← Differentiation methods: past papers

AS 91578 · 2025 paper · the graph, related rates, rectangle optimisation

Question 3(a)(ii) Achieved

Practises Reading a graph: differentiability, f′ = 0, and limits

Reading differentiability, stationary points and a limit straight off a drawn graph, then a log quotient, a related-rates sphere, a parametric gradient, and a rectangle inscribed in a curve.

Find the value(s) of x where f'(x) = 0.

The 2025 paper, page 9 ↗ Marking schedule, page 5 ↗

Work it out

Try it on paper first. Stuck? Take one hint at a time.

  1. 1 Where is the tangent to the curve flat?

    At the top of the peak, x = −4, and the bottom of the trough, x = 6.

  2. 2 Is there a whole stretch where the graph is flat?

    Yes: the horizontal line to the left of x = −6. Its gradient is 0 all the way along, so f′(x) = 0 for every x < −6. Not at x = −6 itself, where the graph jumps.
Answer x < −6, x = −4 and x = 6.

What the marker wanted

  • Achieved Two of the three answers to (i), (ii) and (iii) correct.

The schedule marks (i), (ii) and (iii) together as one Achieved item: any two of the three correct earns it.

In our words, from NZQA’s 2025 marking schedule.

How did it go?

Got it moves it further out. Review brings it back tomorrow.