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I am curious why a curriculum in the US includes four courses on calculus. Why isn't this just part of maybe three analysis courses for undergraduate studies?

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My university had quarters, not semesters, but:

1. Derivatives

2. Integrals

3. Applications — function approximation; solids of rotation; vectors; etc.

4. Multivariate — partial derivatives, multiple integrals, etc.

Glancing at a nearby community college’s course catalog, they have similar split.

Does this include the full machinery to prove the necessary theorems or are these basically practical courses?

Those are practical courses, meant to prep for linear algebra and differential equations — and various STEM tracks. As I recall, there were some proofs (eg, limits showing derivative rules; limits showing sums for integral rules).

We covered all the proofs in real analysis 1 (derivatives; sequences) and 2 (integrals; measure).

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