Differentiation in Mathematical Investigations

Paper 3 may introduce a new representation and ask you to develop it using familiar AA HL mathematics. Read each definition and retain the restrictions established in earlier parts.

A curve given using a parameter

If a question supplies \(x=x(t)\) and \(y=y(t)\), the chain rule gives \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{\mathrm{d}y/\mathrm{d}t}{\mathrm{d}x/\mathrm{d}t}\), provided \(\dfrac{\mathrm{d}x}{\mathrm{d}t}\ne0\). This is a tool for the introduced representation in the following IB investigation.

For a horizontal tangent, check \(\mathrm{d}y/\mathrm{d}t=0\) and \(\mathrm{d}x/\mathrm{d}t\ne0\). A simultaneous zero requires further investigation, not cancellation without justification.

Proof and interpretation

Use a numerical sketch to explore, then give the algebraic argument requested. Keep “show that” conclusions justified, check every candidate against its domain, and explain what each result means for the graph or model.

These complete questions also use complex numbers, polynomial roots, motion, differential equations and probability. They are intended for synthesis after those topics have been taught.