Item — HN Reader

I don't know about MIT and Stanford, but I went to Caltech, and while there are first-year courses with names like 'Calculus', they are categorically nothing like the first-year courses at UCs or CSUs.

To put things into perspective, there is zero AP credit offered at Caltech. The only way to skip classes is to pass Caltech's own placement exams. I took the first placement exam (after finishing the entire AP math curriculum, as well as the entire math curriculum at the local community college—through linear algebra, diff eqns, and multivariable calc), got a 96% on Caltech's placement exam, and was recommended to take an extra, remedial mathematics course in addition to the regular freshman load to help bring me up to speed.

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In what way are those first year Calculus courses different at Caltech? I'm quite curious.

As another (recent) Caltech grad, I'll take a stab this. The Caltech intro calc class is very heavily proof-based. We started with easy problems that had nothing to do with calculus like proving that 1/sqrt(3) is irrational, went into epsilon-delta proofs for limits, and continued on to showing that the Dirichlet function is discontinuous everywhere. I don't think we evaluated a single integral in that class.

A strange feature of the class was that the weeks where we studied derivatives were a lot harder than the weeks when we studied integrals. Although derivatives are a lot easier to evaluate than integrals, it's a lot harder to prove that they exist. Most of our problems were proving the existence of derivatives and integrals, not evaluating them, so it was a weird inversion in what is normally considered difficult in calculus.

If you want a sample of the problems, here's one of the problem sets from this year's class:

http://www.math.caltech.edu/%7E2011-12/1term/ma001a1/homewor...

Oh, that's just elementary analysis. That's still "calculus" of course, but in most colleges you might not be expected to take that unless you were a math major. I can respect that.

(Note: I studied math as well as chemistry and CS at a state college so my background is a little unusual.)