We just wrapped up our applications of derivatives unit in AP Calc and I wrote this little activity to introduce local linearization and tangent line approximations. It's one of the simplest but most important ideas in all of AP Calc- that if we look close enough, a tangent line at a particular point will be almost the same as the function since functions are locally linear.
I knew my kids were more than capable of discovering this phenomenon and explaining it to each other without much help from me, so I wrote up this very simple activity for them:
We wrapped up with a discussion about what students had discovered, focusing heavily on part (f). I love seeing kids have these "duhhhh" moments- when something seems so simple that it's barely worth mentioning. It reminds me of those moments in college when a professor calls something trivial that was anything but trivial to you. When my students can start to view concepts in Calculus as so obvious they barely deserve explanation, I'm a happy teacher.
Here's a link to the document!