This is the third of our three WitCHes on VCAA’s Specialist Mathematics Exam 1 Sample Questions. Newcomer aps was first to comment on the error, in regard to VCAA’s webinar, where the same question is “solved” (and see the subsequent comments on that post). Once again, it seems worthwhile to encourage a prominent discussion on an unfamiliar topic, and there is more to say than has come up so far.

### UPDATE (14/02/23)

Most of what needs to be said has been pointed out, in the comments here and below, and in particular in edderiofer’s very good summary. It is still worth highlighting the major error, and there is another error that no one has noted.

The major error is in part ~~(b)~~(a), where referring to and attempting to find “the maximum number of bacteria” is simply wrong. The model predicts a limiting value of the bacteria, and nothing else. (Interestingly, the preamble says the population “satisfies” the DE, but ~~(b)~~(a) refers to a prediction of the “model”.)

Note also that the limiting value being an upper bound for P requires knowing the initial population: if the initial population is too large, then P will decay to the limiting value (which is then a lower bound). Note, however, that the webinar “solution” of ~~(b)~~(a) nowhere uses the initial population, and thus has to be wrong, even on its own terms.

The second error is implicit in (c). The webinar makes clear that was intended was to solve the initial value problem. But, that is not what it means to “solve the differential equation”. Solving a differential equation is comparable to determining the antiderivative of a function: it does not include specifying the + C.

That’ll do. There are other nitpicks, mostly relevant to the webinar.

Probably this isn’t a main issue. But is it weird that P is treated as both a function, P(t), and a variable? Honestly have no idea whether this is accepted notation or not.

No, that notation is fine. They don’t really need the (t) in the introduction of P, but it’s reasonable to do it.

Ah ok good to know

I feel like most of my criticisms for these are nitpicks (some of which are directed at the solution on the slides), but there are a fair number of nitpicks:

* The question says that the population *satisfies* the differential equation, but population is discrete, so clearly the differential equation cannot be completely accurate to reality. Better to say that it *is modeled by* the differential equation (which is consistent with a)), and then reword b) to also make it clear that we’re asking about the model and not the actual Petri dish.

* For that matter, maybe it should say “in a certain Petri dish”.

* Why express the differential equation in that way? Why not write it as “P(12 – P/4000)” instead of having a random coefficient of 2 on the outside? In fact, to make it clear that we aren’t evaluating P(x) at x = 12 – P(t)/4000, the differential equation should really be written as “P(t)(12 – P(t)/4000)”.

* a) is done by solving for dP/dt = 0, which gives that P = 0 or that P = 48000, and clearly the latter is the intended answer. But this maximum is never attained except “at infinity”. Can this really be called the “maximum”, in that case?

* b) is perhaps worded confusingly, but is done by solving for d^2P/dt^2 = 0. So the answer is 24000 bacteria. The slides attached to this sample question obtain this answer another way, by taking the average of the solutions to dP/dt = 0, but they don’t properly justify why this is valid (e.g. by stating that dP/dt is a quadratic function of P, so its vertex’s P-coordinate is at the average of its roots). Is it intended that students do this by graphing alone?

* c) requires integrating 4000/(P(t)(48000 – P(t))) with respect to P. I assume this is covered in the VCAA syllabus, since the solution does just that, but the solution also should state the assumption that 0 < P(t) < 48000 in order to justify removing the absolute value bars from inside the logarithm.

* All that and not a single graph of what P(t) looks like, or any sort of a discussion on what makes the differential equation "logistic". Why even mention that it's a "logistic" differential equation?

Thanks, edderiofer. Yes, in the main (but not entirely) you could regard them as a bunch of nitpicks, but it’s more like an infestation of fleas. I think you have missed one major aspect, which I think is a clear error, and there are for me (at least) two other irritations. Other than that, here are my thoughts on your nitpicks:

*) Yes, “models” is the proper term. I hadn’t thought of this one, but you are correct.

*) Yes, the factor 2 out the front is a little odd. Such a factor might be considered natural, in the way you might justify or set up the logistic equation for the model, but of course the nice numbers are just cooked anyway.

*) I disagree on the P(t). Simply writing P in an ODE is standard and preferable.

*) Yes, there is no maximum, and this is very much not a nitpick. The question is plain wrong.

*) I loathe expressions such as “fastest rate”, but VCAA’s quadratic proof of (b) is fine, and clear enough. (The audio/transcript makes it clearer.)

*) Yes, the removal of the absolute values needs to be justified, and this is not done in either the slides or the audio/transcript.

*) Yes, the lack of a graph, of any proper consideration of the model, is absurd. All of VCAA’s expressed concern for the “real world” is simply performative.

Since the question says “predicted” is it OK though to use the asymptote as a prediction, even though such a P value is never actually realised?

I’m trying to look at this question in a way that a student might if it were an actual exam.

Scary.

No.

OK. Let me rephrase.

Do you think someone at VCAA thought it was OK to use the limiting value as a valid “prediction” even though the model predicts the population will never reach this limit?

No idea, and it doesn’t matter. Words have meaning.

It doesn’t matter in the sense that the question is wrong and the answer given in the presentation is wrong.

It does matter in the sense that, if this were an exam question, would 48000 be marked as correct?

The question is worse than wrong – I’m still trying to work out how to guide students through the November exams which will be written by the same organization that brought us these questions.

OK. I don’t think of it as my job to guide teachers and students how to interpret VCAA stupidity. My job is restricted to pointing out the stupidity. But, clearly VCAA was, and presumably is, willing to use “maximum” in the sense you suggest.

I’ve updated the post, including a not yet noted error.

Marty in your update do mean part (a) rather than part (b)?

And I agree with eddy that the coefficient of 2 is silly, it would be more sensible if the coefficient was such that the equation had the form dP/dt = rP(1 – P/K) because then the growth parameter and the carrying capacity are obvious. (Are these terms/definitions that students need to know now that logistic DE is in the study design?) Who wrote these sample questions?

Thanks, A. Yep, I meant (b): corrected now.

I don’t know whether “growth parameter” and “carrying capacity” are official VCE terms or not. A couple texts use those terms, but a third does not.

As for who wrote the sample questions, the questions are clearly linked to the sample exam questions, discussed here. So, presumably whoever writes the standard exams, also wrote the sample questions. I have no idea who that is.