This is an interesting question, especially because the question of units in trigonometric and inverse trigonometric functions is rarely discussed in mathematics textbooks.

If you’re interested, think this problem through on your own. Then check Problem of the Day 86 at my other site, on or after 23 April 2013, for a discussion.]]>

However, the problem with this explanation is that some planes nevertheless manage to fly upside-down! How could these planes fly both right-side up and upside-down? Could the top of the wing and the bottom of the wing both be shorter than the other when depending on the plane’s orientation? This is absurd, and so there must be something wrong with the explanation.

For the correct explanation (briefly, the angle of attack of the wings provides overwhelmingly more lift than any lift produced because of asymmetry in the cross-section of the wing), check out the following references:

http://danielmiessler.com/blog/why-planes-fly-what-they-taught-you-in-school-was-wrong

http://en.wikipedia.org/wiki/Lift_(force)]]>

The main purpose of QED Infinity is to prepare students to make the difficult transition from high-school mathematics to college/university mathematics. Many students need to take a little bit of mathematics, and it’s often scary and difficult. Others need a lot of mathematics. In my teaching career, I’ve been saddened by the number of students who have been unnecessarily stymied by mathematics issues, sometimes having to drop out of their chosen programs, and other times feeling dumb or being stressed out unnecessarily.

At QED Infinity, I intend to help students master as much mathematics as they need to flourish in the college/university program of their choice. There are daily exercises, problems, and thoughts to encourage students to do consistent daily work. There are other resources on the site, and I’ll be adding resources gradually as I produce them. For example, my online textbook will be posted to the site later this year.

Please visit QED Infinity if you’re interested, and please tell others about it.

If you have feedback, or would like to submit a favourite exercise, problem, or thought, please contact me here.

Thanks!

]]>Religion at its best is complementary to science. At its worst, it is stupidly contradictory. It is disheartening that the leadership of the Republican Party in the United States is currently dominated by arrogant, anti-intellectual, religious bigots. The Conservative Party in Canada is also burdened by similar types. David Suzuki laments here. Some quotes from the article follow.

Rick Santorum just seems out of touch on every issue, from rights for women and gays to the environment. He’s referred to climate change as a “hoax” and once said, “We were put on this Earth as creatures of God to have dominion over the Earth, to use it wisely and steward it wisely, but for our benefit not for the Earth’s benefit.”

This amounts to saying that playing with fire is for my fun, not for my house’s fun, and ignoring the fact that if I accidentally burn my house down I will suffer the consequences.

Some of these people put their misguided beliefs above rational thought. Republican senator James Inhofe, one of the more vocal and active climate change deniers in U.S. politics, recently said, “God’s still up there. The arrogance of people to think that we, human beings, would be able to change what He is doing in the climate is to me outrageous.”

What is arrogant is the belief that God will save us no matter what we do. Jumping off a cliff in the belief that God will save you is irresponsible, and so is continuing to destroy our environment without a second thought, expecting that God will protect us from the consequences.

That statement is in keeping with the Cornwall Alliance’s Evangelical Declaration on Global Warming, which has been signed by a range of religious leaders, media people, and even some who work in climate science, such as Roy Spencer, David Legates, and Ross McKitrick.

It says, in part, “We believe Earth and its ecosystems — created by God’s intelligent design and infinite power and sustained by His faithful providence — are robust, resilient, self-regulating, and self-correcting, admirably suited for human flourishing, and displaying His glory. Earth’s climate system is no exception.” It also states that reducing atmospheric carbon dioxide and fossil fuel use will “greatly increase the price of energy and harm economies.”

I certainly agree with part of this statement. I do believe that earth’s ecosystem is robust enough and resilient enough that it will survive no matter what we do. What is at issue is whether human society will survive in its present form, or whether we will be relegated to a few bands of hunter-gatherers again. Contemplate for a moment the magnitude of human suffering that would accompany a collapse of human population and the state of the economy will plunge down the priority list.

And then there was this claim from Arizona Senator Sylvia Allen: “This Earth’s … been here 6,000 years, long before anybody had environmental laws, and somehow it hasn’t been done away with. …”

We all wake up every morning, year after year, until one day we don’t. And that’s the way human society is headed if we don’t get our act together soon. And understanding a bit about science is essential to getting our act together, which underlies how important education, particularly science education, is for our collective survival.

]]>Very strong winds jammed ice against the mouth of the river near Buffalo, damming the flow for almost two full days, starting on 30 March 1848.

You can read more about the strange events of the day here and here.

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In light of current controversies around testing and teacher evaluation, let’s do a little thought experiment. How would Miss Snug have handled this lesson if it were occurring just before a round of standardized testing? Would she not have had to interrupt the children’s speculations and instructed them that actual circumstances in word problems must be completely disregarded, because the point is to arrive at the answer the test designers have in mind? After all, how could test designers anticipate the lines of thought that spontaneously erupted in her classroom? Real life, and real thought, are too complicated to be foreseen – and so need to be put aside at testing time.

We … are concerned about education because our adult citizens need to be flexible thinkers, ready to adapt to the ever-changing circumstances of the global marketplace. But making standardized tests the center of our curriculums tells children the most important thing they need to learn in school is how to arrive at predetermined answers on the tests. …

So in fact the test doesn’t reflect at all what kids should learn in school. What they really need to master is the kind of imaginative, adaptive thinking Miss Snug encourages in the passage from “The Trumpet of the Swan” – skills that cannot be assessed in any way other than actually knowing the children.

This little episode captures what volumes of education research have shown: we are born curious, and the best education models do not proceed on the basis of “what we want them to learn,” as Mr. Bloomberg correctly describes the goal of test-oriented education, but on the assumption that our job is to foster children’s ability ultimately to shape a world different from what we leave to them.

Now I agree that basic skills in mathematics are important. To learn mathematics effectively, you need to have your multiplication facts at your fingertips, and also your trig identities (if you’re learning at a higher level). However, the problem is that our assessment systems are skewed towards what is easy to test, especially what is easy to test using standardized tests. When resources are strained, as they are now, what is easy to assess ends up dominating the entire educational enterprise.

The best way to engage students is to ask them what they wish to learn. What is it that makes their hearts sing? (However, we should begin asking them early in life; if you wait until 12 or 16 years of the standard education system has crushed the spirit of our children, then you might not get a very satisfying answer to what makes their hearts sing.) And then create an environment where they are lovingly supported in pursuing their own goals. This is the only way to nurture creative and critical problem-solvers that we need so much to confront the serious problems facing civilization. Assessing their work can never be done using standardized tests.

In pursuing their own goals, many students will have need for some mathematical technique or other. With 21st century technology, it should be possible to create a system, a repository, of lessons that teach all of the basic techniques. (I’m sure that the internet is already chock full of such material, albeit scattered and of uneven quality.) When students need them, they can be guided and supported in learning what they need. Making students sit through 12 years of mathematics through high school, and then even more afterwards, in case they should need it, is a terrible waste of effort. Students would be better off *doing something* of value.

I cringe every time I think about the excellent teachers out there spending precious class time teaching 500 students how to work the product rule for derivatives and such things. Such things are best learned by doing, with feedback, in a combination of small groups and alone. I would much rather see our excellent lecturers spending precious class time on modelling the kind of critical and creative thinking that can’t easily be transmitted in other ways.

A final note: The inhumane standardized testing that is consuming the education landscape like a plague is infecting teacher evaluation as well. Tim Clifford’s article at the same SchoolBook blog is a powerful argument against this horrible practice.

]]>and

is to integrate by parts twice. If you haven’t seen the standard method, I’ll show you how to do the first one; the second one is similar. Later in the post I’ll show you the neat trick for determining both integrals at once.

It doesn’t much matter whether you let *u* represent the exponential function or the trigonometric function in the first integration, but you have to be consistent in the second integration. (That is, if you let *u* stand for the exponential function in the first integration, then also let *u* stand for the exponential function in the second integration. Alternatively, if you let *u* stand for the trigonometric function in the first integration, then also let *u* stand for the trigonometric function in the second integration.) Otherwise, after two integrations by parts you will end up with 0 = 0, which is true but not helpful.

So, let and , so that and . Then, calling the integral to be determined “*I*,” for reasons that will become clear shortly, we have

In the integral on the right side of the previous equation, integrate by parts again, letting and , so that and . The result is

We seem to be going around in circles, because the integral on the right side of the previous equation is the same as the one we started with. However, if we just replace it by its label *I*, the previous equation is seen to be an algebraic equation that we can solve for *I*. (This is the motivation for introducing the label *I*.) Doing this, we obtain

You can check the result by differentiating it to arrive at the original function. Some people enjoy this method as it seems as if we got something for nothing. We never really “finished” the integration by parts (after two iterations, we were still left with an integral), and yet the final result somehow popped out.

And of course, if you need to determine the other integral, then you have to go through the process once more, integrating by parts twice again. Try it for practice, if you wish; the result is

Now for the trick, which relies on you knowing about complex numbers, including Euler’s formula:

The strategy is to multiply the second of the original integrals by and then add it to the first of the original integrals. It turns out that combining them in this way results in an integral that is quite easy to determine; no integration by parts four times is needed. Then we just separate the final result into a sum of real and imaginary parts; the real part is the result for the first integral and the imaginary part is the result of the second integral.

Let’s see how it works: First multiply by and add it to and combine the integrals into one integral:

(using Euler’s formula)

Now combine the exponential functions on the right side of the previous equation and antidifferentiate:

The final stage is to express the right side of the previous equation as the sum of a real part and an imaginary part. Part of this process is to multiply the numerator and denominator of the fractional factor by the complex conjugate of the denominator; the other part is to separate the exponential functions on the right side of the equation:

Matching real and imaginary parts on both sides of the previous equation gives us the final results:

Conclusion: Does the trick save work? The integration step in the trick method is very easy, so we’re trading four integrations by parts plus some algebra for a simple integration and some algebra with complex numbers. It’s a trade I’d make any day. But of course, to each his own, so try both ways and decide for yourself which way you like better.

If the integrals are a bit more complex, then the savings in the trick method are even greater. For example, you might try using both methods to determine the integrals

and

Integration by parts is even more of a pain, but the trick method is hardly more difficult. If you do try them, you can check your final results against these:

I just did these two integrals with pencil and paper, and the trick method is much faster. Even if you only have to work out one of the integrals (the method is the same, you just ignore one of the final results), I think the trick method is still a time-saver and the probability of making an error is reduced, because you avoid the messy integrations by parts.

]]>Here is an excerpt (the entire article is highly recommended for anyone interested in the issues; further discussion is here and here):

No high-performing nation in the world evaluates teachers by the test scores of their students; and no state or district in this nation has a successful program of this kind. The State of Tennessee and the city of Dallas have been using some type of test-score based teacher evaluation for twenty years but are not known as educational models. Across the nation, in response to the prompting of Race to the Top, states are struggling to evaluate their teachers by student test scores, but none has figured it out.

All such schemes rely on standardized tests as the ultimate measure of education. This is madness. The tests have some value in measuring basic skills and rote learning, but their overuse distorts education. No standardized test can accurately measure the quality of education. Students can be coached to guess the right answer, but learning this skill does not equate to acquiring facility in complex reasoning and analysis. It is possible to have higher test scores and worse education. The scores tell us nothing about how well students can think, how deeply they understand history or science or literature or philosophy, or how much they love to paint or dance or sing, or how well prepared they are to cast their votes carefully or to be wise jurors.

Of course, teachers should be evaluated. They should be evaluated by experienced principals and peers. No incompetent teacher should be allowed to remain in the classroom. Those who can’t teach and can’t improve should be fired. But the current frenzy of blaming teachers for low scores smacks of a witch-hunt, the search for a scapegoat, someone to blame for a faltering economy, for the growing levels of poverty, for widening income inequality.

**Update**: Further interesting links are provided by Japheth Wood here.

09/02/12

Yesterday was 20 February 2012, and my question is: had the best-before date passed yet?

In Europe, it is typical to write dates in the form “day/month/year,” so if this were Europe, then the best-before date would have passed, and I perhaps ought not to eat the bar. (Although the best-before date would not have passed by too much, and usually the time at which the food is spoiled is beyond the best-before date, so perhaps eating it would be OK.)

In North America, the year is usually last, but the day and month could be in either order. But there are six possible permutations of the six data, so although only two of the permutations are in common use (“year/month/day” is also used, but less commonly), there is still room for confusion. For something as important as best-before dates, shouldn’t we have an unambiguous usage convention? Clarity is courtesy, but when it comes to best-before dates, clarity could also save someone from illness.

But it is difficult to reach consensus on conventions, particularly in this case with products being produced around the world. So here is my proposal for ensuring that best-before dates will never be misinterpreted:

1. Always write all four digits of the year.

2. Always use a day that is at least 13.

Then it doesn’t matter which permutation is used. The dates 13/02/2012, 02/13/2012, 2012/13/02, etc., can only be interpreted in one way. And if a packaged food is good for a year or more, then altering the day by a few to ensure that it is at least 13 will cause no problems. In my situation, this would have saved me wondering whether the best-before date was 9 February or 2 September, or even some time in 2009 (2002 seemed unlikely).

If the food product is only good for a few days after packaging (which is the case for meats, bread, and some other foods), then it is unlikely that it will travel far, and the word for the month could be safely used.

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