Witch 108: A Mean Trig

This one, which has been discussed a bit here, comes from the 2023 NHT Methods Exam 2. It is a little strange. There are aspects of the question we like, or at least there are some interesting ideas underlying the question. Nonetheless there is no shortage of crap, and so here we are.

23 Replies to “Witch 108: A Mean Trig”

  1. One obvious comment from me:
    Questions are asked about \displaystyle f.
    \displaystyle f(x) is defined but \displaystyle f is not.
    (Yes, we all understand what is meant by \displaystyle f. But why can’t they actually say what they mean? Notation is important [as VCAA is fond of reminding us]).
    Ditto with \displaystyle g.

    1. I think there is an issue there, but I don’t think you’ve grabbed it correctly. I can see nothing wrong with the wording of (a) and (b), although using “State” and then “Find” jars a bit.

      1. Do we assume the domain of f is Reals?

        Again, it seems obvious, but…

        I’ve also not found give anywhere in VCAA’s glossary of command terms.

        1. Thanks, RF. I’m fine with assuming the maximal domain, but it’d be nice to have the word “function” somewhere. It’s too casual, particularly coming from such officious pedants. And, yes, “give” is a gratuitously poor choice.

      1. On its own, yes, the wording is… awkward, poor, unusual, odd, …

        Relative to the rest of the exam though… it fits.

        The wording fits that is. The definition of g(a) is a bit weird and unnecessary in my (unprofessional) opinion.

  2. I feel like allowing a=0 and a=\frac{\pi}{2} is problematic as either will lead to a \frac{1}{0} in d iii.

    I *think* the examiner is trying to get students to find the value of a for which two average values are equal which, in and of itself is not a bad idea. The setting up of the whole question though I feel is rather problematic.

    It also seems (and has been noted before by other commenters) that parts *c* and *e* are meant to relate to each other. If time were more plentiful, I’d like to see if CAS can ruin this…

    1. Thanks, RF. It’s a little clumsy, but I think (d)(iii) is ok. The endpoints simply can’t be the solution.

      There is something worth thought underlying (c) and (e), which doesn’t change the fact that this is an atrocious exam question. I’ll let others ponder.

      1. Yeah… that is clunky, but I’m not sure it is the source of the problem.

        Circular functions, by nature of their infinitely many solutions (or roots, depending on which exam question you are reading…) are problematic when a question asks for some-parameter-relating-to-horizontal-translations. Domain restrictions might be the solution to this problem, but why not just choose a different function???

  3. I haven’t looked really, so this question comes without any thought. But I’ll ask anyway.

    Do students not know how to evaluate a definite integral? Why do they need to say such a weird think like “Let this integral be this other function” when the “other function” is actually just the definite integral?

    It’s like saying “suppose your name is Marty” and being like… uhh ok, my name is Marty anyway but whatever.

    1. Yep, the wording is nuts. I wonder if when originally drafted the question required students to do the integral, and then they did some clumsy surgery for some reason.

  4. Ok I did it. It’s so WEIRD. Is this question testing reading comprehension and examiner mind-reading?

    Why do people think (c) and (e) are connected? I might be missing something obvious but I don’t see anything there. Sorry.

    It just seems like somebody thought “oh let’s get them to do this” and just wrote it down, without even bothering to wonder how that would work as an exam question.

    1. Thanks, Glen. I goofed (don’t know if RF goofed): I meant that (d) and (e) are connected, not (c) and (e). And yes, it is very difficult to understand the purpose of the question, particularly for a CAS exam. I have little doubt that students would have blown the question, although we’ll never know.

      1. Yes and no on the goofing. I think the general solution idea was in the mind of the examiner when they wrote part (e), there just seem to be hints of it.

        It doesn’t really matter though, because the question should stand on its own.

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