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Introduction of Applets
Instructions
and a detailed help menu are provided with each applet. This Introduction
module illustrates the use of the two of them: the Java Math Pad
Applet and the PlotSolve Applet.
Java Math Pad Applet
This applet
allows the user to evaluate mathematical expressions. A mathematical
expression is made of numbers, variables, builtin and user defined
functions together with operations. For more information on the
use of this applet, click on the scroll down menu under "Help
Topics". The following examples will illustrate some of the features
of the applet.
Example 1 Entering
and Evaluating Mathematical Expressions
Evaluate the
expression _{}when _{}
Solution
Begin by assigning
values to the variables. On the Math Pad screen,
_{}
Next
we type the expression using * to indicate multiplication and
^ for exponents and press enter. Java Math Pad will return the
numerical value of our expression. See Figure 1.
Figure
1
Example 2
Defining and Evaluating Functions
Evaluate the
function _{}
Solution
Click on the
Clear button to clear the Java Math Pad screen. Then type
f(x)=3*x^2+17*x33
and press enter to define the function. Now type f(2.40) and
enter. Java Math Pad will return the value of the function. See
Figure 2.
Figure
2
The function
f(x) will be stored in Math Pad’s memory until clear it. So, you
can evaluate the function for as many different values of x
as you desire. To clear a defined function from Math Pad’s memory
type “clear function name”. For example, to clear the function
f(x) that we just defined, type, “clear f ”. You can also simply
define a new function named f.
Example 3
Evaluating Trigonometric Functions
Find the value
of each of the following trigonometric expressions.
(a) sin(2.14)
(b)
_{}
(c)
arcsin(0.5)
Solution
The trigonometric
and inverse trigonometric functions are builtin functions in
Java Math Pad. In the case of the sine and cosine functions, the
argument must be a number expressed in radians and must be enclosed
in parenthesis. For the inverse sin, the argument should be a
number between –1 and 1 and should also be enclosed in parenthesis;
Math Pad will give the answer in radians. See Figure 3. For more
information about the builtin functions, go to “builtin functions”
in Java Math Pad’s help menu.
Figure
3
Example 4
Combining Functions to Form a New Function
Given the
functions _{}, evaluate each of the following:
(a) _{}
(b) _{}
(c) _{}
Solution
Click the
Clear button to clear the Math Pad screen. Next define the three
functions. Use sqrt to denote the square root. Type f(x)=2*x+5
and press enter, then type g(x)=3x^2+2 and press enter, and finally
type h(x)=sqrt(x1) and press enter once again.
(a) Simply type f(2)+g(2) and press enter.
Math Pad will return the numerical value.
(b) For this problem, type f(1)/g(1) and press enter.
(c) To evaluate the composite function h(f(x))
when x = 6, you must first assign the value 6 to the variable
x. To do this type x = 6 and press enter. Then type h(f(x)) and
press enter.
See
Figure 4.
Figure
4
PlotSolve Applet
The purpose
of this applet is to plot functions. It can plot up to 10 different
functions at the same time. It can also be used to solve equations
numerically. This demo includes Instructions for using the applet
and examples that will illustrate some of the features.
 The Viewing
Window Parameters area is used to set the area of the graph
the user wishes to see. Use it as follows:
 Specify
the desired values for XMin, XMax, YMin, YMax. The applet
will do basic error checking. If an entry other than a number
is typed in, an error will be displayed, and the focus will
remain in the field which caused the error.
 When
Use yrange is checked, the supplied values for the
yrange will be used. Otherwise, the applet will find YMin
and YMax from the supplied functions.
 Any
change in this area will take effect only after the Plot
button is pressed.
 The Function
Information area is where the functions to be plotted are
entered. Use this area as follows:
 To
enter a new function, always press the New Function
button. Then, enter the expression defining the function.
The syntax is similar to the syntax used in the Java Math
Engine. For example, to defines the sine function, you would
type in sin(x). Make sure you use x for the variable in
your definition.
 A maximum
of 10 functions can be defined at the same time.
 Checking
Active means the function will show on the graph.
Unselecting it means the function will not show.
 Use
the + and  buttons to scroll through the list of defined
functions.
 Each
defined function has a different color assigned to it. The
selection is automatic.
 When
a function is defined, it is automatically assigned a name
of the form fi where i is a number which starts at 1 and
is incremented every time a new function is defined. The
name a function will be saved under appears to the left
of the field where it is defined.
 Once
a function is defined, its name can be used in the definition
of other functions. For example, if two functions have been
defined, the definition of function 3 could be f1(x)+x*f2(x)
Note that we use f1(x) and not just f1.
 The
functions defined will plot only after Plot has been
pressed.
 Del
Function will delete the function currently showing.
When deleting a function, make sure that it is not used
in the definition of another function. An error would occur
in this case. For example, if f2(x) = x + f1(x) and f1 is
deleted, then the definition of f2 contains an unknown symbol,
f1.
 The Zoom
and Trace area is where zooming and tracing take place.
Use this area as follows:
 To
zoom in, picture in your mind the rectangular region you
would like to zoom in. Left click on one of the corners
of this imaginary region. While holding the mouse button
down, move it to the opposite corner, then release it. As
you move the mouse, a rectangle will be drawn to help you
visualize the region. Once the mouse button is released,
the graph will redraw, XMin, XMax, YMin, YMax will be updated.
 To
go back to the original view (XMin = 10, XMax = 10, YMin
= 10, YMax = 10), press Reset Zoom.
 To
trace, simply single left click in the graph area. A point
will be generated by taking the xcoordinate of the point
where you clicked and the ycoordinate on the function currently
showing.
 To
move the point, use the Right or Left buttons.
The point will move on the function currently showing. By
selecting a different function, you can select which function
you want your point to follow.
 The
coordinates of the point will be displayed under X:
and Y:.
 The Control
Buttons area contains buttons which have a global effect
for the applet.
 The
Clear All button erases all the function definitions.
 The
Plot button is used to update the plot area after
changes in the other areas have been made.
 The Messages
area is where the applet communicates with the user. Error messages
as well as user information is displayed there. Errors are displayed
in red, while information is displayed in black.
Example 1
Graph the
function _{}.
Solution
Step 1) On
the Plotsolve applet, click the New Function button and type
the expression for the function (do not type f(x)=) using * to
indicate multiplication and ^ to indicate exponents, just as in
the Java Math Pad applet.
Step 2) Set
the viewing window. The default window is xmin –10, xmax 10,
ymin –10,
ymax 10. You can change these values to give the best view of
the function. In general, you can either specify the ymin and
ymax or you can let the applet choose appropriate yvalues for
the function. If you wish to specify the ymin and ymax, “Use
yRange” must be checked. For this example, set the following
values:
xmin –5,
xmax 5, ymin –5, ymax 12.
Step
3) Click the Plot button.
Figure 1 below
shows the graph.
Figure
1
Example 2
Solve the
equation _{} graphically.
Solution
Begin
by graphing both the left side and the right side of the equation
on the same coordinate axes. To do this, click the New Function
button and enter x. Next click New Function again and enter x^3+1.
Click the Plot button. The resulting graph is shown in Figure
2. Notice that the default window, xmin –10, xmax 10, ymin
–10, ymax 10 gives a good view of the graphs and their point
of intersection.
Figure
2
The
graph in Figure 2 shows one point of intersection. The xcoordinates
of this point is the solution to the equation. Follow the steps
outlined below to estimate this solution.
Step
1) Place the cursor on the point of intersection and click the
left mouse button. The coordinates of the point will appear in
the Zoom and Trace portion of the applet. See Figure 3.
Figure
3
Step
2) Figure 3 shows that the xvalue of the point is approximately
–1.27. You can get a better estimate of this value by zooming
in on the portion of the graph containing that point. To zoomin
on the graph, place the cursor on the graph near the point, but
a little to the left and above it; then hold the left mouse button
down and drag it to the right and down to form a “box” around
the point of intersection. When you are satisfied with the “box”,
let go of the mouse button and the graph will be redrawn in that
viewing window. See Figure 4.
Figure
4
Step
3) Notice that the closeup view shows that we were not actually
at the point of intersection. Now move the point closer to the
actual point of intersection using the left (or right) button
under Point Motion. See Figure 5.
Figure
5
Just
as before, the coordinates of the point are displayed by the applet.
You can see that a good estimate for the solution to this equation
is x = 1.32. You can repeat this procedure as often as necessary
to get closer and closer estimates for the point of intersection.

