
2) Set up a difference quotient     D.Q.  =   f(x+h) - f(x)
                                                                           h
3) Simplify the difference quotient using algebra operations.
4) Approximate the limit of the difference quotient by substituting smaller and smaller values of  h , starting with  h =.1 , h =.01 , h =.001 ,etc.
        The purpose of this module is that students notice what happens to the value of the
        difference quotient as  h  becomes smaller and smaller.
 
        Note:    Before using this module, teachers should read the following description and statement of purpose.
 

Learning Activity        This module allows the student to investigate how taking the limit of a difference quotient involving a polynomial function yields a formula for finding the slope of the polynomial function at any point in its domain set.
            Do your students possess the background skills required to complete the module?
            Key topics: 1) What is a polynomial function?
                               2) What is a difference quotient?
                               3) What algebra skills are needed to simplify a difference quotient?
 

Part  1    To the teacher

                         Part  2      To the student:
Please read the statement of the 'Learning Activity' and the 'Goals for the module' before proceeding.

Goals for the module
       1) Students should set up a difference quotient for a polynomial function.
        2) Students should simplify the difference quotient.
        3) Students should evaluate the difference for smaller and smaller values of  h .
        4) Students should realize that the result of this process will give an exact formula for finding the slope of the polynomial function.

Example 1: Suppose that we wish to find the slope of the function  f(x) = x^2 when the value of  x  = 4  .First  construct a difference quotient for f(x) and make  h  be .1, .01, .001, .0001 successively.
     Then  record the value of the difference quotient in each case.
 
 D.Q. = (x+h)^2 - x^2    when x = 4 and  h = .1  becomes
                    h

 D.Q = (4+.1)^2 - 4^2      =  8.1
                  .1

 D.Q. = (4+.01)^2 - 4^2   =  8.01
                    .01

D.Q.  = (4+.001)^2 - 4^2  =  8.001
                    .001

D.Q. =  (4+.0001)^2 - 4^2 = 8.0001
                    .0001

Note that the sequence  8.1, 8.01, 8.001, 8.0001, ... etc., formed by successive inputs into the difference quotient, seems to be approaching  8 .

        Now simplify the difference quotient using algebra and then let  x  equal 4 .
 
D.Q. =  (x+h)^2 - x^2  =  (x^2 + 2xh + h^2) - x^2  =  2xh+h^2  = 2x+h
                      h                                    h                           h
 

If  x = 4 and h = .1 , then 2x+h = 2*4+.1 = 8.1 .
If  x = 4 and h = .01 , then 2x+h = 2*4+.01 = 8.01 .
If  x = 4 and h = .001 , then 2x+h = 2*4+.001 = 8.001 .
If  x = 4 and h = .0001 , then 2x+h = 2*4+.0001 = 8.0001 .

Again as  x  tends toward zero, the value of the difference quotient tends toward  8 .
         Is it clear that as  h  tends toward zero, the value of the difference tends toward  2x  ?

Thus,  f '(x) = 2x  is the derived function that gives the slope of the polynomial  f(x) =(x^2)  at every value of its domain..
 

Part 3
To the teacher
        After completing the module:
        1) Do you believe that the module addressed the goals stated in part one?
        2) Do you believe that the module presented the material clearly and at an appropriate level for your students?  Explain..
        3) Do you use this module to supplement or to replace your teaching of this topic?

To the student
        1) Does the module improve your understanding of using the difference quotient to find the derivative of a polynomial function?
        2) Does the module make its point clearly?
        3) Do the examples used in the module make the topic clearer?
        4) Can you now find the derivative of a simple polynomial function using a difference quotient?
 

Acknowledgment
        I wish to express my gratitude to the Mellon Foundation and the Associated Colleges of the South for making the production of this module possible.