We have the good fortune of having "lab sections" here at my college. I'm interested in conducting some activities in the spirit of this talk. However, even in my stash of inquiry-based learning resources I can't easily find a large number of natural questions outside the usual suspects (falling bodies and other physics examples) that aren't somehow artificial-looking. I'm mostly interested in questions that lead into topics before related rates and optimization...since those topics have been around so long in calculus courses that the relevant ideas can be found "canned" easily.
Question: What are some very good inquiry-based activities for calculus that lead students to the discovery of basic definitions in the subject via natural examples?
One more point of clarification: Ideally, I'd like to see problems that lead to calculus topics through a "Martin Gardner" feel...if this helps...rather than some contrived artifice meant to jam the definition of derivative down the throats of unwilling students...
Remark: Perhaps this will be helpful: http://www.iblcalculus.com/. The bottle graph activities are nice (things like this can be found in Hughes-Hallett) and the traffic camera activity is pretty good, too. You can imagine a student wondering, outside of a mathematics class, how traffic cameras can measure a car's speed. That is another litmus test for the quality of activity...would a student come up with it on his or her own under minimally invasive circumstances?
Here is a "lowbrow" example that seemed to work well today: I drew a circle on the board, then the tangent line to the circle at a point and a radius of the circle connecting the origin to that point. Then I asked the students to use the definition of derivative to verify that the angle formed by any radius and the tangent line to the circle at the terminus of that radius is a right angle. This question was very simple to state, as well as straightforward and natural, but incorporates and reinforces several concepts from the course.