I just looked up a sample AP test, part AB and part BC online and scanned it. It strikes me as both trickier and more elementary, (in the sense of being less abstract and less rigorous), than a course I would usually teach in college. It consists almost entirely of computations, but perhaps trickier than most I would ask. But there is almost no theory at all, which I always include. There are a few questions that require the student to know the statement of a standard theorem or two, like the Rolle theorem or Fundamental theorem of calculus, but there are no definitions required and no theorem statements, and no proofs. Even at the second tier public state university I taught at, many non honors classes did require a student at least to know the definition of a limit and of a derivative, and often to prove something easy like the product rule for derivatives. In my honors classes we would often prove the existence of a global maximum for a continuous function on a closed bounded interval. The language is also not entirely universal. The term "Maclaurin" series to denote the Taylor series set at x=0 has no historical justification, at least according to my Harvard calculus professor, and hence is often not used. The term "relative maximum" is also not universal, since it is not specific (relative to what?), and the term "local maximum" is preferred, by me at least. I would often prove something like the fact that all monotone functions on a bounded interval are integrable, whether continuous or not, (due to Isaac Newton), and give discontinuous examples to check understanding of this. So students who jump from this level of high school preparation into my second college course are seldom prepared for the level of abstraction they will encounter. In later years I found myself reteaching the material from the first course quickly in the beginning of calc 2, from a higher point of view, but it was hard for the students anyway.
Simple but abstract things are hard to grasp, like the idea that the product rule has two parts: a theoretical part: 1) the product fg is differentiable if both f and g are; and a computational part: 2) in that case, the derivative of fg is f'g+fg'. Some of my students thought the second part was all there is, so potentially they would try to apply the second formula even if one of f or g was not differentiable. They could learn to parrot that the integral of any continuous function exists and is differentiable with respect to its upper limit, and they knew that e^(x^2) is continuous, but some would still say that the integral of e^(x^2) does not exist, confusing existence of a function with its expressibility in "elementary terms". This misses the whole point that one can use integrals of known continuous functions to define new differentiable functions. Such theoretical subtleties were apparently not sufficiently addressed in AP courses, and take some getting used to. This idea of course is used to give a rigorous definition to the natural logarithm function as the integral of 1/x, and may be used to give a rigorous definition of the inverse trig functions, as integrals of functions like 1/(Sqrt(1-x^2)), and later on may be used to define so called "elliptic functions" by using cubic polynomials in place of quadratic ones. This is part of the struggle to have students appreciate that a "function" f is any way at all of specifying a value f(p) at each point p of a domain, and f need not be expressible as a familiar formula. Failing to grasp this blocks the way to much of mathematics.
https://secure-media.collegeboard.o...ple-questions-ap-calculus-ab-and-bc-exams.pdf
