Calculus
AA HL Topic 5 — Calculus
Calculus is where AA HL stops rewarding memorised algebra and starts rewarding process. Every mark scheme in this topic is built around specific words — show that, hence, determine, justify — and examiners are trained to withhold marks the instant your working skips a step, even when your final answer is correct. The four subtopics here build on each other in a strict chain: limits define continuity, continuity is the gatekeeper for differentiability, derivatives feed every application (tangents, optimisation, kinematics), and integration — the reverse process — feeds differential equations. Learn it in that order and the HL-only material (implicit differentiation, integration by parts, volumes of revolution, and the whole of differential equations) stops looking like extra content and starts looking like the natural next question.
Overview — the shape of HL Calculus
Why this topic is structured the way it is
Everything in Calculus rests on one idea: a limit. Continuity is defined using limits; the derivative is defined as a limit of a difference quotient; the definite integral is defined as a limit of a sum. HL adds machinery on top of the SL toolkit at every stage — more derivative rules (inverse trig, implicit), more integration techniques (parts, substitution into arctan/ln forms), and an entirely new HL-only strand: differential equations, where you use derivatives and integrals together to model how quantities change.
- Limits & Continuity — the language and conditions everything else is built from.
- Derivatives & Applications — differentiation rules, then what you do with a gradient function (tangents, optimisation, motion, related rates).
- Integration & Applications — the reverse process, then areas and volumes it produces.
- Differential Equations (HL only) — equations involving itself, solved by separating, substituting, or using an integrating factor.
The shape of the chapter
Command terms that decide your marks in this topic
| Command term | What it demands | AO | Mark-earning move |
|---|---|---|---|
| Show that | Derive the exact given result using visible algebraic/logical steps — the answer is given, so marks come entirely from the method. | AO2 | Jumping from the unsimplified expression straight to the stated answer scores 0 — every intermediate line matching the target value must appear. |
| Hence | You must use the result or method from the previous part — a correct answer from a different method can score 0. | AO2 | Re-deriving from scratch instead of building on the previous part loses the method mark even if the final value is right. |
| Determine | Find the value, but justification/working is expected, not just a GDC output. | AO1/AO2 | A bare final answer with no supporting line typically earns only the answer mark, not the method marks. |
| Justify / Prove | Give a complete logical chain — every claim needs a reason. | AO3 | Stating a true conclusion without the reasoning that gets you there scores 0, even if the statement itself is correct. |
| Sketch | Show the correct shape with key features labelled (intercepts, asymptotes, turning points) — not to scale, but not blank either. | AO2 | A correctly-shaped curve missing a labelled feature (e.g. the turning point coordinates) drops a mark even if the sketch 'looks right'. |
Key point
Overview