Algebraic manipulations of polynomials and rational functions provide the foundation for a lot of the work we do in calculus. In taking limits, for example, we often encounter rational functions — quotients of polynomials — that need to be analyzed and simplified before we can evaluate a limit.
More specifically, here are some basic skills that come up over and over again in calculus:
In addition to polynomials and rational functions, trigonometric functions — sine, cosine, tangent, cotangent, etc. — appear both theoretically and in application.
Again, for a broad overview, KA has a whole section on Trigonometry, all of which appears regularly in calculus. Specific relevant skills include: