WitCH 15: Principled Objection

OK, playtime is over. This one, like the still unresolved WitCH 8, will take some work. It comes from Cambridge’s Mathematical Methods 3 & 4 (2019). It is the introduction to “When is a function differentiable?”, the final section of the chapter “Differentiation”.

Update (12/08/19)

We wrote about this nonsense seven long years ago, and we’ll presumably be writing about it seven years from now. Nonetheless, here we go.

The first thing to say is that the text is wrong. To the extent that there is a discernible method, that method is fundamentally invalid. Indeed, this is just about the first nonsense whacked out of first year uni students.

The second thing to say is that the text is worse than wrong. The discussion is clouded in gratuitous mystery, with the long-delayed discussion of “differentiability” presented as some deep concept, rather than simply as a grammatical form. If a function has a derivative then it is differentiable. That’s it.

Now to the details.

The text’s “first principles” definition of differentiability is correct and then, immediately, things go off the rails. Why is the function f(x) = |x| (which is written in idiotic Methods style) not differentiable at 0? The wording is muddy, but example 46 makes clear the argument: f’(x) = -1 for x < 0 and f’(x) = 1 for x > 0, and these derivatives don’t match. This argument is unjustified, fundamentally distinct from first principles, and it can easily lead to error. (Amusingly, the text’s earlier, “informal” discussion of f(x) = |x| is exactly what is required.)

The limit definition of the derivative f’(a) requires looking precisely at a, at the gradient [f(a+h) – f(a)]/h as h → 0. Instead, the text, with varying degrees of explicitness and correctness, considers the limit of f’(x) near a, as x → a. This second limit is fundamentally, conceptually different and it is not guaranteed to be equal.

The standard example to illustrate the issue is the function f(x) = x2sin(1/x) (for x≠ 0 and with f(0) = 0). It is easy to to check that f’(x) oscillates wildly near 0, and thus f’(x) has no limit as x → 0. Nonetheless, a first principles argument shows that f’(0) = 0.

It is true that if a function f is continuous at a, and if f’(x) has a limit L as x → a, then also f’(a) = L. With some work, this non-obvious truth (requiring the mean value theorem) can be used to clarify and to repair the text’s argument. But this does not negate the conceptual distinction between the required first principles limit and the text’s invalid replacement.

Now, to the examples.

Example 45 is just wrong, even on the text’s own ridiculous terms. If a function has a nice polynomial definition for x ≥ 0, it does not follow that one gets f’(0) for free. One cannot possibly know whether f’(x) exists without considering x on both sides of 0. As such, the “In particular” of example 46 is complete nonsense. Further, there is the sotto voce claim but no argument that (and no illustrative graph indicating) the function f is continuous; this is required for any argument along the text’s lines.

Example 46 is wrong in the fundamental wrong-limit manner described above. it is also unexplained why the magical method to obtain f’(0) in example 45 does not also work for example 46.

Example 47 has a “solution” that is wrong, once again for the wrong-limit reason, but an “explanation” that is correct. As discussed with Damo in the comments, this “vertical tangent” example would probably be better placed in a later section, but it is the best of a very bad lot.

And that’s it. We’ll be back in another seven years or so.

32 Replies to “WitCH 15: Principled Objection”

  1. Just a question (genuine, I promise!) Mathematica gives “ComplexInfinity” as the limit for this example. Is the word “complex” just dressing, or does it convey additional meaning? (Anyone is free to assist me here)

    1. Example 47? Roots can always be considered as multiple-valued complex things, but isn’t there a way to tell Mathematica to not be dumb?

  2. Of course, the specification Reals can be used in any solve command (amongst other things) but I was asking whether “complex” infinity was any different to the Aleph-Null infinity that comes to mind when using the word “infinity” more loosely.

    1. Ah, I see. Yes, complex infinity is a thing, and it is different. In the reals you can go to +∞ or -∞. In complex numbers you can think of going out along any radial direction, or weird spirals or whatever, but it turns out to be useful to think of just one point ∞ at the end.

  3. In Example 46 they say:
    “f ‘(0) is not defined as the limits from the left and the right are not equal.”
    I think that it is fair to assume that they mean that the expression for f'(x) when x>0 is not equal to the expression for f'(x) when x0. And I think that it is fair to assume that they assume that this is equivalent to the very first line of this WitCH, where they define what it means for a function to be differentiable. But it’s not.

    1. Thanks, Damo. You’ve nailed the central problem, though there’s lots more there. The whole discussion is inconsistent and strange, and it is worth noting the “informal treatment” in section 9D to which the text refers (Example 19). But there is no question that the “limits” in Example 46 refers to limits of derivatives. As you point out, that simply does not give an equivalence for differentiability.

  4. I do not feel confident about Example 47, but I’m going to give it a go. To begin with, to show the graph of the derivative of f(x) and not the graph of f(x) is pretty crap. There should at least be some discussion of the fact that at x=0 there is a vertical tangent line. Also, returning to their initial definition of differentiability, I’m not sure why they didn’t use limits here. I mean, they kinda do, but they don’t. And actually, that limit is defined as being equal to infinty. So where does that leave us? You can either extend the reals to include +infinity and -infinity, in which case the function is differentiable, or you need to return to the original definition and say that it is only differentiable if the limit exists and is finite. But to gloss over it and not even relate it back to the original function? Not ideal.

    1. Not ideal? Damo, you’re a master of understatement. It’s a godawful mess. To take it bit by bit:

      1) Yes, of course they should show the graph of f as well as that of f’. The solutions to Examples 45 and 46 are similarly flawed.

      2) Yes, it is a missed opportunity to not bring in the notion of vertical tangent here. True, that concept is only introduced (poorly) in the next chapter, but this simply demonstrates that Example 67 is misplaced. What does it do in this section other than confuse?

      3) It is standard to discuss “infinite” limits without extending the reals (though it’s better to bite the bullet and extend). But yes, flippantly throwing in an infinite limit here is close to meaningless.

      4) Yes, they correctly look at the correct limits (give or take infinity), but only as an afterthought, as an “explanation” of their (invalid) “solution”.

      1. Re: 4).
        In their “explanation”, Cambridge gives the \displaystyle wrong reason why the limit \displaystyle \lim_{h \rightarrow 0} h^{-2/3} (and hence \displaystyle h'(0)) Does Not Exist.

        \displaystyle \lim_{h \rightarrow 0^+} h^{-2/3} = +\infty whereas \displaystyle \lim_{h \rightarrow 0^-} h^{-2/3} = -\infty and so

        \displaystyle \lim_{h \rightarrow 0^+} h^{-2/3} \neq \lim_{h \rightarrow 0^-} h^{-2/3}.

        That’s why \displaystyle \lim_{h \rightarrow 0} h^{-2/3} DNE.

        For the same reason, \displaystyle \lim_{x \rightarrow 0} \frac{1}{x} DNE but \displaystyle \lim_{x \rightarrow 0} \frac{1}{x^2} = +\infty does exist. Although some might argue that if the limit is not finite then it DNE …

        1. My mistake. I take it all back. The left and limit is also +oo. So the limit exists and is +oo. So the derivative is not defined at x = 0 because we want a finite value.

  5. I am going to try and be a little generous here – while their approach to this is a shambles, this specific example and the explanation of why we can or can’t differentiate when x=0 is done universally badly. Even when reference is made to a vertical tangent, the argument that is often made for it not being differentiable is that the slope of this tangent is undefined or simply that f'(x) is not defined when x=0. They have at least made some attempt to refer back to the original definition, although, as you point out, it is done as an afterthought and with no
    respect for the significance of the result.

    On another matter, the casual use of smoothness in the original notes and the questions leaves me anxious.

    On another matter, this sudden influx of WitCHs has left me feeling giddy, like it’s Christmas. At some point, I’m going to have to begin engaging with my family again, but for now, this is too much fun. Thank you.

  6. Damo, you set a good example, and I can also be generous here. But I choose otherwise.

    I have sympathy for the writers. It is very tricky to write correctly and clearly on informal limits, and infinite and non-existent limits are trickier still. If the “explanation” for Example 47 had appeared in isolation, I’d be fine with it. But, it’s *not* in isolation, and it’s not an “explanation”, and it’s not an “also”.

  7. Just discovered your Witch series, in particular Witch 15.

    Re Examples 45 and 46:
    The Cambridge explanation is begging for the use of one-sided derivatives. The same issue came up in Marty’s analysis of question 9 of the Maths Methods 1 exam of 2017. One-sided derivatives are not in the syllabus – or are they?

    Technical Teacher’s College in the 60s was unfairly known as Bonehead College by my class of graduates studying for a teaching certificate. The one thing that has stayed with me is the claim “a syllabus is defined by the examinations”. Clearly true for Maths Methods; I could write two vastly different exams within the framework defined by the study design. So if the 2017 examiner has decided that one-sided derivatives are in, maybe that becomes a fact…

    And it’s easy. If the student understands the full derivative it takes just another minute to cover the one-sided variety. Is there any point to it? Well yes – one place in which this tedious piece-wise guff is actually useful is in analysing polynomial splines. A simple approach could be to use one-sided derivatives. I leave the readers to see exactly how.

    1. Thanks, Tom. As you say, there are obvious uses for one-sided derivatives, and it’s not like it’s a difficult extra step. On the other hand, I have no trouble if Methods avoids one-sided derivatives. What is troublesome is the current neither-in-nor-out fogginess.

    2. Tom, the Examination Report for the 2021 Maths Methods Exam 1 clearly says that using the second derivative test to determine the nature of a stationary point is OK. This is not stated in the Study Design. But the writer of the Report has decided that this test is “in”, and so it has now become “fact”.

      I agree with you that
      “a syllabus is defined by the examinations”. Clearly true for Maths Methods”.
      And personally, I think this is a truly damning indictment on the Study Design.

  8. It occurred to me today during the teaching of my Maths Methods class that the removal of differentiation from first principles in the new VCE Mathematics Study Design means that differentiability at a point can no longer be (correctly) taught.

    Misconceptions such as:
    1) Continuity of f'(x) at x = a is a \displaystyle necessary condition for f'(a) to exist.

    2) Existence of \displaystyle \lim_{x \rightarrow a} f'(x) is a sufficient condition for f'(a) to exist.

    will flourish. With regards to misconception 2), I wonder how, within the scope of the \displaystyle new Study Design, VCAA would prove that the following function is NOT differentiable at x = 0:

    \displaystyle f(x) = -x - 1 when \displaystyle x \leq 0 and \displaystyle f(x) = -x + 1 when \displaystyle x > 0.

    I had a student ask during class why this function is not differentiable at x = 0 – they appealed to misconception 2) above that f'(x) was always equal to -1 including at x = 0.
    And NO, you’re are NOT allowed to ‘cheat’ by saying
    “Well … a necessary condition for f'(0) to exist is that the function is continuous at x = 0. Now look at the graph of y = f(x)”.
    In fact, I wonder how, within the scope of the new Study Design, you can actually prove when a function is continuous at x = a, apart from appealing to a pen moving along a graph and never leaving it … a pretty tough gig for a pen when \displaystyle f(x) = x^2 \sin \left( \frac{1}{x}\right) (for x≠ 0 and with f(0) = 0).

    1. Thanks, John. I still haven’t had a chance to look at the new SD. (So much crap …) Of course if first principles has been removed, whatever that even means, then no fringe examples, or anything, can be done properly.

      In general, I’m not fussed if the fringe examples are not done in high school (even if FP is there). What pisses me off is that the textbooks and the exams and the exam reports do the fringe examples, and invariably do them incorrectly. If these people would just understand that they don’t understand, and then shut the hell up about such examples, I’d be content.

      1. Re: “In general, I’m not fussed if the fringe examples are not done in high school”

        I understand what you’re saying, Marty. But I’m in two minds on this. On the one hand, one can reasonably argue that the average Maths Methods student doesn’t need to see them. It can just lead to confusion and distract from the main idea. But on the other hand, and I know I don’t need to tell you something that you already know:

        1) ‘Fringe examples’ can create greater awareness and insight, and so can be (are) useful for preventing misconceptions (like 1) and 2) mentioned above).

        2) The ‘counter-intuitive’ nature of ‘fringe examples’ can be exciting, provocative and intriguing for stronger students and hence make the subject more interesting (at least for a few moments).

        3) As you say – “textbooks and the exams and the exam reports do the fringe examples, and invariably do them incorrectly.” Doing one – correctly! – in class can be a pre-emptive measure of sorts (particularly when it comes to the textbook).

        As a less high-powered example, considering a function f with a rule like \displaystyle f(x) = \frac{x}{x^2 + 1} is useful in Maths Methods Unit 1 because it (might) prevents the misconception that a function can never touch or cross an asymptote. Preventing/mitigating this misconception is important for students who do Specialist Maths. (And I’d argue it’s important in general). I really wish teachers of Methods Units 1/2 would do this, but it rarely happens. Having said this, half the class will yawn and say “So what” (in which case the example is not for them …)

        It’s a fine line.

        1. I’m certainly not against a (competent) teacher incorporating such examples, and obviously I can see the benefit. But I don’t see much value in the school curriculum including differentiability of “hybrid” functions.

          1. Could it be that piecewise (“hybrid”) functions provide better examples for proving when a function is not differentiable at a point than what would otherwise be available? Or would you consider examples such as f(x) = 1/x as being sufficient at this level?
            I think the value of including piecewise functions in this context is that they provide good and simple examples for showing things like:

            a) f(x) differentiable at x = a implies f(x) continuous at x = a, but not the converse.

            b) f(x) not differentiable at a ‘jump’ discontinuity.

            Using the fundamental definition is not too difficult (and often exposes other underlying – sometimes ‘simple’ – things that students don’t understand that I thought they did).

            I suppose it depends on how much ‘technical detail’ is wanted/required – the Study Design is not clear on this. I’d be interested to know how much time and detail other teachers invest in teaching limits, continuity and differentiability. I guess this will all be moot after 2022.

            1. Thanks, John. Of course I agree with your basic point, and you’re right that I undersold y = |x| as an important example. That function should definitely be in the syllabus, for its own sake, and as the easiest example of differentiability ≠ continuity. But when school kids are learning the basic idea and techniques of differentiation, I don’t want them to get too worried by or tangled in the conceptual difficulties.

              In any case, what is happening now is insane. Ask pretty much any VCE student or teacher what it means for a function to be differentiable, and the answer you get is absurd. Maybe not exactly wrong, but absurd.

              1. I agree. And it’s even worse, since limits, continuity, differentiability etc. are \displaystyle meant to be taught in Methods Unit 2. This stuff should then be \displaystyle revision in Methods Unit 3, with a couple of ‘fringe’ (*) examples added (or re-visited) to round things out.

                * That is, counter-examples to some common ‘intuitive’ (to students) but wrong beliefs.

      2. Hi Marty
        I gather your site is primarily aimed at (i) exposing errors in maths textbooks, exams and exam reports, and (ii) the decline of competence in teaching, syllabus design and examining generally. I hold the hope that pressure from professional associations, university mathematicians and experienced teachers will one day turn this around. Possibly, all it would take would take would be one student who sues the examiners for an egregious question, or one not in the syllabus.

        So it would be nice to have a site that takes an error as a starting point and then discusses the best way to deal with the topic, fringe examples and all. Somewhere to present positive suggestions. Yes, you are already insanely busy; your site is swamped by the regular discovery of errors and poor exposition, so you should maintain your focus. I did promise to set up something along the lines suggested but am currently frying bigger fish. Sorry.

        1. Hi Tom.

          “I hold the hope that pressure from professional associations, university mathematicians and experienced teachers will one day turn this around.”

          Hope springs eternal.

          i) Professional associations appear to have unhealthily close and cosy relationships with VCAA, so don’t hold your breath. See https://mathematicalcrap.com/2020/02/09/the-troubling-cosiness-of-the-vcaa-and-the-mav/

          ii) Very few university mathematicians care. I am aware of exactly TWO who care enough to try and discuss these sorts of things with VCAA. (And the second may well exclude themselves from the category of university mathematician). There may be more, but I doubt it.

          iii) “Experienced teachers” … Aha ha ha ha …. You’re killin’ me, Tom. Teachers don’t care squat. Yes, they complain among themselves (at least those who have a clue that there actually IS something to complain about) but I strongly doubt there is even a single teacher that cares enough to actually get involved.

          As for students … Yes, I’d love to see a student sue VCAA for “an egregious question, or one not in the syllabus.” However, having said this, the past is the past. VCAA has a new Maths Manager and he deserves the opportunity to turn things around – at least the things that are within his control and influence. No easy task. This would start with the 2022 mathematics exams and exam Reports.

        2. Thanks, tom. A lot to which I should reply, and your comment has encouraged me to move a couple intended posts to my (insanely long) shortlist. Without preempting those posts, a couple remarks:

          1) What is the purpose of blog? In some sense to just give me a place to howl, so I don’t go crazy (er). But what am I howling about? On the surface it’s about errors and idiocies, of exams and textbooks and whatnot. But, more fundamentally, it is about institutional and institutionalised corruption, and the loss of culture.

          I don’t point out VCE exam errors because they are bad errors, although they are, and that is sufficient and good reason. I point out these errors, en masse, to make the point that VCAA is psychopathically unconcerned for standards or truth. And, that pretty much everyone, teachers and students, idiotically steps along.

          2) Should I be making “positive suggestions”? Well: (a) It’s not my job; (b) I do; (c) a few others others do, and it makes no tangible difference.

          On (a), I am a critic. Who the Hell else is doing it? A very few, and not nearly loudly enough. I have my job, and I do not feel at all guilty about not being a creator. Also, I am not the best person to create such material. I could not, by myself, write a decent school textbook.

          On (b), and not contradicting (a), I do try to offer “positive suggestions”, or at least coherent truth. The WitCHes, for example, are there not only to ridicule high-profile crap, but to include updates which indicate how best to think clearly and properly about the topic. And, yeah, I’m a million miles behind on the updates, but that’s because I sweat so much on getting those updates as clear and correct as possible.

          Could I do entirely separate “positive” posts, or a whole website? Yes and yes, and I do and I do. Yes, I could do more and have explicit plans to do more. But, as you note, I am busy. And it’s not top priority because of (c).

          On (c), there are people like David Treeby and Anthony Harradine doing absolutely brilliant work, but their work is almost invisible. As far as I am aware, these guys get absolutely no support from AAMT or AAS or AMSI or AUTSMS or Cambridge or AMT or AMSI or MERGA or MAV. Or anyone. All the while Cambridge churns out their updated shit, and AAS corrupts themselves to support ReSolve shit, and AMT plays kissy kissy with ACARA’s enquiry halfwits. It is obscene, and it raises the question of why anyone should bother.

          If Michael “Shakespeare” Evans or AMT’s grandstanding twats would get off their fat asses and support David Treeby, we would have a brilliant textbook. If AAMT’s and AAS’s idiotic leaders would get off their fat asses then Harradine’s Numerical Acumen and MathsCraft would be properly funded and would be everywhere. None of this will happen. And it won’t happen because, fundamentally, the self-aggrandising half-brains with the power to make change care about themselves and their stature way, way more than they care about doing genuine good.

          I’ll leave the good and great work to Treeby and Harradine. My job is to hammer the scum culture that is impeding them. I am proud of my work.

  9. I don’t think the following statements at the end of the introduction to the examples have been properly bashed:
    “Some piecewise-defined functions are differentiable everywhere. The smoothness of the ‘joins’ determines whether this is the case”.

    a) What is meant by “smoothness”?
    b) If, by “smoothness”, Cambridge means that the derivative is continuous at the ‘joins’, then the statement is false. The standard example given by Marty (\displaystyle f(x) = x^2 \sin \left( \frac{1}{x}\right) for x≠ 0 and f(0) = 0) is the counter-example.

    It’s just an awful terrible section that requires significant re-writing. But we’ll (probably) never know if a re-write would have happened. Because a re-write is unnecessary for the new edition. Because the new Study Design has removed differentiation from first principles from VCE mathematics. In throwing out the bath water, all the babies in the bath (such as the formal definition of differentiability at a point) have also been thrown out.

    So Marty. Re: “Indeed, this is just about the first nonsense whacked out of first year uni students.” I assume the whacking stick will start at nonsense #2 from 2023 …

    1. Thanks, John. You are correct, I should have whacked that statement, although I’ll probably leave it at this stage.

      I don’t see it primarily as different nonsense, but an informal statement of the main nonsense that I whack. It’s again expressing the idea that the differentiability of f at a comes down to (maybe is defined as) the limit of f'(x) as x approaches a. Yeah, there’s more, and the word “smoothness” is wonky jargon. But I think I’ve bashed enough. If anybody at Cambridge reads this post as is and comes to the conclusion that the text is fine, then nothing else I might write would help.

  10. Dear Colleagues.

    I came across what I think are better solutions to Cambridge’s Examples 45 and 46 – attached. Two other examples are included. All four examples could be considered “fringe”, but they are useful to me and might be useful to you.

    Please feel free to point out the numerous errors (most of which the author will undoubtedly claim are deliberate).

    Continuity and Differentiability examples

    1. Thanks, John. They’re a nice exposition. The first line of page 2 is presumably not quite what was intended, and the Mathematica at the end was, um, unfortunate. But a good coverage of the concepts and the issues.

      1. Thanks, Marty. I see what you mean. I suppose the first line is a bit ambiguous, and the continuity of the function should be explicitly mentioned (rather than implied from earlier). I think the intent is that if the derivative is continuous at x = a then the function is differentiable at x = a. I’ll make sure the author makes the necessary changes.

        The CAS at the end is unavoidable if the limit is to be calculated (it’s not too hard to – stealthily – use the sandwich theorem to get the limit. But that could be a bridge too far for many).

        1. If a function is continuous at a then of course the function is defined at a. That’s all the first line of page 2 is claiming (in regard to the function f’). And of course the CAS is avoidable. But this is also my point: getting into the details of non-differentiability takes a level of thought and care beyond the standard material.

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