For a couple years in Pre-Cal, I had a lot of success with design projects. Students would take an aspect of the curriculum we had done recently, and complete an assignment that showed me a lot of aspects of that topic (Vectors, Polynomials, Polar Conics). What they used to accomplish it was up to them. It was a way to defeat identical project syndrome.

Adapting that idea to Calculus has taken some time as I tried to come up with ideas that suited the format. A few weeks ago I had students complete the first one. We had spent time discussing continuity and limits, so I had students design a piecewise function that demonstrated a lot of aspects of the topic. The function had to have five continuity problems (left/right disagreement, removable discontinuity, and unbounded), they had to demonstrate they could find the limit at various points on their function, and they had to explain the situation.

The lower one has a lot more detail to offer since I can see the functions used and all the boundary points are labeled. Students had a good opportunity to play with function restrictions in Desmos and could use a wide variety of function types to accomplish the goal.

Every time I do one of these, the students spend of a time thinking about how to make the requirements happen. Common problems throughout this project were multiple functions in the same domain, figuring out how to translate unbounded functions, and getting a handle on restrictions. As I keep my students in groups, usually one student is able to crack the issue and spread the knowledge around.