Beginning College Level Microworlds

(Arranged alphabetically)

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Applications of integration
Author(s): ravinder kumar
Topics: integration, area,arc length of a curve, area under a curve,surface area, volume of revolution (Level: Beginning College)

Description: This 9-page microworld explores arc length of a curve, area under a curve, and surface area and volume of revolution. For simplicity we explore only those surfaces of revolution that can be obtained by revolving a curve about x-axis

Applications of the derivative
Author(s): samuel masih
Topics: limits, difference quotients, derivatives, second derivatives, monotonicity, concavity, inflection point, tangent lines, secant lines, slope, piecewise defined functions (Level: Beginning College)

Description: This microworld examines derivatives of functions. The reader may supply functions, choose points on the graph, and the tangent and secant line approximations are drawn while she chooses small increments for the independent variable from the base point.

Bernoulli trials
Author(s): kyle siegrist
Topics: probability, bernoulli trials, law of large numbers, distributions (Level: Beginning College)

Description: This microworld gives the opportunity to experiment with bernoulli trials. The term Bernoulli trial refers to a sequence of trials, each of which has two possible outcomes, called success and failure. The probability of success is the same for each trial.

Calculus in action: chapter0: section 1: precalculus introduction
Author(s): james white
Topics: extrema, maxima, minima, inflection points, monotonicity, graphing functions, derivative (Level: Beginning College)

Description: This precalculus introduction to calculus in action covers the following topics. free fall, ballistic trajectories, inclined plane, energy conservation, momentum conservation, pendulum, escape velocity, brachistochrone, symbolic calculator.

Calculus in action: chapter1: section 1: ballistics
Author(s): james white
Topics: extrema, maxima, minima, ballistic trajectories, monotonicity, graphing functions, derivative (Level: Beginning College)

Description: This microworld studies the derivative as a means of approximation. But galileo used another method, which was due to archimedes, to calculate the exact distance fallen by a uniformly accelerated object. This method will be called integration.

Calculus in action: chapter1: section 2: Newton's method
Author(s): james white
Topics: limit, newton, equations, iterative sequence, recursive sequence, graphing functions, derivative (Level: Beginning College)

Description: This microworld contains an illustration of the conventional picture of newton's method. At each step, the student sees a ray trace the tangent line to the x axis, and determine a new value of x. The student defines the function and starting point.

Calculus in action: chapter1: section 3: Newton's method and Chaos
Author(s): james white
Topics: chaos, logistic map, equations, iterative sequence, recursive sequence, graphing functions, derivative (Level: Beginning College)

Description: Newton's Method leads to a sequence of numbers that may converge to a solution. This sequence may lead to stable oscillations that do not converge to a solution from almost any starting point but oscillate among several values. And it may lead to chaos.

Calculus in action: chapter2: section 1: polar coordinates
Author(s): james white
Topics: polar coordinates, moving frames, central force, centripetal force, satellite orbits, ellipse, conic section (Level: Beginning College)

Description: In this microworld, we introduce a style of description that will serve us well throughout the book. It is called polar coordinates. Any satellite moving under the influence of a central force (like Newtons law of gravity) must remain in a plane.

Calculus in action: chapter2: section 2: conservation of energy
Author(s): james white
Topics: conservation of energy, conservation of angular momentum,radius of curvature, satellite orbits, moving frames (Level: Beginning College)

Description: In this microworld, we discover another conserved quantity for satellite motion. That is the energy of the satellite. For that, we must make Newton's assumption about the actual form of the gravitational law.

Calculus in action: chapter3: section 1: plane curvature
Author(s): james white
Topics: plane curvature, fundamental theorem of calculus,arc length parameterization, gauss map, winding number, integral curvature (Level: Beginning College)

Description: In this microworld, we study plane curvature. We take up the notion of invariance under parameterization. We study arc length parameterization, the gauss map, and the fundamental theorem of calculus.

Calculus in action: chapter3: section 2: logarithmic spirals
Author(s): james white
Topics: initial value problems, euler number,natural exponential, exponential functions, natural logarithms, eulers method, ordinary differential equations, compound interest (Level: Beginning College)

Description: In this microworld, we study the equiangular spiral, and develop the general techniques of differential equations that are needed to analyze it. We develop first eulers number and natural exponentials and logarithms and continue to apply eulers method.

Calculus in action: chapter4: section 1: geometry of planetary orbits
Author(s): james white
Topics: keplers second law, gravity, acceleration, fundamental theorem of calculus, planetary orbit, space curves, vector geometry, angular momentum, frenet frames (Level: Beginning College)

Description: In this microworld, we will show that planetary orbits are actually plane curves, and we establish the existence of the first conserved quantity, the angular momentum to establish keplers second law.

Calculus in action: chapter4: section 2: the grand deduction
Author(s): james white
Topics: keplers first law, keplers second law, keplers third law, gravity, acceleration, fundamental theorem of calculus, planetary orbit, space curves, vector geometry, angular momentum (Level: Beginning College)

Description: In this microworld, we follow isaac newton in the deduction of the first and third laws of planetary motion around the sun of john kepler from his single hypothesis of universal gravitation.

Derivatives and graphs of functions
Author(s): james white
Topics: maxima, minima, critical points, inflection points, graphing (Level: Beginning College)

Description: This microworld experiments with some general applications of the derivative to the study of the behavior of functions. This includes their critical behavior, local extrema, inflections. The reader may work with built-in examples or supply her own.

Discrete mathematics and computational structures I
Author(s): james white
Topics: set theory, logic, propositional calculus, boolean algebra, relations, function composition, permutations, demorgan's laws, order, cardinality, digraphs (Level: Beginning College)

Description: This is the first half of a 14-week course in discrete mathematics. The lectures cover: sets,functions sets and logic,composition of functions,set operations,permutations of sets,boolean algebra,graphs and directed sets,relations,order,cardinality.

Discrete mathematics and computational structures II
Author(s): james white
Topics: set theory, logic, propositional calculus, boolean algebra, relations, iteration, recursion, counting, critical paths, prolog, automatic theorem proving, ruleset, inference engine function composition, permutations, demorgans laws, order, cardinality, (Level: Beginning College)

Description: This is the second half of a 14-week course in discrete mathematics. The lectures cover: iteration and recursion,search,sets defined by propositions,critical path analysis,counting,automatic problem solving,relations and functions,graphs and logic.

Epsilon
Author(s): dan kalman
Topics: graphing, limits, epsilon-delta definition of limit (Level: Beginning College)

Description: This microworld presents the epsilon-delta definition of limit as a game, using a geometric picture of the usual inequalities. Say the graph of a function "fits in a rectangle" if it can be graphed in the rectangle without going off the top or bottom.

Graphs and their functions
Author(s): k.p. satagopan
Topics: graphing, composition of functions (Level: Beginning College)

Description: This microworld steps you through a series of 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.

Hifi: personal household finance manager
Author(s): james white
Topics: financial management, escrow, accounting, paying bills, budgets, interest (Level: Beginning College)

Description: HiFi is a fully functional object-oriented, LISP-based Expert System that can make managing and planning your household finances easy and fun.

Implicit surfaces
Author(s): james white
Topics: surfaces, implicit surfaces, equations (Level: Beginning College)

Description: Implicit Surfaces is a utility that students may use to draw the solution of an equation: f(x,y,z) = 0 in 3 dimensions. They may draw any pair of surfaces simultaneously and view them solid or wireframe.

Inclined planes
Author(s): samuel masih
Topics: force diagrams, friction coefficient, trigonometry of inclined planes (Level: Beginning College)

Description: This 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 various parameters.

Introduction to linear programming
Author(s): ravinder kumar
Topics: linear programming, simplex, simplex method, simplex algorithm, linear optimization, slack variables, objective function, pivot (Level: Beginning College)

Description: The seeds of linear programming were sown during world war II. Linear programming deals with optimization. Learn the simplex algorithm.

Limits and continuity
Author(s): samuel masih
Topics: left-hand limits, right-hand limits, continuous functions (Level: Beginning College)

Description: This 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 piecewise. There is a versatile function grapher on each page.

Linear approximations
Author(s): ravinder kumar
Topics: differential approximation, linear approximation, differential, derivative, tangent line (Level: Beginning College)

Description: Linear approximation is the easiest way to approximate the value of a function. This method is also the simplest way to interpolate data. The method uses the value f(a) and derivative at the point x=a to approximate value of f(c), where c is close to a.

Magnify
Author(s): dan kalman
Topics: graphing, limits, derivative, differentiability, tangent lines (Level: Beginning College)

Description: This workbook explores two fundamental ideas of calculus, the limit and the derivative, visually in an activity of successively zooming in on one point of a graph. If the graph eventually becomes a straight line, the derivative is the slope of that line.

Mathscript author toolkit
Author(s): james white
Topics: equations, graphing, factorization, tutorial, command line,differentiation, symbolic algebra,function composition (Level: Beginning College)

Description: This microworld has a command-line that teachers can provide for their students. Students may use the book to type commands to do graphics, decimal arithmetic, matrix or symbolic algebra, for example. It provides a tutorial for the mathscript language.

Matrix equations
Author(s): dan kalman
Topics: matrices, linear algebra, row reduction (Level: Beginning College)

Description: This is a web presentation of three interpretations of a linear system: as simultaneous scalar equations, as a single vector equation, and as a matrix-vector equation. For each view there is an interactive visualization exercise, embedded in a webpage.

Matrix workbench
Author(s): dan kalman
Topics: matrices, linear algebra, row reduction (Level: Beginning College)

Description: This microworld calculates exact rational matrices. All operations are exact rational operations,unless the reader enters decimals in the matrices. The reader has a control panel to create and edit new matrices, and a command line for matrix expressions.

Method of bisection
Author(s): kwok-wai mok
Topics: nonlinear equations, bisection method (Level: Beginning College)

Description: The method of bisection is one of the simplest and most reliable ways to find numerical solutions to equations. Use it to solve an equation in the form: f(x) = 0 for x. We assume that f is a continuous function on an interval of numbers [a, b].

Mwmilton calculator
Author(s): william waller
Topics: graphing, slopes, tangent lines, scatter plots, linear regression, systems of equations (Level: Beginning College)

Description: This microworld performs a variety of numerical and symbolic computations: m.w. milton graphs and tabulates functions of one variable, plots data points,fits curves to data points,solves systems of linear equations and graphs functions of two variables.

Optimize
Author(s): samuel masih
Topics: maxima, minima, critical points, differentiation, graphing, geometry, area and volume calculations (Level: Beginning College)

Description: This 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.

Pollution of lakes
Author(s): ravinder kumar
Topics: difference equations, dynamical systems, ecology, modeling (Level: Beginning College)

Description: In this microworld there are three lakes. The inflow pattern for the three lakes is defined by difference equations. The task is given to engineers to figure out how many years it will take to clean lake 3. You experiment by varying flow parameters.

A primer on derivatives
Author(s): ravinder kumar
Topics: limits, derivative, continuity, derivative formulas, implicit differentiation, implicit curves, slopes, tangent lines, calculus (Level: Beginning College)

Description: This microworld is an active excursion into one of the most basic concepts of the calculus: the derivative. It develops this idea with many simulations, illustrations, examples and exercises. Throughout the book, readers may ask their own questions.

Progeny
Author(s): ravinder kumar
Topics: fibonacci,golden ratio,difference equations (Level: Beginning College)

Description: In this 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 use linear difference equations.

Rational matrix calculator
Author(s): james white
Topics: matrices, linear algebra, row reduction (Level: Beginning College)

Description: This microworld calculates exact rational matrices. All operations are exact rational operations,unless the reader enters decimals in the matrices. The reader has a control panel to create and edit new matrices, and a command line for matrix expressions.

Row reduction
Author(s): dan kalman
Topics: matrices, linear algebra, row reduction (Level: Beginning College)

Description: This microworld has a tool that one may use to row reduce matrices. All operations are exact rational operations, unless the reader enters decimals in the matrices. There are two "help" pages that explain how to operate the book.

Shortest paths
Author(s): kanchan manaktala and ravinder kumar
Topics: exploring shortest distance, formula for finding the distance of a point from a line, finding shortest distance between a point and a curve (Level: Beginning College)

Description: This microworld explores the distance from a point to a line or a curve. The user can explore characteristics of the shortest distance. Shortest distance from a point to a line is studied using a formula in geometry as well as optimization techniques.

Simultaneous equations
Author(s): dan kalman
Topics: college algebra, linear function, lines, graphing, relations, inequalities, equations, slope, intercept (Level: Beginning College)

Description: This is a web presentation of three interpretations of a linear system: as simultaneous scalar equations, as a single vector equation, and as a matrix-vector equation. For each view there is an interactive visualization exercise, embedded in a webpage.

Spherical logo
Author(s): james white
Topics: logo, spherical logo, spherical geometry, riemannian geometry, parallel translation, 3d graphics (Level: Beginning College)

Description: Logo has long been a favorite way of introducing students to computational geometry and logic. Well, Terra, the turtle in this microworld lives in a sphere. Her geometry is Riemannian (spherical) geometry and she has a great many things, to show you.

Stay afloat!
Author(s): james white
Topics: optimization, extrema, archimedes' principle, bouyancy, implicit differentiation (Level: Beginning College)

Description: this playbook explores archimedes' bouyancy principle and optimization problems in a story in which players build a boat by cutting and assembling planks from a board and putting them together. They may view their product in three dimensions.

Surfaces of revolution
Author(s): james white
Topics: graphing, surfaces, level sets, partial differentiation, directional derivatives, maxima, minima, extrema (Level: Beginning College)

Description: This microworld discusses surfaces of revolution. Each topic is developed through an interactive exploration that allows the reader to experiment with the ideas and to visualize its consequences.

Symbolic calculator
Author(s): james white
Topics: equations, graphing, factorization, tutorial, command line,differentiation, symbolic algebra,function composition (Level: Beginning College)

Description: Students may use this microworld to type commands to do graphics, decimal arithmetic, matrix or symbolic algebra. The online documentation provides a tutorial for the mathscript language, the over 270 mathscript commands and built-in functions.

Synthetic division
Author(s): james white
Topics: college algebra, polynomials, gcd, synthetic division, symbolic algebra (Level: Beginning College)

Description: In this command-line workbook, students explore synthetic division of polynomials with rational coefficients. There is, among others, a command called synthetic that returns the quotient of one rational polynomial by another together with the remainder.

Triangle optimization
Author(s): ravinder kumar
Topics: optimization, area of triangles, isosceles triangles (Level: Beginning College)

Description: This microworld helps readers see 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 proof depends upon multivariable calculus. The second is simpler.

Vector equations
Author(s): dan kalman
Topics: matrices, linear algebra, row reduction (Level: Beginning College)

Description: This is a web presentation of three interpretations of a linear system: as simultaneous scalar equations, as a single vector equation, and as a matrix-vector equation. For each view there is an interactive visualization exercise, embedded in a webpage.

A world of curves
Author(s): ravinder kumar
Topics: Parametric curves, envelopes, evolutes, involutes, pedals, negative pedals (Level: Beginning College)

Description: This microworld studies envelopes, evolutes, involutes, pedals, negative pedals etc. These ways of determining curves not only shed light on the curves and their characteristic properties, but also produce some fantastic curves.

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