The M.W. Milton Mathwright microworlds are designed to simulate some of the major features of standard graphing calculators. In addition to carrying out a variety of numerical and symbolic computations, M.W. Milton: 1) graphs and tabulates functions of one variable; 2) plots data points; 3) fits selected curves to data points; 4) solves simultaneous systems of linear equations, up to 6 equations by 6 unknowns; 5) graphs functions of two variables; 6) performs row operations with matrices; and 7) graphs polar functions and parametric curves. They also can calculate average rates of change, instantaneous rates of change (derivatives), and areas under the curve (definite integrals) for functions of one variable. The authors hope the microworlds will prove useful for teachers and students in College Algebra, Precalculus, Applied Calculus and Calculus courses, and may provide an alternative to the purchase of expensive graphing calculators. The microworlds look best when viewed at resolutions 800 by 600 pixels or greater. You may want to enlarge the applet window to fill the entire screen by toggling the F11 key. Most of the following documentation applies to volume 1. The authors may be contacted at the following email address:
We would be glad to hear any comments, criticisms, or suggestions.
CONTENTS
i) Starting M.W. Milton
ii)
Graphing a Function of One Variable
iii) Tabulating
a Function of One Variable
iv) Plotting
a Set of Data Points
v) Fitting
a Curve to a Set of Data Points
vi) Solving
Simultaneous Systems of Linear Equations
vii)
Computing Average Rates of Change
viii)
Computing Instantaneous Rates of Change
ix) Computing
Areas Under the Curve
x) Using the
Calculator Text Box
xi)
Graphing a Function of Two Variables
xii) Performing
Row Operations
xiii) Graphing
Polar Functions and Parametric Curves
xiv) Printing
i)
Starting M.W. Milton
An identical off-line version of M.W. Milton is available
for users who do not have continuous Internet access. This version
requires installation of a reader program different from the plug-in
used to operate the Internet-based version. Contact the authors for
details of downloading and starting this version.
ii) Graphing a Function
of One Variable
This feature is offered on the Graphing Calculator page
of the microworld (page 2), which is accessible through a button
on the title page. M.W. Milton can graph a function of one variable
whose expression (formula) is entered into the Function Expression
text box. For instance, to graph the function f(x) =
x3-2x+1, one would enter x^3-2*x+1 in the Function
Expression text box, then click the Graph Function button
with the mouse. The function expression need not use x as the
input (independent) variable. Any variable can be used; just enter the
correct input variable in the Input Variable text box before graphing.
(Mathwright is not case-sensitive.) When a function is graphed, the
function is stored in memory as f. The function can then be evaluated
at numerical and symbolic inputs using regular function notation through
the Calculator text box (see instructions below).
***Important Note*** You must use the * character to indicate multiplication
at all times.
The portions of the horizontal and vertical axes to be shown on the graph screen can be designated through the Graphing Window text box controls, as follows:
Input From - lower bound for horizontal axis (x-axis or
input axis) To - upper bound
Output From - lower bound for vertical axis (y-axis
or output axis) To - upper bound
(If the Auto Fit check box is checked, the lower
and upper bounds for the vertical axis will be determined by the
program, by sampling outputs from the given function from the lower
bound of the horizontal axis to the upper bound of the horizontal
axis.)
Input Label - label for horizontal axis Output
Label - label for vertical axis
The graph screen is cleared by clicking the Clear Graph button.
Graphs are superimposed on the graph screen until this button is pushed.
The graph screen is also cleared when the graphing window is changed.
The graph screen is cleared and the graph window is set to the standard
default by clicking the Standard Window
button.
The color used to draw the curve and labels is shown in the small window to the right of the Graph Color button. Each time this button is clicked, a new color is selected. There are 16 possible colors.
If you wish to graph a function over a restricted domain (application domain), check the Application Domain check box. Enter the lower bound of the application domain in the From text box, and the upper bound in the To text box. Using the application domain option allows you to graph a piecewise-defined function, by graphing the different pieces of the function one after the other before clearing the graphing window.
Many more settings that control the appearance of graphs are available
through various pop-up dialog boxes. To access these dialog boxes,
click on the graph screen with the right mouse button. Then choose
Settings… from the resulting pop-up menu. One setting that
you may want to change from time to time is the spacing used for the
tick marks drawn along the horizontal and vertical axes. Mathwright will
not draw graphs with more than 1,000 tick marks either horizontally or
vertically. These spacings are accessed through the Rulings... command.
There are a couple of other especially useful auxiliary features that
can be activated by clicking with the mouse on the controls located along
the bottom of the graph screen. If you click on the Trace check box and then move the mouse
over the graph screen, the coordinates of the point at the crosshairs
are displayed. This feature simulates the popular "Trace" feature
available on virtually any graphing calculator. Another nice action
is "zooming in" on a graph. To activate this action, click on
the In button. The user can then
reset the graphing window to a rectangle drawn by hand on the graph
screen. This rectangle is drawn by clicking the left mouse button on
the graph screen and holding it down, dragging the mouse to another
point on the screen, and then releasing the mouse button. The user
can then reverse his or her actions by clicking on the Out button a corresponding number of
times.
By clicking the Big Graph button, you can see an enlarged version of the graph screen drawn on a different page (page 4) of the microworld. This page also provides access to three other M.W. Milton features through buttons at the bottom of the page: Average Rate of Change, Instantaneous Rate of Change, and Area Under the Curve. The button labeled << returns you to the main Graphing Calculator page.
Some common functions (and constants) are predefined and can be entered or evaluated using regular function notation. These include:
sin asin - Arcsin
cos acos
- Arccos
tan atan
- Arctan
csc sinh
- hyperbolic sine
sec cosh
- hyperbolic cosine
cot tanh
- hyperbolic tangent
ln
exp - natural exponential function
pi
e
log - log base 10
lg - log base 2
abs - absolute value sqrt
- square root cbrt - cube root
fourthrt - fourth root fifthrt -
fifth root factorial
combinations(n, r) - binomial coefficient n over
r mod(m, n) - m modulo n
stdnormal - the standard normal probability density
function
normal(x, m, s) - the normal probability density
function with mean m & standard dev. s
fv(p, i, n) - the future value of investment p
at rate i with n compounding periods
pv(a, i, n) - the present value of amount a
at rate i with n compounding periods
fvannuity(p, i, n) - the future value of an annuity
with payment p at rate i with n payments
pvannuity(p, i, n) - the present value of an annuity
with payment p at rate i with n payments
amortize_payment(a, i, n) - the amortized payment
for loan amount a at rate i with n payments
sinking_payment(a, i, n) - the sinking fund payment
for amount a at rate i with n payments
effective_rate(r,m) - the effective interest rate
for annual interest r with m compounding periods per
year
iii) Tabulating a Function of
One Variable
This feature is offered on the Graphing Calculator page
of the microworld (page 2), which is accessible through a button
on the title page. M.W. Milton can tabulate a function of one variable
whose expression (formula) is entered into the Function Expression
text box. For instance, to tabulate the function f(x)
= x3-2x+1, one would enter x^3-2*x+1 in the Function
Expression text box. The function expression need not use x
as the input (independent) variable. Any variable can be used; just enter
the correct input variable in the Input Variable text box before
graphing. (Mathwright is not case-sensitive.)
***Important Note*** You must use the * character to indicate multiplication
at all times.
***Important Note*** You must click on the pen icon to change it to a
triangle before updating, clearing, or plotting the data table.
The table of input-output pairs is displayed in the data window shown
on the left-hand side of the graphing calculator page. To clear the inputs
column in the data window, click on the Clear Inputs button
with the mouse. Likewise, to clear the outputs column, click
on the Clear Outputs button. However, due to certain limitations
of Mathwright neither of these buttons functions optimally. Instead of actually
clearing slots in the table, values are replaced by 0, which may not be
desirable. One has to restart M.W. Milton to completely clear the values
in the data window.
The list of inputs at which the function is evaluated can be entered manually into the inputs column, or the inputs can be generated automatically. To enter inputs manually, make sure the Auto Inputs check box is turned off. The Auto Outputs check box must be turned on. An input can be entered by clicking on the correct cell of the inputs column and then keying. Symbolic inputs cannot be used. Once the inputs are entered, the matching outputs are computed by clicking the Update button.
To have the inputs generated automatically, first make sure the Auto
Inputs check box is turned on. The Auto Outputs check box
must also be turned on. The inputs generated range from the parameter
entered in the First text box, up to the parameter entered in the
Last text box. The number of inputs generated is the same as
the parameter entered in the Number text box. These inputs are equally
spaced between the first and last inputs. After the automatic input parameters
are entered, click on the Update button to compute the inputs
and matching outputs.
iv) Plotting a Set of Data Points
***Important Note*** You must click on the pen icon to change it to a
triangle before updating, clearing, or plotting the data table.
To plot a set of data points, the coordinates are first entered as input-output
pairs in the data window on the left-hand side of the graphing calculator
page. First coordinates and matching second coordinates should be
entered side-by-side in the inputs and outputs columns,
respectively. Both the Auto Inputs and Auto Outputs check
boxes must be turned off. An input can be entered by clicking on the
correct cell of the data window and then keying. Symbolic coordinates cannot
be used. Once the data points are entered, click on the Update button.
The plot can then be drawn by clicking the Plot button. The graphing
window specified by the Graphing Window controls on the right-hand
side of the page is used (see Graphing a Function of One Variable above).
v) Fitting a Curve to a Set
of Data Points
Eight types of curves can be fit to a set of data points
stored in the data window on the left-hand side of the Graphing Calculator
page (see Plotting a Set of Data Points above). Least squares regression
is the fit technique. A particular type curve is fit to the set of data
points by clicking on the corresponding button. The curve is also graphed
on the graph screen using the graphing window specified by the Graphing
Window controls on the right-hand side of the page (see Graphing
a Function of One Variable above). The formula for the curve is stored
in memory as a function named by an identifier suggestive of the fit
type. The fitting function can then be evaluated at numerical and symbolic
inputs using regular function notation through the Calculator
text box (see instructions below). The eight types of curves and their
corresponding identifiers are:
Linear - linearfit(x) = mx + b
Quadratic - quadraticfit(x) = ax2
+ bx + c
Cubic - cubicfit(x) = ax3
+ bx2 + cx + d
Quartic - quarticfit(x) = ax4
+ bx3 + cx2 + dx + f
Power - powerfit(x) = bxa
Exponential - expfit(x) = beax
Ln - logfit(x) = m*ln(x)
+ b
Logistic - lgstfit(x) = L/(1
+ beax) [approximate best fit]
The formula for the fitting function is printed on the Output Screen, along with either the correlation coefficient of the fit or the index of determination (a.k.a. coefficient of determination) of the fit. The correlation coefficient is printed for the linear-based fits, while the index of determination is printed for the others (i.e. the higher-order polynomials). The Paste Fit button is used to paste the memory identifier for the fitting function mentioned earlier into the Function Expression text box. The fitting function can then be graphed in different graphing windows, tabulated, or used for computing average rates of change, instantaneous rates of change, or areas under the curve.
vi) Solving Simultaneous Systems
of Linear Equations
This feature is offered on the System of Equations Solver
page of the microworld (page 3), which is accessible through
a button on the title page. Just follow the instructions (carefully!)
printed on the output screen at the top of the page. Linear systems in
standard form up to 6 equations by 6 unknowns can be solved. The button
labeled << returns you to the title page.
vii) Computing Average
Rates of Change
After a function has been graphed on the Graphing Calculator
page, it can then be used for computing average rates of change (see
Graphing a Function of One Variable above). This feature is located
on page 5. To access this page, click on the Big Graph button
located at the bottom of the Graphing Calculator page. This brings you
to page 4, which displays an enlarged version of the function graph from
page 2. Then click on the Average Rate of Change button at the
bottom left-hand side of the page. The Average Rate of Change page also
displays the function graph from page 2. To compute the average rate of
change of the function over a given interval, enter the lower bound of
the interval in the From Input text box, and the upper bound of
the interval in the To Input text box. Then click on the Calculate
button. The average rate of change value and step-by-step computations
involving the difference quotient are printed on the Output Screen.
Some text interpreting the average rate of change value is printed also.
If the Secant Line check box is checked, the corresponding secant
line to the function graph is drawn on the graph screen. If many average
rates of change are computed, the corresponding secant lines are superimposed
on the graph screen until the Clear Secants button is pushed.
Symbolic values can entered in the From Input and To Input text boxes. In this case, make sure the Secant Line check box is turned off before calculating the average rate of change, and likewise make sure the Exact / Symbolic check box is turned on. In this case, any numbers used must be either integers or expressed in exact rational form. Decimal numbers cannot be entered. The final result displayed for the average rate of change may not be simplified, but it can be highlighted and copied from the Output Screen, pasted into the Calculator text box at the top of the Graphing Calculator page of the microworld (page 2), and then simplified by clicking the Simplify button (see Using the Calculator Text Box below).
The button labeled << returns you to the previous page (page 4).
viii) Computing Instantaneous
Rates of Change
After a function has been graphed on the Graphing Calculator
page (see Graphing a Function of One Variable above), it can then be
used for computing instantaneous rates of change (i.e. derivative values).
This feature is located on page 6. To access this page, click on the
Big Graph button located at the bottom of the Graphing Calculator
page. This brings you to page 4, which displays an enlarged version of
the function graph from page 2. Then click on the Instantaneous Rate
of Change button at the bottom middle of the page. The Instantaneous
Rate of Change page also displays the function graph from page 2. To
compute the instantaneous rate of change of the function at a given input,
enter the input in the At Input text box. Then click on the Calculate
button. The instantaneous rate of change value and step-by-step computations
involving the limit of the difference quotient are printed on the Output
Screen. Some text interpreting the instantaneous rate of change value
is printed also. If the Tangent Line check box is checked, the corresponding
tangent line to the function graph is drawn on the graph screen. If many
instantaneous rates of change are computed, the corresponding tangent lines
are superimposed on the graph screen until the Clear Tangents button
is pushed.
Symbolic values can entered in the At Input text box. In this case, make sure the Tangent Line check box is turned off before calculating the instantaneous rate of change, and likewise make sure the Exact / Symbolic check box is turned on. In this case, any numbers used must be either integers or expressed in exact rational form. Decimal numbers cannot be entered. The final result displayed for the instantaneous rate of change may not be simplified, but it can be highlighted and copied from the Output Screen, pasted into the Calculator text box at the top of the Graphing Calculator page of the microworld (page 2), and then simplified by clicking the Simplify button (see Using the Calculator Text Box below).
The button labeled << returns you to the previous page (page 4).
ix) Computing Areas Under
the Curve
After a function has been graphed on the Graphing Calculator
page (see Graphing a Function of One Variable above), it can then be
used for computing areas under the curve (i.e. definite integrals).
This feature is located on page 7. To access this page, click on the
Big Graph button located at the bottom of the Graphing Calculator
page. This brings you to page 4, which displays an enlarged version of
the function graph from page 2. Then click on the Area Under the Curve
button at the bottom right-hand side of the page. The Area Under the Curve
page also displays the function graph from page 2. To compute the signed
area between the curve and the input axis (x-axis) over a given
interval, enter the lower bound of the interval in the From Input
text box, and the upper bound of the interval in the To Input text
box. Symbolic values cannot be used. Turn the Approximate check box
off. Then click on the Calculate button. The area value is printed
on the Output Screen and the corresponding region is shaded on the
graph screen (this may take a few seconds). The area is estimated by using
10,000 trapezoids. The Clear Tangents button will clear the shaded
area from the graph screen.
On the other hand, if the Approximate check box is checked, the area is estimated by the total area of the number of rectangles entered in the Number text box. Each rectangle has equal width. The height of each rectangle depends on which of the three check boxes Left, Right, or Midpoint is checked. (These check boxes are independent, so more than one can be checked, or all can be unchecked. The leftmost checked box is assumed to be the selected check box. If none are selected, the Left check box is assumed to be selected.) If the Left check box is checked, the height of each rectangle is computed by evaluating the function at the left endpoint of each partition of the interval. If the Right check box is checked, the height of each rectangle is computed by evaluating the function at the right endpoint of each partition of the interval. Finally, if the Midpoint check box is checked, the height of each rectangle is computed by evaluating the function at the midpoint of each partition of the interval.
The button labeled << returns you to the previous page (page 4).
x) Using the Calculator Text
Box
Routine calculations, operations on expressions such as
expansion and factoring, evaluation of functions, etc. can be performed
by entering the appropriate expression in the Calculator text
box at the top of the Graphing Calculator page of the microworld
(page 2), which is accessible through a button on the title page. Symbolic
computations are supported. Once an expression is entered, it can be
evaluated or simplified in different ways by clicking one of the four
buttons shown to the right of the Calculator text box. Mathwright
will attempt to factor expressions (including integer constants) when
the Factor button is clicked. Conversely, Mathwright will attempt
to expand products in expressions when the Expand button is clicked.
The Calc and Simplify buttons both cause expressions to
be evaluated and simplified, albeit in somewhat different ways. But the
Simplify, Expand, and Factor buttons require any numbers
used to be either integers or expressed in exact rational form. Decimal
numbers cannot be entered. Results are displayed in "pretty print" form
on the Output Screen. Results displayed on the Output Screen can be highlighted and
copied back into the Calculator
text box for extended series of calculations. The Output Screen can be cleared by clicking
on the Clear Output button
to its right.
***Important Note*** You must use the * character to indicate multiplication
at all times.
xi) Graphing a Function
of Two Variables
The 3D Graphing page (page 8) is accessible through the
3-Dimensional button located on the bottom right-hand side
of the Graphing Calculator page of the microworld (page 2). This
very nice page was supplied courtesy of James White at Bluejay Lispware
and therefore looks somewhat different from the other pages in M.W.
Milton. Instructions for using this page can be opened by clicking on
the Instructions button found
at the top of the page.
The button labeled << returns you to the previous page (page 2).
xii) Performing Row Operations
The Row Operations page (page 9) is accessible through the
Row Operations button located on the bottom right-hand side
of the System of Equations Solver page of the microworld (page
3). This page is not yet documented or extensively tested.
The button labeled << returns you to the previous page (page 3).
xiii) Graphing Polar Functions
and Parametric Curves
These features are not yet documented or extensively tested,
but are available in volume 2.
xiv) Printing
While Mathwright does not contain any native printing facilities,
printing any page from M.W. Milton can be easily accomplished by
using the Print Screen command from Windows, which is activated
by pressing the Print Scrn function
key contained on any standard Microsoft keyboard. A bitmap image of
the screen is copied to the Windows clipboard, and can then be pasted
into virtually any word processor (such as MS Word) and printed. If
preferred, the screen image can also be cropped using a bitmap editor
such as MS Paint prior to printing.