Level 3 Differentiation Walkthrough

2024 NCEA Level 3 Differentiation Question 3(e)

2024 Paper

← Back to paper

Question

The diagram below shows part of the graph of the function \[ f(x)=e^{-x^2},\qquad x\ge 0. \]

O P Q y = e^{-x^2} x y

The point \(P\) lies on the curve and the point \(Q\) lies on the \(x\)-axis so that \(OP=PQ\), where \(O\) is the origin.

Prove that the largest possible area of the triangle \(OPQ\) is \(\frac{1}{\sqrt{2e}}\).

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

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

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

First walkthrough idea

Hint to try first

Notice the triangle is isosceles because \(OP=PQ\). That means \(P\) sits above the midpoint of the base.

Step 1

Write the area function

The symmetry means the base is \(2x\), so the half-base-times-height calculation collapses to \(xe^{-x^2}\).

Show the first step’s working

The symmetry means the base is \(2x\), so the half-base-times-height calculation collapses to \(xe^{-x^2}\).

Key result

\[ x \cdot e^{\left(-x^{2}\right)} \]

Walkthrough overview

What this question practises

This 2024 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Forming an area function and proving its largest possible value.

This is Question 3(e) from the 2024 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 .

Learning summary

Review the method, not only the answer

This walkthrough helps you practise forming an area function and proving its largest possible value. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Check intersections, signs, and whether the question asks for signed area or total geometric area.

Continue practising

← Question 3(d)