Calculus

What happens to a function as x → ∞? End behaviour at a glance.

Sneak up on a value — see why limits are not just substitution.

ε–δ — the airtight definition of a limit.

The slope of a curve at a point, animated.

A function that has a derivative everywhere — smooth, no corners.

Power, product, quotient and chain — the four moves you really need.

When y is mixed with x — differentiate both sides and solve for dy/dx.

Why d/dx(sin x) = cos x — proved with limits and the squeeze theorem.

The derivative of the derivative — measures curvature and acceleration.

Equations where the unknown is a function — and a derivative is in there too.

When a function has many inputs, slope it one variable at a time.

Slope vs area, instantaneous vs accumulated — the two halves of calculus.

From a to b — turn an antiderivative into an exact area.

Sum tiny strips to find an area — Riemann sums live.

Why integration is just the reverse of differentiation.

The basic rules — power, sum, constant — that handle most integrals.

Reverse the chain rule — let u replace the inside function.

∫u dv = uv − ∫v du. Reverse the product rule to integrate.

Trapezoids and Simpson's rule — when antiderivatives won't budge.

Slope vs area, instantaneous vs accumulated — the two halves of calculus.

Equations where the unknown is a function — and a derivative is in there too.

Get all the y's on one side, all the x's on the other, then integrate both.

Approximate any nice function with a polynomial — add more terms, get closer.

Decompose any periodic function into a sum of sines and cosines.

Twice the input, twice (or half) the output — two flavours of proportion.

Slide m and b on y = mx + b and watch the line move.

f(3) means: drop 3 in for x, simplify, read the answer.

Mirror across the y-axis (even) or rotate around the origin (odd).

The matrix that undoes another — like 1/x but for matrices.