Topic
Calculus
The math of change. Derivatives describe rates, integrals add up tiny pieces, and limits hold both ideas together.
Two big ideas
Differential calculus asks: how fast is something changing? Integral calculus asks: if I add up infinitely many small things, what do I get? Surprisingly, they're inverses of each other.
Introduction
What is calculus? Tiny changes, instant rates and infinite sums — a tour.
Limits
Sneak up on a value — see why limits are not just substitution.
Limits at infinity
What happens to a function as x → ∞? End behaviour at a glance.
Limits — formal
ε–δ — the airtight definition of a limit.
Continuity
A function with no jumps, holes or breaks. Pencil never lifts.
Derivatives
The slope of a curve at a point, animated.
Difference quotient
(f(x+h) − f(x))/h — the formula behind every derivative.
Differentiable
A function that has a derivative everywhere — smooth, no corners.
Differentiation rules
Power, product, quotient and chain — the four moves you really need.
Power rule
d/dx(xⁿ) = nxⁿ⁻¹. The most-used differentiation rule.
Product rule
d/dx(uv) = u'v + uv'. For when two functions are multiplied.
Chain rule
Differentiate the outside, multiply by the derivative of the inside.
Implicit differentiation
When y is mixed with x — differentiate both sides and solve for dy/dx.
Trig derivatives proof
Why d/dx(sin x) = cos x — proved with limits and the squeeze theorem.
Maxima & minima
Where a function peaks or bottoms out — set the derivative to zero.
Stationary points
Points where the derivative is zero — peaks, troughs, and saddles.
Inflection points
Where the curve changes its bend — concave up flips to concave down.
Concave up & down
Cup-up or cup-down — and what the second derivative tells you.
Second derivative
The derivative of the derivative — measures curvature and acceleration.
L'Hôpital's rule
When 0/0 or ∞/∞ shows up, differentiate top and bottom.
Integrals
Sum tiny strips to find an area — Riemann sums live.
Integration introduction
Why integration is just the reverse of differentiation.
Integration rules
The basic rules — power, sum, constant — that handle most integrals.
Definite integrals
From a to b — turn an antiderivative into an exact area.
Integration by substitution
Reverse the chain rule — let u replace the inside function.
Integration by parts
∫u dv = uv − ∫v du. Reverse the product rule to integrate.
Integration by approximation
Trapezoids and Simpson's rule — when antiderivatives won't budge.
Solids of revolution
Spin a curve around an axis — find the 3D volume it traces.
Arc length
How long is a curve? Pythagoras + integral = answer.
Fundamental theorem of calculus
Differentiation and integration are inverses — the link that ties everything together.
Differential equations
Equations where the unknown is a function — and a derivative is in there too.
Separation of variables
Get all the y's on one side, all the x's on the other, then integrate both.
First-order linear DE
dy/dx + P(x)y = Q(x) — the integrating-factor recipe.
Second-order DE
y'' + by' + cy = 0 — characteristic equations and three solution families.
Taylor series
Approximate any nice function with a polynomial — add more terms, get closer.
Fourier series
Decompose any periodic function into a sum of sines and cosines.
Partial derivatives
When a function has many inputs, slope it one variable at a time.
Derivative vs integral
Slope vs area, instantaneous vs accumulated — the two halves of calculus.