[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.