It’s true that most people will probably drown if you throw them into Rudin first thing (although I’ve known a couple who held their breath, swam the whole length, jumped out and asked for more).
The thing is, “analysis” in (English) mathematical vernacular covers a lot more than were dreamt of in Newton’s philosophy, and that in turn is a lot more than is habitually included in a course entitled “calculus”.
On the other hand, most calculus courses cover (badly and shallowly) many things that are properly from other places (commutative algera [field axioms], order theory [Dedekind completion], general topology [limits and opens], set theory [cardinals]) but just can’t be avoided when talking about the reals.
So, what do you not get in a standard first course in calculus that still goes under “analysis” (but is not a research topic)?
- Filters and/or nets (a coherent viewpoint on all the limits)
- (Multiple) derivatives as objects of (multi)linear algebra (no more horrific “Jacobians” and “Hessians”)
- Implicit / inverse function theorem (local normal form under smooth change of coordinates, cf Morse’s lemma as well)
- All of that in the infinite-dimensional setting (for a decent theory of ordinary differential equations)
- Exponential / trig functions as solutions of ODEs (all other definitions obtained from various solution approximations, requires previous point to be nice and unforced)
- Fourier-Laplace decomposition (take previous point up to eleven, solve all linear ODEs in existence at once, including every passive electric circuit)
- Distributions aka generalized functions (you can, technically, do the previous point without that, but it’s a complete mess; this instead requires a rather advanced theory of infinite-dimensional spaces)
- Differentiation and integration as continuous and smooth operators on infinite-dimensional spaces of functions, infinite-dimensional-vector-valued integrals (you can make do with the classical theory of “differentiating under the integral sign”, but it’s Lovecraftian levels of horrible, better not)
- Integration by residues (together with the previous point, makes the two most powerful methods for computing indefinite integrals when the definite one is intractable and/or inexpressible)
- Functions of a complex variable (required for the preceding to even make sense, unlike mere complex-valued functions is essentially a completely different theory closer to algebra if anything)
- Power series (don’t make sense without the preceding point even if you’re interested in the reals; why I called exponentials and trig the same thing above)
- Lebesgue integration (because Riemann integration sucks for all of the above even if you can make do)
- Stokes theorem (the theorem of multidimensional integration, like Barrow/Newton–Leibnitz is for the one-dimensional case; you did learn multilinear algebra didn’t you?)
- More?
I’m not saying Rudin covers all of that, but no one book does. I’ve also omitted (a lot of) hooks into what are usually considered other disciplines (manifolds, speed of convergence, solution in radicals, probability measures, ...).