# Physical devices for exploring calculus or pre-calculus

I saw this partial derivative machine yesterday, and it got me excited about other physical devices for exploring calculus concepts in a "lab" setting (e.g. make a prediction, collect data, etc.) Do you use any devices like this?

There are certainly some nice objects for visualizing graphs, and great computer apps, but I am looking for physical devices that can be used to explore calculus (or precalculus).

Here is a question about interesting physical implements for the classroom, but it was specifically about wall-mountable, display items. I want great ideas for things to build and bring to class that assignments/labs can be built around.

• I went to the doctors office yesterday for a cough, and they had me blow into a tube to measure my lung capacity. What they actually measured was the rate of change in the volume with respect to time. They produced a $V'$ graph, and then the computer numerically integrated to produce the $V$ graph. It would be pretty cool to get one of those machines for a calc class. – Steven Gubkin Mar 27 at 1:13
• – Joel Reyes Noche Mar 27 at 4:59

A planimeter is a wonderful little device that measures the area surrounded by a simple closed curve by tracing it. It is based in the version of Green’s Theorem that computes area by integrating vertical/horizontal displacements along the curve.

• So, I built a planimeter (and also bought an old one off Ebay) for demonstration in class, but I couldn't think of a nice way to use it (the planimeter) to explore calculus. We used Green's theorem to prove why it worked, but it didn't provide much of an "exploration" -- more a "cool thing that calculus proves should work". How might you use one in an exploratory setting? – Nick C Nov 22 at 23:45

In chemistry, it is still common to integrate some test results by cutting the curve out and weighing the paper.

• Do you have a current resource for this? – Nick C Jun 4 at 20:19

I have a browser-based app I use where students wiggle the mouse and see graphs of position, velocity, and acceleration. These are the tasks I use with this.

Your goal is to produce an x-t graph that looks like a staircase going down and to the right. Discuss with your group (a) how you would need to move the mouse in order to accomplish this, and (b) how the v-t graph would look.

Pass out blank graphs, one per group, and do laps around room.

As each group agrees on a prediction, tell them to type in the URL they see on the demo screen and try it.