College Math Teaching

February 19, 2012

Divergent Improper Integrals: change of variables to an unbounded integrand.

Filed under: calculus, change of variable, improper integrals, integrals, integration by substitution, pedagogy — collegemathteaching @ 10:41 pm

This post was motivated by a student question: my student wanted help with the following problem:

\int^{\infty}_{1} \frac{x^2}{\sqrt{x^3 +1}} dx
Of course the idea is to do a substitution: u = x^3 + 1 which transforms the integral into \frac{1}{3} \int^{\infty}_{2} \frac{1}{\sqrt{u}} du which diverges. So far, so good. But then I told him one of my calculus tips: “it is often a good idea to try to guess the answer ahead of time” and then pointed out that for large values of x, \frac{x^2}{\sqrt{x^3 +1}} \approx \frac{x^2}{\sqrt{x^3}} = \sqrt{x} and of course \int^{\infty}_1 \sqrt{x} dx diverges because the integrand does not go to zero (in fact, is unbounded!) as x tends to infinity.

Then I realized that a change of variables had taken an unbounded function to a bounded one…though one which did not produce a convergent improper integral.

That lead to the natural question: if one has an integrand which is positive but monotonically decreasing to zero on [1, \infty ) , is there a change of variables which will change the integrand to either an unbounded function on [1, \infty ) or at least one that does not decrease to zero?

I admit that I have not answered this question yet, nor have I looked it up. But I can answer this question for a certain class of functions:

Theorem

Given \int^{\infty}_1 \frac{1}{x^r} dx
If 0 < r < 1 , let k > \frac{1}{1-r} . Then the change of variable u = x^{\frac{1}{k}} transforms \int^{\infty}_{1} \frac{1}{x^r} dx to k \int^{\infty}_1 u^{k(1-r) -1} du and of course u^{k(1-r) -1} is unbounded on [1, \infty) .

If 1 < r let k < \frac{1}{1-r} < 0 . Then \int^{\infty}_1 \frac{1}{x^r} dx is transformed into |k|\int^{1}_{0} u^{-1+k(1-r)} du which is an integral of a bounded function over a bounded region.

In short, one class of functions whose improper integral diverges can be transformed to functions that tend to infinity and the class of functions whose integrals converge can be transformed into functions which are bounded over a bounded interval.

Here is such an example: We show the equivalent integrals \int^{1.5}_{1} 3x^{\frac{1}{2}} dx and \int^{(1.5)^3}_1 \frac{1}{\sqrt{u}} du . The transformation is accomplished by using u =x^3 . Note how the transformation stretches the interval of integration to account for the function “shrinkage”.

On the other hand, using u^{-2}=x transforms \int^{\infty}_1 \frac{1}{x^2} dx into 2\int^{1}_{0} u du

February 16, 2012

The “equals” sign: identities, equations to be solved and all that…

Here is the sort of thing that got me thinking about this topic: a colleague had a student complain about how one of her quiz problems was scored. The problem stated: “show that \sqrt{2} + \sqrt{3} \neq \sqrt{5} “. She was offended that her saying “\sqrt{x} + \sqrt{y} \neq \sqrt{x+y} ” wasn’t enough to receive credit and would NOT take his word for it. In fact, she took this to the student ombudsman!!!

But that raised the question: “what do we mean when we tell our students “\sqrt{x} + \sqrt{y} \neq \sqrt{x+y} “?

Of course, there are some central issues here. The first issues is that our “sure of herself” student thought that “\sqrt{x} + \sqrt{y} \neq \sqrt{x+y} ” meant that this relation is NEVER true for any choice of x, y , which of course, is false (e. g. let y = 0 and x \ge 0 .) In fact, \sqrt{x} + \sqrt{y} \neq \sqrt{x+y} is the logical negation of the statement \sqrt{x} + \sqrt{y} = \sqrt{x+y} ; the latter means that “this statement is true for ALL x, y and its negation means “there is at least one choice of x, y for which the statement is not true. “Equal” and “not equal” are not symmetric states when it comes identities, which can be thought of as elements in the vector space of functions.

So, \sqrt{x} + \sqrt{y} \neq \sqrt{x+y} means that \sqrt{x} + \sqrt{y} and \sqrt{x+y} are not equal in function space, though they might evaluate to the same number for certain choices in the domain.

So, what is the big deal?

Well, what about equations such as x^2 + 3x + 2 = 0 or y^{\prime \prime} + y = 0 ?
These are NOT equalities in the space of functions; the first means “what values in the domain does f^{-1}(0) take given f(x)=x^2 + 3x + 2 and the second asks one to find the inverse image of 0 for the operator D^2+1 where the domain is the set of all, say, twice differentiable functions.

But, but…would the average undergraduate student understand ANY of this? My experience tells me “no”; hence I intentionally allow for this vagueness and only address it as I need to.

February 12, 2012

Mathematical Research at “Teaching Institutions”

Filed under: academia, editorial, mathematics education, pedagogy, research — collegemathteaching @ 1:03 am

I read the blogs of a few academics and I had to wince a bit when I read that the push to make professors teach a higher load was happening even in Canada.

Now as a professor, I don’t have a “dog in this hunt” so to speak because I teach at a 10-12 hour per semester load institution and I am grateful to have the job. But as a citizen and as a mathematician, I think that the research intensive universities have a place and that it would be a colossal mistake to turn them into “teaching institutions”. We need places that generate knowledge, and one isn’t going to be able to generate top-level mathematical knowledge (say, at the level that gets published in the Annals of Mathematics or in Inventiones mathematicae if one isn’t devoted to keeping current and active on a full time basis.

Now my institution does have a research requirement for tenure and promotion and I’ve developed a modest publication record and am still working to add to my publication list. And yes, many of my colleagues have published far more than I have and I salute them for it.

But let’s face facts: mathematical research at institutions like ours consists of
1. tackling spin-off problems from areas opened some time ago
2. working with another medium level scholar in another discipline to solve some of the mathematical problems related to that discipline
3. working on “nice to know” things that one discovers (or rediscovers) when preparing for class.

As a colleague at a similar institution said: “I dabble here and there; there just isn’t time to learn something that takes 5 years to master”.

The fact is that teaching classes, doing service and meeting with perplexed students soaks up the vast majority of one’s time. Add to that the fact that one’s “upper division” class might be a class that one has never taught or one that you last taught a decade ago; in reality one has to relearn much of the material that has faded from memory.

Then, there is the basic brain rot that occurs from mostly dealing with trying to explain to students that \int e^x dx \neq \frac{e^{x+1}}{x+1} + C . One also has the baby-sitting of getting the poorer performing students to not text in class and to explain to them why their course average of 65 doesn’t entitle them to a B (or even a C) and to do so in a way that doesn’t have their parents complain to the department chair.

Then there is the brain atrophy that comes from not reading anything hard for months at a time; then when you try to read something hard you often only have a few minutes of uninterrupted time to do so.

Hence the research that you can do, while it can require cleverness, really can’t require that you master the new sophisticated techniques.

On a side note: this is part of the purpose of my writing this blog; it encourages me to learn stuff that is “new to me” or “what I should have learned a long time ago.” After this next round of exams, I hope to talk about quartic splines that produce increasing, convex curves.

February 7, 2012

Forgotten Basic Algebra: or why we shouldn’t rely on the “conjugate trick”

Filed under: basic algebra, calculus, derivatives, elementary mathematics, how to learn calculus, pedagogy — collegemathteaching @ 7:01 pm

I’ll admit that, after 20 years of teaching at the university level, I sometimes get lazy. But…as I age, I must resist that temptation even though at times I find myself muttering “I don’t have 30 extra f*cking minutes to figure out how to do this…”

But often if I stick with it, it doesn’t take 30 “f*cking” minutes. 🙂

Here is an example: I was trying to remember how to calculate lim_{z \rightarrow w} \frac{z^{1/3} - w^{1/3}}{z - w} and was trying to remember instead of think. I looked at an old calculus book…no avail…then I was shamed into thinking. About 2-3 minutes later it struck me:
“you know how to simplify \frac{u - v}{u^3 - v^3} don’t you?”

Problem solved…shame WIN.

of course things like lim_{z \rightarrow w} \frac{z^{7/8} - w^{7/8}}{z - w} are easily converted to things like \frac{u^7 - v^7}{u^8 - v^8} , etc.

This leads to another point. Often when we teach lim_{h \rightarrow 0} \frac{\sqrt{x + h} - \sqrt{x}}{h} we use the “conjugate trick” which only works for square roots. The above method works for the other fractional powers.

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