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Calculus 1
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Cujo:
I have highlighted the topics in the Calculus 1 class at my school. I just want to know if those topics are normally taught in Calculus 1?
ataradov:
I'm not sure how different program names map to each other. In my university (in Russia) this stuff was in the first and second semester.

Taylor series is specifically important to get in early, since it is used in a ton of theorem proofs later on. I'm not sure if it is necessary in all those cases, but mathematicians like to use that stuff.
boB:

L'Hopital's rule was the 1st thing in Calculus 2 here I think.

Pretty much all other highlighted was Calculus 1
Cerebus:
Presumably we're talking about an undergraduate (university) level course here?

The last two, differential equations, used to be the first thing you'd hit in a university level course here in England, with most of the rest being stuff you'd expect people to have learned at secondary school (as long as A-level maths was a course entry requirement). You might get a break-neck speed revision over a couple of lectures of the rest. L'Hopital wasn't taught in my A-level course, Taylor was. Some A-level syllabi covered differential equations too, but that wasn't universal - mine didn't except in very rudimentary form.

I've heard people saying that universities were finding that they had to teach a lot of maths to 1st year students that they used to take for granted. I thought it was just old men moaning that "the kids of today know half of what we did" and I'm genuinely surprised to see a university syllabus that covers what was in my A-level syllabus with just a few extras.
mathsquid:
I don't know about Australia, but at most universities in the US calc I, doesn't include any of the topics you highlighted (apart from really basic initial value problems), or the review stuff up through continuous functions.

Calc I is pretty much limits, derivatives, applications of derivatives (mainly rates of change, optimization and related rates, but maybe linearization and newton's method), and then the basics of integration/antiderivatives up through u-substitution.

Calc II is split into four parts: applications of integration (mostly areas and volumes), integration techniques, parametric and polar calculus, and sequences and series.  Calc II texts usually have a chapter on ODEs, but I've never covered it in calc II.

This is based on my experience at a few universities, but I know that it varies--particularly at engineering-heavy schools such as MIT, Carnegie Mellon, or Ga Tech.
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