We’re back, at least sort of. Apologies for the long silence; we were off visiting The Capitalist Centre of the Universe. And yes, China was great fun, thanks. Things are still tight, but there will soon be plenty of time for writing, once we’re free of those little monsters we have to teach. (Hi, Guys!) In the meantime, we’ll try to catch up on the numerous posts and updates that are most demanding of attention.
We’ll begin with a couple new WitChes. This first one, courtesy of John the Merciless, is a multiple choice question from the 2017 VCE Specialist Mathematics Exam 2:
The Examiners’ Report indicates that 6% of students gave the intended answer of E, and a little under half the students answered C. The Report also comments that
f”(x) does not change sign at a.
Have fun.
What an unnecessarily convoluted question.
It looks like it’s a question purely constructed and purposefully “weird” to throw students off, but to also maybe “show off” that VCAA can “do complicated maths”.
And then they stuff up the Examiner’s Report by forgetting a negative in front of the pronumeral a.
That report has been online for the better part of a whole 2 years.
If an amended version of the Report appears, it might indicate that VCAA does pay attention to public comment on its errors …. (or at least errors that can be attributed to typos rather than incompetence and negligence. Once comments on this blog have run their course, it will be very interesting to see what amendments VCAA makes to the Report ….)
This is an abomination.
I’ll start with questioning whether (I doubt it) g(x)<0 is at all relevant.
@RF: In my opinion the inclusion of g(x) is unnecessary and ridiculous. But once you do include it, the requirement that its sign never changes *is* relevant.
@Steve: Nice observation about the missing negative in front of the a. I hadn’t even noticed that, I was distracted by something else (not mentioned in Marty’s introduction) in the Examiner’s Report comment (take a look at the full comment in the report).
@GW: lol! I think you’re being too kind. In my opinion, disgraceful abomination gets closer.
If g(x) is omitted, so that we have f”(x) = (x+a)^2*(x-b), I don’t think that an f(x) would exist that could meet the other conditions (although I’m happy to be proven wrong). I assume that this is why g(x) was included in the first place. And I assume that it was placed on the denominator so that it would not create any potential additional points of inflection. And made negative all the time so that we wouldn’t get a denominator equal to 0.
I assume that they didn’t want to include anything in the question that might unintentionally muddy the waters.
Damo, you’re correct (except for the two special cases a = -b and b = -a). So in the interests of clarity, surely the sensible thing would have been to use different conditions so that the g(x) was not required. But once g(x) is included it must be made either negative or positive all the time so that there is no sign change in f(x) (or does it ….? After all, the final function is |f(x)| not f(x) ….)
What I find frustrating is that the question provides no insight into a student’s understanding of either points of inflection or the modulus function. It’s interesting that the most common (incorrect) response was option C (45% of the state) which has negative y-coordinates …. Now, were students who chose this option bamboozled by all the crap or did they not understand that |f(x)| cannot be negative ….? What makes those students different to the students who chose A ….?
The question tries to do way too much (for 1 pissant mark) and, inevitably, fails spectacularly:
1. It provides no insight into student thinking. 6% chose the correct option – I wonder what proportion of that was a guess. How many students who chose option C would draw a graph of y = |x^2 – 1|, say, that went below the x-axis …?
2. There is no correct option (depending on the definition of point of inflection one uses).
3. There are two correct options under SR’s definition (each special case above leads to a different option).
I feel like in this case, it’s way more likely that a large proportion of students missed the modulus signs around f(x). I know I did too when I first looked at this question.
You could be right. It’s certainly the final sting in the crap-filled tail.
And Steve’s point is hilarious about the missing negative!
Thanks JF – I realised that once I actually played with the functions a bit in the general sense. However, the more I played around with it sans CAS, the more I come to think it is a stupid multiple choice question which, given the value and time pressures, is probably best guessed-at by all but the very best students.
As stated, it’s entirely possible for a = -b to be true, which means that the examiner’s statement is not necessarily true even after correcting for the typo.
Question: Do points where the left-hand and right-hand limits of the second derivative disagree in sign (e.g. as in x = 1 of f(x) = |ln(x)|) count as points of inflection even if the second derivative at that point is not defined? Because if so, it’s entirely possible that there exist points of inflection of |f(x)| between a and -a, or between b and -b (and wherever else f(x) = 0), even if no such points exist for f(x). After all, it’s f(x) that has its derivatives defined, not |f(x)|.
edderiofer, that’s the $64 question …. There are differing opinions on the answer, so ideally one should be able to go to the Mathematics Study Design for advice …. (Good luck with that ….)
Hi,
Perhaps the definition of a point of inflection should be defined precisely in the Study Design to avoid confusion
Eg https://en.m.wikipedia.org/wiki/Inflection_point
Concavity changing sign on a continuous smooth curve in neighbourhood of the point
Steve R
On the other hand:
http://mathworld.wolfram.com/InflectionPoint.html
A necessary and sufficient condition for a function g to have a point of inflection at x = c is that g”(c – e) and g”(x + e) have opposite signs in the neighbourhood of x = c (that is, there is a change in concavity).
(Bronshtein and Semendyayev 2004, p. 231).
The first and second derivatives g'(c) and g”(c) are not required to exist under this definition ….
There’s no perhaps about it, SR …. The definition of a point of inflection SHOULD be defined precisely in the Study Design to avoid confusion. But it’s not. That is one of its many defects. A really important part of mathematics is precise definitions.
JF,
fully agree that a clear definition of an inflection point c is needed
to answer question unequivocally
to exclude asymptotes and/or bumps etc where f”(c) is undefined
to be fair to wolfram they mention f”(c) =0 as a necessary condition in their link
Steve R
Yes SR, but the sufficient condition subsequently given makes no mention of this.
I think Wolfram should have added that a necessary condition when f”(c) exists is that f”(c) = 0 ….