Change and Motion

What you will learn

• Explain derivatives and integrals in words and estimate both from tables of numbers
• Explain why the fundamental theorem of calculus links speed and accumulated distance
• Describe how a differential equation states a law of change and what its solution tells you
• Explain why chaos limits prediction even when the governing equations are known

Contents

1. Module 1 Calculus 2 lessons

   This module covers the branch of mathematics that measures change and accumulation. You'll see how a puzzle about speed at a single instant, and another about the area under a curve, turned out to be two sides of one idea, and how that idea now drives physics, engineering, and economics. You'll do the reasoning with tables of numbers, the way the ideas began, without any of the rules that make school calculus feel like drills.

   Module overview and further reading

   1. 1.1Calculus: Instantaneous speed and areas under curves: limits, derivatives, and integrals1h 3m
   2. 1.2Calculus: Optimization, physics, and economics, and the questions calculus leaves open1h 33m
2. Module 2 Differential Equations and Dynamical Systems 2 lessons

   This module covers the branch of mathematics in which science writes its laws of change. You'll see how equations that relate a quantity to its own rate of change describe cooling coffee, growing populations, spreading diseases, and the weather, and why some of those systems are chaotic, so that tiny errors grow until prediction fails. You'll finish with the Navier–Stokes problem, where an AI-generated proof and a dispute over credit are being worked through right now.

   Module overview and further reading

   1. 2.1Differential Equations: Laws of change: differential equations from planetary orbits to population growth1h 3m
   2. 2.2Differential Equations: Weather, epidemics, chaos, and the Navier–Stokes problem1h 33m

Course project