The 500 Page Interactive Web Story Calculus on Action develops the subject around the theme of gravitation. And it is written from the point of view of the dynamics of reader interaction. The entire static hypertext version (500 pages) is available to any visitor to Project WELCOME. Also, the dynamic version of the Pre-Calculus introductory Chapter (42 pages with 9 interactive explorations) is yours to explore here at the Project WELCOME site as a demonstration of the new idiom that we are developing. The whole story may be found at the Mathwright Library.

Interactive Web Course

This 10-Chapter Interactive Web Course is a complete College Algebra Course that was written by Samad Mortabit, A Project WELCOME Co-PI. It differs from other textbooks in significant ways. It is a genuine effort to provide students with the right tools and the appropriate level of discussion that are necessary for a successful learning experience. Students can interact with the text, pose their own questions, and are provided the tools to discover the answers to the questions they pose.

The Preview Version, Chapter 2 of this book (70 printed pages with 9 embedded explorations) is available here at Project WELCOME as a demonstration of the new idiom that we are exploring. The entire text is available for purchase at another website.

Interactive Web Stories (Advanced)

Marden's Theorem

Marden's theorem says that the foci of the ellipse inscribed in the triangle of roots of a cubic complex polynomial p(z) and tangent to its midpoints are precisely the roots of the derivative, p'(z). In this 10-page Microworld, you may experiment with this, and a more general version of Marden's Theorem. And you may review properties of complex numbers and the complex plane, both visually and algebraically.

Symmetry

This 8-page Microworld 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.

Congruences

This 12-page Microworld introduces the reader to number theory. 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.

A World of Curves

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.

Interactive Web Stories (Intermediate)

Applications of Integration

This microworld explores the arc length of a curve, area under a curve, and surface area and volumes of revolution. For simplicity we explore only those surfaces of revolution that can be obtained by revolving a curve about x-axis. The theory will be briefly explained on the help pages that can be viewed by pressing the button math for this page. Often an example or two may be used to explain the theory. When a page of the microworld contains a button named instructions, you can press it to view instructions for using the interactivity of the page in order to make explorations.

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.

Progeny

In this 10-page Microworld, we study various forms of growth, considering: Propagation of Plants, Fibonacci's Rabbit Problem, Golden Ratio, and Golden Rectangle

Lakes

This 9-page Microworld constructs an interactive model of 3 lakes 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.

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.

Triangle Optimization

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.

Interactive Web Stories (Elementary)

Exploring Lines

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.

Shortest Paths

In this 4-page microworld 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 ways of finding the shortest distance.

Graphs of Functions and Symmetry

This 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.

Stand Alone Demos (Intermediate)

Inclined Planes

This 5-page Microworld tells a story about Newtonian Force Diagrams in the contexts of inclined planes and pulley systems.

Stand Alone Demos (Elementary)

Graphs and their functions

This 13-page Microworld 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.

Applications 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.

Trigonometric functions

This 6-page Microworld 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.

Visualization and Tools (Advanced)

Dynamic Programming

Dynamic programming has many applications, some of which we'll investigate in this workbook. One of these is in modelling the cortical surface of the brain. The first question that might come to mind when hearing about the brain is why is it shaped the way it is, and why is it so folded?

The brain is shaped the way it is simply because it has to fit into your skull, and it is folded because in order to be as large as possible (have the greatest surface area) it must fold in on itself, similar to crumpling a piece of paper to get it to fit through a small hole that it wouldn't fit into if it were flat.

Deformable Templates

Have you ever wondered about the patterns in the world around you? Like the patterns in the arrangements of molecules that make up our genes, or the patterns of behavior of people in crowds? Pattern theory proposes the idea that the world can be understood in terms of patterns.

The goal of pattern theory is to build a mathematical framework to describe these patterns. This chapter may seem like a deviation from our path to understanding the basics of metric matching, but it will actually be introducing the ideas that are central to our final goal. Imagine the set of all the possible images of a stomach. By assuming that it is possible to define how "close" or "far" every stomach shape is to/from every other stomach, we are assuming that they possess a common pattern.

Metric Mapping

The goal of this book is to explain how an algorithm for metric mapping in 1-dimension works. Only the 1-d algorithm will be explored because it is the easiest to explain, and the 1-d approach can be applied to multiple dimensions for 2-d or 3-d problems. A map is a function. It is used here as a transformation function between one coordinate system and another. A 1-d map can be represented on a set of coordinate axes for ease of viewing.

Introduction to Group Theory

This microworld will hopefully open your eyes to the wonderful world of group theory. The microworld is by no means a complete course, it's just something that might whet your appetite for group theory. I attempted to be as thorough as possible, while being as succinct as possible. I made sure to give a description of the activities on each page at the bottom of the Page Info, which is mainly to introduce the concepts of each page.

Lie Groups

A Lie Group (pronounced "Lee") is a Differentiable Manifold, which satisfies the properties of a Group, and which also has the property that its group operations are differentiable.

A sphere and a torus are just two examples of smooth (infinitely differentiable) manifolds. The earth, as it is a sphere, is locally "flat" and so it appears flat when we are on it, but at a distance we can see that it is actually a sphere. A population living on a torus would encounter a similar phenomenon.

Matrix Groups

A Matrix Group is a collection of square matrices that satisfies the group properties. The group composition is matrix multiplication, and the group inverse is the matrix inverse. These properties are in addition to those that define all groups (closure, associativity, existence of identity, and existence of inverses). See the Microworld, "Introduction to Group Theory", for a review of these properties.

Group Actions and Orbits

Applying an action to a point results in a transformed point. Similarly, applying an action to a group of points results in a transformed group. In the two examples to the left, the point and the group of points are part of the same space, the 2-D real space. The scale and shear actions can be thought of in both examples as acting on the entire space. The concept is a simple one, but is an important starting point to understanding how one member of an anatomical "family" (orbit) can be transformed into another member of that "family".

Visualization and Tools (Intermediate)

Evolutes

This 4-page Microworld provides a way to visualize the construction of evolutes of the graphs of functions, and of parametric curves.

Optimize

This 9-page Microworld 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.

Piecewise Defined Functions

This 9-page Microworld is designed to allow a student to visualize the graphs of functions, piecewise continuous or not, and to explore limits, continuity, and derivability.

Best Linear Fit

This 4-page Microworld develops tools for solving discrete models that depend upon proportionality. 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.

Nonlinear Equations

In the 14 pages of this Microworld, we explore various methods for finding approximate numerical solutions of the equation f(x) = 0 when f(x) is a nonlinear function.

Cycloids

This 6-page 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.

Linear Approximations

This 8-page Microworld illustrates the technique of linear approximation, which is the simplest way of approximating the value of a function by using a bare minimum of conditions.

Visualization and Tools (Elementary)

Limits of functions

This 5-page Microworld 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.

Playing with Points

This 6-page Microworld 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.

Pictures of Functions

This 6-page Microworld 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).

Transforms of Functions

This 5-page Microworld 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: Shifts, Reflections, Stretching / Shrinking.

Zeros of Polynomial Functions

In this 3-page Microworld, 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).

Cubic Splines

This 3-page Microworld was a student project designed to explore cubic splines. 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.