AS 91578 · 2024 paper · sec, graph, cone, triangle area
Question 3(b)(i) Achieved
Practises Reading a graph: differentiability, f′ = 0, and limits
A product with sec, three readings off a drawn graph (continuity, stationary points, a limit), stationary points of a rational function, a related-rates cone, and an optimisation that has to be proved to a given exact value.
The graph shows the function y = f(x).
For the function above, find the value(s) of x where f(x) is continuous but not differentiable.
Work it out
Try it on paper first. Stuck? Take one hint at a time.
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1 Continuous means unbroken. Where does the graph break?
At x = −1 the two pieces meet at an open circle, so f(−1) is not defined.
At x = 3 the curve stops at a filled dot, (3, 1), and the flat line starts from an open circle at (3, −1): a jump.
Neither is continuous, so neither is the answer. -
2 Where is the graph unbroken, but with a sharp corner?
At x = 5, where the flat line at y = −1 meets the rising line. No break, but a gradient of 0 meets a positive one.
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 2024 marking schedule.
How did it go?
Got it moves it further out. Review brings it back tomorrow.