Level 3 Differentiation Walkthrough

2022 NCEA Level 3 Differentiation Question 1(d)

2022 Paper - Parametric differentiation

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Question

A curve is defined parametrically by the equations:

\[ x=2+3t \qquad \text{and} \qquad y=3t-\ln(3t-1), \qquad t>\frac{1}{3} \]

Find the coordinates, \((x,y)\), of any point(s) on the curve where the tangent to the curve has a gradient of \(\frac{1}{2}\).

You must use calculus and show any derivatives that you need.

A later walkthrough visual becomes available with JavaScript; the complete question and first learning steps remain below.

First walkthrough idea

Tip to try first

For parametric curves, the gradient is found from \(\frac{dy}{dx}=\frac{dy/dt}{dx/dt}\).

Step 1

Find \(\frac{dx}{dt}\)

Reveal the explanation and working for this step.

Show the first step’s working

Differentiate \(x=2+3t\) with respect to \(t\).

Worked result

\[ 3 \]

The derivative of \(2+3t\) with respect to \(t\) is just \(3\).

Walkthrough overview

What this question practises

This 2022 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Parametric differentiation.

This is Question 1(d) from the 2022 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 parametric differentiation. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Check each step against the original condition, preserve signs and restrictions, and confirm that the final result answers the question asked.

Continue practising

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