AS 91578 · 2023 paper · gradient of normal, turning points, a proof
Question 3(b)(ii) Achieved
Practises Reading a graph: differentiability, f′ = 0, and limits
A log chain rule, three readings off a drawn graph, a parametric normal on a Ferris wheel, stationary points of a reciprocal pair, and a catenary to verify by substitution.
Find the value(s) of x where f'(x) = 0 and f''(x) < 0 are both true.
Work it out
Try it on paper first. Stuck? Take one hint at a time.
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1 f′(x) = 0 and f″(x) < 0 together describe what kind of point?
A flat point where the curve bends downwards: a maximum. -
2 Which points on the graph are maxima?
Only the peak at x = −4. The flat line from −2 to 2 has f′(x) = 0, but it is straight, so f″(x) = 0 there, not negative. The trough at x = 4 is a minimum.
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 2023 marking schedule.
How did it go?
Got it moves it further out. Review brings it back tomorrow.