Does my teacher have any right to do this?

I just learned some new stuff yesterday in calculus III that will be on my calculus III test tomorrow.

Yes, this is normal.

You say that they could just move everything forward one day, but could they? If you move everything forward one day, all of a sudden you have a lesson taking place prior to the start of the semester.

A single class period in college is equivalent to 2-3 high school class periods.

The first sections in a math book aren’t bs. Just because you feel that it doesn’t need to be covered, it does not mean you can assume that everyone else feels the same way. Some people need to have that structure with the basics. It’s important to develop ideas from the ground up.