This demo illustrates matrix-vector multiplication Ax, computed as linear combinations of the columns of A. Animation is done using a Java applet.
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In most introductory linear algebra courses, prominent applications of transformations or mappings are moving (rotating, reflecting, translating), resizing (contracting, expanding), changing the shape (shearing, projecting) objects in the plane. A typical problem in such courses is to write the matrix of a transformation that changes points in such prescribed ways. The goal of this demo is to help students more easily visualize such transformations on a wide class of plane objects.
The purpose of this demo is to illustrate the use of the method of cross sections to find estimates of the volume of physical objects. The objects in this demo are not solids of revolution. The demo presented here utilizes Mathematica. Mathematica 5 notebooks are provided. The notebooks can be modified to approximate volumes of other objects for which a cross section approach is appropriate and a digital photograph is available.
This demo incorporates combines the visualization of approximating rectangles together with a graph of the approximate areas as a function of the number of rectangles. Thus, the limiting behavior of the approximating sums can be observed on the graph.
This demo estimates the area of a circle or triangle using a probability experiment employing the Monte Carlo technique. We also indicate how to use our approach to estimate the area of a polygonal region.
This demo illustrates two methods for computing the area of a triangle (and other regions) using the coordinates of the vertices. Both methods lead to interesting extensions which can provide topics for student investigation at a variety of academic levels.
This demo provides students with a concrete understanding of the average rate of change for physical situations and for functions described in tabular or graphic form. Animations in gif and Quicktime format are available, as well as Excel programs for classroom or student use.
This demo develops a procedure for randomly allocating bacteria to a fixed number of droplets from a total of 100 droplets using a probability experiment employing the Monte Carlo technique.
This demo provides a visual foundation and geometric intuition for best fit (least squares) models of data sets of ordered pairs using lines or parabolas. MATLAB routines are included.
THis demo provides an application of linear combinations to construct colors displayed on computer screens. This colorful demo provides a realistic example of linear combinations in action. BOth MATLAB and Java routines are included.
This demo demonstrates how to draw a circle using a carpenter's square, also known as a steel, framing, or rafter square.
This demo illustrates how a carpenter can draw an ellipse on wood or a sheet of wall board using simple tools. A "jig" can be used to demonstrate the technique and there are software animations to illustrate the use of the "jig".
This demo develops a function that measures the power of the signal of a cell phone as a user moves in a cellular network and then determine the position in the network when the signal is a maximum.
This demo illustrates a simple physical method for determining the centroid of an irregular region.
This demo develops a procedure for randomly allocating chocolate chips to cookies using a probability experiment employing the Monte Carlo technique.
This demo shows the development of the sine and cosine functions and their graphs by 'wrapping' around a circle. A MATLAB routine is included.
This demo illustrates the use of statistical simulation to find a rule which will lead to the prediction of probabilities.
This demo collection approaches probability through demos and experiments by using simulations via the Monte Carlo method. The collection includes interactive experiments for estimating the area of a circle, simulation of a network, allocation problems, creation of a rainbow, and choosing segments & drawing circles within a circle.
This is a gallery of animations for illustrating selected families of conic sections. Included are GSP and Excel files.
This demo provides a visual development for the "locus of points" definitions of the conic sections.
This demo provides a toolbox of visual aids for geometrically oriented word problems. These visual tools are designed to help students develop equations that provide an algebraic model for the problem. The visual aids are presented in a gallery. No particular software package is required. Support for a viewer of gif or mov files is required. Viewers within a browser, Windows media player, Quicktime, or a commercial program can be used.
Students in calculus need to be proficient in working with functions in a variety of ways: graphical, numerical, analytic, or verbal, and to understand connections among these representations. This demo provides a set of visualizations designed to help students better understand what it means for a piecewise function to be continuous at a particular domain value.
This demo shows how a craftsman such as a carpenter, cabinet maker, or sheet metal worker can divide a rectangular piece of material into a number of regions of equal size using a pencil and straight edge. The technique illustrated in this demo avoids cumbersome and sometimes inconvenient direct measurements. This demo is appropriate for a geometry class, an introductory modeling class, or possibly a general mathematics class for industrial arts.
Take a solid cube with rods attached at diagonally opposite vertices. Hold the rods horizontally and rapidly spin the cube. (See Figure 1.) You should see a curved outline formed by the spinning cube. The objective of this demos is to discover how the straight edges of the cube become curved. The demo is physically based, but can be simulated within various software packages.
This demo provides a gallery of visual aids that illustrate fundamental concepts for understanding and developing equations that model optimization problems, commonly referred to as max-min problems. Animations, MATLAB routines and Java applets are included.
This demo provides a toolbox of visual aids that illustrate fundamental concepts for understanding and developing equations that model related rate problems. A gallery of animations for various problems is included.
This is a gallery of demos for illustrating the disk method for volumes of solids of revolution. The animations can be used by instructors in a classroom setting or by students to aid in acquiring a visualization background relating to the steps of disk method. Three file formats, gif, mov, and avi are available.
The definition of the derivative at a point x = a involves the limiting behavior (as h approaches zero) of the slopes of the secant lines passing through through (a,f(a)) and (a+h,f(a+h)). This demo provides a visualization of the secant lines as they approach the tangent line.
THis demo provides a toolbox of aids for teaching students about the volume of solids by the disk method. We include a variety of approaches which can be easily adapted to different levels of instruction. In addition, a gallery of animations is included which can be run on a number of platforms.
This demo is designed to help students use graphical representations of functions to determine the domain and range. A set of interactive Excel spreadsheets and animations are included and can be downloaded.
This demo provides visualizations of the effect on conics by varying the eccentricity.
This demo provides an early introduction to related rates of change using escalator motion and average rates of change.
This is a gallery of animations for illustrating selected families of log and exponential functions. Included are GSP and Excel files.
This demo provides a toolbox of aids for teaching students about families of functions. The toolbox includes collections of animations that illustrate how functions change when certain parameters are varied.
An Easy way to access the galleries of animations within the project Demos with Positive Impact.
This is a gallery of demos for visualizing common max-min problems. These animations can be used by instructors in a classroom setting or by students to aid in acquiring a visualization background relating to the steps for solving max-min problems. Two file formats, gif and mov are available.
This is a gallery of demos for visualizing common related rate situations. These animations can be used by instructors in a classroom setting or by students to aid in acquiring a visualization background relating to the steps for solving related problems. Two file formats, gif and mov are available.
This is a gallery of demos for visualizing common geometric word problems that involve a measurement. These animations can be used by instructors in a classroom setting or by students to aid in acquiring a visualization background relating to constructing algebraic equations that model a word problem. Gif and Quicktime formats can be downloaded.
This demo provides a collection of examples of physically significant vector fields that can be used to illustrate important topics in the study of vector calculus