Cubic polynomials and tangent lines

Just read an article in the most recent NCTM Mathematics Teacher magazine called “Students’ Exploratory Thinking about a Nonroutine Calculus Task” by Keith Nabb. I really, really enjoyed this article. Maybe for some this isn’t new, but I didn’t know this fact:

Average two of the roots of a cubic polynomial. Draw a tangent line to the cubic at this point. Did you know it will always pass through the third zero?? Incredible!

Here’s a nice site that I just googled that goes through one proof. However, the charm of the article mentioned above is that there are many interesting proofs that students came up with, some of which are more or less elegant (brute force algebra with CAS, Newton’s Method, just to name two of the four strategies mentioned in the article).

I wish I could give you the whole article, but you have to have an NCTM membership to see it. Here’s the link, but you’ll have to log in to actually see it.

Integration by parts and infinite series

I was teaching tabular integration yesterday and as I was preparing, I was playing around with using it on integrands that don’t ‘disappear’ after repeated differentiation. In particular, the problem I was doing was this:

\int x^2\ln{x}dx

Now this is done pretty quickly with only one integration by parts:

Let u=\ln{x} and dv=x^2dx. Then du=\frac{1}{x}dx and v=\frac{x^3}{3}. Rewriting the integral and evaluating, we find

\int x^2\ln{x}dx = \frac{1}{3}x^3\ln{x}-\int \left(\frac{x^3}{3}\cdot\frac{1}{x}\right)dx

=\frac{1}{3}x^3\ln{x}-\int \frac{x^2}{3}dx

= \frac{1}{3} x^3 \ln{x} - \frac{1}{9} x^3 + c .

But I decided to try tabular integration on it anyway and see what happened. Tabular integration requires us to pick a function f(x) and compute all its derivatives and pick a function g(x) and compute all its antiderivatives. Multiply, then insert alternating signs and voila! In this case, we choose f(x)=\ln{x} and g(x)=x^2. The result is shown below.

\int x^2\ln{x}dx = \frac{1}{3}x^3\ln{x}-\frac{1}{12}x^3-\frac{1}{60}x^3-\frac{1}{180}x^3-\cdots

= \frac{1}{3}x^3\ln{x} - x^3 \sum_{n=0}^\infty \frac{2}{(n+4)(n+3)(n+2)(n+1)} +c

If I did everything right, then the infinite series that appears in the formula must be equal to \frac{1}{9}. Checking with wolframalpha, we see that indeed,

\sum_{n=0}^\infty \frac{2}{(n+4)(n+3)(n+2)(n+1)} = \frac{1}{9} .

Wow!! That’s pretty wild. It seemed like any number of infinite series could pop up from this kind of approach (Taylor series, Fourier series even). In fact, they do. Here are just three nice resources I came across which highlight this very point. I guess my discovery is not so new.

Why Calculus still belongs at the top

AP Calculus is often seen as the pinnacle of the high school mathematics curriculum*–or the “summit” of the mountain as Professor Arthur Benjamin calls it. Benjamin gave a compelling TED talk in 2009 making the case that this is the wrong summit and the correct summit should be AP Statistics. The talk is less than 3 minutes, so if you haven’t yet seen it, I encourage you to check it out here and my first blog post about it here.

I love Arthur Benjamin and he makes a lot of good points, but I’d like to supply some counter-points in this post, which I’ve titled “Why Calculus still belongs at the top.”

Full disclosure: I teach AP Calculus and I’ve never taught AP Statistics. However I DO know and love statistics–I just took a grad class in Stat and thoroughly enjoyed it. But I wouldn’t want to teach it to high school students. Here’s why: For high school students, non-Calculus based Statistics seems more like magic than mathematics.

When I teach math I try, to the extent that it’s possible, to never provide unjustified statements or unproven claims. (Of course this is not always possible, but I try.) For example, in my Algebra 2 class I derive the quadratic formula. In my Precalculus class, I derive all the trig identities we ask the students to know. And in my Calculus class, I “derive” the various rules for differentiation or integration. I often tell the students that copying down the proof is completely optional and the proof will not be tested–“just sit back and relax and enjoy the show!”

But such an approach to mathematical thinking can rarely be applied in a high school Statistics course because statistics rests SO heavily on calculus and so the ‘proofs’ are inaccessible. I’d like to make a startling claim: I claim that 99.99% of AP Statistics students and 99% of AP Statistics teachers cannot even give the function-rule for the normal distribution.

Image used by permission from Interactive Mathematics. Click the image to go there and learn all about the normal distribution!

Image used by permission from Interactive Mathematics. Click the image to go there and learn all about the normal distribution!

In what other math class would you talk about a function ALL YEAR and never give its rule? The normal distribution is the centerpiece (literally!) of the Statistics curriculum. And yet we never even tell them its equation nor where it comes from. That should be some kind of mathematical crime. We might as well call the normal distribution the “magic curve.”

Furthermore, a kid can go through all of AP Statistics and never think about integration, even though that’s what their doing every single time they look up values in those stat tables in the back of the book.

I agree that statistics is more applicable to the ‘real world’ of most of these kids’ lives, and on that point, I agree with Arthur Benjamin. But I would argue that application is not the most important reason we teach mathematics. The most important thing we teach kids is mathematical thinking.

The same thing is true of every other high school subject area. Will most students ever need to know particular historical facts? No. We aim to train them in historical thinking. What about balancing an equation in Chemistry? Or dissecting a frog? They’ll likely never do that again, but they’re getting a taste of what scientists do and how they think. In general, two of our aims as secondary educators are to (1) provide a liberal education for students so they can engage in intelligent conversations with all people in all subject areas in the adult world and (2) to open doors for a future career in a more narrow field of study.

So where does statistics fit into all of this? I think it’s still worth teaching, of course. It’s very important and has real world meaning. But the value I find in teaching statistics feels VERY different than the value I find in teaching every other math class. Like I said before, it feels a bit more like magic than mathematics.**

I argue that Calculus does a better job of training students to think mathematically.

But maybe that’s just how I feel. Maybe we can get Art Benjamin to stop by and weigh in!



*In our school, and in many other schools, we actually have many more class options beyond Calculus for those students who take Calculus in their Sophomore or Junior year and want to be exposed to even more math.

** Many parts of basic Probability and Statistics can be taught with explanations and proof, namely the discrete portions–and this should be done. But working with continuous distributions can only be justified using Calculus.

Happy Mean Girls day

In my Calculus class today I showed just a short clip from this video (“the limit does not exist!”).

Apparently, completely and totally without my knowledge, I showed this video today, on October 3rd, which just happens to be Mean Girls Day. So..happy Mean Girls Day! (happy/mean…that sounds a little funny)

Kudos to the folks who put these videos together!

Can’t touch this?

Here’s a popular t-shirt design:

But I have a mathematical problem with it. It’s certainly true that THIS particular function never touches its asymptote. I think the t-shirt suggests that this is true of any asymptote, though. As if to say, “hey I’m an asymptote, and as an asymptote, you can’t ever touch me!” However, functions in general CAN touch their asymptotes, sometimes an infinite number of times. (I’ve talked at great length about this issue.)

I also have typesetting-issues with this design (notice the italicized “lim” and the unitalicized variables).

Am I being too picky?

Summer Odds and Ends

I promise I’ll start blogging again. But as followers of this blog might know, I like to take the summer off–both from teaching and blogging. I never take a break from math, though. Here are some fun things I’ve seen recently. Consider it my own little math carnival :-).

I love this comic, especially as I start my stat grad class this semester @ JHU. After this class, I’ll be half-way done with my masters. It’s a long road! [ht: Tim Chase]

Speaking of statistics, my brother also sent me this great list of lottery probabilities. Could be very useful in the classroom.

These math dice. Honestly I don’t know what I’d do with them, but you have to admit they’re awesome. [ht: Tim Chase]

These two articles about Khan academy and the other about edX I found very interesting. File all of them under ‘flipping the classroom.’ I’m still working up the strength to do a LITTLE flipping with my classroom. My dad forwarded these links to me. He has special interest in all things related to MIT (like Khan, and like edX) since it’s his alma mater.

I’ll be teaching BC Calculus for the first time this semester and we’re using a new book, so I read that this summer. Not much to say, except that I did actually enjoy reading it.

I also started a fabulous book, Fearless Symmetry by Avner Ash and Robert Gross. I have a bookmark in it half way through. But I already recommend it highly to anyone who has already had some college math courses. I just took a graduate course in Abstract Algebra recently and it has been a great way to tie the ‘big ideas’ in math together with what I just learned. The content is very deep but the tone is conversational and non-threatening. (My dad, who bought me the book, warns me that it gets painfully deep toward the end, however. That’s to be expected though, since the authors attempt to explain Wiles’ proof of Fermat’s Last Theorem!)

I had this paper on a juggling zeta function (!) sent to me by the author, Dr. Dominic Klyve (Central Washington University). I read it, and I pretended to understand all of it. I love the intersection of math and juggling, and I’m always on the look out for new developments in the field.

And most recently, I’ve been having a very active conversation with my math friends about the following problem posted to NCTM’s facebook page:

Feel free to go over to their facebook page and join the conversation. It’s still happening right now. There’s a lot to say about this problem, so I may devote more time to this problem later (and problems like it). At the very least, you should try doing the problem yourself!

I also highly recommend this post from Bon at Math Four on why math course prerequisites are over-rated. It goes along with something we all know: learning math isn’t as ‘linear’ an experience as we make it sometimes seem in our American classrooms.

And of course, if you haven’t yet checked out the 90th Carnival of Mathematics posted over at Walking Randomly (love the name!), you must do so. As usual, it’s a thorough summary of recent quality posts from the math blogging community.

Okay, that’s all for now. Thanks for letting me take a little random walk!


This article about the saddle-shape of Pringles is a joy to read [ht: Prisca Chase]. I’ll give you an excerpt, but I encourage you to read the whole thing. It’s both mathematically stimulating and extremely funny:

Saddle up for maximum snack satisfaction (mathematically speaking)

Stephanie V.W. Lucianovic

My husband is a calculus professor and one who brings food items into the classroom with surprising regularity. No, he doesn’t bring pies on Pi day – though he can recite the string up to a couple dozen digits – but he does bring Pringles. As a teaching aid.

This afternoon when I walked into his study, I nearly tripped over a plastic Safeway bag filled with six red cans of Pringles. “Is it Pringles Day already?” I asked, nudging the bag. Pringles Day is the day Dr. Mathra lectures on the classification of critical points in multivariable calculus, and he uses the saddle-shaped Pringles to illustrate his points.

After class, the students get to eat his illustrations. It’s their favorite day.


Later in the article, the fact that a Pringle can’t be made from a sheet of paper is mentioned. For a normal sheet of paper, this is true. But you can fold paper in such a way as to approximate a hyperbolic parabaloid. I’ve mentioned this before here and here. So go try it!

Using math to get out of a ticket

By now perhaps you’ve seen this floating around the internet. It was reported here and here and here and here, at least.

Physicist Dmitri Krioukov got a $400 ticket for not making a full stop at at stop sign. He wrote a paper explaining why the police officer could have been wrong, went to court, and got the fine lifted.

If you haven’t read the paper, I encourage you to do it. It’s fairly short and only requires knowledge of a Calculus. Here is a direct link to the pdf. Here’s the abstract:

We show that if a car stops at a stop sign, an observer, e.g., a police officer, located at a certain distance perpendicular to the car trajectory, must have an illusion that the car does not stop, if the following three conditions are satisfi ed: (1) the observer measures not the linear but angular speed of the car; (2) the car decelerates and subsequently accelerates relatively fast; and (3) there is a short-time obstruction of the observer’s view of the car by an external object, e.g., another car, at the moment when both cars are near the stop sign.

What do you think? Is Professor Krioukov just trying to buffalo the court, or does he have a legitimate case? I guess if there’s any doubt at all about his guilt, then he should be forgiven the fine. And that’s what the court did rule.

However, there is one particular assumption that he makes which is absolutely way-off (look at the paper and notice the key on Figure 3, labeling the blue curve). But the court certainly didn’t catch that. Like I said, he may have been trying to pull the wool over the eyes of the court with lots of math and physics.

Despite the fact that he published his paper on April 1st, I do think this story is true. Like I said, though, the paper does contain an error. So despite all of his good effort, I think he should have been given the ticket. The court didn’t notice it, but he pulled a fast one on them! (pun intended!)

Pi R Squared

[Another guest blog entry by Dr. Gene Chase.]

You’ve heard the old joke.

Teacher: Pi R Squared.
Student: No, teacher, pie are round. Cornbread are square.

The purpose of this Pi Day note two days early is to explain why \pi is indeed a square.

The customary definition of \pi is the ratio of a circle’s circumference to its diameter. But mathematicians are accustomed to defining things in two different ways, and then showing that the two ways are in fact equivalent. Here’s a first example appropriate for my story.

How do we define the function \exp(z) = e^z for complex numbers z? First we define a^b for integers a > 0 and b. Then we extend it to rationals, and finally, by requiring that the resulting function be continuous, to reals. As it happens, the resulting function is infinitely differentiable. In fact, if we choose a to be e, the \lim_{n\to\infty} (1 + \frac{1}{n})^n \, not only is e^x infinitely differentiable, but it is its own derivative. Can we extend the definition of \exp(z) \, to complex numbers z? Yes, in an infinite number of ways, but if we want the reasonable assumption that it too is infinitely differentiable, then there is only one way to extend \exp(z).

That’s amazing!

The resulting function \exp(z) obeys all the expected laws of exponents. And we can prove that the function when restricted to reals has an inverse for the entire real number line. So define a new function \ln(x) which is the inverse of \exp(x). Then we can prove that \ln(x) obeys all of the laws of logarithms.

Or we could proceed in the reverse order instead. Define \ln(x) = \int_1^x \frac{1}{t} dt . It has an inverse, which we can call \exp(x) , and then we can define a^b as \exp ( b \ln (a)). We can prove that \exp(1) is the above-mentioned limit, and when this new definition of a^b\, is restricted to the appropriate rationals or reals or integers, we have the same function of two variables a and b as above. \ln(x) can also be extended to the complex domain, except the result is no longer a function, or rather it is a function from complex numbers to sets of complex numbers. All the numbers in a given set differ by some integer multiple of

[1] 2 \pi i.

With either definition of \exp(z), Euler’s famous formula can be proven:

[2] \exp(\pi i) + 1 = 0.

But where’s the circle that gives rise to the \pi in [1] and [2]? The answer is easy to see if we establish another formula to which Euler’s name is also attached:

[3] \exp(i z) = \sin (z) + i \cos(z).

Thus complex numbers unify two of the most frequent natural phenomena: exponential growth and periodic motion. In the complex plane, the exponential is a circular function.

That’s amazing!

Here’s a second example appropriate for my story. Define the function on integers \text{factorial (n)} = n! in the usual way. Now ask whether there is a way to extend it to (some of) the complex plane, so that we can take the factorial of a complex number. There is, and as with \exp(z), there is only one way if we require that the resulting function be infinitely differentiable. The resulting function is (almost) called Gamma, written \Gamma. I say almost, because the function that we want has the following property:

[4] \Gamma (z - 1) = z!

Obviously, we’d like to stay away from negative values on the real line, where the meaning of (–5)! is not at all clear. In fact, if we stay in the half-plane where complex numbers have a positive real part, we can define \Gamma by an integral which agrees with the factorial function for positive integer values of z:

[5] \Gamma (z) = \int_0^\infty \exp(-t) t^{z - 1} dt .

If we evaluate \Gamma (\frac{1}{2}) we discover that the result is \sqrt{\pi} .

In other words,

[6] \pi = \Gamma(\frac{1}{2})^2 .

Pi are indeed square.

That’s amazing!

I suspect that the \pi arises because there is an exponential function in the definition of \Gamma, but in other problems involving \pi it’s harder to find where the \pi comes from. Euler’s Basel problem is a good case in point. There are many good proofs that

1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + ... = \frac{\pi^2}{6}

One proof uses trigonometric series, so you shouldn’t be surprised that \pi shows up there too.

\pi comes up in probability in Buffon’s needle problem because the needle is free to land with any angle from north.

Can you think of a place where \pi occurs, but you cannot find the circle?

George Lakoff and Rafael Núñez have written a controversial book that bolsters the argument that you won’t find any such examples: Where Mathematics Comes From. But Platonist that I am, I maintain that there might be such places.