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One way to use CalcPlot3D, a versatile Java applet, is to demonstrate new concepts during multivariable calculus lectures. I often develop a new concept on the chalk board first and then take a couple minutes to make the concept come to life using the applet. Students find these demonstrations helpful and fun, and they bring variety to my presentations, helping students process the new concepts in a new way.

An even more exciting way to use CalcPlot3D in class is to engage in a visual exploration of new concepts using “What if…” types of questions. An example of a topic for which I find this approach works especially well is exploring the variety of possible parameterizations of a plane/space curve, paying special attention to the behavior of the velocity and acceleration vectors. Using these sorts of visual demonstrations in class improves student learning, but to fully engage students in the exploration and discovery process and give them the best chance of learning the geometric nature of the calculus concepts, I feel it is vital to give students opportunities to “play” with the concepts visually themselves. This article focuses on one way this can be done: by requiring students to visually verify solutions to particular homework problems and turn these in for a grade. [Another way this project supports student engagement and “play” is with the guided explorations being developed for various concepts. See the main project website to explore these. At this writing, there are explorations for Dot Products, Cross Products, Velocity & Acceleration Vectors, and Lagrange Multiplier Optimization.]

Below is a list of example topics where I often assign this type of visual verification exercise to my students to get them to begin using the applet on their own. Once they start using the applet in this way (because they have to), students often report using the applet more often on their own to explore additional exercises they complete from the textbook and on other assignments. Before giving a visual verification assignment involving new skills with the applet, I always demonstrate using CalcPlot3D to visually verify a similar problem we worked on the board. Once students have seen one example using the applet, most have little trouble completing the exercise on their own.

Without further discussion, let’s look at some examples of how this can be done! As you develop your own examples, please send them along to me! I would love to develop a library of useful ways to use this applet.

- Plane through Three Non-Collinear Points
- Line of Intersection of Two Planes
- Intersections of Other Types of Surfaces
- Contour Plots
- Level Surfaces
- Tangent Planes
- Directional Derivatives
- Taylor Polynomials of a Function of Two Variables (1st and 2nd degree)
- Lagrange Multiplier Optimization
- Riemann Sums of a Double Integral

Paul Seeburger (Monroe Community College), "CalcPlot3D, an Exploration Environment for Multivariable Calculus - Using CalcPlot3D to Visually Verify Homework in Multivariable Calculus," *Loci* (November 2011), DOI:10.4169/loci003781