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Slopes and Derivatives
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Teacher Note:
This activity is designed to follow the activity From Secant
Lines to Tangent Lines.
To do this activity, you will need a Sketchpad
Custom Tool we refer to as the Tangent Line Tool. (For
more information about tools, see below). You can download or
create it yourself. To download it, click on the link to the sketch
tangentline.gsp which contains the
Tangent Line Tool as well as illustrates the construction
of a tangent line to a curve. Save this sketch to the Tools Folder
in the Sketchpad Folder on the C drive. As a result, every
time Sketchpad is opened on your computer, the Tangent
Line Tool will be easily accessible by holding the left mouse
button down on the Custom Tools icon (double arrow icon)
in the tool bar. If you want to construct the Tangent Line
Tool yourself, detailed instructions can be found under the
Resources
link above.
What is a tool? Custom Tools are generalized
recordings of sketches. It is sometimes convenient to record the
steps of particular sketches (e.g., tangent lines to curves) that
you have made for subsequent playback. For example, when a future
sketch requires a constructed tangent line, you can just use your
custom tool rather than reconstructing one. Sketchpad can generate
a custom tool of a construction while you are sketching it. You
can use custom tools repeatedly to generate figures, or portions
of figures, while sketching. Individual custom tools can be used
as foundations for larger sketches, leading to potentially more
complex constructions. (This explanation was adapted from: http://www.teacherlink.org/content/math/activities/skpv4-construcisitro/guide.html
)
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Part 1: What is a Derivative?
- A curve is defined
to be a collection of points in the plane. For example, ellipses and
parabolas are curves. What are other examples of curves? Can straight
lines be classified as curves? Explain.
- What are the values of the slopes of
curves in a. and b. below?
a.
b.
c.
d.
- What are the values of the slopes of each of the
pieces in the graphs shown in c. and d. above? What are the values
of the slopes of the curves at the endpoints of each piece? Does the
slope exist at these points? Explain. We will explore this in more
depth at a later time.
- Open a new sketch in The Geometer's Sketchpad. Show the grid.
Graph the function f(x) = x2. Explain how we might find
the slope of this curve at any point.
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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>.
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- The slope of a curve at a point is defined as the slope of
the tangent line to the curve through that point. By viewing the graph
of f(x)= x2 shown in Sketchpad, estimate the slope
of f(x) = x2 at (2, f(2)). Explain your reasoning.
- The Geometer's Sketchpad can be programmed to automatically
construct the tangent line to the curve at any point. Such programs
are called "tools." Use the Tangent Line Tool to
construct a moveable tangent line to f(x) = x2. Then display
the slope of the tangent line. Use the moveable tangent line to answer
the following questions:
a. What is the the approximate slope of the curve at x = 2?
b. For what values of x is the slope of the curve negative, positive,
zero, or undefined?
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How to Use the Tangent Line Tool:
- Hold the mouse button down on the double arrow tool in
the tool bar until a menu appears. Select the Tangent
Line Tool.
- Click on the algebraic form of the function through which
the tangent will pass. As you move the cursor, a point will
appear next to the cursor. Position this point on the x-axis.
(Note: When the point lies on the x-axis, the x-axis will
change color.) Click the mouse button to put the point in
place.
How to Display the Slope of a Line:
- Select the line. Be sure that only the line is selected.
- Under Measure, select Slope.
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- Construct a table using Sketchpad displaying xA,
the value of f(xA), and the slope of the tangent line.
Collect at least 6 pairs of data for the table. What patterns do you
see in the table? What relationship do you see between xA
and the slope of the tangent line? Express this relationship as a
function of x.
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How to create the table needed in Task
7:
- Set up the table:
a. Use the arrow tool to select xA, f(xA),
and the slope of the tangent line. These will be the headings
for the columns in the table.
b. Under Graph select Tabulate. The table will appear on your
sketch.
- To save a pair of values to the table:
a. Note that as you change elements of your sketch, the
values in the bottom row of
the table change to reflect your sketch.
b. Double click on the table to save the bottom row of values
to the table and simultaneously add a new row.
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- The function you constructed in Task 7 is called the derivative
of f(x) = x2. It is denoted f '(x) and provides a formula
for finding the value of the slope of the curve f(x) through any point
(x, f(x)). Use the derivative of f(x) = x2 to answer the
following:
a. f '(x) =
b. f '(-10) =
c. f '(8) =
d. Check your answers to b. and c. by approximating them using The
Geometer's Sketchpad.
e. Discuss the relationship between the slope of the tangent line
to a curve f(x) at a point, the slope of the curve f(x) at that point,
and the derivative of the function f(x) at that point.
- Edit f(x) to read f(x) = ex. By viewing the graph of
f(x)= ex shown in Sketchpad, answer the
following questions and explain your reasoning.
a. What is the slope of the curve at x = 0?
b. For what values of x is the slope of the curve negative, positive,
or zero?
c. Use the moveable tangent line to assess the accuracy of your estimates
in a. and b.
d. Change the precision of xA, f(xA), and the slope to hundred-thousandths.
Then re-assess the accuracy of your estimates in a.
and b.
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How to edit a function:
- Double click on the algebraic form of the function shown
in the sketch.
- Edit the function in the window that appears. Click <OK>.
All dependent items in the sketch will change simultaneously.
How to change a measurement's precision:
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- Construct a new table displaying the abscissa of A, the value of
f(xA), and the slope of the tangent line. Collect at least
6 pairs of data for the table. What patterns do you see in the table?
What relationship do you see between xA and the slope of
the tangent line? Express this relationship as a function of x.
- The function you constructed in Task 8 is the derivative of f(x)
= ex. Use the derivative of f(x) = ex to answer
the following:
a. f '(x) =
b. f '(-5) =
c. f '(20) =
d. Check your answers to b. and c. by approximating them using The
Geometer's Sketchpad.
- Determine f '(x) for the functions shown in Task 2a and 2b. Explain
your reasoning.
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EXTENSION:
- Find the derivatives of other functions such as sin(x), log(x),
or (1/3)x3 by constructing tables as discussed in Task
7.
- Construct tables to find the deriviatives of sin(x) and cos(x).
Then construct tables to find the derivatives of sin(2x) and sin(3x).
Generalize to determine a formula for the derivative of sin(nx).
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Part 2: The Derivative as a Limit
- Describe to a friend how secant lines can be used to determine the
value of the derivative at the point A'(2, f(2)) on the curve f(x)
= x2.
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Hint:
To help in your explanation, you might want to use the
sketch you constructed in
the activity From Secant Lines to Tangent Lines;
Or, construct a secant line to the curve on your existing
sketch. To do this:
- Plot a moveable point B on the x-axis.
- Display the abscissa of B; i.e., xB).
- Calculate the functional value f(xB).
- Plot the point B'(xB, f(xB)). To do this:
a. Click on xB;
b. Click on f(xB);
c. Under Graph select Plot as (x,y).
- Construct the secant line through the points A' and B'.
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- What is the general formula you determined in the activity From
Secant Lines to Tangent Lines for determining the slope of the
secant line through any two points A'(xA, f(xA))
and B'(xB, f(xB)) where A(xA, 0)
and B(xB, 0) are points on the x-axis?
- Using the formula from Task 16, construct a limit statement which
relates the slopes of the secant lines to f(x) through A' and B' to
the value of the derivative at A'. On your own paper, illustrate your
limit statement with a carefully labeled sketch.
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Appendix: Creating
Your Own Tangent Line Tool
- Open a new sketch
and show the grid. Plot a nonlinear function such as f(x) = x2
and change its line width to thick. Plot a draggable point P on the
x-axis. Make sure P is confined to the x-axis. Next plot the point
P' with coordinates (xP, f(xP)) according to
the instructions below.
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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 (or
Point On Axis)
- Check to be sure
the new point is confined to the line.
How to plot P':
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.
In your sketch, be sure a function has been graphed and a point P has
been plotted on the x-axis.
- First, display the
abscissa of P':
a. Select the
arrow tool from the tool bar, then select point P.
b.
Display abscissa of P by clicking on Measure
then Abscissa(x).
This value
is also the abscissa of P'.
- Second, display the
ordinate of P':
a. Again select
point P.
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 P' by clicking
on its value
displayed in the sketch.
d.
Click <OK>.
The ordinate of P' will appear on the sketch.
- Third, plot P':
a. Be sure the
ordinate of P' has been deselected. To deselect it,
click on it.
b.
Select the abscissa of P' first , then the
ordinate of P'.
Under Graph select Plot as (x,y).
c. Label it P'.
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- To construct the tangent line to the curve through P', we'll need
to know the coordinates of a point through which the tangent line
passes and we'll need to know the slope of the tangent line.
What point can we use to construct the tangent line?
How can we determine the numerical value of the slope of the
tangent line through P'? Use The Geometer's Sketchpad to determine
this value.
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How to find f '(x):
- Select f(x).
- Under Graph select Derivative.
How to evaluate a function:
- Click on Measure in the menu bar, then Calculate....
- Enter the function into the calculator
by clicking on the its algebraic form shown in the sketch.
- Enter the value for which the function is to be evaluated
by clicking on this value shown in the sketch. Click <OK>.
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- Using the point and slope you determined in Task 15, construct the
tangent line to the curve at P' in your sketch. Change the tangent
line's color to green and its line width to thick. Be sure the slope
of the tangent line changes as P' moves along the curve.
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How to enter a 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>.
How to change the features of a line:
- Select the line.
- Under Display select either Line Width or Color.
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Teacher Note: Unfortunately the Slope
feature is only available for lines constructed from two points,
and the tangent line was constructed from a slope and a point.
Thus, additional steps are necessary to refine the construction
of the tangent line so that Sketchpad's built-in Slope
feature can be utilized. (These steps are only necessary
if the tool is being contructed for use in parts 1 and 2 of
this activity.)
- Construct another point B on the tangent line by selecting
the line, then selecting Point on Line under Construct.
- Select the tangent line. Under Display select
Hide Line.
- Construct a new line through B and P'. The line BP' will
be the new tangent line constructed from two points.
- Use the Slope feature to display its slope.
- Change the new tangent line's color to green, its line
width to thick, and its label to "tangent line".
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- Use your construction of the tangent line to create a Custom Tool
which, given the algebraic form of any function f(x), will construct
the tangent line to f(x) through P'. The tool should plot the function
f(x), P, P', and the tangent line. It should also display the value
of the slope of the tangent line through P'; i.e., f '(xP).
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How to Create a Custom Tool:
- Create a sketch of an example of the construction you
want the tool to produce. You may use any Sketchpad tools
or menus to create this exemplar.
- Select both the given objects (in this activity, the algebraic
form of the function) and the desired resulting objects
you’d like the tool to produce(in this activity, the
function plot, P, P', the tangent line, and f '(xP)
). The order in which you select the givens determines the
order in which you’ll match the givens when using the
tool.
- Click and hold on the Custom Tools icon (double
arrow icon) in the Toolbox. Choose Create New Tool
from the menu that appears. You may name the tool in the
dialog box that appears, and click OK. Your tool is added
to the Custom Tools menu and is ready to use.
This explanation was adapated from: http://www.teacherlink.org/content/math/activities/skpv4-construcisitro/guide.html
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Teacher Note: So that the tool is
available for use in the Custom Tools menu every time
The Geometer's Sketchpad is opened on the student's
computer, save the sketch containing the tool in the Tools
Folder in the Sketchpad Folder on the computer's
C drive.
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- Test your Tangent Line Tool on a new sketch. If you wish,
edit your tool to use the constant label "tangent line"
for the tangent line every time the tool is used. Also, edit the tool
so that no matter what label the tool chooses to use for the point
on the x-axis you labeled P, the tool will automatically label the
corresponding point (xP, f(xP)) with a prime,
e.g., P'. This is called a variable label.
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To test your new Tangent Line Tool:
- Open a new sketch, show the grid, and plot a nonlinear function
such as f(x) = x2.
- Click and hold on the Custom Tools icon (double arrow
icon) in the Toolbox until the menu appears. To ensure that
your tool is selected for use, be sure it is check-marked in
the menu.
- Click on the algebraic form of the function shown in the sketch.
As you move the cursor, a point will appear next to the cursor.
Position this point on the x-axis. (Note: When the point intersects
the x-axis, the x-axis will change color.) Click the mouse button
to put the point in place.
- Click on the arrow tool to stop the tool from repeating.
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How to create a constant label:
- Click and hold on the Custom Tools icon (double arrow
icon) in the toolbox until the menu appears. Select Show
Script View. The script shows the given objects and each
step you took as you constructed the sketch.
- Double-click the step in the script which creates the tangent
line. On window that appears, enter the desired label (i.e.,
"tangent line") and be sure that Use Label in Sketches
is selected.
How to program a variable label for this sketch:
- Click and hold on the Custom Tools icon (double arrow
icon) in the toolbox until the menu appears. Select Show
Script View. The script shows the given objects and each
step you took as you constructed the sketch.
- Double-click the step in the script which creates the object
you want to have a variable label. A window will appear. On
window that appears, enter the following: ={3}'
Note: Inside the curly braces is the line number of the step
which defines the given object on which the variable name is
based. For example, ignoring the line numbers used by The
Geometer's Sketchpad, point P is defined on the third line
of the script in the Tangent Line Tool. The apostrophe
adds a " ' " directly after the label; i.e., the prime
in P'.
- Before clicking <OK>, be sure Use Label in Sketches
is selected.
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Last modified on March 2, 2004
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