IBDP · Maths AI SL
Calculus
AI SL Topic 5 — Calculus
34 min readStandard~30% of AI SL — the single largest topic across Paper 1 and Paper 2
Calculus in AI SL is really two machines bolted together: differentiation (rates, gradients, optimisation) and integration (areas, accumulation, reversing a rate). Every exam question is one of a handful of disguises — a cost function to minimise, a velocity graph to read area from, a population to model. Learn the four moves in this note and you can decode almost any 'in context' wording the examiner throws at you.
Overview
What this topic actually tests
AI SL calculus rewards fluency, not cleverness. You need automatic recall of the derivative and integral rules, then the judgement to know which application — tangent, optimisation, area, kinematics — a wordy context question is hiding.
- Paper 1 tests clean mechanical skill: differentiate this, integrate that, find the equation of a tangent.
- Paper 2 (GDC allowed) tests modelling: a context gives you a function or a rate, and you interpret gradients, areas, and turning points in real units.
- A small pocket of this topic — separable differential equations, logistic growth — is examined at HL only. It's flagged clearly in its own section below so you know exactly what's off your SL paper.
The shape of the chapter
Command terms that decide your method
| Command term | What it demands | AO | Mark-earning move |
|---|---|---|---|
| Show that | Derive the given result with every algebraic step visible — the answer is given, so marks are only for the working. | AO2 | Skipping a line (e.g. combining logs) loses the method mark even if your final line matches. |
| Hence | You must use the result just found — a fresh, unrelated method scores zero even if it's mathematically valid. | AO2 | Markers check your working literally references the previous part. |
| Find | Produce the value/expression — working is expected but the final answer carries most marks. | AO1/AO2 | |
| Determine | Similar to 'find' but usually needs a stated method (e.g. solve ), not just GDC output. | AO2 | |
| Interpret | State what a number means in the context — units and a sentence, not just a number. | AO3 | Bare numeric answer with no context sentence typically loses the mark. |
| Sketch | Show general shape, intercepts, and key features labelled — doesn't need to be to scale, but must be qualitatively correct. | AO2 |
Key point
Every calculus question is graded on process, not just the final number. 'Show that' and 'hence' parts exist specifically to check you can chain results together — write down every algebraic move, even the ones that feel obvious.
Overview