I gave my 6th grade nephew a 90 second "Calculus Made Easy": First, credibility: Ah, I never took the first year of college calculus -- to make faster progress in college math, got a good book and taught myself. After an oral exam at a black board, was admitted to the second year. Majored in math. Took (so called) advanced calculus from Rudin's Principles .... Took applied advanced calculus from a famous MIT book from an MIT Ph.D. Taught calculus at Indiana University. Studied Fleming's Functions of Several Variables, right, through the inverse and implicit function theorems, Stokes formula, exterior algebra, etc. Published some advanced math, essentially advanced calculus. Once, with some calculus, at FedEx pleased the most serious investors on the Board, had them return their airline tickets back to Texas, stay after all, and saved the company. Okay, the 90 seconds: Consider a car, its speedometer and odometer. Calculus has two parts. For the first part, you read the data from the odometer and reconstruct the speedometer readings. Doing this, you take changes in (increments of, differences in) the odometer readings, say, every second. This is called differentiation. For the second part, you read the data from the speedometer and reconstruct the odometer readings. Doing this, you add the speedometer readings, say, every second. This is called integration. Starting with the odometer readings and differentiating to get the speedometer readings and then integrating the speedometer readings will give back the odometer readings -- this is the "fundamental theorem of calculus". ~90 seconds. For more, instead of the 1 second steps could use 0.1 seconds, .... 0.0001 seconds, etc. With really small steps, making them smaller will make no or nearly no difference. So, the reconstructions will have converged, reached a limit. Reaching this limit is mostly what was novel when Newton, Leibnitz, etc. invented calculus. It is fair to say that the first big application was to Newton's law F = ma where have some object -- baseball, airplane, rocket -- with mass m and are applying to the object force F. Then a is the acceleration of the object. Integrate the acceleration and get the velocity v. Integrate v and get distance d. So, can find where the rocket is after, say, 10 seconds. Other early applications were to planetary motion. There are applications to areas, volumes, classical mechanics. fluid flow, mechanical engineering, electricity and magnetism, quantum mechanics, relativity, electrical and electronic engineering, e.g., Fourier theory. Physics and engineering are big users. And there are applications in economics, e.g., work of Arrow, Hurwicz, and Uzawa on the Kuhn-Tucker conditions. By the early 20th century, calculus was refined, e.g., presented with careful assumptions, definitions, theorems, and proofs, e.g., B. Riemann and, soon, H. Lebesgue. By then there was the idea of the highly irregular Brownian motion and the observation that differentiation wouldn't work there -- Brownian motion was differentiable nowhere! Calculus? A pillar of science, technology, and, thus, civilization.