The question below is from the second 2020 Specialist exam (not online), and was flagged by commenter John Friend in the discussion here. John has spelled out the problems, but the question is bad enough to warrant its own post, and there’s arguably a little more to be said.
For convenience, I’ll repeat my comments from the other blog:
This is an improper integral (I wonder if the exam setting panel even understand what these are since they keep appearing) but no calculation is required so you might suppose it can go through to the wicket keeper. However …
The integral doesn’t converge! So we have an improper integral (not on the course) that doesn’t even converge.
Maybe this was deliberate so that students couldn’t compute the value with a CAS and then compare with the value from the options … A cheap trick if true.
I keep saying it: You don’t try to test skills that are trivialised using CAS technology in Exam 2. Test those skills in Exam 1.
Verdict: A good question in principle. IF it was not improper and IF it was on Exam 1.
Marty has alluded that there’s more to say, so I’ll stop and give other commentators a chance.
JF,
The question could easily be fixed by changing the sign of the -3 tanx to + 3 tanx in the denominator to make the integral proper but I agree it would make sense to have the question in the Non CAS section.
Otherwise a fair question perhaps as it tests integral substitution and knowledge of trig derivatives and partial fractions
Steve R
Steve, that “otherwise” is doing a lot of work.
Fixing the integral so that it either is no longer improper or converges will trivialise the question. So I get why it’s non-convergent. What I detest is that to non-trivially test “integral substitution and knowledge of trig derivatives and partial fractions” in the MCQ section of Exam 2 you have to make it improper and non-convergent. And I think trying to test those three things is too much for a 1 mark MCQ.
Trying to test “integral substitution and knowledge of trig derivatives and partial fractions” in Exam 2 (and particularly in Exam 2 MCQ) is inappropriate and really stupid.
And trying to do it by using concepts that are not on the course is immoral. I think it’s reasonable to assume that some students might have thought the question was defective because of the non-convergence and been put off by this.
A pinhead in VCAA has made this rod for everyone’s back whereby there is too little time in Exam 1 to test all that should be tested, and too much time in Exam 2 so that the questions used to pad it out are trivialised by the CAS technology.
Testing “integral substitution and knowledge of trig derivatives and partial fractions” would not be clumsily attempted in Exam 2 if it was shorter and Exam 1 was longer.
Strong constant feedback with consistent messaging from teachers and schools CAN make a difference. (A case in point is the roll-BACK of Mathematica: The state-wide licencing arrangement for Wolfram is ending on 31 December 2020 (which by the way leaves approved participants in the VCAA Mathematical Methods computer-based examination program potentially up shit creek). Personally I’m disappointed – I saw it as a choice between using a real CAS and using a ridiculous CAS-calculator that has no relevance outside of school. But VCAA shot itself in the foot by consistently over-promising and under-delivering. The Pilot was poorly rolled out. VCAA lacked credibility and quickly lost trust. Word got around. Maybe there were hacks who had reputations invested in the CAS-calculator. The tribe has spoken).
CAS technology (but not Mathematica) is here to stay. And *sigh* SACs are here to stay. So I think the constant loud and clear feedback given to VCAA should be:
Exam 1 (CAS-free): 2 hours.
Exam 2 (CAS-assisted): 1 hour, NO multiple choice, no bullshit.
One SAC: Done midyear, written by schools, 3 hours.
Exam 1 (CAS-free): 3 hours. Done.
I don’t “accept” that CAS is here to stay, and I don’t “accept” that SACs are here to stay. I don’t “accept” lunacy.
I don’t write this blog to change things. I write this blog to point out the idiocy of our education system, and the idiocy more broadly. How people want try to change things is their business. But seeking to compromising with bad-faith lunatics, hoping for a tweak or two, is a losing strategy.
JF,
You make a number of sensible suggestions and moving the question to non CAS rather than doctoring the values to make the integral improper to try to neuter the less competent CAS button pressers and making it worth more than one point makes sense to me.
Surely Mathematica CBT users should be scaled separately IMO as they appear to have a massive potential advantage with suitably designed preloaded functions ,programs etc
Steve R
Steve R, I totally agree that there should be separate scaling for the CAS-calculator and Mathematica. But for some unfathomable reason there isn’t.
The field may or may not become level some time soon, since Mathematica is being rolled back. The state-wide licencing arrangement for Wolfram is ending on 31 December 2020 which may leave approved participants in the VCAA Mathematical Methods computer-based examination program up shit creek in having to quickly return to the CAS-calculator after spending three or more years phasing it out and re-writing curriculum to embed Mathematica.
But if Mathematica remains in schools that currently use it, even if only for the next 3 years (to allow a phased withdrawal), I don’t see that VCAA has any choice but to introduce separate scaling – the inequity is inarguable (https://mathematicalcrap.com/2020/07/16/how-to-play-vce-with-mathematica/) and schools not currently using Mathematica no longer have a chance to change their minds and adopt it.
All things considered, VCAA’s bungled Pilot of Mathematica almost rivals the ‘words-cannot-describe-the-shitfull’ roll out of the NBN or the Ultra-Farce *ahem* I mean the Ultra-Net. (Given that VCAA is part of the DET, DET can now proudly own two disasters in the last 10 years. And if you want to include the ‘*Every* student will be considered for disadvantage’ twaddle that came out of someone’s big ignorant gob ….)
Where is the “this question makes no sense because the thing you are manipulating doesn’t exist” option? This question is beyond fucked.
@all: the substitution rule only works on integrals that exist.
Thanks, Glen. Of course the question is screwed, and the manipulation of a divergent improper integral is not kosher for an exam where improper integrals aren’t in the syllabus. But it feels to me like substitution could extend to divergent integrals. Is there an example of a substitution on a divergent integral that will change it from infinite to undefined, or something of the sort? Didn’t try hard but I couldn’t think of one.
Of course that couldn’t save the exam question. But I’m a bit curious.
(As I’m sure you know…) The substitution rule is exactly an integrated form of the chain rule. Combine the chain rule with the fundamental theorem of calculus and you have exactly the substitution rule.
I haven’t got an example in mind, but what I was thinking when I wrote that is that the validity of the chain rule and the FTC (for say Riemann integration) requires things like integrand being absolutely continuous.
In this question the integrand diverges at one of the endpoints, and diverges quite badly too. So the proof doesn’t work. The proof not working means the theorem doesn’t apply, and that’s why I said it doesn’t work on integrals that don’t exist.
Your question about extending it to integrals that are +\infty or otherwise divergent is interesting though. As a first reaction, I’d try to use the results on re-arranging (conditionally) convergent series and getting multiple answers (Riemann’s theorem? I think). IIRC this can be done so that the new series is convergent to any number or even divergent. This might result in quite a pathological substitution, which might end up proving more that substitutions should be differentiable instead of integrals converging. Of course for the chain rule to work it needs to be differentiable anyway.
Hmmmm. Measure-theoretically, if we are talking about positive functions, the integral is the measure of the region bounded by the graph of the function and the
axis. So, I think the substitution rule works in that setting.
Oh! Right. So we can integrate a function that oscillates, perhaps that might give us a counterexample. I’ve gotta get back to work but if I do come up with any explicit example I’ll add another comment.
Thanks, Glen. That was exactly the kind of maybe/maybe-not pondering I was going through. I can imagine some oscillating guy could be switched from convergent to infinity or undefined by a clever substitution. But, given that substitution requires differentiability, i can also imagine it’s not possible. I haven’t had time to sit down and look more carefully.
Yeah. Something like
for 
large will make the speed that we traverse 
quite large at infinity, so if the oscillation does something good (or bad) at infinity, that can be emphasised by this kind of sub, But yeah, I don’t have an example yet.
Sorry, what might work is if we have an odd function that is fast at one end and slow in the other, e.g.
for 
and 
for 
. We could make positive parts dominate on one side and negative on the other; then by changing "how much" of each side of zero we are considering how fast, that should allow us to get what we want.
OK! I really should work :).
Yes, a doubly improper integral is vulnerable to such attacks. But I’m not sure a single improperness, as in the exam question, is vulnerable.