Level 2 Calculus Walkthrough

2025 NCEA Level 2 Calculus Question 2(b)

2025 Paper — Find the original function from \(f'(x)\)

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Question

\[ f'(x)=2x^3-6x^2+2x+4 \]

The graph of \(f(x)\) passes through \((2,5)\).

Find the equation of \(f\).

Antidifferentiate first, then use the point \((2,5)\) to work out the constant.

First walkthrough idea

Tip to try first

Antidifferentiate each term in \(f'(x)\) separately and remember to add \(+c\).

Step 1

Antidifferentiate term by term

Reveal the explanation and working for this step.

Show the first step’s working

Antidifferentiate \(f'(x)=2x^3-6x^2+2x+4\) term by term.

Worked result

\[ \frac{1}{2} x^{4} - 2 x^{3} + x^{2} + 4 x + c \]

Integrating term by term gives \(\frac{1}{2}x^4-2x^3+x^2+4x+c\).

Walkthrough overview

What this question practises

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

Method: Antidifferentiating and using a point on the graph to find the constant.

This is Question 2(b) 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 antidifferentiating and using a point on the graph to find the constant. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Include the constant of integration, then use any given point or initial condition to determine it.

Continue practising

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