Bernd Schröder

MathCAD

 

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Fundamental Instructions.

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.

 
How to compute limits in MathCAD.

 
How to compute derivatives in MathCAD.

 
How to compute sums in MathCAD.

 
How to compute indefinite integrals in MathCAD.

 
How to write simple programs in MathCAD.

 
How to plot two dimensional parametric curves in MathCAD.

 
How to plot three dimensional parametric curves in MathCAD.

 
How to produce surface and contour plots of functions of two variables in MathCAD.

 
How to plot parametric surfaces in MathCAD.

 
How to plot a two dimensional vector field in MathCAD.

 

 

Advanced Tools.

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.
Slicer Specify the function and the plane and view the slice.
  Normal Slicer 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.
Rotating Slicer   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.

1D-zoom   Displays a function of a single variable, and a shrinking zoom  window that is enlarged periodically.
2D smooth zoom  Displays a function of two variables and then smoothly enlarges the neighborhood of the center point. More reminiscent of a continuous deformation.
2D stepped zoom  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.
Tangent lines.
Tangent 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.
Secant 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.
  Newton's method 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. No animation.

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 circle   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.
Checking solutions of DE's with MathCAD   brief instructions, trivial example
z-alpha ... 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.
Numerical integration   This file contains the setup for approximating integrals using the midpoint, trapezoidal and Simpson's rule.