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.

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    $\begingroup$ 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. $\endgroup$ – Steven Gubkin Mar 27 '19 at 1:13
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    $\begingroup$ Related: matheducators.stackexchange.com/q/936/77 $\endgroup$ – Joel Reyes Noche Mar 27 '19 at 4:59

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.

task 1

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.

task 2

Imagine -- but don't do it yet -- that you repeatedly wiggle the mouse up and down, doing it fast but smoothly. (You will actually find it physically smoother to move the mouse rapidly in a circle; the horizontal part of the motion will be ignored.)

As a group, predict what the x-t, v-t, and a-t graphs would look like.

It takes some physical practice to get good results, e.g., for task 2 they often do it too slowly at first. The quality of the results is limited by the poor resolution of the mouse.

There are also sonar sensors that can be used for this. They're expensive and have their own difficulties and limitations.


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.

  • $\begingroup$ 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? $\endgroup$ – Nick C Nov 22 '19 at 23:45

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

  • $\begingroup$ Do you have a current resource for this? $\endgroup$ – Nick C Jun 4 '19 at 20:19

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