I am having calculus classes that are titled as Calculus for Mathematicians, for the rest of the students who are studying calculus, they use Stewart's book. In our classes, we're having something roughly similar to Courant's Introduction to Calculus and Analysis. I am not looking for calculus or analysis, I'm looking for a hybrid course that is about calculus, but there is an addition of some topics in analysis.
I'm not sure if such calculus syllabus are standard, but if they are, can you recommend me some video lectures on it?
The Massachusetts Institute of Technology has a whole library of lectures, I'm sure you'll find something you like:
http://ocw.mit.edu/courses/#mathematics (MIT Open CourseWare)
Not all of them have videos, and I'm not sure if they will have exactly what you need, but hopefully that will help.