Like the previous post, this one comes from Maths Quest Mathematical Methods 11, and is most definitely a WitCH. It can also been seen as a “contrast and compare” with WitCH 15.
Subsection 13.2.5, below, is on “differentiability”. The earlier part of chapter 13 gives a potted, and not error-free, introduction to limits and continuity, and Chapter 12 covers the “first principles” (limit) computation of polynomial derivatives. We’ve included the relevant “worked example”, and the relevant exercises and answers.
17 Replies to “WitCH 58: Differently Abled”
Would like to take another look at this later, but initial reactions.
1. The “a+” and “a-” notation is awful. Normally I’m not super picky about notation (especially because I often struggle to choose the most lucid notation), but I found this egregiously bad.
2. The example tests whether the derivative, if it were to exist, would be continuous, it does not test the derivative’s existence (additional assumptions could be used here to justify the test, but since the text is silent on this, that is a source of crappiness).
3. “Smoothness” is not defined. One might attempt to rescue some of the examples / exercises by saying that we are testing for smoothness-defined-as-existence-of-continuous-first-derivative, but the examples / exercises explicitly state that we are testing for differentiability tout court.
Oh, and if you really want to see some crappiness, go back a few pages in the same book to page 669 for the “kick off with CAS”introduction to the topic.
1. I agree, this usage of this notation isn’t standard. They should write this out with limits, as in “lim[x→a-]((f(x) – f(a))/(x-a)) = lim[x→a+]((f(x) – f(a))/(x-a))”; as it stands now, it’s easily misread as meaning “lim[x→a-](f'(x)) = lim[x→a+](f'(x))”, which is very much wrong as it assumes these derivatives exist and that these limits exist.
2. I’m not immediately sure I understand the exact nature of the error in the text (primarily due to the ambiguous use of the “a-/a+” notation), so I’m not too sure whether your complaint applies here. I’m interpreting the test as “find the two one-sided derivatives at a and check if they agree” (the first interpretation I listed above; Q21b and c seem to suggest this is the intent, at least). If that’s the case, then I’m pretty sure the derivative is defined as this value if they agree. However, I see that it’s also interpretable as “find the *one-sided limits* at a of the derivative and check if they agree” (the other interpretation I listed above), which is of course wrong, so at the very least this should be more-clearly stated.
3. I’ve always heard “smooth” to mean “infinitely differentiable”. This is much too strong a condition for the statement “This means that only smoothly continuous functions are differentiable at x = a”, when what they really mean is “This means that only functions that have a derivative at x = a are differentiable at x = a”, which is of course such a simple tautology they may as well have not said it.
edderiofer, one of the main issues is it’s impossible to tell what they mean. With the Cambridge WitCH, they started with the correct definition and then fucked it up. Here, it’s unclear, which is certainly enough to declare it crap. But I also suspect they are fucking it up, simply in effect taking the limit of the derivative. Except for the last, weird exercise. What is the purpose of that exercise, and what does it suggest about what comes earlier?
And yes, “smoothly continuous” is weird and wrong. For MQ it means nothing more than differentiable.
Thanks, SRK. The a+ notation is not just awful, it is undefined and thus meaningless. And I think you are right: they are looking for continuity of the derivative, meaning my reply to edderiofer is incorrect. It also means, as you suggest, they are fundamentally not looking at differentiability: they are looking at limits of the derivative.
I think the author of this text could educate themselves on the existence of differentiable but not continuously differentiable functions.
Also, the grammar and sentence structure of this thing is gross… might be just me though.
Wow, sorry, this is completely fucked. They are literally teaching the wrong thing. What is this crap here indeed.
Humpty Dumpty would know what smooth means.
May I have a rant here about language in problems?
Today I was helping a Year 11 student in General Mathematics with a problem on statistics. It involved a survey of what people thought of capital punishment. My student, whose first language is not English, had no idea of what capital punishment meant. This is a constant problem: setting applications of mathematics in a context of which some students have no understanding.
Should Year 11 students know the meaning of capital punishment? Where will they learn this? It has not been part of Australian traditions for more than 50 years. Will students who were born and raised in Australia during the last 18 years know the meaning of capital punishment? Ironically part of our school used to be a prison and there were three executions here in the 19th century.
Language is interesting. The student informed me that they did not have a computer at home. However, he went on to say that they have 4 laptops at home.
Would we ask a question about the price of “wireless”?
That feels better. Thanks for listening if you read this far.
Terry, you’re correct. I’ve met students who did not know the make up of a deck of cards and were confused by questions asking for the probability of 3 spades chosen without replacement etc.
I remember a question on a Maths Methods Exam 2 many years ago about a farmer describing the layout of his farm to a surveyor and about two roads being adjacent to each other. The question gave no map (diagram). I queried the lack of map (diagram) at a Meet the [ insert appropriate noun] session (yes, I stupidly attended one). I asked whether it would have killed the exam writer(s) to have included a diagram so that students, especially ESL students, who could not decode the word adjacent in this context, were not disadvantaged. I said that in real life the farmer would surely have given the surveyor a rough map. The [insert appropriate noun] said that all questions were ESL-vetted and there was no issue with this question. Let’s move on.
There is complete ignorance at best and total disregard at worst for the culture many students bring with them. And of course, when you’re imposing contrived ‘real-life’ contexts onto a mathematics question, this problem becomes magnified.
I concede that questions involving surveys etc. must include a context. But you would think the contexts would be carefully chosen and carefully vetted by the editors.
On a related note, idiot teachers over the years have caused quite a stir for the contexts of their maths questions:
Here’s one to ponder:
Terry, whilst I do sometimes worry about language, I worry about notation a lot more. In this particular case, the jumping between and just seems… not needed at best and confusing to perhaps even dead wrong at worst.
I agree that writers should be careful with notation and endeavour to be consistent. On the other hand, students should strive to be fluent in the language. I have met many university students who are confused by the equations and . “Which is correct?” they ask. If I try to explain that there is no difference, they want to know why mathematicians use different notations for the same thing.
*Sigh* And I suppose if they were given they would boggle at the idea that it represented a line (“But where’s the gradient!?” I can hear them asking. Before deciding that the gradient must be ).
But this is more an indictment on either whoever taught the student or on the mathematical intelligence of the student (probably the latter) than a textbook. And maybe an indictment of the entry criteria used by the university. (I can imagine this will be the least of the problems faced by universities who have set Further Maths as the minimum entry requirement for Engineering).
And, as a particular frustration, what does even mean? I have a feeling they mean the limit as is approached from the negative side, but this is not what they seem to have written.
Which (I do not know which, I am no mathematician) is either completely wrong or insane and completely wrong.
RF, it is entirely unclear what they mean by that notation. It is never defined, and none of the discussion or examples or exercises make properly clear what they want.
…final airing of frustrations (for now… should have put them all in the one comment I know)
“the derivative from the left of must equal the derivative from the right of ”
How far to the left/right? Would it kill them to actually mention limits in this sentence? (Possibly. It is as good an explanation as any I can think of)
Is it just me, or does Q21c/d, which involves the limit seem misplaced? I’m curious as to what the worked solutions say because students would most definitely not have the techniques in order to evaluate the limit. Is it just a calculator question? Aside from that, the gravest mistake in my opinion is the word smoothness, which has a related definition ( where the function is infinitely differentiable…) or it could also be , that is that the first derivative is continuous. It’s not clear what the textbook is going for here…
No, Sai, it’s not just you.
Exercise 21 is simultaneously the only reasonableness there, and completely insane. Parts (b) and (c) are the only clear indication of the required limits for such a “hybrid’ function. So, why do these limits only appear in the very last exercise? And, as you note, why needlessly choose a function whose derivative the kids don’t have a hope in hell of evaluating?
You are also correct, that “smooth” is simply the wrong word, and it’s also for the wrong concept. To the extent that “smooth” has an accepted, precise definition, it generally means, as you suggest, infinitely differentiable. Outside of ignorant textbooks, I have never seen “smooth” used to mean continuously differentiable, which appears to be the concept that the textbook is, incorrectly, trying to capture.