I am due to teach Calculus II in the fall at an American state university. Our calculus sequence is somewhat slow, due to the fact that many of our students come with limited backgrounds. Most of our typical syllabus consists of the following.
Techniques of integration. We teach the students to memorize and apply complicated techniques for evaluating obscure integrals --- all of which Wolfram Alpha cando in a fraction of a second.
Why do they work? Well, that requires a lot of mathematical maturity to understand, and for the most part we don't even try. For example, Stewart's calculus book makes no attempt to explain why partial fractions work, not even a handwavy one. Just "A theorem in algebra guarantees that it is always possible to do this," and the rules are presented for memorization, in all of their complicated glory.
Sequences and series. This chapter is a huge, long slog in every calculus book I've ever seen. Stewart's book starts with lots of scary theorems ("Every bounded monotonic sequence is convergent", proved with epsilon-delta), so that by the time you write $1 + \frac{1}{2} + \frac{1}{4} + \cdots = 2$ on the board, or even $.9999\dots = 1$, the students are made to feel that it secretly means something complicated and technical which they half-admit they don't really understand.
From there it gets even worse. Typical problem in Stewart: Does $\sum_{n = 1}^{\infty} \frac{n + 5}{\sqrt[3]{n^7 + n^2}}$ converge or diverge? Well, there is a rule for that -- one of very many -- and for the comparison and integral tests you can at least give a fairly convincing simple explanation for why they work. But I shudder to ponder how many of our students understand that the Latin word converge means equals a number -- this fact is obscured because usually there is no good way to evaluate what this number is.
In the meantime -- where do these series come from? The students will have seen few or none, and typically applications are de-emphasized in the book (and also feel a bit artificial or canned). The textbook does not provide the material with which to convince the student that such a series is a natural thing to write down, except if one is looking for convenient fodder for homework or an exam.
Now, I could revise the material to do it "my way", but if I really did it my way, I'd end up teaching something radically different from your average Calculus II course. (And my department expects me to stay fairly close to the book.)
I'm feeling unprepared to make the case to my students that this course is interesting, useful, or important. What can be done with this material?
