Yeah, that's pretty much the standard approach to teaching calculus and I would guess that the majority of people with any understanding of calculus understand

Yeah, that's pretty much the standard approach to teaching calculus and I would guess that the majority of people with any understanding of calculus understand it roughly that way. It probably is the most intuitive and easiest way to get students from zero to being able to solve problems and use the calculus that they might encounter in many other fields (basic physics, statistics, etc). It's not really a "wrong" interpretation of calculus, but it has limitations. Eventually the geometric intuition breaks down. You can use it pretty well as long as you are working in standard euclidian spaces with up to three dimensions. Past that, you really can't visualize things and your intuition will lead you to incorrect conclusions or at least just get in the way. A course in Analysis will tear down that intuition and (hopefully) replace it with a more rigorous one based on limits of infinite series. It's a longer and harder road, but leads to an approach to calculus that carries over to N dimensions almost trivially (vector and tensor analysis). It also extends to limits of series of things other than just Real numbers, eg, calculus on complex numbers, limits of series of functions (harmonic/Fourier analysis), stochastic processes, or variations in functions (calculus of variations). Not being inherently tied to our normal ideas of euclidian spaces even allows for some really weird things like being able to integrate the Dirichlet function (a function where rational numbers have the value 1 and irrational numbers have the value 0). Many of these turn out to be very important and useful in advanced physics and other fields.