Curriculum
Calculus
Ages 16+ · Limits, derivatives, integrals, differential equations, Taylor series.
Limits
Under Calculus
Derivatives
Under Calculus
- Derivatives
The slope of a curve at a point, animated.
- Differentiable
A function that has a derivative everywhere — smooth, no corners.
- Differentiation rules
Power, product, quotient and chain — the four moves you really need.
- 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.
- Second derivative
The derivative of the derivative — measures curvature and acceleration.
- Differential equations
Equations where the unknown is a function — and a derivative is in there too.
- 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.
Integrals
Under Calculus
- Definite integrals
From a to b — turn an antiderivative into an exact area.
- 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.
- 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.
- Derivative vs integral
Slope vs area, instantaneous vs accumulated — the two halves of calculus.
Differential equations
Under Calculus
Infinite series
Under Calculus
Functions
Under Algebra
- Direct & inverse proportion
Twice the input, twice (or half) the output — two flavours of proportion.
- Functions & graphs
Slide m and b on y = mx + b and watch the line move.
- Evaluating functions
f(3) means: drop 3 in for x, simplify, read the answer.
- Odd & even functions
Mirror across the y-axis (even) or rotate around the origin (odd).
- Matrix inverse
The matrix that undoes another — like 1/x but for matrices.
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