I teach AP Calculus BC at my high school and we have AP Calculus AB as a pre-req for taking BC. So most of my students are coming in with a strong calculus foundation, and I can spend less time on the basics (like product and quotient rule). This made me think about some more challenging topics I could include for the class. So now I'm curious as to what topics could appear in a standard college level Calculus I and II course that are left out of the AP Calculus curriculum (pages 27-30 of this pdf). Here's a preliminary list I've already brainstormed, but I'd love to see other topics that AP Calculus is missing out on.
- Delta-Epsilon definition of limits
- Logarithmic Differentiation
- Newton's Method of finding roots
- Simpson's Rule
- More work with trig integrals (like integrating $\int\sin^{2}{(x)} dx$)
- Trig substitutions for integrals
- Volumes by the shell method
- Surface area of revolution
- Root Test for series convergence (I think this might have been part of the AP Curriculum in the past, but it's not currently).
Personally, I don't think I would show my high school class the delta-epsilon definition, but I think the other topics I listed would be fair to include. Any other good topics I left off?
EDIT: I teach in the US. "AP" calculus refers to "Advanced Placement," which means that at the end of the school year, the students take a standardized exam. Some colleges offer college credit for students that reach a certain score threshold. There are two AP Calculus classes: AB and BC (they don't stand for anything that I'm aware of). You can see specifically what AB and BC cover in the pdf I linked. But roughly speaking, AB covers all of Calc I plus a bit of calc II (like volumes of rotation and some easy DiffEQ stuff) while BC covers most of Calc I and most of Calc II.