Level 2 Calculus Walkthrough

2025 NCEA Level 2 Calculus Question 1(c)

2025 Paper — Rebuild the original function from its gradient

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Question

\[ f'(x)=2x^2+2x-24 \]

Find the equation of the original function \(f(x)\) if the \(y\)-value of the local minimum is \(-46\).

Use the derivative to find the critical points, use the second derivative to identify the minimum, then integrate and solve for the constant.

First walkthrough idea

Tip to try first

Solve \(f'(x)=0\) first to find the critical points.

Step 1

Find the critical points

Reveal the explanation and working for this step.

Show the first step’s working

Set the derivative equal to zero and solve for both critical x-values.

Worked result

\[ -4,\ 3 \]

\(2x^2+2x-24=0\) simplifies to \(x^2+x-12=0\), which factors to \((x+4)(x-3)=0\).

Walkthrough overview

What this question practises

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

Method: Finding critical points, identifying the minimum, and solving for the constant.

This is Question 1(c) 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 finding critical points, identifying the minimum, and solving for the constant. 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 the argument; justify that it is the required minimum and respect the domain.

Continue practising

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