When using a computer algebra
system, certain elementary questions frequently arise. Though they are not hard
to answer, they can present an annoying threshold for students. To overcome
these thresholds, the collection of files below shows how to solve certain tasks
in MathCAD. These instructions will be live links to MathCAD files in the pdf
version of my modular materials.
The tools below are meant to
produce animations and visualizations with user-specified
functions and user-specified domains, center points, etc. They
can also be used as exploration tools by students.
Development of these tools was sponsored by
the Louisiana Board of Regents Fellow of Excellence in Engineering Education grant "Mathematical Support
for an Integrated Engineering Curriculum".
No guarantee for correctness of the code is made or implied.
Comments are appreciated.
|Slicers for functions of two variables. These tools intersect the graph of a given function f(x,y) with a vertical
plane specified by the user. The intersection can be viewed as the graph
of a function of one variable.
||Specify the function and the plane and view the slice.
||Specify the function and the plane. Animation moves the plane
in the direction of one of its normal vectors and displays the slices as the plane moves along.
||Specify the function and the plane. Animation rotates the
graph about a given center point and displays the intersection with the plane.
Zoomers in one and two variables. These tools allow to zoom in on a specific point on a function of one
or of two variables. Should help in visualizing differentiability of functions
of one or two variables. Just specify the function and the x- or (x,y)-coordinate
of the point and animate. The zoomer automatically shrinks the neighborhood
about the point for the specified number of steps.
||Displays a function of a single variable, and a shrinking zoom
window that is enlarged periodically.
||Displays a function of two variables and then smoothly enlarges the neighborhood of the center point. More
reminiscent of a continuous deformation.
||Displays a function of two variables, shrinks the display (like a zoom window for a function of one variable) and
periodically enlarges the remaining picture.
lines and osculating circles
||This tool displays a function
of one variable and its tangent line at a point. In an animation the point can be
moved through the domain showing how the tangent line changes as we move across the domain. A second animation allows the
same thing for the osculating circle.
lines approach the tangent line
||Choose the base point for the tangent
line and a starting point for the secant. Animate to display how the secant line approaches the tangent line as the second point
approaches the base point.
||Choose a starting point for Newton's method and animate to
display how the zeroes of the tangent line approach the zero of the function (or how a situation arises in which Newton's method fails).
Integral tools. These tools are to visualize the integral as an area and to visualize
the convergence of Riemann sums.
|One variable Riemann sums
||Displays a function of one variable and the specified number of Riemann rectangles with specified sample
points. Animation increases the number of Riemann rectangles by a specified number.
|Two variable Riemann sums
||Displays a function of two variables and the specified number of Riemann blocks with specified sample points.
Animation increases the number of Riemann blocks by a specified number.
|Visualizing definite integrals
||Displays a function of one variable and colors the part over which an integral is computed with a different color.
Visualization tools for parametric curves and parametric surfaces.
|Follow the particle
||Parametric curves are paths of particles. This tool allows you to
view the particle as it moves along the path without the path itself being drawn.
|Velocity, acceleration and the osculating
||This tool draws the parametric curve and puts a particle on it. Two animations are possible. One moves the particle along its
path and plots the velocity and acceleration vector at every point. The other moves the particle along the path and plots the osculating
circle at each point.
| Other tools.
solutions of DE's with MathCAD
||brief instructions, trivial example
||... is the point such that to the right of it lies alpha of the area
under the standard normal curve with mean 0 and standard deviation 1. Input alpha to get z-alpha or animate.
||This file contains the setup for approximating integrals
using the midpoint, trapezoidal and Simpson's rule.