L’Hopital’s rule.

Two photos from a recent trip up north. Major bonus points for knowing which of New England's many trails this was taken on.

L’Hopital’s rule is really how every student of calculus (and I believe Leibniz, though I cannot find a reference) wishes the quotient rule worked.  Specifically, that

\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}.

Of course, it can’t be that easy.  We also need that f and g are differentiable in a neighborhood of a, that both function approach 0, or they both approach \infty, or they both approach -\infty as x approaches this point a, and finally that the limit on the right hand side exists (though we all recall that if it does not work the first time, we may continue to apply L’Hopital until the limit does exist, which then justifies using L’Hopital in the first place).

I was thinking of this rule in relation to generating interesting examples of limits.  In particular, if we are in a situation where L’Hopital’s applies, then we can apply the rule in two ways:

\lim_{x\to a}\frac{f'(x)}{g'(x)}=\lim_{x \to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{\left(\frac{1}{g(x)}\right)'}{\left(\frac{1}{f(x)}\right)'}.

Proceeding informally (i.e., I’m not going to keep track of hypotheses), the right hand side of this evaluates to
\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f(x)^2}{g(x)^2}\frac{g'(x)}{f'(x)}.

This is all well and good- the right hand side looks appropriately ugly, but now the trick is picking f and g to get interesting limits.  I have worked out two reasonable examples:

1. Choosing f(x) = sin(x)g(x) = x, we get

\lim_{x \to 0} \frac{\sin{x}}{x^2}\tan{x} = 1.

Also, moderate amounts of bonus points for naming (at least) two universities in the northeast with this mascot.

2. Choosing f(x) = e^x-1 and g(x) = \log{x}, and applying (hopefully correctly!) a number of logarithm rules, we can get

\lim_{x \to 1} \frac{(e^x-1)^2}{\log{(x^{\log{(xe^x)}})}} = 0.

What would be interesting is to find an example where it is difficult/impossible to evaluate without recognizing that it was created using this process.  This second example might fit the “difficult” bill, as I would not want to take the derivative of the denominator directly, but factoring, you might recognize it as xe^x (\log{x})^2, and then be able to reverse engineer this process, somehow.

As usual, just a thought I’ve been playing with.


On teaching about teaching

Just went to a fantastic talk by Susan Loepp, who won a Haimo Award for teaching this year.  Was struck by her observation that only a very small percentage of a mathematician’s training goes towards teaching, while a much higher percentage of (many) mathematician’s time will be spent teaching.  In my experience, good teachers have cared, both about the subject, and in making sure to impart information on the subject to the student.  The first is only helped by learning more about the subject (though, I suppose there may be those who turn their nose up at teaching *sniff* calculus), and the second is hard to train.

I still have a soft spot in my heart for the results of microeconomics, in particular those where you seek to align goals with incentives.  I think it is hard to do in this case, though.  It is a great thing to have smart people researching hard problems and moving science along.  To reward such a person for putting aside cutting edge research in favor of making an extra worksheet for abstract algebra seems silly.  Also, while I am glad that awards like the Haimo Award exist to celebrate the best of the best, I can’t imagine many teachers decide that they care about teaching because of a desire for this award.

I am not an administrator by any means, but it doesn’t seem like a bad idea for a department to offer financial incentives for those professors who either

1) Perform above a certain level according to student evaluations (whether numeric or prose)

2) Improve from the previous year, to encourage professors to continue to work at getting better

I’d imagine, but would be interested to hear, that the main objection over such a system is the capricious nature of student evaluations.

Some actual math posts coming soon, as the Joint Meetings are drawing to a close, and I prepare to jet off to Palo Alto for a week long conference on “Mapping Theory in Metric Spaces”.

Generating research, interest

I have taken Stephen Semmes‘s topics course for each of the past six semesters, and think he has a fantastic teaching style, partly because I could never see myself using the same one.  During a typical class, he will make a definition or prove a theorem, and then ask “Does anyone have anything to say about this?”  The path the class takes is then largely controlled by answers to this question.  I am constantly impressed by how many answers to this question (often early in the morning!) Dr. Semmes and the students in the class have.  More specifically, I am interested in the process of formulating such answers.

This is, of course, a huge topic (that is to say, how are scientific ideas/good research questions generated).  Just to put down a few observations of good questions to ask:

Statement of a definition (“this is an X“):

  • An example of the object/phenomena being described (“x is an X”)
  • A characterization of the object/phenomena (“x is an X if and only if x is a Y,Z,…”
  • The object/phenomena’s relationship to previously defined objects (“every object that is X is also Y” or “every object that is Y is also X“)

Statement of a theorem (“If A, then B“):

  • Is the converse true? (“If B, then A“)
  • What is true with weaker hypotheses? (“If a, then b“)
  • What is true with stronger hypotheses? (“If A, then B“)
  • Is the converse true with stronger hypotheses? (“If B and C, then A“)
  • Is there a stereotypical example of the situation described? (“X is A, so X is also B“)
  • Why is this surprising or interesting? (“If almost A, then not B“)