Secant/Tangent Exercise

The purpose of this exercise is to observe that while the difference quotient  is undefined for h = 0, this quotient does “settle” upon a finite number as  or .

Click here to display the applet in a separate window.      Smaller applet

Introduction

Drag the point a as close as possible to the point 3.^*  Click the Show Secant button to display the line through points  and .  Move the point a+h as close as possible to 5 and observe that if your points are accurately positioned, the value of h is 2 and the slope of this secant line is 0.44.  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 part of the image. 

1.

Move a+h as close as possible to 4 so that h = 1 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 lists h = 0 and m [sec] as undefined?  Record this smallest non-zero value for h > 0 and the secant slope in the table.  Why does the graph of the secant line disappear when h = 0?

4.

Click the button to display the tangent at a = 3, and observe the slope of the tangent.  Is it equal or very close to the slope of the secant for your smallest value of h?

5.

By keeping a+h to the right of a so h > 0 we found what is called the “right hand limit” written .

Now keep a+h to the left of a so h < 0.  Then let  and observe the “left hand limit.”  Record a few of your values in the table at the right, such as h = −1, h = −0.5, and h = −0.1.  How close to 0 can h < 0 become before the slope is undefined? Record the values for this h < 0 closest to 0 and m[sec].  Is this m[sec] close to the slope of the tangent line?

6.

Choose another value for a such as a = 4 or a = 5 and investigate  and .  In the tables below record the values of h > 0 and h < 0 as close to zero as possible.