Level 3 Differentiation Walkthrough

2016 NCEA Level 3 Differentiation Question 1(b)

2016 Paper

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Question

The height of the tide at a particular beach today is given by the function

\[h(t)=0.8\sin\left(\frac{4\pi}{25}t+\frac{\pi}{2}\right),\]

where \(h\) is the height of water, in metres, relative to the mean sea level and \(t\) is the time in hours after midnight.

At what rate was the height of the tide changing at that beach at 9.00 a.m. today?

First walkthrough idea

Focus to try first

Differentiate the sinusoidal tide model with the chain rule, then use \(t=9\) for 9.00 a.m.

Step 1

Identify the inner angle

The sine input is a linear function of time.

Show the first step’s working
\[u(t)=\frac{4\pi}{25}t+\frac{\pi}{2},\qquad \frac{du}{dt}=\frac{4\pi}{25}.\]

The amplitude \(0.8\) stays as a constant multiplier.

Walkthrough overview

What this question practises

This 2016 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Trigonometric chain rule and evaluating a tide-height rate.

This is Question 1(b) from the 2016 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 trigonometric chain rule and evaluating a tide-height rate. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Do not stop after differentiating the outside function; include the derivative of the inside function as a factor.

Continue practising

← Question 1(a) Question 1(c) →