College Math Teaching

January 22, 2014

Mean Value Theorem for integrals and it’s use in Taylor Polynomial approximations

Filed under: Uncategorized — collegemathteaching @ 3:12 am

First, what is the Mean Value Theorem for integrals? There are two common versions (calculus level versions; one can make these more general):

Version one: if $f$ is continuous over $[a,b]$ then there is some $c \in (a,b)$ such that $f(c)(b-a) = \int^b_a f(x) dx$. The proof is pretty easy: Let $F(t) = \int^t_af(x)dx$ then the Fundamental Theorem of Calculus says that $F$ is continuous on $[a,b]$ and differentiable on $(a,b)$ so the Mean Value Theorem applies: there is some $c \in (a,b)$ where $F'(c)=f(x) =\frac{F(b)-F(a)}{b-a} = \frac{1}{b-a} \int^b_a f(x)dx$ and the result follows.

This version has a nice geometric interpretation. If one interprets the definite integral as an area bounded by the graph of the function, the $x$ axis and the lines $x=a, x =b$ then the Mean Value Theorem says that there is a rectangle whose base is $b-a$ and whose height is $f(c)$ whose area is equal to the integral.

But there is a second version:

Version Two: if $f$ is continuous over $[a,b]$ and $g$ is positive and continuous (actually, integrable and positive is enough) on $[a,b]$ then there is some $c \in (a,b)$ where $f(c)\int^b_a g(x) dx = \int^b_a f(x) g(x) dx$.

The proof of this result is not much harder than the first one. Since $f$ is continuous over $[a,b]$, $f$ attains both a maximum and a minimum value on the interval; say the maximum is $f(x_M)$ is the maximum; $f(x_m)$ is the minimum. So we have for all $x \in [a,b], f(x_m)g(x) \le f(x)g(x) \le f(x_M)g(x)$ Now integrate: $f(x_m)\int^b_ag(x)dx \le \int^b_af(x)g(x)dx \le f(x_M)\int^b_ag(x)dx$ Now divide through by $\int^b_a g(x) dx$ and note that the integral is positive. So $f(x_m) \le \frac{\int^b_af(x)g(x)dx}{\int^b_a g(x) dx} \le f(x_M)$. Now because $f$ is continuous, every value between $f(x_m), f(x_M)$ is attained by $f$ so there is at least one $c$ such that $f(c) = \frac{\int^b_af(x)g(x)dx}{\int^b_a g(x) dx}$ and the result follows.

So, what does this have to do with Taylor polynomials?

In a previous post we went over how to use integration by parts to obtain a Taylor polynomial for a function that has a sufficient number of derivatives. Here is the plan:

for our purposes, let $f$ have as many derivatives at $0$ as desired.

Now compute $\int^x_0f'(t)dt = f(x)-f(0)$. Now compute by using integration by parts: $u =f'(t), dv= dt, du =f"(t), v = t-x$ (if the assignment of $v$ seems strange, remember we can use ANY anti-derivative of $dv$.)

So we have $f'(t)(t-x)|^x_0 - \int^x_0 f"(t)(t-x)dt = f(x)-f(0)$ so by substitution we obtain $f(0)+f'(0)x - \int^x_0 f"(t)(t-x) dt =f(x)$

I’ll repeat the process so you can see what happens to the sign in front of the integral: we use parts again: $u =f"(t), dv = (t-x), du = f^{(3)}(t), v = \frac{1}{2}(t-x)^2$ and so we obtain (for the integral) $\frac{1}{2}f"(t)(t-x)^2|^x_0 -\frac{1}{2}\int^x_0 f^{(3)}(t)(t-x)^2dt =-\frac{1}{2}f"(0)x^2-\frac{1}{2}\int^x_0 f^{(3)}(t)(t-x)^2dt$ Now substitute this for the integral in the order 1 expression, and remember the negative sign: we obtain $f(0)+f'(0)x \frac{1}{2}f"(0)x^2+\frac{1}{2}\int^x_0 f^{(3)}(t)(t-x)^2dt =f(x)$

So we can proceed inductively; the important thing here is the remainder term after $n$ steps is $\pm \frac{1}{n!} \int^x_0 f^{(n+1)}(t)(t-x)^n dt$

A word on the sign: the negative occurs when $n$ is odd and positive when $n$ is even. So one can remove this ambiguity by replacing $(t-x)$ by $(x-t)$ and so the remainder formula becomes: $\frac{1}{n!} \int^x_0 f^{(n+1)}(t)(x-t)^n dt$.

Now this is still not the usual Lagrange or Cauchy remainder formula that many texts give. But we can get that from our Mean Value Theorem for integrals. Since the integrand is continuous over the interval $[0,x]$ the First Mean Value Theorem for Integrals says that there exists a $\zeta \in (0,x)$ where $\frac{1}{n!}f^{(n+1)}(\zeta)x(x-\zeta)^n$ is the remainder; that is the Cauchy form.

The Lagrange form comes from the Second Mean Value Theorem for Integrals: we know that there is a $\omega \in (0,x)$ where $\frac{1}{n!} \int^x_0 f^{(n+1)}(t)(x-t)^n dt = f^{(n+1)}(\omega)\frac{1}{n!}\int^x_0(x-t)^n dt = \frac{1}{(n+1)!}f^{(n+1)}(\omega)x^{(n+1)}$

That is the Lagrange version of the error term that one usually sees.

January 20, 2014

A bit more prior to admin BS

One thing that surprised me about the professor’s job (at a non-research intensive school; we have a modest but real research requirement, but mostly we teach): I never knew how much time I’d spend doing tasks that have nothing to do with teaching and scholarship. Groan….how much of this do I tell our applicants that arrive on campus to interview? đź™‚

But there is something mathematical that I want to talk about; it is a follow up to this post. It has to do with what string theorist tell us: $\sum^{\infty}_{k = 1} k = -\frac{1}{12}$. Needless to say, they are using a non-standard definition of “value of a series”.

Where I think the problem is: when we hear “series” we think of something related to the usual process of addition. Clearly, this non-standard assignment doesn’t related to addition in the way we usually think about it.

So, it might make more sense to think of a “generalized series” as a map from the set of sequences of real numbers (or: the infinite dimensional real vector space) to the real numbers; the usual “limit of partial sums” definition has some nice properties with respect to sequence addition, scalar multiplication and with respect to a “shift operation” and addition, provided we restrict ourselves to a suitable collection of sequences (say, those whose traditional sum of components are absolutely convergent).

So, this “non-standard sum” can be thought of as a map $f:V \rightarrow R^1$ where $f(\{1, 2, 3, 4, 5,....\}) \rightarrow -\frac{1}{12}$. That is a bit less offensive than calling it a “sum”. đź™‚

January 18, 2014

Fun with divergent series (and uses: e. g. string theory)

One “fun” math book is Knopp’s book Theory and Application of Infinite Series. I highly recommend it to anyone who frequently teaches calculus, or to talented, motivated calculus students.

One of the more interesting chapters in the book is on “divergent series”. If that sounds boring consider the following:

we all know that $\sum^{\infty}_{n=0} x^n = \frac{1}{1-x}$ when $|x| < 1$ and diverges elsewhere, PROVIDED one uses the “sequence of partial sums” definition of covergence of sums. But, as Knopp points out, there are other definitions of convergence which leaves all the convergent (by the usual definition) series convergent (to the same value) but also allows one to declare a larger set of series to be convergent.

Consider $1 - 1 + 1 -1 + 1.......$

of course this is a divergent geometric series by the usual definition. But note that if one uses the geometric series formula:

$\sum^{\infty}_{n=0} x^n = \frac{1}{1-x}$ and substitutes $x = -1$ which IS in the domain of the right hand side (but NOT in the interval of convergence in the left hand side) one obtains $1 -1 +1 -1 + 1.... = \frac{1}{2}$.

Now this is nonsense unless we use a different definition of sum convergence, such as the Cesaro summation: if $s_k$ is the usual “partial sum of the first $k$ terms: $s_k = \sum^{n=k}_{n =0}a_n$ then one declares the Cesaro sum of the series to be $lim_{m \rightarrow \infty} \frac{1}{m}\sum^{m}_{k=1}s_k$ provided this limit exists (this is the arithmetic average of the partial sums).

(see here)

So for our $1 -1 + 1 -1 ....$ we easily see that $s_{2k+1} = 0, s_{2k} = 1$ so for $m$ even we see $\frac{1}{m}\sum^{m}_{k=1}s_k = \frac{\frac{m}{2}}{m} = \frac{1}{2}$ and for $m$ odd we get $\frac{\frac{m-1}{2}}{m}$ which tends to $\frac{1}{2}$ as $m$ tends to infinity.

Now, we have this weird type of assignment.

But that won’t help with $\sum^{\infty}_{k = 1} k = 1 + 2 + 3 + 4 + 5.....$. But weirdly enough, string theorists find a way to assign this particular series a number! In fact, the number that they assign to this makes no sense at all: $-\frac{1}{12}$.

What the heck? Well, one way this is done is explained here:

Consider $\sum^{\infty}_{k=0}x^k = \frac{1}{1-x}$ Now differentiate term by term to get $1 +2x + 3x^2+4x^3 .... = \frac{1}{(1-x)^2}$ and now multiply both sides by $x$ to obtain $x + 2x^2 + 3x^3 + .... = \frac{x}{(1-x)^2}$ This has a pole of order 2 at $x = 1$. But now substitute $x = e^h$ and calculate the Laurent series about $h = 0$; the 0 order term turns out to be $\frac{1}{12}$. Yes, this has applications in string theory!

Now of course, if one uses the usual definitions of convergence, I played fast and loose with the usual intervals of convergence and when I could differentiate term by term. This theory is NOT the usual calculus theory.

Now if you want to see some “fun nonsense” applied to this (spot how many “errors” are made….it is a nice exercise):

What is going on: when one sums a series, one is really “assigning a value” to an object; think of this as a type of morphism of the set of series to the set of numbers. The usual definition of “sum of a series” is an especially nice morphism as it allows, WITH PRECAUTIONS, some nice algebraic operations in the domain (the set of series) to be carried over into the range. I say “with precautions” because of things like the following:

1. If one is talking about series of numbers, then one must have an absolutely convergent series for derangements of a given series to be assigned the same number. Example: it is well known that a conditionally convergent alternating series can be arranged to converge to any value of choice.

2. If one is talking about a series of functions (say, power series where one sums things like $x^n$) one has to be in OPEN interval of absolute convergence to justify term by term differentiation and integration; then of course a series is assigned a function rather than a number.

So when one tries to go with a different notion of convergence, one must be extra cautious as to which operations in the domain space carry through under the “assignment morphism” and what the “equivalence classes” of a given series are (e. g. can a series be deranged and keep the same sum?)

January 17, 2014

The New Semester: Spring 2014

Filed under: academia, advanced mathematics, algebraic curves, analysis, knot theory, research — Tags: — collegemathteaching @ 11:34 pm

The new semester is almost upon us here; our classes start up next Wednesday. I am ashamed to report that I am delinquent with a referee’s report; I’ll work some weekends to catch up.

Of course, we come in with “new ideas” which include evaluating things like this:

â€śMost people like to talk about how in college we need to develop critical thinking skillsâ€ť, said Mike Starbird near the beginning of this talk yesterday, â€śbut really, who wants to hear â€śOh, yeah, Soandso, heâ€™s really criticalâ€ť?â€ť. This, Starbird says, is what led him and coauthor Ed Burger to coin the phrase â€śeffective thinkingâ€ť. Because that is something one would like to be called.

The talk was affected by some technical difficulties, which meant that the slides Starbird had prepared with mathematical examples were unavailable to us. But, following his own advice, Starbird rose to the challenge and gave a talk, without slides, and using the overhead projector for the examples he needed to draw. As usual, his delivery and demeanor were both charming and informative (I am lucky enough to have both taken a class from him and taught a class with him), and the message on what strategies to follow for effective thinking, and to get our own students to be involved in effective thinking, was received loud and clear.

The 5 elements of effective thinking, as Starbird and Burger describe in their eponymous book, are the following: understand simple things deeply, fail to succeed, raise questions, follow the flow of ideas, and everything changes. The first couple he described by using examples of mathematics in which each strategy led to deep insights about a problem. For â€śunderstanding simple things deeplyâ€ť, Starbird showed us a new, purely geometric, way of proving that the derivative of sin(x) is cos(x).

Note: Professor Starbird was one of my professors at the University of Texas. I took a summer class from him which involved the class going over his technical paper called A diagram oriented proof of Dehn’s Lemma

(Roughly speaking: Dehn’s Lemma says that if a polygonal closed curve bounds an immersed polygonal disk whose self intersections lie in the interior of the disk, then that given curve also bounds an embedded polygonal disk (e. g. one without self intersections). Dehn’s Lemma is especially interesting because the first widely accepted “proof” proved to be false; it wasn’t rigorously proved true into years later.)

Ed Burger was a Ph. D. classmate of mine; I consider him a friend. He has won all sorts of awards and is now President of Southwestern University.

I have to chuckle at the goals; at my institution we mostly teach calculus, which is mostly for engineers and scientists. The engineering faculty would blow a gasket if we spent the necessary time for finding deeper proofs that the derivative of sine is cosine.

And yes, we are terribly busy with this or that: on the plate, right off of the bat, is a meeting on “reforming” (read: watering down) our general education program, a visit day, among other things (such as search).

It has gotten to the point to where things like a “department lunch” went from being something fun to do to being “yet another frigging obligation”.

I’ll have to find a way to keep my creative energy up.

So, what I’d like to “think about”:

1. I have a couple of papers out about limits of functions of two variables. Roughly speaking: I gave new proofs of the following:

1. A real valued function of two variables can be continuous when evaluated over all real analytic curves going through the origin and yet still fail to be continuous. (see here)
2. If a real valued function of two variables is continuous when evaluated over all convex $C^1$ functions running through a point, then that function is continuous at that point. This result does NOT extend to $C^2$.
(see here)

So, what is so special about $C^1$? Is this really a theorem about curves through a planar set of points with a limit point? Or is more going on….can this result extend to results about differentiablity?

Then there is something that sparked my interest.

There is this very interesting result about Bezier curves and their control polygons in 3-space: it is known that a Bezzier simple closed curve can be unknotted but have a knotted control polygon. What else is there to explore here? Can only certain differences appear (say, in terms of crossing number or other invariants?) Here is another reference.

I’d like to sink my teeth into this. It doesn’t hurt that I am teaching a numerical methods course. đź™‚

January 15, 2014

Applying for a math job at a “teaching with some research required” institution

Filed under: academia, editorial — Tags: , — collegemathteaching @ 2:47 am

I am on the search committee this year and have read a LOT of applications. I ranted just a bit here. Here are some comments to applicants.

Note: our job’s ad is this:

Applications are invited for two Tenure-Track Assistant Professor positions in the Department of Mathematics beginning August 2014. Candidates must possess a Ph.D. in mathematics or statistics by the start date. Preference will be given to candidates who are broadly trained, have a strong commitment to undergraduate teaching, and have college-level teaching experience. Preference will be given to those applicants whose research specialty is in statistics, applied mathematics, or other areas of current interest in the department. An active research program and scholarly publication are required for tenure and advancement. Candidates must be able to work in the U.S. without sponsorship. For full consideration, applications should be complete by December 16, 2013. Other positions may become available in the future.

Please submit all application materials electronically. Post the AMS cover letter, a letter of application, vita, a copy of each graduate transcript, description of research, statement of teaching philosophy, and three current letters of recommendation (at least one of which addresses teaching and one research) on the MathJobs.org website. Additional information regarding xxxx University and these positions may be found at http://www.xxx.edu or obtained by email from xxx yyy at www@whatever.edu

Employment with xxxx University is contingent upon satisfactory completion of a criminal background check. xxxx University is an equal-opportunity, affirmative action employer. The administration, faculty and staff are committed to attracting qualified candidates from underrepresented groups.

Note: our usual load is 11 hours per semester, along with the usual admin BS. And yes, we teach a lot of calculus.

So here are a few statements from me. I can’t say if anyone else on the search committee thinks this way but I can tell you what catches my eye and what *I* would recommend.

1. In your cover letter, it is useful to highlight facts. Everyone says that they are “a great teacher, researcher, etc.” This is what I am talking about: more than one candidate took pains to tailor a cover letter just for our school. They said some generic stuff. But the left out the fact that they won a teaching award in grad school….and according to the recommendation letters, this award only goes to a small percentage of grad students. That’s a nice thing to highlight!

2. Give the letter a little thought. Saying “one thing that sets me apart from the other applicants with strong degrees and credentials is that “I work hard to be the best I can be.” Sure…and the rest of the applicant pool consists of slackers??? Seriously, get a grip.

3. If you spent several years at an institution that was NOT a visiting position or a post-doc and are leaving, please say a bit as to WHY you are leaving, or at least have a reference writer bring up the point. Seriously; no one wants to hire someone else’s problem. Hey, if you didn’t get tenure that is not always a “kiss of death”; in one case a person with a good research record didn’t get tenure due to the university trying to switch to being a research institution and that applicant’s research record would be fine at our place.

If you are trying to get closer to family: that is fine too. If you are think that our place is a step up from your current place: fine. If you are at a research place and want to teach more and spend less time writing grant proposals for research, that is fine too.

But don’t leave questions unanswered; remember we have a lot of applications and it is easy to go to the bottom of the pile.

4. Remember our job requirements. If you state that it is your goal to become an international class researcher, you’ll be miserable here. Forget it.

On the other hand, if you had 3 years at a post doc (at places like MSRI and Cal Berkeley) and haven’t done squat in terms of research, you won’t publish here either…and you won’t get tenure.

5. We have an engineering college here and therefore teach a LOT of calculus. Seriously; it is great that you taught manifold theory and algebraic topology, but here you’ll be doing a couple of calculus sections (engineering or “business/life science) almost every semester. Make sure you highlight your calculus teaching experience.

6. If you ask for a teaching reference, you might ask the reference if they can write a honest, positive letter. I read one letter in which the reference said that the applicant doesn’t explain things well, tries to explain again and often has no more success the second time around.