It had to be done. Feel free to transcribe your comments from the Exam 2 discussion post.
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UPDATE (14/11/21)
The first rule of writing: if you cannot write then do not write.
One can list the various specific issues with this question, and we shall do so. Such a list, however, is beside the point. The overall awfulness of the question is much greater than the sum of its awful parts. Most obviously, the question is painfully and self-indulgently wordy. You want to ignore air resistance? Fine. Ignore it. You’re not the first. But don’t simultaneously bore the crap and stress the crap out of the poor students by providing some pointlessly contrived rationalisation. You hadn’t noticed that the introduction to the question was already painfully long?
Anyway, here are the point-by-point criticisms:
- Forget the stupid “rear wheels” thing and just admit you’re treating the car as a point object. It’s fine. We won’t tell your parents.
- Part (a) is ok, but learn to write. What is in the air? The path? The wheels? The car? Lose the commas. (Which, coming from us, is saying, something.)
- Part (b) is ok but, as for (a), the writing is appalling. Simplify!
- For (c), it would be nice if at some point the VCAA looked up the meaning of the word “smoothly”. You guys supposedly come from a tradition and you supposedly respect that tradition. You don’t get to make up your own meanings.
The question itself amounts to pointless CAS garbage, which apparently didn’t work for some students, because “you have to restrict the variables” or some such who-gives-a-shit shittiness. Seriously, Heads of Maths are fighting over whether it is reasonable or unreasonable to expect students to readily know how to get CAS to do this stuff, while not a word is spoken about the craziness of having this anti-mathematical garbage appear on a Specialist exam in the first place. You have all lost your minds.
- In Part (d) we have a car taking off with infinite acceleration. That’s great. That’s really great.
- It’s not clear anybody ever made it through Part (e). We keep finding bodies along the way.
In sum, a mix of pointlessness and incomprehensiblity. In its own way, a masterpiece.
If I had to transcribe I’d be here for hours. I’ll transcribe my final thoughts:
Still with the car that should be a motorcycle that should have a centre of mass that experiences no air resistance due to a miraculous tail wind whose path is having a love affair with a smooth join …
The preamble to part (d) tells us that this car accelerates from rest with an initial acceleration that is
. Wotta car! No wonder the ramp is having a passionate joining with its path.
To solve the DE, the technique of separating the variables must be used. This requires that
. And yet, the initial condition v = 0 when t = 0 and s = 0 must be used to calculate the value of the arbitrary constant of anti-differentiation. Does anyone see a way of resolving this technical infelicity? Perhaps
Will VCAA require the use of limits or does it conveniently ignore the v = 0 issue?
In fact, I don’t think this stunted car can recover from an acceleration that is infinite at t = 0. The acceleration should have been something like
or whatever …
JF,
I guess the tail wind is designed to eliminate variable aerodynamic drag proportional to speed say Which would make the ODE more complicated and presumably beyond the specialist curriculum
Infinite initial acceleration is more limited to Star Trek episodes. Here is an example of might happen in reality
Steve R
Indeed. I know why they had the tail wind. My point (raised in the Exam 2 blog) is that it’s a very amazing tail wind that can exactly cancel the drag. Then again, in a question that has an infinite initial acceleration, I suppose we have to suspend our disbelief. Except, unlike a James Bond movie, you don’t do it willingly and you don’t sit back and have a good time. A stunted car in an intellectually stunted question.
The use of “smoothly” in part c. is asking for trouble. When I first learned that word, it meant infinitely differentiable. But I think the car’s path does not have a second derivative at x = 16.
I know we are supposed to interpret “smoothly” as continuity of (i) the car’s path, and (ii) the first derivative. But the only reason I know we are supposed to do that is because this misunderstanding appears in so many textbooks.
I don’t think smooth always means infinitely differentiable – more that it has the necessary number of derivatives for the situation, so it depends on context. [I couldn’t find a discussion looking at the prevalence of different uses]
Mathworld defines it as “A smooth function is a function that has continuous derivatives up to some desired order over some domain.” So basically it is at least differentiable –
, not just 
.
As for what VCAA meant… in the study design there are two relevant dot points in Methods:
– interpretation of graphs of empirical data with respect to rate of change such as temperature or pollution levels over time, motion graphs and the height of water in containers of different shapes that are being filled at a constant rate, with informal consideration of continuity and smoothness
– informal concepts of limit, continuity and differentiability
Neither of these define smoothness vs differentiability… so either way, if the exam was using the word casually or formally, it should be clarified.
In effect VCAA use “smooth” to mean differentiable. It’s non-standard and annoying, pointless and pretending an understanding and sophistication that isn’t there.
A function is smooth if and only if it is infinitely differentiable at each point in its domain. It may or may not be also of class
(these mean different things).
It is very common terminology.
… except in VCE mathematics.
Hi,
A good example if y = mod(x) which is continuous everywhere but neither smooth nor differentiable at 0 but with x/mod(x) in the limit as x approaches 0
Steve R
I would say that “smooth” is not a well defined term in mathematics. One should be more precise.
It’s well-defined enough, and in a manner nothing to do with VCAA’s usage. In any case, it’s up to VCAA to define their terms.
While a CAS calculator would not be the tool of choice out in the real world (I used geogebra to get the intersection / numerical solution (and I’m also not saying that is necessarily a wonderful tool)) it is conceivable that students may at some point need to find a(n approximate, numerical) solution to a system of non-linear equations.
Given that VCAA is determined to run ‘CAS Assisted Mathematics’ subjects rather than actual mathematics subjects, you would think it could at least do it properly and use a CAS that, at a minimum:
1) Is fit for purpose (the hand held pieces of cagle* are anything but fit for purpose),
2) Is consistent across all schools,
3) Prepares students for what they’ll be using post-secondary school (no-one in industry, university etc uses those hand held pieces of cagle).
The previous Maths Mangler, a permanent resident of the magic faraway tree, completely bungled the implementation of Mathematica across Victoria.
Teachers who
in using a particular brand of hand CAS calculator with a 
weakness for handling systems of (highly) non-linear equations and don’t teach their students how to work around this weakness are idiots. I have no sympathy for their mewling. This whole issue reeks of teachers trying to absolve themselves of guilt. I have no sympathy for this sort of stupidity but I feel sorry for their students. The hilarious thing in all of this is that I’ve heard anecdotally that the chief writer for the Specialist Maths Exam 2 is a HoD at a school that uses the Casio!! Hoist with their own petard.
* With a nod to the Stainless Steel Rat.
Alternatively, the students could have no nuclear-powered machine at all, and students and teachers could spend a little time thinking about mathematics.