Derivative Definition Exercise

The purpose of this exercise is to familiarize the student with alternate forms of the definition of the derivative, both useful as the course is developed.  Exercises are included to help develop the functional ability to use both forms of the difference quotient to determine derivatives algebraically.

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Introduction

Drag the point a as close as possible to the point 2.^*  Click the button labeled   to display the secant line through points  and  .  Move the point a+h as close as possible to 3 and observe that if your points are accurately positioned, the value of h is 1 and the calculated slope of this secant line is 0.7.  These values have been entered into the table below.

^*It may be easier to accurately position points if you enlarge the graph by moving the unit point (1, 0) further away from the origin then repositioning the origin to see the important parts of the curve. 

1.

Move a+h closer to a so that h = 0.5 then record the value for m[sec] in the table.

2.

Choose several smaller values for h and record the values of h and the slope of the  secant, but keep a+h to the right of a so h > 0.  Let h become as small as 0.1 and 0.05.

3.

How small can h become before the applet displays h = 0 and m[sec] as undefined?  Record this smallest value for h>0 along with the slope of the secant.

4.

Click the button to display the tangent at a = 2, and observe the slope of the tangent.  It should be very close to the secant slope for your smallest value of h.

5.

We will now explore a different yet equivalent form of the difference quotient.

Click the button labeled  to display the secant line through points  and .  While keeping a at the point 2 move x as close as possible to 3 and observe that, as you would expect, the value of x  a is 1 and the  slope of this secant line is 0.7.  Move x as close as possible to 2.5 and record your values.

6.

Choose several more values for x even closer to 2 and record these values in the chart at the right.  It should become apparent that if , then  

7.

We will now exercise algebraic skills instead of viewing a geometric representation.

Consider the function .  Find an algebraic expression for  when a = 2, and simplify this expression.  It will be necessary to know that .  Then determine .

8.

Using the same function , find an algebraic expression for  when a = 2, and simplify this expression.  The simplification will require polynomial division.  Then evaluate .

9.

Using the function , setup and simplify algebraic expressions for both forms of the different quotient  and .  To simplify you will have to “rationalize the numerator.”  Then find  and .  Of course, the two limits must be equivalent.  Is one form easier to deal with than the other?

10.

Another form of the difference quotient commonly used to approximate the slope of a curve at x = a is , called the “symmetric difference.”   Is this expression equivalent to the other difference quotients?  Explain why in your own words.

Let , a = 0 and  h some very small number such as h = 0.01,  Evaluate .  What is ?  Express in your own words how this explains why older graphing calculators erroneously determined that  for  instead of identifying  as undefined.