Wikipedia works (somehow)

February 14, 2009 at 5:14 pm | In Internet, Mathematics | Leave a Comment

Start from this 2006 revision of an article and step forward until you get to the current day. At no point did anybody know what was going on but somehow it went from a joke to a real article.

Australian cricket

February 7, 2009 at 3:07 pm | In Cricket | Leave a Comment

The team sucks at the moment. But I don’t really have any answers aside from picking Katich in the one day team (surely in current form he’s more likely to score a run a ball century than any of Warner, Hussey, Hussey, White, Ferguson, Voges…)

I agree entirely with the test team to tour South Africa. It’s not a particularly good team but it’s the best they could have done.

What we need are these people who have dominated state cricket for years to come in and dominate international cricket: D. Hussey, White and Haddin just haven’t been the world beaters that M. Hussey and Clark were immediately upon coming into the team.

Alternatively Ponting, Clarke, Johnson etc need to start winning games for us with amazing personal performances.

Alternatively, the Aussies just need to play as well as they did when they lost those games to SA. New Zealand probably can’t chase 270 but we haven’t asked them to yet.

p.s. lol at England

Teaching Calculus

February 7, 2009 at 2:54 pm | In Mathematics, Uni | 1 Comment

I’m teaching Calc I this semester. This course (after a review of precalc, limits and continuity) covers derivatives and their applications and the beginning of integrals and antiderivatives. The University of Arizona was at the forefront of calculus reform so we’re using a reform textbook. Hughes-Hallett and McCallum are both in the department so if I want to complain about the text I know where to go.

I’m finding it difficult to decide on the level of rigour that I want to provide. I didn’t even write down the epsilon-delta definition of limit (my exact words were “you can read the definition of limit in the text but I wouldn’t recommend it”). But the other day we calculated the derivative of x^2 and x^3 using the definition of derivative and then said that `obviously’ d/dx(x^n) = nx^{n-1}…

Then I said: “in fact this is true for all real n”. After that we did some examples and class ended. I’m not too happy with this way of teaching – there is no reason for the students to believe me.

I was thinking about how to prove this: for positive n it’s true by the binomial theorem (I don’t know if my students know the binomial theorem though), for negative n you can prove it by the product/quotient rule, for fractions you can prove it with implicit differentiation, then it follows for real exponents by continuity of something (and the density of the rationals in the reals).

But a better proof (for positive x anyway) is by the definition/fact that x^n = e^{n \ln(x)}. Then d/dx(x^n) = e^{n \ln(x)} d/dx(n \ln(x)) = x^n n/x = n x^{n-1}.

But then how do you define e and ln anyway? The `best’ way is to wait until you have integrals and define \ln(x) = \int_1^x 1/t \, dt and \exp(x) as the inverse of this function. But that’s not a reasonable idea for people that haven’t seen calculus before. Also I’m not so convinced that the logical way to teach things is better than the way people already know from high school or whatever – last semester I taught trigonometry using the unit circle and it just confused everybody.

So: how much rigour should be in a calculus course for the general population?

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