Graphs, Signs and Limits NCEA Practice Questions | Calc.nz

Skill overview

What these questions practise

Choose a question, attempt it independently, then compare the reasoning and restrictions in its walkthrough.

Relevant standards: NCEA Level 3 Differentiation — AS91578 · NCEA Level 2 Calculus — AS91262

This collection contains 20 matching walkthroughs across 9 examination years.

The grouping is based on the method recorded for each walkthrough. Use the linked NZQA standard and paper material for authoritative wording, diagrams, and assessment information.

Before you start

Common mistakes in these methods

Check one-sided behaviour at joins and excluded points. The sign of the derivative describes increasing or decreasing, not the sign of the function.

Related practice

Graphs, Signs and Limits questions from different years

Open a question for hints and a progressive worked solution, or use the standard and paper links to return to its assessment context.

NCEA Level 3 · AS91578 2025 Differentiation 3 questions

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Question 1(e) — Parametric Differentiation Parametric differentiation with stationary inflection points. Question 2(b) — Product Rule Product rule and deciding whether a model is increasing or decreasing. Question 3(a) — Stationary Points and Optimisation Reading differentiability, stationary points, and a limit from a graph.

NCEA Level 3 · AS91578 2024 Differentiation 4 questions

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Question 1(c) — Function Behaviour Differentiating, solving f'(x) = 0, and finding where a function is decreasing. Question 1(e) — Tangents Locating a point of inflection and finding the tangent there. Question 2(d) — Quotient Rule and Inflection Finding a point of inflection on ln x divided by x. Question 3(b) — Graph Analysis Reading continuity, differentiability, and limits from a graph.

NCEA Level 3 · AS91578 2023 Differentiation 2 questions

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Question 2(d) — Logarithms Second derivatives and finding a logarithmic point of inflection. Question 3(b) — Graph Analysis Reading continuity, concavity, and limits from a graph.

NCEA Level 3 · AS91578 2022 Differentiation 2 questions

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Question 1(e) — Second Derivative and Concavity Proving there are no points of inflection. Question 3(b) — Graph Analysis Reading differentiability, derivatives, and limits from a graph.

NCEA Level 3 · AS91578 2021 Differentiation 2 questions

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Question 1(b) — Stationary Points and Optimisation Reading stationary points, concavity, and limits from a graph. Question 3(c) — Function Behaviour Using the derivative sign to find where a function is increasing.

NCEA Level 3 · AS91578 2019 Differentiation 2 questions

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Question 1(d) — Product Rule Product rule and finding where a function is decreasing. Question 3(b) — Graph Analysis Reading derivative conditions and limits from a graph.

NCEA Level 3 · AS91578 2018 Differentiation 1 question

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Question 2(c) — Graph Analysis Function values, limits, derivatives, and continuity from a graph.

NCEA Level 3 · AS91578 2017 Differentiation 1 question

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Question 3(c) — Graph Analysis Derivatives, continuity, limits, and concavity from a piecewise graph.

NCEA Level 3 · AS91578 2016 Differentiation 1 question

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Question 2(c) — Stationary Points and Optimisation Continuity, differentiability, stationary points, concavity, and limits from a graph.

NCEA Level 2 · AS91262 2025 Calculus 2 questions

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Question 2(a) — Stationary Points and Optimisation Sketching the derivative from stationary points, signs, and changes in concavity. Question 3(d) — Stationary Points and Optimisation Differentiating, factorising, classifying critical points, and finding where the function is decreasing.

Keep practising

Page updated 24 September 2026.