AS 91578 · 2025 paper · the graph, related rates, rectangle optimisation
Question 3(a)(i) 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.
The graph shows the function y = f(x).
Find the value(s) of x where f(x) is not differentiable.
Work it out
Try it on paper first. Stuck? Take one hint at a time.
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1 Where is the graph broken: a jump, a hole, or a point that sits off the curve?
x = −6: the line at y = −2 ends in a filled dot, and the curve starts from an open circle at y = 2. A jump.
x = −2: both pieces head for the open circle at (−2, 2), but f(−2) is the filled dot at 7.
x = 3: both pieces meet at an open circle, so f(3) is not defined at all. -
2 Is there anywhere else, such as a sharp corner on an unbroken piece?
No. The peak at x = −4 and the trough at x = 6 are smooth, and the pieces between the breaks are unbroken. A break is enough on its own: a function that is not continuous at a point cannot be differentiable there.
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.