Level 2 Calculus Walkthrough

2025 NCEA Level 2 Calculus Question 2(a)

2025 Paper — Sketching \(f'(x)\) from the graph of \(f(x)\)

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Question

The graph below shows \(y=f(x)\). Use it to sketch a possible graph of \(y=f'(x)\).

x f(x)

Sketch \(f'(x)\) here:

x f'(x)

The key idea is that stationary points of \(f\) become x-intercepts of \(f'\). Then use where \(f\) is increasing, decreasing, and changing concavity to shape the derivative.

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First walkthrough idea

Tip to try first

Start with the stationary points of \(f\). Those are the places where the tangent is horizontal, so they become the zeros of \(f'(x)\).

Step 1

Find the stationary points of \(f\)

Reveal the explanation and working for this step.

Show the first step’s working

Look for the places where the tangent to \(f\) is horizontal.

Key result

\[ x\approx -3,\ 0,\ 4 \]

The graph of \(f\) has two local minima and one local maximum, at about \(x=-3\), \(x=0\), and \(x=4\).

Walkthrough overview

What this question practises

This 2025 walkthrough is part of AS91262 — Apply calculus methods in solving problems.

Method: Sketching the derivative from stationary points, signs, and changes in concavity.

This is Question 2(a) from the 2025 NCEA Level 2 Calculus paper for AS91262 — Apply calculus 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 AS91262.

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Learning summary

Review the method, not only the answer

This walkthrough helps you practise sketching the derivative from stationary points, signs, and changes in concavity. 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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