Level 3 Differentiation Walkthrough

2025 NCEA Level 3 Differentiation Question 1(e)

2025 Paper

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Question

A curve is defined by the pair of parametric equations \[ x=3t^2+6t+1 \quad \text{and} \quad y=2t^3. \]

Find the coordinates of any points of inflection on this curve that are also stationary points.

You may assume that any inflection point(s) found are actually inflection points.

You must use calculus and show any derivatives that you need to find when solving this problem.

First walkthrough idea

Hint to try first

For parametric equations, start with \(\frac{dy}{dx}=\frac{dy/dt}{dx/dt}\).

Step 1

Find the first derivative

Start with \(\frac{dy}{dx}=\frac{dy/dt}{dx/dt}\).

Show the first step’s working

For parametric equations, differentiate both \(x\) and \(y\) with respect to \(t\), then divide.

\[ \frac{dx}{dt}=6t+6,\qquad \frac{dy}{dt}=6t^2 \] \[ \frac{dy}{dx}=\frac{dy/dt}{dx/dt}=\frac{6t^2}{6t+6}=\frac{t^2}{t+1} \]

Walkthrough overview

What this question practises

This 2025 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Parametric differentiation with stationary inflection points.

This is Question 1(e) from the 2025 NCEA Level 3 Differentiation paper for AS91578 — Apply differentiation methods in solving problems. Use the guided hints to practise the method before revealing the full working.

Calc.nz is an independent learning resource. Compare questions, diagrams, and assessment information with the official NZQA resources for AS91578.

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Learning summary

Review the method, not only the answer

This walkthrough helps you practise parametric differentiation with stationary inflection points. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

After solving the derivative condition, check the point's nature and answer the conclusion the question actually asks for.

Continue practising

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