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Collection
Of Mathwright Interactive Explorations
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 A
Primer on Derivatives
This
36-page Microworld
is an active excursion into one of the most basic
concepts of the Calculus: the Derivative. It develops
this seminal notion with many simulations, illustrations,
examples and exercises. Throughout the book, readers
may ask their own questions and study their own examples.
It
begins by illustrating the graphical representation
of derivative as slope with lively animations: physical
notions of speed, steepness of ascent, and growth
of geometric objects. Here, the author describes in
an informal way, the common-sense ideas behind the
constructions.
Next,
it defines the derivative in a formal way in terms
of limits, and illustrates this limiting process for
two-sided and left- and right-hand derivatives. After
that, the book goes on to explore the relationship
between continuity and differentiability. In particular,
it studies some interesting counterexamples to the
erroneous hypothesis that continuity implies differentiability.
Moving
into the algebraic content of the subject, the next
section presents a number of algebraic rules for calculating
derivatives of functions. This begins with the usual
formulas for specific classes of functions: polynomial
functions, trigonometric, exponential, logarithmic,
etc. But then it backs up and explores where these
rules come from with several exploratory exercises
that offer a way to visualize the rules and to compare
the algebraic procedures with their pictorial and
graphical correspondents.
Having stated and illustrated the basic differentiation
rules, the Chapter on "Arithmetic of Derivatives"
puts it all together in an interesting new way. In
that chapter we explore step by step such rules as
"The Sum Rule", "The Product Rule",
"Quotient Rule" by allowing the student
to supply her own examples and then giving a step-by-step
application of the relevant rule to calculate the
derivative. In this way, the student can see the rule
applied to problems that she supplies, and therefore,
the student will be more likely to understand the
calculation.
Next, follows a discussion of the "Chain
Rule" within the context of the same pedagogic
strategy outlined above. This is very rich, because
the student can easily supply examples whose calculation
will lead to surprises. Since the calculations are
always explained step by step, and there is an unbounded
set of possibilities...
After presenting a "general strategy"
for differentiation, the book moves to its Center:
the Exercises. The Exercise section can be used to
generate thousands of problems that will give practice
in solving the derivative problems based on the basic
derivative laws listed earlier. When the student clicks
on any of the four buttons (Sum or Difference of functions,
Product of Functions, Quotient of Functions, or Composition
of Functions), a function of the listed type is printed.
The student attempts to solve the problem by hand
or with the Symbolic Calculator which is available
from most pages. Then they may check their answer
using the visual and symbolic "Derivative Checker."
As
in previous examples, students may attempt the randomly
generated problems, or make up their own problems
of each type. They may of course pick problems on
finding derivatives from any source for further practice.
The book finishes with explanation and examples of
implicitly defined curves. It then allows the student
to supply example equations of the form f(x,y) = 0
for which it shows, step by step, how to calculate
the implicit derivative.
Author: Ravinder Kumar Alcorn State University
E-mail: rkumar@lorman.alcorn.edu
 
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Best
Linear Fit
This
workbook develops tools for solving discrete models
that depend upon proportionality. Typically what happens
in such models is that some parameters or their powers
are proportional to each other. The problem then reduces
to determining a suitable line y = kx that fits the
(modified) data best so that the constant k of proportionality
can be determined. This can be done either manually,
as suggested in Giordano's book on mathematical modeling,
or one can apply the method of least squares. Regression
lines do not help since a regression line may turn
out to be of the form y=mx+c where c is not zero.
This book lets the user find the constant of proportionality
both manually as well as using the method of least
squares. Besides two built in examples of models,
there are three examples suggested to the user with
enough hints. All examples are taken from Giordano's
book on Modeling Theory.
Author: Ravinder Kumar Alcorn State University
E-mail: rkumar@lorman.alcorn.edu
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Congruences
The
simplest example of congruence arithmetic comes from
an analog clock. In this book we consider the following:
congruences, Fermat theorem, solution of congruence
equations, systems of congruence equations, and Cayley
tables.
The user is invited to conjecture Fermat's Theorem.
The user is also challenged to explore some simple
groups. This book can be used for a course in number
theory. Help pages are given that, among other things,
explain some concepts which a user may not be familiar
with. This book may be used in college algebra, number
theory, or examples of a group.
Author: Ravinder Kumar Alcorn State University
E-mail: rkumar@lorman.alcorn.edu
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Evolutes
This
4-page workbook provides a way to visualize the construction
of evolutes of the graphs of functions, and of parametric
curves. This construction how these evolutes are the
locus of the centers of the "osculating circles"
at points along the curve. These osculating circles,
and the radii from their centers are drawn interactively
as the Reader supplies examples.
The Microworld explores these themes in 3 interactive
pages + a Table of Contents.
Author:
Sam Masih, Albany State University
E-mail: smasih@asurams.edu
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Limits
of functions
This
5-page workbook assembles a variety of tools for visualizing
left, right, and two-sided limits of functions of
a single variable. The reader may define functions
with algebraic forms, or may define functions piecewise.
There is a versatile function grapher on each page
of the exploration that allows the reader to zoom
in or out around a chosen point, and then to select
points along the graph to see the function values,
and to learn the conventions that associate function
graphs with sets of ordered pairs. This latter important
and powerful heuristic reinforces the visualization
encouraged in later pages.
Author:
Sam Masih, Albany State University
E-mail: smasih@asurams.edu
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Optimize
This
9-page workbook, which is based on the earlier Mathwright
WorkBook: The Classic Box Problem, by Charles
and Rosanne Hoffman (the former of Villanova University),
takes as its theme: The visualization of maximization
problems. It presents a sequence of problems that
are masterfully chosen to help the reader see what
optimization means in the context of a lively and
interactive environment.
Author:
Sam Masih, Albany State University
E-mail:
smasih@asurams.edu
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Piecewise
Functions
This
workbook is designed to allow a student to visualize
the graphs of functions, piecewise continuous or not,
and to explore limits, continuity, and derivability.
This book can be used to
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Symmetry
This
mathwright workbook is an attempt to offer visualization
of the symmetric group S3 of degree 3. Rotations and
reflections of an equilateral triangle are shown visually
using animations. Calculator for the symmetries helps
compute various combinations of the symmetries. The
user is encouraged to use the tools developed in exploring
the group S3; for example, order of an element, cyclic
subgroups, and lattice of subgroups of S3. Although
different basic concepts related to a group are defined
and some basic properties developed, it will be better
if the user is exposed to the definition of a group
and some basic properties
Author:
Ravinder Kumar Alcorn State University
E-mail:
rkumar@lorman.alcorn.edu
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Progeny
In
1202 Fibonacci posed and solved a problem on the population
of rabbits, which gave rise to the famous Fibonacci
sequence. There are many population models, such as,
population of plants, population of blood cells etc.
Fibonacci sequence in turn gives rise to 'golden ratio'.
In this 10-page self-contained Microworld, we consider
the following:
- Propagation
of Plants
-
Fibonacci's
Rabbit Problem
- Golden
Ratio
- Golden
Rectangle
A
paper-folding activity is given to construct a golden
rectangle. The modeling procedures involve linear
systems of difference equations, higher order difference
equations, and eigen values. This book can be used
for modeling in biology, and also exploring Fibonacci
sequence. The users are advised to explore carefully
the methods and then attempt to solve the problem
of movement of population that also depends upon linear
system of difference equations.
Author:
Ravinder Kumar Alcorn State University
E-mail: rkumar@lorman.alcorn.edu
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Nonlinear
Equations
When
f(x) is a nonlinear function, there is often no general
procedure for finding exact solutions to the equation:
f(x) = 0. Various methods are explored, in the 14
pages of this Microworld, for finding approximate
numerical solutions of the equation in this case.
In particular the following methods are explored and
illustrated: Bisection method, Secant method, Newton's
method, and Fixed Point Iteration Method. Newton's
method is also used for determining complex roots.
This book can be used both as a solver as well as
an exploratory tool. The users are asked carefully
to explore the methods and then describe the methods
in their own words.
Author:
Ravinder Kumar Alcorn State University
E-mail: rkumar@lorman.alcorn.edu
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Inclined
Planes
This
5-page Microworld tells a story about Newtonian Force
Diagrams in the contexts of inclined planes and pulley
systems. It does this in a series of four interactive
and animated examples that allow the reader to experiment
with such parameters as weight, inclination of the
plane, coefficient of friction, load on the pulley,
etc.
In
each example, the author asks a question to which
the reader may discover the answer by experimenting
with values for the parameters, and also by making
calculations in the built-in calculators.
Author:
Sam Masih, Albany State University
E-mail: smasih@asurams.edu
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Lakes
There
are three lakes: L. Audrey, L. Bob, and L. Cecil.
Inflow pattern for the three lakes is best described
by an accompanying picture: Scientists estimate that
there are currently 12000 kg of pollutants in L. Audrey,
5000 kg in L. Bob, and 20000 kg in L. Cecil. The percentage
of water replaced each year in the three lakes is
respectively 30%, 25%, and 16%. Under an agreement
with environmentalists, it is agreed that no new pollutants
will be dumped into the lakes. The task is given to
engineers to figure out how many years it will talke
top completely clean L. Cecil. The Army Corps of Engineers
comes up with an idea. The inflow into L. Bob is diverted
into L. Audrey. The total outflow from Lake Audrey
is then routed to Lake Bob through a canal that Army
Corps of Engineers construct. Lake Bob then empties
into Lake Cecil. As a result of this rerouting 62.5%
of Lake Audrey and 50% of Lake Bob is replaced every
year. Rate of discharge from Lake cecil remains the
same 16%. This Microworld constructs an interactive
model where the amounts of pollutants and the rates
at which the water is replaced can be changed. Obviously,
such an interactive model will be quite useful for
solving a variety of similar problems.
Author:
Ravinder Kumar Alcorn State University
E-mail: rkumar@lorman.alcorn.edu
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Cycloids
This
Microworld explores a number of parametric curves
with sprightly animations and plenty of opportunity
for readers to practice graphing or to graph their
own curves. the curves of principal interest are various
types of cycloids, including hypocyloids, epicycloids,
prolate cycloids, and curate cycloids.
Author:
Sam Masih, Albany State University
E-mail: smasih@asurams.edu
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Application
of the derivative
This
6-page Microworld presents a series of explorations
that examine the derivative of a function. The reader
may supply functions, and choose points on the graph,
and the tangent and secant line approximations are
drawn while she chooses small increments, h, for the
independent variable away from the base point. The
symbolic derivative is supplied, and step-by-step
procedures for calculating the limit of the difference
quotient for each example or student selection are
supplied. The reader may also define functions piecewise
and explore points of non-differentiability, zooming
in and out, and so on.
Author:
Sam Masih, Albany State University
E-mail: smasih@asurams.edu
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Graphs
and their functions
This
WorkBook steps you through a series of 6 demonstrations,
each of which shows some relation between functions
and graphs: the role of parameters, shifts, translations,
and stretches and compressions. The reader may supply
his own functions to see these effects, or may view
the examples given.
Author:
K.P. Satagopan, Shaw University
E-mail:
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Trigonometric
functions
This
book illustrates the effects of the parameters 'a'
, 'b' , and 'c' on the graph of the function f(x)
= a*sin(b*x+c) which respectively represent the amplitude,
period or frequency, and phase shift of a trigonometric
function. The activities are designed to understand
the relationship between the parameters and the graph
of the function.
Author:
K. P. Satagopan, Shaw University
E-mail:
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Demoivre's
theorem
This
8-page Microworld has for its theme the calculation
of complex roots of complex numbers. The exercises
introduce in gradual steps,
The
representation of complex numbers in the plane in
polar and Cartesian form
Euler's representation of complex numbers in complex
exponential form
The calculation of products of complex numbers
The extraction of roots of complex numbers using De
Moivre's Theorem
Author:
Sam Masih, Albany State University
E-mail: smasih@asurams.edu
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Linear
Approximations
Linear
Approximation is the simplest way of approximating
the value of a function by using a bare minmum of
conditions. This method is also the simplest way of
interpolating data. The method depends upon using
the value (f(a)) and rate of change (derivative) at
the point x = a to approximate value of f(c), where
c is in close proximity of a. Graphically, this method
approximates the value of the function at a point
by replacing the curve by a tangent to it at a nearby
point.
Note:
Requires Version 2.0 MathwrightWeb Control (4/ 18/
02)
Author:
Ravinder Kumar Alcorn State University
E-mail: rkumar@lorman.alcorn.edu
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Exploring
Lines
Exploring
Lines Book Description:
 This
25 page microworld is a module on the topic of lines
as in a high school algebra course or a college intermediate
algebra course. We have included theoretical considerations
as well as a historical note. However, the main focus
is on learning to find equation of a line under various
conditions.
Each
topic has a page devoted to examples. In almost all
cases hundreds of examples are generated randomly.
This page is followed by one to two pages of practicing
the relevant problems. These problems, hundreds and
sometimes thousands, are generated randomly. You can
check your answer too.
You
are advised to follow instructions strictly, particularly
when solving questions using the suggested commands.
Please click on Help button on a page, if available,
to read how to conduct that page or execute certain
commands relevant to the page. Some important instructions
are printed in red as a reminder on some pages. It
is important to follow the instructions.
Note:
Requires Version 2.0 MathwrightWeb Control (4/ 18/
02)
Authors:
Ravinder Kumar and Kanchan Manactala, Alcorn State
University
E-mail: rkumar@lorman.alcorn.edu, kanchan@lorman.alcorn.edu
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Shortest
Paths
Shortest
Paths
Book Description:
In
this book we explore the concept of shortest distance
from a point to both a line and a curve in general,
developed in separate sections. The user is given
an opportunity to explore consequences/characteristics
of the shortest distance. Shortest distance from a
point to a curve is obtained using optimization technique
in single variable calculus. Shortest distance from
a point to a line is studied using the formula in
Cartesian geometry as well as optimization techniques.
At the end of each section the user can practice finding
shortest distance using well-explained commands by
randomly generating problems.
Authors:
Ravinder Kumar and Kanchan Manaktala, Alcorn State
University
E-mail: rkumar@lorman.alcorn.edu, kanchan@lorman.alcorn.edu
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Graphs
of Functions and Symmetry
Graphs
of Functions and Symmetry
Book Description:
This
delightful 6-page Microworld is a gentle introduction
to the symmetries of a graph. It approaches this idea
through the metaphor of reflection, as in reflection
through a mirror. The basic reflections that it considers
are: reflections through the x-axis and y-axis, and
through the line y=x.
Beginning
with reflections of points
through these lines, it moves on to let the reader
experiment with the reflections of graphs
and curves through these lines,
and gives a visual tour of the various notions of
symmetry associated with these geometric operations.
Author:
Kanchan Manaktala, Alcorn State University
E-mail: kanchan@lorman.alcorn.edu
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Playing
with Points
Playing
with Points
Book Description:
This
book is an introductory module on the concept of a
point.
The
reader learns
- to
plot a point if coordinates are given
- to
read the coordinates if the point is given
- to
determine the distance between two points
- to
determine collinearity of three points
Two
pages in this book help students plot points and determine
distance either individually or in groups of two or
more. These exercises invite the students to play
a game while learning how to plot points!
Author:
Kanchan Manaktala, Alcorn State University
E-mail: kanchan@lorman.alcorn.edu
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Triangle
Optimization
Triangle
Optimization
Book Description:
Equilateral
triangles have some interesting properties. This 8-page
Microworld provides visualization of why among triangles
of fixed perimeter equilateral triangles are the ones
that have maximum area. Two proofs of this fact are
also discussed. The first explanation depends upon
multivariable calculus. The second proof depends essentially
on single variable calculus.
We
believe that this latter proof is a new proof and
the power of Mathwright is to make visualization of
this not so straightforward proof possible.
Author:
Ravinder Kumar Alcorn State University
E-mail: rkumar@lorman.alcorn.edu
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Pictures
of Functions
Pictures
of Functions Book Description:
This
book is a module about functions and how to picture
them. In general, when a function is defined, its
domain and range are not given explicitly. It is defined
as a relation between two variables x (represents
the elements in the domain and is called the independent
variable) and y (represents the elements in the codomain
and is called the dependent variable). In order to
show that y depends on x, we write y = f(x), and f
is called a function. For each x, f(x) is called the
image of x, and x is called a pre-image of f(x). The
collection of all f(x) is called the range of f.
Author:
Mohammed Karim, Alabama A&M University
E-mail: mrkarim@aamu.edu
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Transforms
of Functions
Transforms
of Functions Book Description:
This
book is a module about standard ways in which functions
may be modified algebraically, and about the concomitant
geometric changes in their graphs. We study and experiment
with the following transforms of functions.
(1)
Shifts
(2)
Reflection
(3)
Stretching / Shrink
Author:
Mohammed Karim, Alabama A&M University
E-mail: mrkarim@aamu.edu
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Zeros
of Polynomial Functions
Zeros
of Polynomial Functions
Book Description:
In
this book we study the Zeros of Polynomial Functions.
Let f be a polynomial function and c be a real number.
Then x = c is a zero of the function f if x = c is
a solution of the polynomial equation f(x) = 0, i.e.,
f(c) = 0. In that case, (x - c) is a factor of the
polynomial f(x), and the graph of f(x) crosses the
the x-axis at the point (c,0).
The
book has a command line that will allow you to define
arbitrary polynomials and graph them. You may trace
points along the curve, also. The Factor command will
attempt to factor the polynomials that you supply,
and you may also solve polynomial equations using
the Expert system.
Author:
Mohammed Karim, Alabama A&M University
E-mail: mrkarim@aamu.edu
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Cubic
Splines
Note:
This Microworld makes use of Microsoft Access databases.
You DO NOT have to have Microsoft Access on your personal
computer, but you do have to configure your "data
source (ODBC)" to enable buit-in MS Access tools.
To do so, go to the
configuration page.
Cubic
Splines
Book Description:
Natural
Cubic Splines are used for creating a model that can
fill in the holes between data, in effect, approximating
a trend. They are therefore useful for making observations
and inferences about a pattern existing in the data.
Cubic
splines have three basic properties.
- They
pass through all given data points with a unique
one between each set of points.
- They
are smooth, meaning that at the points where they
merge their first and second derivatives are equal.
- And
finally, since this book deals with natural splines,
the second derivative at the two endpoints is always
zero.
This
project was developed as a requirement for Numerical
Methods course during fall 2001 (Instructor Dr.
Ravinder Kumar). This interactive module provides
an effective tool for interpolating a data set using
the method of cubic splines.
Author:
Shomari Mosi, Senior, Alcorn State University
E-mail: queperknuckle@yahoo.com
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A
World of Curves
Graphing is a very important aspect of learning of
mathematics at all levels, particularly at the undergraduate
level. The world of curves is full of wonders. But,
with the advent of technology, we find ironically,
that the culture of studying this beautiful world
is getting pushed into oblivion. This is despite the
fact that these days books, particularly calculus
books, emphasize the graphical aspect of concepts.
Books, however, often do not talk about envelopes,
evolutes, involutes, pedals, negative pedals etc.,
any more. Historically, these and other ways of determining
curves not only shed light on the curves and their
characteristic properties, but also produce some fantastic
curves, otherwise difficult to create and visualize.
It
is the objective of this 14 page microworld to provide
some ways of gaining insight into the world of curves.
This effort is by no means exhaustive or comprehensive.
Here, we explore curves defined by parametric equations
only. We also provide mechanism to understand and
explore the envelope, pedal, negative pedal, and contrapedal.
For a list of historically famous curves and their
properties we refer to the website: MacTutor of History
of Mathematics.
This
book may be used for
- Calculus
I
- Calculus
II
Note:
Requires Version 2.0 MathwrightWeb Control (4/ 18/
02)
Author:
Ravinder Kumar Alcorn State University
E-mail: rkumar@lorman.alcorn.edu
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