Change, made visible

Two ideas§

Calculus has a reputation for being a pile of rules. It is really two ideas. The first is slope: how fast something is changing at one instant, found by measuring the slope between two points and letting them merge. The second is accumulation: how much something adds up to, found by cutting it into thin rectangles and letting them shrink. The fundamental theorem of calculus says the two are inverses. The slope of the accumulated area is the height of the curve; the area under a rate is the total change.

Each chapter takes one step along that path, with figures you operate rather than look at. The writing is for anyone who wants to understand what the symbols mean, whether you are meeting calculus for the first time, revising for an exam, or returning to it years later. You can read the chapters in order, or go straight to the one your question is about.

How this is made and checked§

Every figure is driven by a small numerical engine, written for this site and tested against exact answers: derivatives and integrals known in closed form, worked examples from textbooks, and limits whose values are proved. The calculators in each chapter use the same engine: the exact differentiation rules applied to your formula, checked against extrapolated difference quotients, for derivatives; adaptive Simpson quadrature for integrals; and extrapolated one-sided samples for limits. They return numbers with error estimates, not formulas, and each one says how it works and when it can be fooled. A separate test suite, written from the published methods without reading the engine’s code, checks its answers independently.

Every equation, number and historical claim on the site is recorded with its source, and every figure says what it simplifies. The history draws on the scholarship of Judith Grabiner and on primary sources such as Heath’s translation of Archimedes. If you find an error, the about page says how to report it.