Tangents and Normals 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 32 matching walkthroughs across 10 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

Use the original function for the point and its derivative for the tangent gradient. A finite nonzero tangent gradient gives the negative reciprocal for the normal.

Related practice

Tangents and Normals questions from different years

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NCEA Level 3 · AS91578 2025 Differentiation 4 questions

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Question 1(c) — Product Rule Product rule and the gradient of a normal. Question 1(d) — Tangents Finding a tangent and its x-intercept. Question 2(c) — Quotient Rule Quotient rule and tangents with a given gradient. Question 2(d) — Parametric Differentiation Parametric gradients and tangent equations.

NCEA Level 3 · AS91578 2024 Differentiation 1 question

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Question 1(e) — Tangents Locating a point of inflection and finding the tangent there.

NCEA Level 3 · AS91578 2023 Differentiation 3 questions

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Question 1(c) — Tangents Using derivatives to find a parallel tangent on a rational curve. Question 1(d) — Parametric Differentiation Parametric differentiation and building a tangent equation. Question 2(c) — Quotient Rule Quotient rule differentiation and horizontal tangents.

NCEA Level 3 · AS91578 2022 Differentiation 3 questions

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Question 1(c) — Normal Lines The normal line and its x-intercept. Question 2(b) — Chain Rule Chain rule and finding a tangent gradient. Question 2(e) — Implicit Differentiation Implicit differentiation and tangent-line distance.

NCEA Level 3 · AS91578 2021 Differentiation 4 questions

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Question 1(d) — Parametric Differentiation Parametric differentiation and evaluating a tangent gradient. Question 2(c) — Product Rule Product rule and finding the equation of a normal. Question 2(e) — Tangents Tangent geometry on y = square root of (2x - 4). Question 3(d) — Quotient Rule Quotient rule with a parameter and a given tangent gradient.

NCEA Level 3 · AS91578 2020 Differentiation 4 questions

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Question 1(b) — Tangents Trig derivatives and evaluating a tangent gradient. Question 1(d) — Product Rule Product rule and proving a tangent equation. Question 3(b) — Tangents Finding where a tangent has a given gradient. Question 3(c) — Normal Lines Normal gradient and an x-intercept.

NCEA Level 3 · AS91578 2019 Differentiation 3 questions

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Question 1(c) — Quotient Rule Quotient rule differentiation and tangent gradient. Question 2(b) — Tangents Differentiating y = tan(2x) and evaluating a tangent gradient. Question 3(e) — Tangents Using a tangent intercept to find the coordinate of P.

NCEA Level 3 · AS91578 2018 Differentiation 1 question

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Question 3(d) — Product Rule Product rule and the equation of a tangent.

NCEA Level 3 · AS91578 2017 Differentiation 3 questions

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Question 1(c) — Normal Lines Finding a normal and its second intersection with a parabola. Question 2(c) — Tangents A tangent to a square-root curve and its x-intercept. Question 3(b) — Power Rule Negative powers and evaluating a gradient at a point.

NCEA Level 3 · AS91578 2016 Differentiation 3 questions

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Question 1(c) — Parametric Differentiation Parametric differentiation and a tangent gradient. Question 1(d) — Tangents Perpendicular tangent gradients on a parabola. Question 3(b) — Tangents Solving a horizontal-tangent condition with an exponential.

NCEA Level 2 · AS91262 2025 Calculus 3 questions

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Question 1(a) — Power Rule Using the power rule term by term and evaluating the gradient at a point. Question 1(b) — Tangents Finding the point of tangency, the gradient, and the tangent equation. Question 3(a) — Gradients and Curve Points Differentiating, solving for the given gradient, and finding the coordinate on the curve.

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Page updated 24 September 2026.