Level 3 Differentiation Walkthrough

2024 NCEA Level 3 Differentiation Question 1(d)

2024 Paper

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Question

Find the \(x\)-value(s) of any stationary points on the graph of the function below, and determine their nature. \[ y=(2x-1)e^{-2x} \]

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

First walkthrough idea

Hint to try first

We need both the first and second derivatives here.

Step 1

Differentiate the product

The product rule simplifies nicely to \(e^{-2x}(4-4x)\).

Show the first step’s working

The product rule simplifies nicely to \(e^{-2x}(4-4x)\).

Key result

\[ e^{\left(-2 x\right)} \left(4 - 4 x\right) \]

Walkthrough overview

What this question practises

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

Method: Stationary points, the second derivative test, and classifying the point.

This is Question 1(d) 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.

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Learning summary

Review the method, not only the answer

This walkthrough helps you practise stationary points, the second derivative test, and classifying the point. 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

← Question 1(c) Question 1(e) →