Level 3 Differentiation Walkthrough

2021 NCEA Level 3 Differentiation Question 2(c)

2021 Paper

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Question

A curve has the equation

\[ y=(x^2+3x+2)\cos3x. \]

Find the equation of the normal to the curve at the point where the curve crosses the \(y\)-axis.

You must use calculus and show any derivatives that you need to find when solving this problem.

First walkthrough idea

Focus to try first

Find the point on the \(y\)-axis, differentiate to get the tangent gradient, then take the negative reciprocal for the normal.

Step 1

Find the point on the y-axis

On the \(y\)-axis, \(x=0\).

Show the first step’s working
\[ y=(0^2+3(0)+2)\cos0=2 \]

The point is \((0,2)\).

Walkthrough overview

What this question practises

This 2021 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Product rule and finding the equation of a normal.

This is Question 2(c) from the 2021 NCEA Level 3 Differentiation paper for AS91578 — Apply differentiation 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 AS91578.

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Learning summary

Review the method, not only the answer

This walkthrough helps you practise product rule and finding the equation of a normal. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Differentiate both factors in turn and keep both product-rule terms.

Continue practising

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