Dynamic Visualization of PreCalculus and Calculus
Concepts with Geometer's Sketchpad
for AMATYC Workshop
November 5, 1998
Martin Flashman
Department of Mathematics
Humboldt State University
flashman@axe.humboldt.edu
http://flashman.neocities.org
What to do when you have
 a question: Ask!
 an idea on how to do something: Try It!
Abstract: Traditional approaches to precalculus and calculus
emphasize graphs and functions in static visualizations. Those attending
this presentation will discover how Geometer's Sketchpad's dynamic features
allow the instructor and the student to create an active environment for
exploring functions, graphs, roots, extremes, differentiability, tangent/vector
fields, and integration.

Introduction.

Who are we?

Getting familiar with Geometer's Sketchpad layout.

Overview of this workshop.

Coordinates and Graphs from Formulae.

Use the Graph menu to Create Axes.

Plotting Points

Select point plotting tool and plot points with mouse.

Plot free point in the plane.

Plot point on Xaxis.

Select select and translate tool.

Select the Xaxis with your mouse.

Use the Construct menu to construct
a point on the object (the Xaxis).

Select the Label tool.

Label one of the points you have constructed on the X axis.

By "double clicking" on the point, change the label of the
point to x.

(optional) Change the font and size of the label.

Finding Coordinates and Calculating with Measurements.

Select a point (or two hold the shift key and use the mouse
left button).

Use the Measure menu to obtain the coordinate(s)
of the selected point(s).

Use the Measure menu to calculate
the first coordinate of a point on the Xaxis. After the screen calculator
appears, click on coordinates of the point. Then select OK on the
screen calculator or hit the enter key.
[Quicky: Triple click with the left mouse
button on the coordinates of the point. Select OK on the screen
calculator or hit the enter key.]

Use the Measure menu to calculate
other values as functions of the first coordinate of the point on the Xaxis
from the following list:

x^{2}

x^{3 } 4x

x^{2 } 3x + 2

e^{x}

ln(x)

sin(x)

Exercise: By selecting and dragging the point on the
X axis, estimate the value(s) of x for which each of these functions
individually has its value equal to

0

.5

1

Optional: Find the coordinates of a free point in
the plane.Use those coordinates to calculuate following values as functions
of both of the coordinates of that point.

x + y

x^{2} + y^{2}

x^{2}  y^{2}

e^{x} e^{y }
e^{x+y}

ln(x^{2} + y^{2 })

sin(x) cos(y)

Plotting points from measurements and calculations.

Select a pair of measurements, x and f (x)
of part C.4, or your own favorite. Using the Measure menu Tabulate
this pair. By moving the point on the xaxis, and then "double clicking"
on the table, you can add other pairs to the table. (Note here undo/redo
on the Edit menu)

Select the table of pairs you have made in D.2. Plot the
points that correspond to these pairs by using Plot table data
from
the Graph menu.

Select a pair of measurements, x and f (x)
[ in that order, remember to hold the shift key down to select the second
point] of part C.4, or your own favorite.Using the Graph menu
plot a single point from the selected pair of measurements.

Playing with controlled points.

Display menu featuresTrace
and Animate  These will be discussed in the presentation.

[Hold down the shift key and ] Select the point on the xaxis,
the xaxis, and the point (x, f (x)). Using
the Construct menu, the graph of the function f can
be created using Locus.

Using the graph of a function for investigating the derivative
and the integral.

The derivative.

Plot points on the locus.

Construct a line (segment) between two distinct points on
the locus.

Use the measurement menu to find the slope of the line.

By moving one point on the curve closer to the locus point
for the curve estimate the slope of the curve (the derivative) at that
point.

Plot the estimated derivative point (x , f
'(x)) and then use that point and the Locus construction to graph
the estimated derivative. Use the display menu to change the color of the
derivative locus.

Exercises for the derivative:

Notice the usual features of the relation between the derivative
and the shape of the curve.

For one of the functions in the previous exercises, create
points with coordinates (1,0) and (1, f (x)) and construct
a thick line segment between these two points. Similarly construct a thick
line segment between (2,0) and (1, f '(x)). Describe the
motion of these segments as you move the original pont along the x
 axis.

Plot the second derivative for one of the functions in the
previous exercises and examine its relation to the graph of the function.

Some Bells and Whistles: Action Buttons and other Edit and
Display features:

From the Edit menu create Action
buttons that hide/show some of the figures constructed
so far.

From the Edit menu create Action
buttons that animates the point on the x axis
that controls f (x) for some of the figures constructed so
far.

The Definite Integral.

Plot and mark numerous points on the graph (locus) of a selected
function.

Plot points on the x axis vertically above or below
the far left and far right points on the curve. [Select the point and the
xaxis,
then the construct perpendicular line using the Construct
menu.]

Use the points from the previous two constructions to Construct
a polygon interior. Measure the Area
of this polygon which is an estimate for the definite integral of the function.

Measure the first coordinate of the right point of the estimating
polygon. Construct (using Locus) the graph of the areas of the polygons
as a function of the right hand endpoint for the region.

Vectors and Vector Fields Just a start.

Using and making Scripts.

From the File menu choose Open.
Go to the samples subdirectory, then scripts, and finally
utilities.
Select arrowopn.gss and open that script.

Construct and mark two points on the sketchpad. Now (1) use
the STEP button, (2) use the PLAY button, and (3) use the FAST button in
the Script window to construct an open arrow (1) step by step, (2) all
at once but seeing the steps, and (3) all at once but FAST!

[This will be discussed in the presentation.]
Use the Display menu to change the Preferences
for "More" to determine a Script Tool Directory. Now use the Script Tool
to construct an open arrow.

Construct a free point in the plane. Find its coordinates.
Now use those coordinates (x, y) to construct the point (x
+ y, y  2x). Construct an open arrow with tail (x,
y)
and head (x + y, y  2x). You have constructed the
vector ( y,  2x).

Select the tail of the vector and move it in the plane. Notice
how the vector changes its length and direction.

Exercise: By placing a trace on the tail of the vector constructed
in part d, trace an integral curve for the vector field determined by this
vector that passes through the point (2,1).

Graphs from measurements of Sketches.

Draw a sketch of a problem figure to be measured.
Keep the figure dynamic.
Be sure you can control the parameters that interest you.

Problem: Given a line l and two points A and B not
on the line, determine the position of a point P on the line where the
total of the distances between P and the points A and B will be smallest.

Problem: Given a line l and three points A, B, and
C not on the line, determine the position of a point P on the line where
the total of the distances between P and the points A, B, and C will be
smallest.

Problem: Find the area for the rectangle of largest area
that can be inscribed in a given right triangle using the right angle vertex
of the triangle for one of the vertices of the rectangle.

Problem: Find the area for the rectangle of largest area
that can be inscribed in a given right triangle using the hypotenuse for
one side of the rectangle.

More challenging: Given a length AB, determine the dimensions
of an isosceles triangle of maximal area that will be enclosed by using
the length AB to determine the total length of two sides of the triangle.

More challenging: Given a length AB, determine the dimensions
of a rectangle of maximal area that will be enclosed by a using the length
AB to determine the toal length of three sides of the rectangle.

Optional label some/all of your objects. Write text.

You can show and change labels.Use the Text tool (the hand)
to show the label of an object. This tool also is used to change the label,
write text on the sketch, or change the label on any button, etc.

Use labels for objects that makes sense.

Use the Measure menu to measure the objects
you wish compare.

Notice units that appear. ( You can change the units by going
to the Display menu and using the Preferences.)

Use the Graph menu to plot a
point
as (x,y) with the coordinates determined by two of the measurements
you have selected. (Select the value of the controlling variable first
for the first coordinate of the point.)

Play with the controlling measurement on the figure.

Tables Measure (Note again here undo/redo)

Plot from Tables Graph

Playing with controlled points.

Display menu featuresTrace
and Animate

Construct the graph of a function (x, f (x))
using Locus.