Calculus
AI HL Topic 5: Calculus
Calculus in AI HL is graded on translation, not just mechanics: can you turn a cooling cup of coffee, a spreading population or a drug leaving the bloodstream into an equation, solve it properly, and read the answer back into the real situation? These notes follow the four moves examiners actually test — building and solving differential equations, using limits to justify what a derivative even is, applying derivatives to real optimisation and rate problems, and using integration to accumulate area, volume and displacement.
Overview: How the Four Strands Fit Together
One skill, four costumes
Every calculus question is secretly asking whether you can translate a real situation into one of four operations and back again: differentiate to describe rate of change, integrate to reverse that process and accumulate it, use limits to justify why any of this is legitimate, and set up a differential equation when the rate of change IS the physical law you've been given.
- New at HL beyond the SL toolkit: implicit differentiation, related rates, integration by substitution and by parts, volumes of revolution, and the full differential equations strand (separation, integrating factor, Euler's method, logistic model).
- The GDC is powerful but does not replace algebra on 'show that' or 'verify' questions — command terms decide how much working you must physically write.
- Almost every context question (drug concentration, population, cooling, harvesting) recycles the same four or five differential equation structures — recognising the structure is half the battle.
The shape of the chapter
Command terms that decide your marks
| Command term | What it demands | AO | Mark-earning move |
|---|---|---|---|
| Show that | Derive the given result using every algebraic step — no jumping straight to the answer even if your GDC confirms it instantly. | AO2 | Marks are awarded for intermediate lines, not just for matching the given final answer. |
| Hence | You must use the result just obtained in the previous part — a correct answer reached by a different method scores 0. | AO2 | Follow-through only applies if the earlier result is visibly used in the working. |
| Verify | Substitute the given expression into the equation or condition and confirm both sides match — don't re-derive it from scratch. | AO2 | One mark for the substitution, one for the explicit confirming statement. |
| Determine / Find | Any valid method including GDC is acceptable, but the equation you solved must appear on paper. | AO1/AO2 | The method mark needs the equation written down even when the GDC gives the number directly. |
| Sketch | Show shape, intercepts, and asymptotic/limiting behaviour with labels — axes don't need scale but key features do. | AO1 | Marks lost for missing turning-point coordinates or unlabelled asymptote equations. |
Key point
Overview