Witch 107: Infer a Pounding

This has a lot in it: it’s more of a coven than a WitCH. We couldn’t see what else to do.

As with our recent PoSWW, this WitCH comes from the Logic and Proof chapter of VicMaths, Nelson’s Specialist Mathematics Year 12 text. This is a new VCE topic, for which the summary from VCAA’s study design (Word, idiots) is,

This summary is also given as the prompt of Nelson’s chapter. Then, the extended excerpt here is Nelson’s introduction to the chapter, together with a few of the associated exercises + answers. (The textbook then continues with its coverage of “conjecture” and so forth.)

Readers who lose sight of land may wish to refer to a solid introduction to argument to get their bearings.

Continue reading “Witch 107: Infer a Pounding”

Integration By Substitution

This is just a straight post, framed around answering the question:

How does one introduce-explain integration by substitution to high school students?

That is the question, but I’ll declare from the outset that I cannot answer it. What I will do is explain as clearly as I can why integration by substitution works in the form(s) in which we use it. It is then up to the teacher to decide how much of this “why” message, if any, is required or helpful for their students. (It is not at all clear to me that delving into the proper “why” of substitution will have much meaning or benefit for more than a few school students.)

The post was motivated by a related request on a recent WitCH. Also, having pondered and hunted through the blog, I notice that frequent commenter SRK made a similar request long ago, and there was a related WitCH. The extensive discussion on those posts may be of interest.

THE BASIC MEANINGS

Just so were all on the same page, the only thing we’re considering on this post are antiderivatives: there is no calculation of areas, no fundamental theorem of calculus. I shall use the term “integration” and integral notation because it is common to do so, but the word and notation properly refer to the summing up of bits, which is not what we’re doing here.

So, the function \boldsymbol{F} is an antiderivative of \boldsymbol{f} if \boldsymbol{F' = f}. We then use the integral sign to represent the general antiderivative:

    \[\boldsymbol{\int f \ = \ \int f(x)\, {\rm d}x \ = \ F + c\,.}\]

(Just as a function may be referred to as f(x) or simply as f, the dx notation in integrals is optional, and I’ll use it or not as seems to be clearer.)

An alternative name for this general antiderivative is indefinite integral. Then, the definite integral indicates for us the evaluation of the antiderivative at the “endpoints”:

    \[\boldsymbol{\int\limits_a^b f = F(b) - F(a)\,.}\]

Again, there is no “integration” here, no computation of areas. It is almost solely definition and notation. The only substantive point is to recognise that any two antiderivatives of \boldsymbol{f} differ by a constant, which is intuitive but takes a proof. Then, this +c, whatever it is, cancels out in the evaluation of the definite integral, implying it doesn’t matter which antiderivative we happened to choose.

STATEMENTS OF INTEGRATION BY SUBSTITUTION

Integration by substitution in indefinite form is standardly presented as,

    \[\boldsymbol{\int f(u)\frac{{\rm d}u}{{\rm d}x}\, {\rm d}x \ = \int f(u)\, {\rm d}u\,.}\]

Here, on the left hand side, u is some (differentiable) function of x. On the right side, there is a double think: we antidifferentiate thinking of u as any old variable, and then, when done, we think of u as the given function of x.

If we write u = g(x) explicitly as a function of x then substitution takes the form,

    \[\boldsymbol{\int f(g(x))g'(x) \, {\rm d}x \ = \int f(u)\, {\rm d}u\,.}\]

And, the definite integral version takes the form,

    \[\boldsymbol{\int\limits_a^b f(g(x))g'(x) \, {\rm d}x \ = \int\limits_{g(a)}^{g(b)} f(u)\, {\rm d}u\,.}\]

Note that the definite integral form requires no double-think: the u on the right hand side is simply a who-cares variable of integration. We can also do without x and u entirely, writing the definite integral equation more simply, more purely and less helpfully as

    \[\boldsymbol{\int\limits_a^b (f \circ g) g' \ = \ \int\limits_{g(a)}^{g(b)} f\,.}\]

Finally, a quick word on the intermediary, dodgy line:

    \[\boldsymbol{{\rm d}u = \frac{{\rm d}u}{{\rm d}x} {\rm d}x = g'(x) {\rm d}x\,.}\]

Whether or not one permits the dodgy line is really just a detail, since it is immediately followed by a non-dodgy line. It is, however, better to permit the dodgy line, because: (a) it works; (b) it helps; (c) it really annoys people who object to it.

JUSTIFICATION OF INTEGRATION BY SUBSTITUTION

At its heart, of course, integration by substitution is simply the chain rule in reverse. The chain rule for the composition \boldsymbol{F \circ g} is,

    \[\boldsymbol{(F \circ g)'(x) = F'(g(x)) g'(x)\,.}\]

The chain rule can then be written in antidifferentiation form as,

    \[\boldsymbol{\int F'(g(x)) g'(x)\, {\rm d}x \ = \ F(g(x)) + c\,.}\]

Or, with u = g(x), we can write the anti-chain rule as,

    \[\boldsymbol{\int F'(u) \frac{{\rm d}u}{{\rm d}x}\, {\rm d}x \ = \ F(u) + c\,.}\]

But now, given an integral of the form \boldsymbol{\int f(u) \frac{{\rm d}u}{{\rm d}x}}, it is easy to apply the anti-chain rule. All we need is to give a name to the antiderivative of f.

So, let’s write F for the (an) antiderivative of f: that is, F’ = f. Then, by the anti-chain rule,

    \[\boldsymbol{\int f(u)\frac{{\rm d}u}{{\rm d}x} \, {\rm d}x \ = \ \int F'(u)\frac{{\rm d}u}{{\rm d}x} \, {\rm d}x \ = \ F(u) + c \,.}\]

But also, just thinking of F as a straight antiderivative of f, we have,

    \[\boldsymbol{\int f(u) \, {\rm d}u \ =  \ F(u) + c \,.}\]

Combining the two lines, and keeping in mind we think of u = g(x) after antidifferentiating, we have integration by substitution:

    \[\boldsymbol{\int f(u)\frac{{\rm d}u}{{\rm d}x} \, {\rm d}x \ = \int f(u) \, {\rm d}u\,.}\]

The other forms of the formula can be thought of and derived similarly. For example, again setting F’ = f, the definite integral form can be justified as follows:

    \[\boldsymbol{\int\limits_a^b (f \circ g) g' \ = \ \int\limits_a^b (F'\circ g) g' \ = \  \int\limits_a^b (F\circ g)' \ = \ F(g(b)) - F(g(a)) \ = \  \int\limits_{g(a)}^{g(b)} F'\ = \ \int\limits_{g(a)}^{g(b)} f\,.}\]

CONCLUSION

Will this help? Probably not: the introduction of (and then disappearance of) the antiderivative F is not so easy to understand. So, it is not necessarily wrong to take a “looks kinda right” Leibniz shortcut, or to focus upon a specific chain rule or two. But, ideally, teachers should have some sense of why things are true, even if they then decide to not try to convey this sense to their students. And the sense, as best as I can express it, is the above.

Don’t Accept No Subsitutions

As a byproduct of the recent differential equations WitCH, I promised to write something on integration by substitution. I’ve been pondering that, and I will write it, soon. But there’s also a little VCAA story about substitution I’ve been thinking about posting for a long time, and it seems natural to do that first. In order to tell the VCAA story, I’ll also give a little history to frame the story. Because you young ‘un readers of this blog need some history. Continue reading “Don’t Accept No Subsitutions”

Witch 106: Plane Silly

We’re snowed, and we have a couple posts in the works that simply won’t behave. So, we’ll just keep the ball rolling with a WitCH, and have you commenters do the work for now. The following is the introduction to planes (a new VCE topic) in VicMaths, Nelson’s Specialist Mathematics Year 12 text.

Continue reading “Witch 106: Plane Silly”

QCAA Shows Some More Class

Not total class, but a ton more class than we’ve ever witnessed from VCAA.

This is a lot of times to write about one multiple choice question, but it is important. A month or so ago we posted a PoSWW on a 2022 Queensland MCQ exam question, for which the accompanying exam report indicated the wrong answer. In a second post last week, we noted that QCAA had updated their exam report, to indicate the correct answer and including a footnote flagging that the correction had been made. QCAA has now gone further. Continue reading “QCAA Shows Some More Class”

The Cost of VCAA’s Dissembling

Last month we posted a PoSWW on a 2022 Queensland MCQ exam question, for which the accompanying exam report indicated the wrong answer. It is depressingly unclear why QCAA had not been previously alerted to the error, but a couple weeks after our PoSWW appeared, QCAA updated their exam report: in the amended report QCAA indicates the correct answer (p 27), and they also indicate in a footnote that the correction had been made.

For QCAA to have done this was professional and classy. It was also important. The uncorrected report invited, effectively demanded, a mathematical misconception (on inflection points); by correcting the report, QCAA ensured that their exam-report could no longer be relied upon as an authority for this misconception.

In Victoria, it’s different. Continue reading “The Cost of VCAA’s Dissembling”

VCAA’s Greater Literary Offenses

The difficulty of critiquing VCAA mathematics exams is capturing the variety and the frequency and the depth of the flaws, and then summing the overall effect, the fundamentally impoverished approach to mathematics and its testing. Documenting straight out errors is not overly difficult, and even non sequitur questions are manageable: the error or weirdness typically speaks for itself. Capturing the ubiquitous awfulness of the writing, and the intrinsic meaninglessness of many of the questions, however, is harder. Continue reading “VCAA’s Greater Literary Offenses”

In the Realm of the Senseless

When we first met Sandra Milligan, “Enterprise Professor” at the University of Melbourne’s Graduate School of Education, she was ringleading a bunch of school principals in a campaign against the ATAR. The Age‘s Adam Carey gave Milligan and her cronies a free kick article, because of course it’s not the job of an education reporter to question whether their primary source might be a know-nothing ideologue. Now, Milligan is back in the news, partnered with something called Realms of Thinking, with the free kick “exclusive” provided this time by The Educator‘s Brett Henebery. Continue reading “In the Realm of the Senseless”

Witch 99: PseudoPseudoCode

And one more step into Exam 2. To be honest, we couldn’t give a stuff about this garbage topic, other than to note that introducing it into VCE is complete madness. But, clearly some people feel there are specifics to hammer. So, hammer away. (Hammerers may wish to refer to this or this or this or this (Word, idiots), all courtesy of VCAA.)

Continue reading “Witch 99: PseudoPseudoCode”