In most US universities, the courses called "calculus" are mostly about computing integrals and derivatives. Yes, they'll have some theorems (Fundamental Theorem, Mean Value Theorem, etc), but most of the problems are related to computation than proving.
Unfortunately, since engineering students outnumber math majors by a large margin, the departments cater to them and not the math majors. The latter study analysis in later years - usually third or fourth.[1] Universities with strong math programs may offer it in the first or second year.
[1] I just checked my undergrad's program. They begin taking analysis in their 4th year, and it's offered only one semester a year!
I was taught Calculus in high school, years 11 and 12 (two optional senior years for those that want more education prior to university, etc rather than transfer to a trade apprenticeship or leave school altogether).
That was a mixture of basic theory and proof and some computation (eg: derive an expression for the volume of intersection of two pipes at right angles, etc).
University went straight in to heavy analysis and foundations (for Math 100 - math for the serious (math, physics, hard chemistry)), with seperate streams for "casual math" - engineering, business, law, etc.
Engineering math prepped people for calculating dynamics and kinetics with vary loads, masses, thrusts, harmonic forces, network propagation, mesh computations, etc.
> They begin taking analysis in their 4th year, and it's offered only one semester a year!
This is insane to me. In the UK about ten years ago, on literally the first day of my degree, my first class was real analysis. Yes, it was the easy stuff like proving sequences and series converge, various things about continuous functions, but we learnt how to prove it all and the exam was all about proving various things. And we built up to harder stuff as the year went on.
What is even happening if math majors aren't studying analysis until their fourth year?
Sorry I'm so incredulous, it's just that I literally don't know what I would've been studying if analysis had been delayed so much.
Not US, but for GCSE A's further math equivalent, calculus primarily involved applying intermediate rules (chain rules, trig identities, etc.) to evaluate differentiation/integrals on elementary functions, and a bit on geometric and arithmetic series.
Analysis discusses less well behaved functions and spaces than these.
You can do the first part without even relying on analysis in a mathematical sense. You simply define a differential algebra, by introducing a derivation function that just happens to respect the correct rules. Then "calculus" is the topic of how to perform computations in such an algebra. Note however that you do need analysis to rigorously address other parts of a typical "calculus course", especially those dealing with infinities, sequences and etc.
Where is calculus "often taught" without "theorems/proofs that are used to build up calculus"?