WitCH 76: Cos and Effect

Yeah, yeah, we should have done this one a week ago. Feel free to transcribe your comments from the Methods Exam 2 discussion post. You can also check out the discussion on stackexchange, courtesy of Stog the Stirrer.

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UPDATE (15/11/21)

The question is awful, with the final part the pinnacle of awfulness. We’ll consider that part in some detail, but first the other awfulness.

  • Part (a) Part (b) could be interesting since finding the critical points amounts to solving a (nasty) cubic, but it is not interesting here. Here, it is meaningless CAS garbage.
  • The wording in Part (b) Part (c) is atrocious. If you mean the functions f(h-x) and f(x) are equal then damn well use the word “function”. And, why not define a new function, Fh or whatnot? You guys are forever, painfully, defining functions for no good purpose. Why not here, when there is actually a purpose? Of course, it also never occurs to anyone that one might prove that the two functions are equal for whatever h. Nope, just look at the damn picture.
  • When does ga equal f? Seriously?

 

  • Part (e)(i) is fine, but Part (e)(ii) contains possibly the worst sentence in human history. The question itself could be a good test of knowledge of trig symmetry, but of course here it is just a test of pushing buttons.
  • Part (f) is good, although we wonder what students will make of it.

 

And now, Part (g), which is, at best, Magritte garbage. Is it worse than that? Yes, it is.

The question asks students to

Find the greatest possible minimum of ga.

There are (at least) six plausible interpretations of this question:

Interpretation 1 For each a, let Ma be the (absolute) minimum of ga. Find the maximum value of Ma over all possible a.

Interpretation 2 As for Interpretation 1, but find the minimum value of Ma over all possible a.

Interpretation 3 For each a, let La be the set of local minima of ga. Find, for each a, the maximal element of La.

Interpretation 4 As for Interpretation 3, but find, for each a, the minimal element of La.

Interpretation 5 As for Interpretation 3, but find the maximal element over (the union of) all La.

Interpretation 6 As for Interpretation 4, but find the minimal element over (the union of) all La.

Now, for the kind of reasons that commenter Tungsten suggests, it is likely that Interpretation 1 was intended, but it’s no gimme. In particular, Interpretation 2 is quite plausible; it takes a special born-that-way stupidity to use the term “greatest” when optimising negative quantities. Moreover, as John Friend and Glen have suggested, below and on the discussion post, Interpretation 3 is also very natural. Then, Interpretation 4 is not far behind. In any case, this is insane. Students shouldn’t need to engage in an idiotic guessing game at the end of the exam, for 1 mark, simply because the writers cannot write.

Anyway, guessing over, let’s assume Interpretation 1 is correct. What then do students do? Yep, as Tungsten suggests, they just fiddle with their buttons, note that Ma gives a minimum of -√2, and that Ma appears to be decreasing. That’s all they can reasonably do. Well, they can also reasonably scream out “This is meaningless garbage”, but that probably won’t score them the mark.

Note that there is a very nice and natural and easy proof that Ma has a minimum of -√2. See the stackexchange reply. But this is Methods. No one gives a damn.

A stupid, hateful question to end a stupid, hateful exam for a stupid, hateful subject. Utter lunacy.

8 Replies to “WitCH 76: Cos and Effect”

  1. e ii)
    ” … the area bounded … is equal above and below the x-axis …”
    I’m guessing it is meant to describe two separate areas that are equal.

    g) “Find the greatest possible minimum value of g_a.”
    I assumed that this meant the greatest global minimum, greatest taken over all possible a. But after looking at comments in the Methods Exam 2 discussion post, I guess it could mean the greatest (or least) minimum for the various local minima for a fixed a. Either way it is too difficult.

    When I was a Year 12 student the exams were tightly controlled by E. R. Love who had an international reputation for the rigour and clarity of his writing. The early eighties saw a move to give examining power to secondary teachers. This is the result.

    1. I will attempt to guess the likely intended solution (by VCAA) to part g. The meaning of “greatest possible minimum” should be inferred from usage of terminology elsewhere in the question. Then turn to the graphing calculator or exam laptop and look at the graph when a=1 (the lower bound of a) where the global minimum is -√2. Increase the value of a, and watch the graph change. Observe that the graph goes below -√2. Conclude that this is true for all a>1 and therefore that the answer is -√2 for 1 mark.

      VCAA would say that it is not too difficult.

      I assume that prior to the early eighties, the use of calculators and laptop computers in examinations was a little more rare compared to now.

      1. Another thing to note is that I have seen in too many places the suggestion that the answer to part g. is -1 as a becomes large, based on looking at a graph of an example of for instance a=10^4. This appears to be the case for a small section of the graph, but is clearly incorrect if one zooms out far enough. I must admit that I was guilty of this error during the examination and so lost the mark. This sort of thinking and exam strategy, just looking at some graphs with some values plugged in, is really encouraged and a direct result of the effect of CAS and the style of questions such as part g. that can be reasonably answered using only such “tricks”.

    2. Thanks, tom. The current tolerance for incompetence is astounding. On E. R. Love, I have a small post planned about him as soon as I escape these exams.

  2. I don’t think the essence of the crap has been captured at this blog. The true essence is what the hell does “the greatest possible minimum value” mean!!??

    As I’ve commented elsewhere – for a given value of a there is a set of local minima. Each of these sets has a value whose magnitude is smallest and a value whose magnitude is largest. You can take each of these values for each value of a and construct two new sets – the set with the largest value and the set with the smallest values.

    Is the question asking for the largest or the smallest value in each of these two sets.

    The only answer that makes sense once all this crap is processed is \displaystyle -\sqrt{2} corresponding to when a = 1. The limiting value of -2 is never attained and so the question must intend the value with minimum magnitude out of all possible minimum values for all possible values of a.

    Maybe I’ve made heavy weather trying to explain this, but to me this is the true crap:

    What does “the greatest possible minimum value” actually mean. Is it possible to explain what it means using only VCE language (which is what I’ve tried to do above).

    1. Agreed John – it is ambiguous. And it needed your full analysis to make that clear. If the question refers to just some fixed value of a then there are at least 2 meanings. If we allow the optimum over all possible a (not in the syllabus) then we get 2 more possibilities.

  3. Quick correction: in your update, you seem to have written “Part (a)” and “Part (b)” when you mean “Part (b)” and “Part (c)” respectively.

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