Calculus Mind Map

Every calculus visualization on this site on one map. The branches are the main ideas. Each leaf is a page; the grey line under it is the one thing to remember about it. Click a branch to focus on it, click a page to open it.

Recall mode hides the page names. Read the hint, say the name, then click the "?" to check.

1. How the map is organized

Overview of the calculus mind map: Limits and derivatives, Reading derivatives, Integrals, Series, Numerical methods, Differential equations, Chaos. — The map at a glance: 7 branches, 36 pages.

Ask what the page does with a function and you know its branch:

Does it find a rate? Slopes and tangents are limits and derivatives; using f′ and f″ to describe a graph is reading derivatives.
Does it add up pieces? Areas, volumes and totals are integrals; writing a function as a sum of terms is series.
Is the answer computed step by step? Roots and areas are numerical methods; curves from a slope rule are differential equations.
Do tiny changes in the start grow without limit? Then it is chaos.

2. The branches side by side

Branch	Question	Key tool	Example
Limits and derivatives	How fast does f change?	limit of a difference quotient	(x²)′ = 2x
Reading derivatives	Where does f rise, peak, bend?	signs of f′ and f″	f′ = 0 at a peak
Integrals	How much in total?	limit of a Riemann sum	∫₀¹ x² dx = 1/3
Series	Can f be built from simple terms?	partial sums	eˣ = 1 + x + x²/2! + …
Numerical methods	What is the number, to n digits?	iteration, error order	Newton for √2
Differential equations	Which curve has this slope?	small time steps	y′ = y gives eᵗ
Chaos	Can we predict far ahead?	sensitive dependence	Lorenz butterfly

3. Five facts to remember

The derivative is the limit of secant slopes; the integral is the limit of Riemann sums.
The Fundamental Theorem links them: d/dx ∫ₐˣ f(t) dt = f(x), so ∫ₐᵇ f = F(b) − F(a).
A differentiable function is continuous, but a continuous one need not be differentiable (|x|, Weierstrass).
A method of order p cuts the error by 2ᵖ when the step h is halved: Euler 2, Simpson and RK4 16.
Chaos is deterministic: the rules have no randomness, but small errors grow exponentially.

See all calculus pages in the index, or the other mind maps.