Level 3 Differentiation Walkthrough

2022 NCEA Level 3 Differentiation Question 2(d)

2022 Paper - Maximising area with product rule

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Question

A rectangle has one vertex at \((0,0)\) and the opposite vertex on the curve \(y=6e^{1-0.5x}\), where \(x>0\), as shown below.

x y

Find the maximum possible area of the rectangle.

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

You do not have to prove that the area you have found is a maximum.

The diagram is shown in its initial state. JavaScript adds any interactive controls and later walkthrough visuals.

First walkthrough idea

Tip to try first

Start with the rectangle area formula \(A=xy\).

Step 1

Write an area expression

Reveal the explanation and working for this step.

Show the first step’s working

Use width times height to write the rectangle area as \(A=xy\).

Key result

\(A=xy\)

The area of a rectangle is width times height, so \(A=xy\).

Walkthrough overview

What this question practises

This 2022 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Forming an area function and maximising it.

This is Question 2(d) from the 2022 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 forming an area function and maximising it. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Finding a stationary value is only part of an optimisation argument; justify that it is the required maximum and respect the domain.

Continue practising

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