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From Secant Lines to Tangent Lines

Activity Description Activity Guide Resources



Part 1: Constructing a Secant Line Through a Curve

1.      In this portion of the activity, we will construct a secant line through a curve.  Explain in your own words what is a secant line through a curve.

2.      To begin, open a new sketch. Under Edit, change Preferences to ten-thousandths to make the calculations more precise. Plot the function f(x) = x2 and the point C(-2,0).   

How to show the grid:

  • Under Graph select Show Grid.

How to plot a new function:

  •  Under Graph select Plot New Function.
  •  Enter the function using the calculator that appears; e.g., type x^2 for f(x) =  x2. Click <OK>.

How to plot a specific point:

  • Under Graph select Plot Points to bring up the Plot Points window.
  • Type in the abscissa and ordinate of the point you wish to plot.
  • Click Plot to plot the point. Then click Done to close the Plot Points window.

How to label a point or change its label:

  • Select the text tool from the tool bar on the left side of the sketchpad window.
  • Double click on the point to be labeled.
  • Enter the new label in the window that appears.  Also, select Show Label if it is not already selected. Click <OK>.
  • Drag the label to a suitable location next to the point.  When you are finished, select the arrow tool again.

 

 

 

 



 

 

 

 

 

 

 

 

 

3.      Determine the numerical coordinates of the point having the same abscissa as point C and lying on the curve of f(x). We will label this point C'. In general, C' has coordinates (xC, f(xC)).

4.      Although there are different ways to plot C' using The Geometer’s Sketchpad, to complete this activity successfully, please follow the instructions below.

Note: Because of the idiosyncracies of The Geometer's Sketchpad 4.0, the ordinate of C' is not as precise if it is simply constructed as a point on the curve. To ensure that the ordinate of C' is as precise as needed for this activity, it should be constructed by using the abscissa of C as described below.

How to plot C':

In your sketch, be sure a function has been graphed and a point C has been plotted on the x-axis.

  • First, display the abscissa of C':
    a.  Select the arrow tool from the tool bar, then select point C. 
    b.  Display abscissa of C by clicking on Measure then Abscissa(x).  This value
          is also the abscissa of C'.

  • Second, display the ordinate of C':
    a.  Again select point C.
    b.  Click on Measure in the menu bar, then Calculate....
    c.  Enter the function into the calculator by clicking on the algebraic form of the
         function shown in the sketch.  Enter the abscissa of C' by clicking on its value
        displayed in the sketch. 
    d.  Click <OK>.  The ordinate of C' will appear on the sketch.

  • Third, plot C':
    a.  Be sure the ordinate of C' has been deselected.  To deselect it,
         click on it.
    b.  Select the abscissa of C' first , then the ordinate of C'. 
         Under Graph select Plot as (x,y).
    c. Label it C'.



 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5.      Plot a draggable point B on the x-axis. Be sure it is confined to the x-axis.  Next plot the point (xB, f(xB)) according to the instructions above and label it B'.  Construct a secant line through C' and B' (see instructions below).  Color this line red and change its width to thick.

How to plot a draggable point confined to a line:

  • Select the line (x-axis) and be sure that only this line is selected.
  • Under Construct select Point On Object.
  • Check to be sure the new point is confined to the line.

How to construct a line:

  • Select two points that define the line.
  • Under Construct select Line.

How to change the features of a line:

  • Select the line.
  • Under Display select either Line Width or Color.

 

 

 

 

 

  

 

 

 

Part 2: Slopes of Secant Lines and Their Limits

6.      Use the information shown in your sketch to determine a general formula for the slope of the secant line through points B'(xB, f(xB)) and C'(xC, f(xC)).  Next, using the formula you determined, use The Geometer’s Sketchpad to calculate and display the slope of the secant line through points B' and C'.

How to enter a dynamic formula:

  • Under Measure select Calculate.
  • As you enter the formula, rather than typing in specific values, click on the variables shown in the sketch as appropriate.  This allows the value of the formula to change as the dynamic features of the sketch are utilized. 
  • Click <OK>.

 

 

 

 

 

7.     Drag point B to a location along the x-axis to the right of (3,0).

  1. Predict what will happen to the slope of the secant line as B moves toward C.
  2. Using the formula, predict the value of the slope of the secant line when B coincides with C.
  3. Using the graph, predict the value of the slope of the secant line when B coincides with C.
  4. Explain your predictions and discrepancies, if any, between your predictions in 7b. and 7c.

Note:  To view more of the sketch, reposition the coordinate system by dragging the x-axis or the origin to a different location on the screen.

 

 

8.      To further assess your predictions, use the instructions below to create a table showing the abscissas of B and the values of the slope of the secant line as B approaches C from the left and from the right.

How to create the table needed in Task 8:

  1. Set up the table:
    a. Use the arrow tool to select the abscissa of B and the slope of the secant line. These will be the headings for the columns in the table.
    b. Under Graph select Tabulate.  The table will appear on your sketch.

  2. Save a pair of values to the table when B is close to the right side of C:
    a. Move point B as close as you can to the right side of point C.
    b. Note that as you change elements of your sketch, the values in the bottom row of     the table change to reflect your sketch.  Double click on the table to save the bottom row of values to the table and simultaneously add a new row.

  3. Save a pair of values to the table when B is close to the left side of C.

  4. Zoom in so that you can move point B even closer to point C than before. (To zoom in, drag the point at (1,0) to the right.) Repeat steps 2 and 3.

  5. Repeat step 4 at least two more times. You should have at least eight pairs of values in your table.

  6. Zoom back out and reposition your sketch appropriately.  (To zoom out, drag the point at (1,0) to the left.) Note: You will find B in a location to the far left of C.

 

 

 

 

 

 

 

 

 

 

 

 

 

9.  Discuss any patterns you see in the table.  Can you express any of these patterns as a limit statement?  What does the limit statement look like? Compare the value of your limit statement to the values you predicted in 7b and 7c.

10.  Test your predictions from Task 7 using The Geometer’s Sketchpad.  It might be useful to create an action button to move point B to point C.  Were your predictions correct?  Explain. What happens to the secant line when B coincides with point C?  Why? What happens to the slope of the secant line?  Why does this happen? Does the limit from Task 9 exist? Explain.

How to create an action button: 

  • Select the object you wish to move, then its target destination.
  • Under Edit select Action Buttons then Movement.
  • Set the speed to medium.  Click <OK>.

 

 

 

 

Part 3: Tangent Lines

11.  Explain in your own words what is a tangent line to a curve at a point. As B approaches C, what is the relationship between the secant line through B' and C' and the tangent line to the curve f(x) at C'?

12.  To construct the tangent line to the curve at C', we need the coordinates of a point through which the tangent line passes and the slope of the tangent line.  What point should we use to construct the tangent line?  What value should we use for the slope of the tangent line?  Why?

13.  Using the point and slope you selected in Task 12, construct the tangent line in your sketch.

14.  Use the limit statement from Task 9 to relate the slopes of the secant lines to f(x) through B' and C' to the slope of the tangent line to f(x) through C'.

Part 4: Investigating a Cubic Function

Note: If you wish to save the sketch you just constructed, do so now.

 

15.  To clean up the sketch to work with a new function, delete the table, the movement button, and the tangent line and its equation. Edit f(x) to read f(x) = (x +1)3 +1.5.

How to delete objects (and recover accidentally deleted objects): 

  • To delete an object, select it then hit <backspace>.
  • To recover an accidentally deleted object, under Edit select Undo.

How to edit a formula:

  • Double click on the existing formula.
  • Edit the formula in the window that appears. Click <OK>.

 

 

 

 

 

 

16.  Change the coordinates of point C to (-1, 0). Notice that the coordinates of C' change simultaneously since the location of C' depends on the location of C. For this function, is it necessary to revise the general formula for the slope of the secant line through points B'(xB, f(xB)) and C'(xC, f(xC))? Explain your reasoning.

How to edit the coordinates of a stationary point:

  • Right click on the point (point C) and select Edit Plotted Point....
  • Edit the coordinates of the point. Select Edit then Done.

 

 


 

17.  Repeat Task 7, but this time drag point B either to a location along the x-axis to the left of (-2, 0) or to the right of the origin. Repeat Tasks 8 through 14.

Note: If you wish to save the sketch you just constructed, do so now.

 

18.  To work with a new point C, delete the table, movement button, tangent line and its equation. Change the coordintes of point C to (-2, 0). Repeat Task 7, but this time drag point B to a location along the x-axis to the right of the origin. Repeat Tasks 8 through 14.

19. Is the slope of the tangent line to the the graph of f(x) through (-1, 0) the same as the slope to the tangent line to the graph of f(x) through (-2, 0)? Explain.


Part 4: Thinking About Tangent Lines

20.    Explain in your own words what is a tangent line to a circle. As you think about the tangent lines you constructed during this activity, what similarities and/or differences do you notice between a tangent line to a circle through a point and the tangent line to the graph of a function through a point?

21. A curve is can be defined as any collection of points in the plane. Thus, circles, function graphs, and even straight lines can be classified as curves. How can you define a tangent line to a curve through a point?

22. Below is one definition for a tangent line to a curve through a point. How does this definition apply to both a tangent line to a circle through a point and a tangent line to a the graph of a function through a point? Search the web for other definitions of tangent lines. Discuss the definitions you find and which ones are most suitable.

A tangent line to a curve is a line meeting a curve or surface at a point and having at that point the same direction as the curve or surface.

from http://www.brainydictionary.com/words/ta/tangent228030.html (October 6, 2003)

 


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Last modified on February 24, 2004