[dougabug]

The first AI generated feature length film I’d want to see would be, “It’s a Wonderful Life, Calculus,” where we get to see what human history would’ve been like if we’d never stumbled across Calculus.

So the basic idea at the heart of Calculus is that when you break a problem up into tiny enough pieces, in the right way, the pieces become simpler to analyze and approximations become much more accurate. Like, if I break a complex surface into triangles, if the triangles are small enough, many of the physical properties of interest can be computed using the little triangles.

The two main manifestations of this principle are differentiation and integration. With differentiation, the simplification which comes from breaking a function into tiny pieces is that the pieces behave asymptotically like linear functions.

For integration, the simplification becomes that the tiny fragments eventually tend to have approximately uniform density, so the mass of the whole body is the sum of the masses of all the tiny bits, each of which is just the volume of the fragment times its density.

Differentiation and Integration are in some sense inverses, or opposite sides, of single greater idea. In one dimension, the derivative of the (indefinite) integral of a function equals the original function, and the integral of the derivative of a function is that function plus a constant. Almost inverses of each other, but not quite, and this nuance is where some care is required in developing your understanding of the subject.

[randomdata]

That about matches my fuzzy understanding, but the big picture is the struggle. Like as submitted in a comment here about the venerable for loop being rooted in calculus, it makes sense when it's explicitly pointed out, but I haven't established a framework to draw those connections intuitively.

To generalize further, given a solution where the use of calculus is explained then that isolated example is comprehensible, but given a problem there is nothing that sparks a "I know, I can solve this with calculus!" I don't feel the same way about other mathematical disciplines that I have studied, even those of which I have spent far less time studying.

Perhaps it's simply harder to understand than other areas of math, but then I think that goes against the idea that is easy for kids to understand.

[Izkata]

Here's how I see the relationship:

Calculus helps define the underlying rules for the higher-level (simpler by appearance) math we use daily. "I know, I can solve this with calculus!" is unlikely to ever come up, but the vague idea that there's something there you can dig into when you need to can be helpful in rare edge cases, where other people might be lost.

An example using programming languages: If all you've ever been exposed to was python, and no CS, you may never have considered why using "insert" on a list may be slow. Python presents it as a single function call, so you probably think of it as a single operation and don't go any further. That's the equivalent of the higher-level (simpler by appearance) math. But if you've been exposed to something lower-level, like C where you may well have implemented "insert" yourself on an array, or general CS concepts where you had to use big-O notation, you'll probably have in the back of your mind "yeah, that's not a single operation, it's doing more stuff in the background". Usually not something you need to think about, until you hit that edge case where it's suddenly running really slowly.

Remember very early on in education when you had to memorize various equations like area of a circle? Those equations can be generated from basic calculus. One I could never remember was area of a sphere, until one day when I was bored at my part-time job, found a pencil and scrap of paper, and decided to see if I could use what I'd just learned in class to derive it. And it worked, and I've never forgotten that equation since, because instead of it just being a series of numbers and letters to memorize, each part now has meaning.

[randomdata]

That makes sense, but suggests that calculus is perhaps the most difficult concept to wrap one's head around, which flies in the face of the idea that is easy to teach to children. It is not clear where the breakdown occurs here.

[dougabug]

Calculus is not easy to teach to children. We fail to teach it adequately to most college students in their first two years of study. Even a few historically noteworthy mathematicians failed their first contact with the subject.

Elementary linear algebra is far easier to understand and motivate. We can deal with finite, concrete examples without having to delve into the subtle complexities of limits, continuity, and infinity.

[randomdata]

But it was said that "since kids can easily understand it" and it is not clear how that resolves.

[dougabug]

That statement was clearly false. With an amazing teacher, an extremely bright student, focus, patience, time, etc., sure we can have the next Galois… but in the vast majority of cases, we should avoid setting kids up for failure by expecting them to easily grasp things which took humanities greatest minds centuries to grasp. Newton “invented” Calculus in the 17th Century, but these ideas had been percolating since Archimedes and even before going back two millennia.

[dougabug]

Calculus was quite difficult for human civilization to get a logically sound handle on; basically, it took the better part of two centuries from Newton’s original formulation of Calculus in the 17th Century to the work of Cauchy, Gauss, Weierstrass, Dedekind, Riemann, etc working throughout the 19th Century to develop rigorous foundations for Analysis (the modern name for the subject). That’s where all the epsilon-delta business comes in. But this machinery is totally overwhelming for the vast majority of children and teenagers without signicant context, motivation, and guidance.

The analogy to a ‘for loop’ in programming is pretty direct to a mathematical summation. For S := 0, i := 1 .. N { S := S + a[i] } differs only in notation from the standard sigma notation for a finite series.

You don’t need to look far for applications of Calculus. Any simulation of a physical system, such as the Solar System, navigating the DART space vehicle into an asteroid; modeling climate, nuclear explosions, fluid dynamics, structural stability, propagation of sound through matter, modeling and manipulating the properties of semiconductors through controlled diffusion of dopants, chemical engineering, thermodynamics, electrodynamics, optics, quantum mechanics, hypersonic missile flight, even Computer Graphics has the Rendering Equation at its heart …

Mastery of Calculus (and the ability to apply it to model dynamics) is pretty much the dividing line between Classical Ancient civilizations, and the modern world.