Radius of Curvature Worked Solution – 2025 NCEA Level 3 Q2(e) (AS91578)

Question

The radius of curvature of a function is a measure of how much the curve is bending at a given point.

\rho=\frac{\left(1+\left(\frac{dy}{dx}\right)^2\right)^{3/2}}{\frac{d^2y}{dx^2}}

Find the radius of curvature for the curve \(y=\cos^2(2x)\) when \(x=\frac{\pi}{3}\).

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

First walkthrough idea

Hint to try first

Find the first derivative carefully using the chain rule.

Step 1

Find the first derivative

Follow the working to find the first derivative.

Show the first step’s working

The first derivative can be written as \(-4\sin(2x)\cos(2x)\), which simplifies to \(-2\sin(4x)\).

Key result

-2 \sin\left(4 x\right)

Walkthrough overview

What this question practises

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

Method: First and second derivatives in a radius of curvature problem.

This is Question 2(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.

Page updated 9 August 2026.

Learning summary

Review the method, not only the answer

This walkthrough helps you practise first and second derivatives in a radius of curvature problem. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Check each step against the original condition, preserve signs and restrictions, and confirm that the final result answers the question asked.

Continue practising

← Question 2(d) Question 3(a) →