
                      +---------------------------------+
                      |                                 |
                      |   PARAM PROGRAM TUTORIAL FILE   |
                      |                                 |
                      +---------------------------------+


The purpose of this file is to provide a tutorial lesson on most of the basic
features of the program called PARAM.  While it is possible to learn how to
use the program by reading all the built-in help screens contained within
each menu, if you are a first-time user, we suggest you read this tutorial
while first running the program, and then later you can digest all the
information contained in the help functions within each menu in the program.
You can import this file, PARAM.TXT, into any word processor and then print
it so you can read the hard copy output while you run the program.

To begin the tutorial you must have the program file PARAM.EXE.  We assume
this file is in the current directory on the currently selected disk drive.



PRELIMINARIES
=============


KEYBOARD CONVENTIONS
--------------------

As part of this tutorial we need to give directions on which keys to press on
your keyboard.  We will enclose in angle brackets single keystrokes that you
should type.  For example, if we ask you to type the first three letters
of the alphabet we will show <A> <B> <C>.  If we ask you to press the control
key, the alternate key, the backspace key, the space bar, or the enter (or
return) key, we will show <CTRL>, <ALT>, <BACKSPACE>, <SPACE> or <ENTER>.

Each enclosure in angle brackets should refer to exactly one keystroke or one
character.  On most keyboards, to type in a left or right parenthesis requires
using a <SHIFT> key in conjunction with another key.  We will NOT show the
shift key as a separate keystroke in this tutorial because we consider
entering a parenthesis as entering a single character.  Other characters like
<+> or <*> can be entered in two different ways on some keyboards, one way
uses the <SHIFT> key, the other way does not.  In general, we would only show
the single character in angle brackets and we leave it up to you to decide
whether or not the <SHIFT> is needed to enter the character.

These keyboard conventions should make clear exactly how many and which keys
you press.  If it is necessary to press two keys simultaneously we will show
a connecting plus sign between the keystrokes.  This is done primarily with
the Control key <CTRL>, and the Alternate key <ALT>.  For example, when we
show <CTRL>+<W> it means you should press both the Control key and key W at
the same time.  As another example, <ALT>+<X> is used in the Main Menu to
exit from the program.


DISPLAY STRINGS
---------------

We also need to indicate the contents of text strings you might see on the
display screen.  Such text parts will always be displayed in double quotes in
this tutorial file.  You will not see the double quotes on the screen, and the
screen may contain other text parts that we do not show in this file.  The
double quotes are simply a convenient way to indicate parts of what you may
see on the display.


ADVICE FOR NOVICES AND EXPERTS
------------------------------

This tutorial file assumes you have the mathematical background required to
understand the features that will be demonstrated.  You may find some sections
more applicable to novices than experts, or vice versa, depending on your
background and experience.  If you encounter an example that is beyond your
understanding, you can either skip that example, or you can press the keys and
view the results, even though you may not fully comprehend the output.  This
tutorial does not discuss techniques on how to best use or apply the available
features.  It only serves to demonstrate the basic features and capabilities
which you can learn to apply to solve problems that are of interest to you.


GETTING STARTED
===============

To begin running the PARAM program type the command:

                          <P> <A> <R> <A> <M> <ENTER>



A FIRST EXAMPLE
===============

You can't perform any really useful operations (except reading the Main Menu
help screens) without first keying in a function.  So perform the first menu
item by pressing
                                      <K>

to key in a new function.  An edit box will appear on the screen into which
you are to type your mathematical formula.  The first example function we will
enter is the ellipse given by

                             X=5*COS(T) and Y=3*SIN(T).

As in the PARAM program, in this tutorial file we will use the letter T to
denote the parameter.

The asterisk character * is used to to indicate multiplication.  Note that
you do not type in the X= or Y= parts of the formula since this program knows
X and Y are functions of T.

When prompted to enter X as a function of T type

                   <5> <*> <C> <O> <S> <(> <T> <)> <ENTER>

After pressing ENTER a brief message should appear indicating your function
formula for X has been accepted and the new formula should now appear at the
top of the Main Menu display screen.  Then you will be prompted to enter the
formula for Y.

When prompted to enter Y as a function of T type

                   <3> <*> <S> <I> <N> <(> <T> <)> <ENTER>

After pressing ENTER a brief message should appear indicating your function
formula for Y has been accepted and the new formula should now appear at the
top of the Main Menu display screen.  Together both formulas should appear
as


                                "X = 5*COS(T)"
                                "Y = 3*SIN(T)"


We will now go the Graph Menu by pressing key

                                      <G>

You should see a new menu labeled as the "Graph Menu".


In this first example the parameter T represents an angle.  To graph the
complete ellipse we want the angle to complete one revolution, and since
T will represent radians, we want our T values to range between 0 and 2*pi.

Press key
                                      <S>

to change the starting and ending values for T.

When prompted to enter new starting and ending values for T type

                     <ENTER> <2> <*> <P> <I> <ENTER>


The T domain should now be the interval between 0 and 2*Pi which is just over
6.28.

The last value we need to change is called delta-T.  This value determines
how densely the coordinate functions are sampled over their domain.  For
this first example we want to sample about every 1 degree, but since T is
in radians we need to enter the equivalent value in terms of radians.  But
we can enter the expression exactly as PI/180.

Press key
                                      <N>

to change the delta-T value and when prompted type

                    <ENTER> <P> <I> </> <1> <8> <0> <ENTER>


Note the number of T samples is 360.


Then to make the graph press key

                                      <G>

and you should see the graph of an ellipse that corresponds to the above
coordinate functions.

It is normal for the program to sound a short beep to indicate it is finished
making a graph.  After admiring the graph press either

                                <ESC> or <ENTER>

twice and you should return back to the Main Menu.  Entering and graphing
most PARAM functions is about as simple as this first example.




PERFORMING INTEGRATION TO FIND AN AREA
======================================

Instead of going to the Graph Menu we are going to the Integration Menu.
Press key
                                      <I>

and you should see a new menu labeled as the "Integration Menu".

The default limits of integration let T vary from 1 to 10.  We want to
approximate the area in the upper half of the ellipse so we want T to
range between 0 and pi.

Press key
                                      <L>

to change the limits of integration.  When prompted for the new limits type

                          <0> <ENTER> <P> <I> <ENTER>


The limits should now define the interval 0 to Pi.

The number of subintervals should already have a default value of 30 which
should be adequate for our first area example.


The delta-T value in the Integration Menu differs from the delta-T value in
the Graph Menu, but both values play similar roles.  Here delta-T denotes the
sampling density over the interval of integration.  Note that delta-T =
0.1047197551196598.

Now press key
                                      <A>

to calculate the area.  The next graph made should now show the complete
ellipse, and then the area is generated by drawing 30 rectangles that extend
from the top part of the graph down to the X-axis.  The answer should be
displayed as


                           "Trapezoid Rule Area"

                         "Area = -23.518904235222031"


Press
                               <ESC> or <ENTER>

once and you will see the above value re-displayed in the Integration Menu.


The reason for re-displaying the integral result is so you can see the
function formula, the limits of integration and all the integral parameters
in one screen.

The true value is 15*PI/2 which as a pure decimal is approximately
23.5619449019.

If you are curious why the displayed value is negative, it is because the
direction of traversal of the generated rectangles is from right to left
across the screen.  Also, whenever an integral approximation is made, the
rectangles are drawn from the graph to the X-axis.

Press
                               <ESC> or <ENTER>

again to completely return to the Integration Menu.

To better understand the sign of an integral answer, we will change the
limits of integration to approximate the area in the lower half of the
ellipse.

Press key                             <L>

to change the limits of integration, and when prompted type

                 <P> <I> <ENTER> <2> <*> <P> <I> <ENTER>


These new limits make the interval range from PI to 2*PI.

Now press key                         <A>

to approximate a new area.

This time the graph is drawn and the rectangles used in the approximation
traverse from left to right.  But the answer is still negative.


                           "Trapezoid Rule Area"

                         "Area = -23.518904235222031"


The reason this answer is negative is because the heights of the rectangles
are counted as negatives since the rectangles extend below the X-axis.

Lets do one more integral approximation.

Press
                               <ESC> or <ENTER>

twice to return to the Integration Menu.

Now change the interval of integration to [0,2*PI].  Press

                                      <L>
and when prompted type

                      <0> <ENTER> <2> <*> <P> <I> <ENTER>

Change the number of subintervals to 100 by pressing

                                      <N>
and when prompted type
                               <1> <0> <0> <ENTER>

Note the value of delta-T is now 0.0628318530717959.

Then make one more integral approximation by pressing

                                      <A>

and watch the rectangular approximations sweep out the entire area inside the
ellipse.

The integral approximation still turns out to be negative, but of course we
would interpret the resulting area as the absolute value of the value
returned.  The true value is 15*PI = 47.1238898038.


                           "Trapezoid Rule Area"

                         "Area = -47.092889646985032"



Press
                               <ESC> or <ENTER>

twice to return to the Integration Menu.




AN ARCLENGTH APPROXIMATION
==========================

The PARAM program can be used to calculate arc lengths.  Since we still
have the ellipse entered with one complete revolution as represented by the
limits of integration, lets try to calculate its arc length.

From the Integration Menu press
                                      <N>

to change the number of subintervals, and when prompted type

                                <1> <0> <ENTER>

Our first approximation will be with 10 arcs.

Press                                 <R>

to begin the calculation of the arclength around the ellipse.  The screen
should first show the graph and then it will draw the approximating arc
length elements.  As you can see from the graph, there is considerable
space between the straight line segments and the curved sections on the
ellipse.


                         "Arc Length Riemann Sum"

                       "Length = 25.1104523394282166"



Press                          <ESC> or <ENTER>

twice to return to the Integration Menu.

To make a better arc length approximation we will double the number of
subintervals from 10 to 20.

From the Integration Menu press
                                      <N>

to change the number of subintervals, and when prompted type

                                <2> <0> <ENTER>

Press                                 <R>

to calculate the arclength.  You should see a new graph with a better
approximation to the true arclength.



                          "Arc Length Riemann Sum"

                       "Length = 25.4221531512306714"



But we can still do better.  So we will change the number of subintervals to
100.

Press                          <ESC> or <ENTER>

twice to return to the Integration Menu.

From the Integration Menu press
                                      <N>

to change the number of subintervals, and when prompted type

                              <1> <0> <0> <ENTER>

Press                                 <R>

to calculate the arclength.  You should see a new graph with a better
approximation to the true arclength.



                          "Arc Length Riemann Sum"

                       "Length = 25.5228000476012589"



This time you probably can't tell any difference between the approximating
arcs and the graph of the ellipse.

Press

                               <ESC> or <ENTER>

three times to return to the Main Menu.




THE GRAPH/TANGENT/NORMAL TRACE MODES
====================================

When a graph is made, you can enter any one of three Trace Modes to trace
along the graph one point at a time or to see the variation in the tangent or
normal lines to the graph.

Press
                                      <G>

twice, the first time you should go to the Graph Menu and the second time
should cause the graph to be drawn.  While looking at the graph press key

                                      <T>

and you should see a tangent line drawn on the graph, starting at the T-value
which is the midpoint of the formula's current domain.

The coordinates of the current point of tangency are displayed at the bottom
of the screen, and the top of the screen shows the equation of the tangent
line in y=mx+b form.



                 "Y = 38.1940446977229414*X + 190.970223488614707

                T = 3.1415926535897932
                X = -5.                     Y = 1.9515639105E-18"



The rectangular coordinates of the point of tangency are given with the T
parameter value that corresponds to that point, at the bottom of the screen.

These values are subject to interpretation due to the presence of normal
round-off error.  The T-value is really pi, and the tangent point is really
(-5,0).  The true slope of the tangent line is infinite at that point on the
ellipse, but the above slope value of 38.194 represents a very steep slope.


You may press either the left or right cursor arrow keys to cause the tangent
line to move across the graph.  As you move the point of tangency and the
tangent line you should note the coordinates of the point of tangency at the
bottom of the graphics screen get updated, as does the actual equation of the
tangent line that appears at the top of the screen.  The parameter value as
well as the rectangular coordinates are constantly updated.  You should think
of the cursor keys as controlling the T parameter by increasing or decreasing
T.  In fact, you can also use the up and down arrow keys to control T.

Pressing a shift key in conjunction with any of the cursor arrow keys causes
a larger amount of movement, for each keypress.  This applies only to any of
the cursor arrow keys on your numeric keypad.  Try moving the tangent line
along the curve by repeatedly pressing the up cursor arrow key with the
<SHIFT> key.  Then move in the opposite direction by using the down arrow key.

After moving the tangent line back and forth, press

                                      <N>

to switch to showing the normal lines to the graph.  Then continue to move
the current point around the graph using the cursor keys to view the tracking
of the normal lines to the graph.  You can press keys

                                   <T> or <N>

to flip the current line between a tangent and a normal.

Next, press                           <G>

to enter the Graph Trace mode.  This time only a single point is plotted.

Again press the left or right or up or down arrow keys to change the T
parameter and thus move the point along the curve.

Finally, press                 <ESC> or <ENTER>

to exit from the Trace Mode.


Then press                     <ESC> or <ENTER>

two more times to quit viewing the graph and return to the Main Menu.




THE COORDINATE TRACE MODE
=========================

When you use PARAM to graph a parametric curve, the program generates the
graph and then waits for you to press a key while you are looking at the
graph.  If the key you press is either the  ESC  key or the  ENTER  key then
you are simply returned immediately to the Graph Menu.  If instead, you press
key C, you will enter what is called the Coordinate Trace Mode.  This is one
of the most useful modes of the entire PARAM program.

To demonstrate this mode, we assume you still have the function formulas


                                "X = 5*COS(T)"
                                "Y = 3*SIN(T)"


displayed at the top of the Main Menu display screen.  Press key

                                      <G>

to enter the Graph Menu and press
                                      <G>

a second time to actually make the graph of the function.  The program
should sound a short beep when the graph is finished.

Now press key
                                      <C>

and you should be in the Coordinate Trace Mode.  You will know when you are
in this mode because the bottom of the graphics screen should show the X and
Y coordinates of a point that is initially in the exact center of the display
screen.  Now, using your numeric keypad, press key

                                      <3>

and you should see the point cursor move away from the origin down into the
fourth quadrant.  Continue pressing key

                                      <3>

until the trace point is approximately near the point X=1 and Y=-1.  You will
probably not be able to hit this exact point, due to the limited resolution
of your graphics display screen.

Pressing key 3 on the numeric keypad causes the trace point to move in a
southeast direction.  Pressing keys 8, 2, 6, and 4 would cause movement in the
north, south, east, and west directions, corresponding to the direction of the
arrows on those keys.  Pressing key 7 would cause a movement in the northwest
direction, and pressing key 9 would make the movement northeast.  Key 1 moves
southwest.  So the arrangement of the keys on your numeric keypad serves as a
convenient way to represent movements in directions corresponding to the
sides and corners of the rectangular arrangement of the keys.

Pressing key 5 in the center of the numeric keypad causes the cursor to be
centered on the graphics screen.  Sometimes the trace point is hard to see
because it overlaps some other graphic object that is already drawn on the
screen.

If you have a large 101-key keyboard with separate cursor arrow keys, you can
use the extra arrow keys to move the trace point in a direction corresponding
to the direction marked on each key.

Pressing a SHIFT key in conjunction with one of direction keys causes the
trace point to move 5 pixels at a time instead of one pixel at a time.  This
makes for a little faster movement for longer distances.  This SHIFT key
feature works only with any of the numeric keypad keys.

As you continue to press any cursor key, the coordinate trace mode point
cursor should continue to move in the direction corresponding to the key, and
more important, the coordinates at the bottom of the screen will continually
be updated to correspond to the position of the cursor point you see.

Now we are going to try to use the trace point to find one of the y-intercepts
of the graph of the function.  Use the cursor keys to move the trace point
over the y-intercept point on the positive Y-axis.

The true point of intersection should occur where X=0 and Y=3., but the
limited resolution of your display means you will probably not see these exact
values displayed for the point's coordinates.




ZOOMING IN ON A POINT
=====================

Press the key for the lower case character z

                                      <z>

The graphics screen window should change and a new graph will be made which
encompasses an area that is 1/4 of the previous graphics screen.  The new
center of this screen will be the old Coordinate Trace Mode point, and you
will remain in the Coordinate Trace Mode.  Press lower case character z a
second time

                                      <z>

and you will continue to zoom in on the origin.  Each time you zoom in, a
completely new graph is drawn.

Pressing lower case z makes the new graphics window smaller and corresponds to
zooming in on a smaller portion of the graph.  Pressing upper case Z performs
a zoom out operation that makes the size of the XY-plane window larger.  You
should remember the relation between the size of the letters z and Z to help
you decide how to zoom in or out.  In either case, the Coordinate Trace Mode
point remains as the point of central focus, and in fact this point will be
the new center of the new graphics screen after zooming either in or out.

Now press the Escape key

                                     <ESC>

twice.  The first time you exit the Coordinate Trace Mode, and the second time
you return to the Graph Menu.  Note the range of values for the current XY-
plane window X & Y minimums and maximums.  While looking at these values,
press
                                  <CTRL>+<W>

and you should see the window min/max values returned to their default values
where X ranges between -7 and +7 and Y ranges between -5 and +5.

Then press key
                                      <G>

to make a new graph and you should see the graph has returned back to its
original state before we began zooming in.




ZOOMING IN USING A BOXED WINDOW
===============================

Press key                             <C>

to re-enter the Coordinate Trace Mode.

Now use the cursor keys to move the Coordinate Trace Mode point to where
it is near the point X=-2 and Y=+4.  The placement does not have to be exact.
Then press key
                                      <X>

to mark the current point as an anchor point (either upper or lower case X
will do).  Whenever you are in the Coordinate Trace Mode and you press key X
you mark an anchor point.

Now move the cursor in a southeast direction until it is near the point X=+2
and Y=+1.  As the cursor moves you should see a rectangle drawn which has the
anchor point you marked as its upper left corner.  The rectangle which you are
defining can be used to zoom in that part of the graphics screen enclosed
inside the rectangle.  When the lower right corner is near X=+2 and Y=+1,
press key
                                      <Z>

(lower or upper case is not relevant with the boxed rectangle) and a new
graphics window should be drawn which shows the interior of the rectangle,
but this interior now fills the entire graphics screen.  The position of the
new Coordinate Trace Mode point is in the center of the new graph window, and
you remain in the Coordinate Trace Mode with the point's coordinates displayed
at the bottom of the screen.

After marking an anchor point, but while still defining the zoom-in rectangle,
you can press the Escape key to abort both the anchor point and the zoom-in
rectangle.  However, you remain in the Coordinate Trace Mode.

Now press
                                     <ESC>

two times to return to the Graph Menu.  Once in the Graph Menu press

                                  <CTRL>+<W>

to put the window back to its default position and size.




USING THE LINE DRAWING MODE
===========================

Now that you know about the Coordinate Trace Mode and the Tangent Line Mode
we are going to try the Line Drawing Mode to manually approximate the
tangent line to a graph, at a particular point on the graph.  Press key

                                      <G>

to make the graph of the ellipse and then press key

                                      <C>

to go into the Coordinate Trace Mode.  Use the cursor keys to move the trace
point out in the first quadrant until it is near where X=4.0 and Y=1.8, but on
the graph of the ellipse.  The placement does not have to be exact.

To start the Line Drawing Mode at the current cursor point, press key

                                      <L>

and you should see a line drawn on the screen through the trace point.  The
equation of the line, in y=mx+b form, should be at the top of the screen.
When you first enter this mode, the line should have an approximate slope of
m=1.0.

The Coordinate Trace Mode point serves as an anchor point, through which the
line is drawn.  Now you can use either the up or right cursor arrow keys to
increase the slope or use the down or left arrow key to decrease the slope of
the line.

Try decreasing the slope of the line.  Each keypress causes a change in the
angle by about 1 degree, or 5 degrees if you simultaneously press a SHIFT
key.  The SHIFT feature only applies to the arrow keys on your numeric keypad.
You can use either the left & right or the up & down arrow keys to increase or
decrease the slope of the line.

Keep decreasing the slope until the line nearly matches what should be a
tangent line to the graph.  The true tangent line has the equation that is
approximately  y = -0.8X + 5.0.

Once you are satisfied with your tangent line approximation, press key

                                      <P>

to cause a radical change in the line.  Pressing key P makes a perpendicular
line through the anchor point, i.e., the new line has a perpendicular slope
compared to the line slope at the time you press key P.  This should
correspond to the normal line to the graph.  Now press key

                                      <P>

a second time and the previous line should be restored.

As you increase or decrease the slope of the line, the line may become nearly
vertical, and the slope may abruptly change between a large positive number
and a large negative number.  In fact, the slope is usually calculated to be
in the range between -200 and +200, both of which represent nearly vertical
lines.

That is about all there is to the Line Drawing Mode.  If you press either

                               <ESC> or <ENTER>

you will exit the Line Drawing Mode and you can confirm this has happened
because the equation of the line at the top of the graphics screen will
disappear.  However, you will remain in the Coordinate Trace Mode and you
should still see the coordinates of the trace point at the bottom of the
screen.  Press

                               <ESC> or <ENTER>

again, and you will exit the Coordinate Trace Mode and just see the graph.

So press
                               <ESC> or <ENTER>

one more time to return to the Graph Menu.




MULTIPLE GRAPHS AND FINDING POINTS OF INTERSECTION
==================================================

Press key
                                      <G>

to make a new graph, and after you hear the beep when the graph is complete,
press key
                                      <S>

and the picture will be saved in memory.  After seeing a brief message that
the graph has been saved in memory, the graph will re-appear.


Press                            <ESC> or <ENTER>

once to return to the Graph Menu.

The graph of the ellipse has been saved and we are now going to change the
current function formula to the curve represented by X = T*T-3 and Y =
T-T*T*T/3.

Press
                                      <K>

to key in new coordinate functions.  The bottom portion of the screen will
clear.  A hidden feature of the PARAM program is the ability to key in new
coordinate functions from within any menu by pressing key K.  Once you get
past the Main Menu you do not have to return all the way to the Main Menu
just to key in new coordinate functions.

When prompted for X press

                          <T> <*> <T> <-> <3> <ENTER>

When prompted for Y press

                    <T> <-> <T> <*> <T> <*> <T> </> <3> <ENTER>


You should see the new formulas at the top of the Graph Menu screen.


                                  "X = T*T-3
                                   Y = T-T*T*T/3"


We need to change the graphing domain for this new curve.  Press

                                      <S>

to change the starting and ending T-values, and when prompted press

                              <-> <3> <ENTER>

                                <3> <ENTER>

The graphing domain use T in the interval [-3, 3].

Next, press
                                      <G>

to graph the new curve.  You should see a hair-pin curve that goes through the
origin twice.

Then press
                               <ESC> or <ENTER>

to return to the Graph Menu.  The graph we just made was formed on a blank
background because the Background Graph should be turned off.  Press key

                                      <B>

to change the Background Graph value to  "On".


Then press key
                                      <G>

to make a new graph and as the new graph is made, you should note that the
initial graph background is the old ellipse that was saved in memory, and
the new hair-pin loop is then graphed on top of the old background graph that
was saved.

We wish to find the point of intersection of the new curve and the ellipse
that lies in the 1st quadrant, near the point X=2.94 and Y=2.42.  Press key

                                      <C>

to enter the Coordinate Trace Mode.  Move the cursor to the point of
intersection of the two graphs and read the coordinates at the bottom of the
screen.  To two decimal places the coordinates should approximate X=2.94 and
Y=2.42.

Now press
                                <ESC> or <ENTER>

two times to return to the Graph Menu, and once there, press

                                      <B>

to turn the Background Graph  "Off", and press

                                      <G>

to make a new graph which now only shows the hair-pin loop.  The ellipse
remains saved in the background, until you quit the program, or until you
save a new graph which overwrites the existing background graph.  Any number
of parametric graphs may be overlayed on top of each other and saved in the
background.

Now press
                                <ESC> or <ENTER>

twice to return to the Main Menu.




FINDING THE MAX/MIN EXTREMA
===========================

Another feature of this program is the ability to automatically find the
maxima and minima of any parametric curve.  We assume the current coordinate
functions are the same as used in the last example above.  The top of the
Main Menu screen should show


                                  "X = T*T-3
                                   Y = T-T*T*T/3"



Press                                 <E>

to get to the menu labeled the Extrema (Max/Min) Menu.


Press key                             <L>

to set the limits for the search interval.  You will be prompted to enter
the lower T-value and then the upper-T of the search interval.

We want to enter values that correspond to the range between the opposite of
the square root of 3 and the positive square root of 3.  We could use the
decimal 1.732 (George Washington's birthday = 1732) to approximate the square
root of 3, but the program will accept an exact expression for this decimal.

Press
                            <-> <S> <Q> <R> <T> <(> <3> <)> <ENTER>

                              <S> <Q> <R> <T> <(> <3> <)> <ENTER>


The search interval for this example corresponds to the small loop of the
hair-pin curve.

We also need to decrease the number of sample points from 500 to 100.  Press

                            <N> <1> <0> <0> <ENTER>

The visual search attribute should be turned on by default.  Press

                                      <S>

to begin the search.


The program should first graph the curve and then it will show a trace
point move across the graph, but with T restricted to the search interval.
The interval will be subdivided into 100 equally spaced points and the
program will determine the absolute extrema of these sampled points.  The
sampling includes both the X-coordinates and the Y-coordinates.  The program
beeps when the search ends, and will show two Y-coordinates at the bottom of
the graphics screen.



         "Y = 0.66664557122357            Y = -0.66664557122357"



The value on the left is the maximum Y and the value on the right is the
minimum Y.  If you look closely at the graph you should also be able to see
the two points on the graph that are marked as the extreme points.  Also note
the marked points that represent the minimum and maximum X-values.

Press                          <ESC> or <ENTER>

once and you will return to the Extrema Menu where you should see the complete
coordinates of the four points that have been found.


     "Minimum X = -3.                    Maximum X = 2.168404345D-19
              T = -2.1006417092E-19              T = -1.7320508075688773
              Y = -2.1006417092E-19              Y = 1.0842021725E-19

      Minimum Y = -0.66664557122357      Maximum Y = 0.66664557122357
              T = -1.0045894683899488            T = 1.0045894683899488
              X = -1.9908                        X = -1.9908"



In interpreting these values we should say X has a minimum value of -3 where
T=0 and Y=0.  The minimum and maximum Y-values occur where X=-2.

Press                          <ESC> or <ENTER>

again to completely return to the Extrema Menu.


For the second extrema example we will turn off the visual display attribute.
Press
                                      <V>

to turn off the visual display option.  The program will perform the search
without going into its graphics mode.  Press

                                      <S>

to begin the search in text mode.  You should note a small symbol at the
bottom of the screen that looks like it is spinning while the search
continues.  When the search ends the program will sound a short beep and you
should see the same four points returned as shown above.  Press

                                     <ESC>

two times to return to the Main Menu.



THE DISCRETE VERSUS CONNECTED GRAPH TYPES
=========================================

When graphing any parametric curve you can have the program make the plot for
the graph in one of two ways.  The default graph type is called discrete and
this means the individual points on the graph are plotted one by one with
no particular connection between one point and the next.  To see this
attribute in a slightly emphasized form enter the coordinate functions
X = 6*COS(T)  and  Y = 4*SIN(T).

Press                                 <K>

to key in new coordinate functions and when prompted type

                      <6> <*> <C> <O> <S> <(> <T> <)> <ENTER>

                      <4> <*> <S> <I> <N> <(> <T> <)> <ENTER>

The status line at the top of the screen should show

                             "X = 6*COS(T)
                              Y = 4*SIN(T)"

Press                                 <G>

to go to the Graph Menu.

Press                                 <S>

to change the starting and ending T-values and when prompted type

                     <0> <ENTER> <2> <*> <P> <I> <ENTER>

The T-domain interval starts at 0 and ends at 2*Pi = 6.2831853071795865.

Next change the delta-T value by typing

                    <N> <ENTER> <P> <I> </> <9> <0> <ENTER>

The delta-T value should be Pi/90 = 0.0349065850398866.  The number of T
samples should be 180.


Now press                            <G>

to make the graph which should appear in the shape of an ellipse.

If you look carefully at the graph you will see a noticeable amount of space
between one point and the next.  The graph is made up of a series of discrete
points with each point plotted at about every 2 degrees of arc in a circle.
There are 180 plotted points around the ellipse.

We will now change the graph type to the connected type.  Press

                                     <ESC>

to return to the Graph Menu and once there press

                                      <T>

to change the graph type.  You should see the graph type appear as

                                  "Connected"

The next graph will be made by connecting each individual dot to the next
dot with a solid line.  Usually the connecting line segments are so short
they are not noticeable and the parametric graph appears as a smooth curve.

Press
                                      <G>
to see the new graph.

You should notice how this graph looks more solid than the discrete graph.

Sometimes it is more desirable to view a continuous graph in its connected
form.  But there are other times when a graph is more properly made by using
the discrete type.

To see the discrete type one more time press

                                  <ESC> <T>

to change back to the discrete type and press

                                      <G>

to make a new graph.  Notice the denseness of the points on this graph.
Press
                                     <ESC>
to return to the Graph Menu.



CONTROLLING THE SPEED AND DENSITY OF THE GRAPH
==============================================

An alternative to using the discrete graph type is to make the graph samples
more dense by specifying a smaller delta-T value.  The current delta-T value
should be set for every 2 degrees of angle within the ellipse.  We will now
change this to plot a point every 1/4 of a degree.

Press                                 <N>

to change the delta-T value and when prompted type

                     <ENTER> <P> <I> </> <7> <2> <0> <ENTER>

You should see delta-T = 0.0043633231299858 and the number of samples is
1440.


Press                                 <G>

to make a new graph.

This time the graph will take longer to plot because many more points are
being plotted.  The graph will also look more dense because the points are
spaced closer together because this graph is made using a smaller delta-T
value.  Compared to the previous discrete graph we have sandwiched an
additional 8 points between each pair of previously plotted discrete points.
So the graph appears more solid, if not completely filled in.

Thus you can simultaneously vary the speed and density by making a larger or
a smaller delta-T value.  The larger the delta-T value the faster the graph
is made but the less dense the points on the graph.  The smaller the delta-T
value the slower the graph is made but the more dense the points on the graph.

Press                                <ESC>

to return to the Graph Menu and once there press

                  <N> <ENTER> <P> <I> </> <1> <8> <0> <ENTER>

to change the delta-T value back to Pi/180.  When T is used as the radian
measure of an angle in trigonometric functions as in the above ellipse
examples, a delta-T value of Pi/180 means the functions will be sampled
every 1 degree.  The total number of T samples is 360.

Press                                 <T>

to change the graph type back to connected.

Then press                           <ESC>

to return to the Main Menu.




GRAPHING WITH A PARAMETER
=========================

The next feature of the PARAM program we will demonstrate involves the use
of what is called a graph parameter.  This is a new and different parameter
from the program's T-parameter variable which is used with the coordinate
functions.  For this program the graph parameter is an auxiliary variable that
is denoted by the letter P.  By controlling the use of P you can make a series
of related graphs.  We will demonstrate the use of P by keying in the
coordinate functions  X = 2*P*COS(T)  and  Y = P*SIN(T).

Press                                 <K>

to start entering new coordinate functions and when prompted type

               <2> <*> <P> <*> <C> <O> <S> <(> <T> <)> <ENTER>

                   <P> <*> <S> <I> <N> <(> <T> <)> <ENTER>

The status line at the top of the screen should show

                                "X = 2*P*COS(T)
                                 Y = P*SIN(T)"


Now press                             <G>

to get to the Graph Menu and once there press

                                      <P>

to get to the Parameter Menu.

The Parameter Menu shows the starting and ending values for P.  The delta-P
value gives the spacing between P-values over this interval and the number of
P samples tells how many times the parameter P will be sampled over the
given interval.  The number of P-values and the delta-P value go together.
Usually you can ignore the delta-P value and just enter the number of samples
that you want.  As each of these two quantities are edited the other quantity
is automatically changed.  The number of samples will usually be between 5
and 20.

Press                                 <S>

to change the starting and ending values of P and when prompted type

                            <1> <ENTER> <3> <ENTER>

The new P domain interval is defined starting at 1 and ending at 3.

Next press                            <N>

and when prompted type
                              <8> <ENTER> <ENTER>

You should now see the delta-P value is 0.25 and the number of P samples is
8.

The use of the parameter should already be turned on so press

                                      <G>

to make a graph using the P parameter.

You should see a series of 9 elliptical graphs.

When P=1 you get the smallest ellipse and when P=3 you get the largest
ellipse.  The other ellipses occur for each parameter sample between 1
and 3.

By varying the range of the parameter P and the number of parameter samples
you can show any series of related graphs.

The variable P can appear in more than one place in your function formulas,
but there is only one graph parameter.  This parameter is always denoted by P.
Except for showing a series of related graphs, P is not intended for any
other use within this version of PARAM.  Unless P appears at least once in
one of your coordinate function formulas, P will be ignored.

Press                           <ESC> <P> <P>

to return to the Parameter Menu and turn the use of the parameter off.  Now
whenever either coordinate function gets evaluated P will act as a constant
whose value is the starting P-value.

Then press
                                     <ESC>

two more times to return to the Main Menu.



MAKING A TABLE OF FUNCTION VALUES
=================================

The most mundane feature of PARAM is the ability to make a table of function
values.  This is not a significant feature, but since it is a Main Menu item
we will give one example of its use.

Actually, there are times when it is desirable to see a table of function
values.  But perhaps most users will only use this feature as a check on
whether they have correctly entered a long complicated formula.


Press                                 <K>

to begin entering new coordinate functions and when prompted type

                           <T> <*> <T> <-> <3> <ENTER>

                  <T> <-> <T> <*> <T> <*> <T> </> <3> <ENTER>

The top lines in the Main Menu screen should show:


                                  "X = T*T-3
                                   Y = T-T*T*T/3"


Now press
                                      <T>

and you should see the "Table of Values Menu".  To make a table with the
default values, press
                                      <S>

and when prompted, enter the starting value as T=0.  Type

                                  <0> <ENTER>

and the program should make a table of values.

The table is made by starting with the T-value you give, and T is incremented
line by line, using the value of delta-T.  The delta-T value can be a negative
value if you want to make a table with values of T that decrease.  To evaluate
a function at only one exact point make delta-T equal to 0.

The table should appear as shown below



            "=== T ===       === X ===           === Y ===

                0.              -3.                 0.
                1.              -2.         0.6666666666666667
                2.              1.          -0.6666666666666667
                3.              6.                  -6.
                4.              13.         -17.333333333333333
                5.              22.         -36.666666666666666
                6.              33.                -66.
                7.              46.         -107.33333333333333
                8.              61.         -162.66666666666666
                9.              78.                -234.
                10.             97.         -323.33333333333333
                11.            118.         -432.66666666666666
                12.            141.                -564.
                13.            166.         -719.33333333333333
                14.            193.         -900.66666666666666
                15.            222.               -1110.
                16.            253.         -1349.3333333333333"



You make a new table every time you change the starting value.  Now press

                                <ESC> or <ENTER>

and you should return to the Main Menu.




MORE PARAMETRIC CURVE GRAPHING EXAMPLES
=======================================

The following examples are just a few of the many interesting graphs that you
can make using the PARAM program.  By now you should be able to enter the
coordinate function formulas and set the associated graphing parameters on
your own.

For all of these examples you should set the XY-plane window to its standard
size with X in the range between -7 and +7 and with Y in the range between
-5 and +5.


                                         Starting      Ending     Delta-T
       Coordinate Function Formulas          T            T        value
      ------------------------------     ---------     ------     -------

 1.   X = 4*COS(T)-COS(4*T)                  0          2*PI        PI/180
      Y = 4*SIN(T)-SIN(4*T)

 2.   X = 3.6*COS(T)*(1+COS(T))-1.8          0          2*PI        PI/180
      Y = 3.6*SIN(T)*(1-COS(T))

 3.   X = 3*COS(T)-2*COS(4*T)                0          2*PI        PI/180
      Y = 3*SIN(T)-2*SIN(4*T)

 4.   X = 3*T*T                            -1.3          1.3        PI/180
      Y = 2*T*T*T

 5.   X = 1.5*(T*T-3)                      -2.5          2.5        PI/180
      Y = T*(3-T*T)/2

 6.   X = 4*SIN(3*T)                       0.314        2.833       0.005
      Y = 4*COS(5*T)

 7.   X = (5*T)/(1+T*T*T)                   -20          20         0.02
      Y = (5*T*T)/(1+T*T*T)

 8.   X = 2*COS(T)+1.6*COS(3*T)            -PI           PI         PI/180
      Y = 2*SIN(T)-1.6*SIN(3*T)





CONCLUSION
==========

This concludes the PARAM program tutorial.  If you haven't done so already,
you can now read the help information contained within each main program menu
item.  Most of the basic features have been covered here, but you will gain
more insight by reading all the help information available to you.  If after
all this you still have questions, you can contact the author at the address
given below.


From the Main Menu press
                                   <ALT>+<X>

to quit and exit from the entire PARAM program.


The PARAM program is periodically updated to make improvements, add new
features, (and sometimes to correct bugs!).  You may also wish to contact the
author to check if you have the latest version of the program.  The author
also invites your comments about how you liked the program and will consider
any suggestions you may wish to offer for making the program even more useful.




OTHER PROGRAMS
==============

If you enjoy using the PARAM program you may be interested to know there is
a whole suite of mathematical programs made by the author of PARAM.  These
programs are intended to help motivate an interest in mathematics and computer
science.  Some of the titles of these programs and a brief description of each
is given below.


 1. MATRIX - a program that teaches row operations with matrices.  Features
    include fraction mode, decimal mode, solves linear systems, inverses,
    determinants, sets of basis vectors, eigenvectors and eigenvalues,
    Gram-Schmidt orthogonalization, and the simplex algorithm.


 2. YFUNX - a program for graphing and analyzing functions in rectangular
    form, Y=F(X).  Includes coordinate trace and tangent/normal line modes,
    zooming in and out, scalable axes, graph parameter variable. Numerical
    integration features standard algorithms plus Gaussian Quadrature and the
    Romberg algorithm.  Animation features include plane areas, plane arc
    length, 3D volumes (disks & cylindrical shells) and 3D surface areas.
    Newton's method and the method of successive bisections are for solving
    F(X)=0.  Automatically finds maximum/minimum extrema.  All algorithms
    may be demonstrated in either graphics or text modes.


 3. POLAR - a program for graphing and analyzing functions in polar form,
    R=F(@) or R^2=F(@).  Similar to YFUNX, includes coordinate trace and
    tangent/normal line modes, zooming in and out, scalable axes, and a graph
    parameter variable.  Numerical integration for polar areas and arc length.
    Automatically finds maximum/minimum extrema over any section of a curve.


 4. PARAM - a program for graphing and analyzing functions in parametric form,
    X=F(T) and Y=G(T).  Similar to YFUNX, includes coordinate trace and
    tangent/normal line modes, zooming in and out, scalable axes, and a graph
    parameter variable.  Numerical integration calculates plane areas and arc
    length.  Automatically finds maximum/minimum extrema over any section of
    a curve.


 5. POLPM - a program for graphing and analyzing functions in polar
    coordinates, but that have been parametrized, say R=F(T) and @=G(T).
    Similar to the POLAR and PARAM programs, this program includes coordinate
    trace and tangent/normal line modes, zooming in and out, scalable axes,
    and a graph parameter variable.  Numerical integration for plane areas
    and arc length.  Automatically finds maximum/minimum extrema over any
    section of a curve.


 6. DIFEQ - a program related to 1st order differential equations.  Includes
    graphing the direction field and solves initial value problems using
    Euler methods and a 4th order Runge-Kutta method.  Includes coordinate
    trace mode, zooming in and out, and scalable axes.  Algorithms may be
    demonstrated in either graphics or text modes.


 7. CURVE3D - a program for making 3D graphs of curves given in the parametric
    form X=f(t), Y=g(t), and Z=h(t).  The resulting curve may be viewed from
    any position, and the drawing is a true-perspective 3D picture.


 8. SURF3D - a program to graph 3-dimensional surfaces of the form Z=F(X,Y).
    The resulting surface may be viewed from any position, and the drawing is
    a true-perspective 3D picture.  The surface may be displayed using lines
    of constant x, or constant y, or a fishnet.  Included is a hidden line
    algorithm for more realistic pictures.


 9. CFIT - a program which performs curve fits to data.  Includes linear
    regression for linear, exponential, logarithmic, and power functions.
    Graphs scatter diagrams and the fitted function curves and performs
    a statistical analysis, including an automatic best fit selection.  Data
    may be saved to or read from disk files.


10. GALTON - simulates coin tossing experiments related to probabilities and
    demonstrates graphically how the binomial distribution is related to the
    standard normal Gaussian bell-shaped curve.  Also compares stack counts
    with the numbers generated in Pascal's Triangle.  Either coins or
    ping-pong balls may be used in simulated experiments.  Variable number of
    rows of pegs, variable number of objects, variable left-right probability
    for generating skewed distributions.  Includes a single-step mode under
    full user control.


11. BUFFON - simulates needle dropping experiments related to probabilities
    used to approximate the number Pi.  Needles are randomly dropped on a
    grid of equally spaced parallel lines.  The length of each needle is 1/2
    that of the distance between the lines.  After dropping a large number
    of needles a count is made of the needles which cross a line.  Most
    needles do not touch or cross any line, but the ratio of the total
    number of needles dropped divided by the number of needles which cross
    a line approximates Pi.


12. PROPC - a symbolic logic program that calculates truth tables, analyzes
    tautologies, parses infix formulas and displays their Polish notation
    form, and generates Karnaugh maps from either tables or formulas.


13. RPNDEMO - a program which simulates how a calculator with RPN logic works.
    This program includes its own language and is similar in power to the
    HP-41 calculator.  Programs may be animated to show the internal workings
    of the machine.  Can also be used to teach assembly language concepts.


14. CALC - a reverse Polish logic calculator that operates on 5 data types.
    Included are real and complex numbers, fractions, binary integers and
    polynomials.  Special features include factoring integers and
    polynomials, analyzing repeating decimals and working with continued
    fractions.


15. LOAN - a finance program that handles the two standard cases of compound
    interest.  Uses the 5 standard financial variables n i PV PMT FV found
    on most financial calculators.  Can determine payment schedules for
    loans and annuities and can print amortization schedules for loans and
    interest earning schedules for lump sums and periodic payments.


16. FCARD - simple flash card type of program that can be used to memorize
    any simple series of facts, with one item per line of text.  Items can
    be presented in a random order with timing if desired.


17. THANOI - a game known as the Towers of Hanoi.  The game solution uses
    a recursive algorithm and the purpose of the program is to demonstrate
    the validity and simplicity of a recursive solution to a complex problem
    that would otherwise overwhelm a normal human being.


18. TRIANGLE - a simple program which solves triangle problems in which one
    is given 3 facts about a triangle and must solve for all the remaining
    parts.  Handles all 19 cases of triangle inputs and includes the Law of
    Cosines and the ambiguous case of the Law of Sines.  Can automatically
    determine when two valid triangle solutions exist.  Draws all triangle
    solutions to scale on a graphics screen and computes the perimeter and
    and the area in addition to finding and labeling all the sides and angles.


19. EXPMCON - a utility type of program that works with the above MATRIX
    program and the commercial scientific word processor called EXP.  This
    program converts MATRIX files from an ASCII format to the EXP format.


20. BMPLOT - a utility program that makes high resolution monochrome bitmap
    function plots, identical to the kinds of graphs made by the programs
    YFUNX, POLAR, PARAM, and POLPM.  The bitmaps may be read into other
    programs such as paint or drawing or desktop publishing programs which
    can be used to add labels and titles.  The monochrome bitmaps may be of
    any size or resolution so the output is compatible with virtually every
    printer and/or graphics environment.  The file formats supported include
    PCX, TIFF, and BMP.  The HP-GL/2 plotter language is an optional output
    to either a file or any HP-compatible plotter or PCL 5 LaserJet compatible
    printer.


21. XPRES - a program which computes integers with up to 20,000 digits per
    integer.  This RPN calculator is useful for computing exact values of
    factorials, permutations, combinations, and powers of integers.  For
    example, you can compute the exact value of numbers like 1000! or the
    exact value of 2 raised to the 5,000th power.  Integers may be saved to
    or read from ASCII text disk files.


22. TURING - a program which simulates the operation of a Turing Machine
    which is an abstract model of a primitive digital computer.  In fact,
    the model is fundamental to all digital (logical) computations.  Such a
    machine was conceived by the British mathematician Alan Turing in 1935,
    long before digital computers became established.  Turing also worked on
    machines to break codes used by the German Enigma spy machine in World
    War II.  Three sample demonstration programs are included.


23. PTRIPLE - a program which generates and tests Pythagorean Triples.
    Three numbers, say a, b, c are a Pythagorean Triple if a^2 + b^2 = c^2.
    If the GCF among a, b,and c is 1 the triple is called primitive.  Every
    non-primitive triple is a multiple of a primtive triple.  This program
    works with both general and primitive triples and can make ranges of
    tables of triples in ASCII text files.


For more information about any of these programs you may contact the author.

   John Kennedy                 Voice Phone/Messages any time of day or
   Mathematics Department       night: (310) 450-5150  Extension 9721.
   Santa Monica College
   1900 Pico Blvd.              Internet E-Mail: jkennedy@netcon.smc.edu
   Santa Monica, CA  90405
   U.S.A.
