Calculus
What the domain tests
Calculus in AI HL covers limits, differentiation rules, tangents and normals, stationary points and optimisation, kinematics, and integration for areas. The exam tests correct setup from context as well as routine differentiation and definite integration, often with GDC support on Paper 2.
Core rules
- Derivative rules: power, product, quotient and chain rule.
- Tangent gradient is f'(a); normal gradient is -1/f'(a).
- Stationary points satisfy f'(x)=0; use f'' or a sign change to classify.
- Definite integral gives signed area; area between curves is ∫(upper-lower) dx.
- Kinematics: v=ds/dt, a=dv/dt; displacement is ∫v dt.
- Optimisation: build one variable from the constraint, differentiate, solve, then check endpoints and context.
Worked example
Find and classify the stationary points of f(x)=x³-3x²+2. f'(x)=3x²-6x=3x(x-2), so x=0 and x=2. Also f''(x)=6x-6. At x=0, f''=-6<0, so f has a local maximum; f(0)=2. At x=2, f''=6>0, so f has a local minimum; f(2)=8-12+2=-2. The classification follows from concavity, not from the sign of f itself.
Traps
Omitting +c for indefinite integrals; treating signed area as total area; using the normal gradient as f'(a); forgetting the chain rule in composite models; failing to check endpoints in optimisation; mixing units in kinematics; assuming every stationary point is a maximum.