Calculus

Calculus is the mathematics of change. Algebra lets you describe relationships between quantities; calculus lets you describe how those quantities move in response to each other. The derivative tells you how fast something is changing at an instant; the integral tells you how much of something has accumulated over an interval. Almost every quantitative description of the physical world — the speed of a car, the area under a curve, the orbit of a planet, the rate at which a population grows — is a calculus statement underneath.

This section covers single-variable calculus at the level of a first university course or an AP Calculus syllabus. The articles assume you are comfortable with algebra (especially functions and graphs) and basic trigonometry; you do not need to have seen any calculus before.

Articles in this section

• Derivatives, properly explained

  What the derivative actually measures, where the limit definition comes from, the four shortcut rules (power, product, quotient, chain) and how to recognise which one to use without guessing.

  By Sara Lin · 19 min read

• Definite integrals and the area under a curve

  The Riemann-sum definition, the fundamental theorem of calculus, and how to evaluate a definite integral by antidifferentiation. Includes a numerical-integration calculator for cases where the antiderivative is not elementary.

  By Sara Lin · 14 min read

• Limits, and why epsilon-delta is not as scary as it looks

  The intuitive definition, when direct substitution works, the indeterminate forms (0/0, ∞/∞) and the two reliable tools for them (factoring, L’Hôpital). With a calculator that handles the standard cases.

  By Sara Lin · 12 min read

A reasonable order to read them in

Strictly, limits come first — the derivative is defined as a limit, and so is the integral. In practice, almost every student I have taught gets a better feel for what limits are after they have seen them used in derivatives, so my recommendation is to read the derivative guide first and then circle back to limits once you know what you are computing them for. The integral guide is the natural finish, and it pulls together both.

What this section does not cover

Multivariable calculus (partial derivatives, multiple integrals, vector calculus), differential equations beyond the simplest separable cases, and analysis (the rigorous treatment of limits and convergence) are all outside the scope of this section. They build directly on the material here, so the articles below are reasonable preparation for any of them.