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
the function and the plane and view the
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.
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
function of a single variable, and a shrinking zoom window
that is enlarged periodically.
function of two variables and then smoothly enlarges the
neighborhood of the center point. More reminiscent of a continuous
function of two variables, shrinks the display (like a zoom window
for a function of one variable) and periodically enlarges the
lines and osculating circles
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
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
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).
These tools are to visualize the integral as an area and to
visualize the convergence of Riemann
|One variable Riemann sums
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
|Two variable Riemann sums
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
function of one variable and colors the part over which an integral
is computed with a different color. No
Visualization tools for parametric curves and parametric surfaces.
|Follow the particle
curves are paths of particles. This tool allows you to view the
particle as it moves along the path without the path itself being
|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
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.
contains the setup for approximating integrals using the midpoint,
trapezoidal and Simpson's