College Math Teaching

September 23, 2014

Ok, what do you see here? (why we don’t blindly trust software)

I had Dfield8 from MATLAB propose solutions to y' = t(y-2)^{\frac{4}{5}} meeting the following initial conditions:

y(0) = 0, y(0) = 3, y(0) = 2.

homeworkexistanceuniqueness

Now, of course, one of these solutions is non-unique. But, of all of the solutions drawn: do you trust ANY of them? Why or why not?

Note: you really don’t have to do much calculus to see what is wrong with at least one of these. But, if you must know, the general solution is given by y(t) = (\frac{t^2}{10} +C)^5 + 2 (and, of course, the equilibrium solution y = 2 ). But that really doesn’t provide more information that the differential equation does.

By the way, here are some “correct” plots of the solutions, (up to uniqueness)

homeworkexistanceuniqueness2

September 19, 2014

Freshman calculus: they don’t always know the basics….

Filed under: basic algebra, calculus — Tags: , — collegemathteaching @ 5:43 pm

Example one: many students don’t know that \frac{\frac{a}{b}}{c} \ne \frac{a}{\frac{b}{c}} (of course, assume that a, b, c, \ne 0 . ) Where this came up: when we computed lim_{h \rightarrow 0} \frac{\frac{1}{1+h} - 1}{h} we obtained lim_{h \rightarrow 0} \frac{\frac{-h}{1+h}}{h} and a student didn’t understand why this was equal to lim_{h \rightarrow 0} \frac{-1}{1+h}

I ended up asking the student to simplify \frac{\frac{2}{3}}{2} = and ….asking: ok, “if I have 2/3’rds of a pie and I give two people an equal piece of that, how much pie does each person get?

Example two: I gave “find the domain of \frac{1}{\sqrt{x^2 - 9}} and two students didn’t understand why an answer of -3 > x > 3 was logically impossible. One of them told me: “my calculus teacher in high school told me to do it this way”: I am 99.99 percent that this isn’t true, but, well, I stayed with it until the student understood why such a statement was logically impossible. Oh yes, this same “I had calculus in high school” student was sure that I was wrong when I told him that “the derivative of a constant function is zero”; he was SURE that it is “1”.

September 18, 2014

The death of a dream

Filed under: editorial — Tags: , , — collegemathteaching @ 6:02 pm

I chuckled when I posted on social media: it has struck me that I had been lifting weights for 42 years (starting in the 8’th grade and being consistent). I was a bit surprised that, after 42 years of weight lifting, that I am so weak! (*)

So, I found at photo of the gym I used. During that period of my life, I just “knew” that I was going to be a professional football player. So I ran, lifted weights and….well…still got run over by those who were destined to play football at the division I level. I simply do not have an athlete’s body…but I really didn’t come to grips with that until I was a senior in high school.

That was a downer for me. The dream died. I still remember getting mail from the football program at the school I ended up attending…it was a form to order tickets. 😦

The point: though I was consistently considerably slower and weaker than those destined to play at the next level, I stayed in denial. If only I ran more sprints, lifted more weights, etc.

What does that have to do with today?

I just gave back my first set of calculus exams. With every group comes a few students who…well, they are going to be an engineer (in some cases, their parents think so). But, try as they might, calculating: \frac{d}{dx} sin(x^2 +1) is tough for them. \int \frac{arctan(x)}{x^2 + 1} dx = is all but impossible for them. Yes, I am talking about the ones who attend classes, study and come to office hours.

It is like me with my 40 yard dash: I did sprints, I ran hills, but when I tried, the stopwatch still said 5.9. Yes, I was that slow; much slower than a typical college lineman.

The reality is that one’s dreams are often out of reach, and sometimes, students find that out in their first college level calculus class.

(*)disclaimer: I am 55 years old, weigh about 183 pounds and regularly do 5 sets of 10 pull ups, and 3 reps with 180 on the bench press; my lifetime best is 310 (when I weighed 230 lbs), which I can’t even take off of the racks now.

Today’s student: communication and e-mail

Filed under: academia, editorial — Tags: — collegemathteaching @ 5:50 pm

Academia:
This is a good guide on how to NOT e-mail your professor (fictional student; e-mail is a collage of actual e-mails)

The professor says something interesting:

And before you go thinking that Anderson is publicly shaming the student: ‘cartmanrulez99′ a fictional creation, based on “two or three poor emails put together,” explains Anderson on YouTube. “I would never post an email of a student to the Internet nor would I suggest anyone else ever doing that.”

Yep. But this is interesting too (emphasis mine):

Moreover, he adds, he’s not youth-bashing. “In my opinion, each and every generation is smarter than the previous generation,” he writes. “I have seen that first-hand in my twenty years of teaching. If you think that there were no dumb people in the past, think again.”

The emphasis is ABSOLUTELY NOT TRUE FOR ME; that is, the current students don’t appear to be smarter to me. BUT….there are mitigating factors at play here:

1. I went to a very selective undergraduate institution. Then I served in the Nuclear Navy; my fellow officers were taken from the upper 20 percent of graduating classes in engineering and science programs. Then I got my Ph. D. at a division I research place.

Therefore, the average student I see at my “median ACT of 25, median calculus ACT of 29-30” isn’t as talented as the people that I went to college and beyond with.

2. We teach service courses including mathematics for non-technical majors. See point 1.

BUT: it is true that today’s A student is pretty good, at least the A students in mathematics and science are.

And yes, we had presumptuous idiots in my day too…though we didn’t have e-mail until I was almost done with graduate school. 🙂

3. Concerning the 20 plus years I’ve been here: we’ve had ebbs and flows in student quality. The current class appears to be an “up” class. This ebb and flow probably wouldn’t be seen by a professor who teaches at an elite university or at a larger state university.

September 9, 2014

Chebyshev polynomials: a topological viewpoint

Chebyshev (or Tchebycheff) polynomials are a class of mutually orthogonal polynomials (with respect to the inner product: f \cdot g  = \int^1_{-1} \frac{1}{\sqrt{1 - x^2}} f(x)g(x) dx ) defined on the interval [-1, 1] . Yes, I realize that this is an improper integral, but it does converge in our setting.

These are used in approximation theory; here are a couple of uses:

1. The roots of the Chebyshev polynomial can be used to find the values of x_0, x_1, x_2, ...x_k \in [-1,1] that minimize the maximum of |(x-x_0)(x-x_1)(x-x_2)...(x-x_k)| over the interval [-1,1] . This is important in minimizing the error of the Lagrange interpolation polynomial.

2. The Chebyshev polynomial can be used to adjust an approximating Taylor polynomial P_n to increase its accuracy (away from the center of expansion) without increasing its degree.

The purpose of this note isn’t to discuss the utility but rather to discuss an interesting property that these polynomials have. The Wiki article on these polynomials is reasonably good for that purpose.

Let’s discuss the polynomials themselves. They are defined for all positive integers n as follows:

T_n = cos(n acos(x)) . Now, it is an interesting exercise in trig identities to discover that these ARE polynomials to begin with; one shows this to be true for, say, n \in \{0, 1, 2\} by using angle addition formulas and the standard calculus resolution of things like sin(acos(x)) . Then one discovers a relation: T_{n+1} =2xT_n - T_{n-1} to calculate the rest.

The cos(n acos(x)) definition allows for some properties to be calculated with ease: the zeros occur when acos(x) = \frac{\pi}{2n} + \frac{k \pi}{n} and the first derivative has zeros where arcos(x) = \frac{k \pi}{n} ; these ALL correspond to either an endpoint max/min at x=1, x = -1 or local max and mins whose y values are also \pm 1 . Here are the graphs of T_4(x), T_5 (x)

cheby4

cheby5

Now here is a key observation: the graph of a T_n forms n spanning arcs in the square [-1, 1] \times [-1,1] and separates the square into n+1 regions. So, if there is some other function f whose graph is a connected, piecewise smooth arc that is transverse to the graph of T_n that both spans the square from x = -1 to x = 1 and that stays within the square, that graph must have n points of intersection with the graph of T_n .

Now suppose that f is the graph of a polynomial of degree n whose leading coefficient is 2^{n-1} and whose graph stays completely in the square [-1, 1] \times [-1,1] . Then the polynomial Q(x) = T_n(x) - f(x) has degree n-1 (because the leading terms cancel via the subtraction) but has n roots (the places where the graphs cross). That is clearly impossible; hence the only such polynomial is f(x) = T_n(x) .

This result is usually stated in the following way: T_n(x) is normalized to be monic (have leading coefficient 1) by dividing the polynomial by 2^{n-1} and then it is pointed out that the normalized T_n(x) is the unique monic polynomial over [-1,1] that stays within [-\frac{1}{2^{n-1}}, \frac{1}{2^{n-1}}] for all x \in [-1,1] . All other monic polynomials have a graph that leaves that box at some point over [-1,1] .

Of course, one can easily cook up analytic functions which don’t leave the box but these are not monic polynomials of degree n .

September 2, 2014

Using convolutions and Fourier Transforms to prove the Central Limit Theorem

Filed under: probability — Tags: , , — collegemathteaching @ 5:40 pm

I’ve used the presentation in the our Probability and Statistics text; it is appropriate given that many of our students haven’t seen the Fourier Transform. But this presentation is excellent.

Upshot: use the convolution to derive the density function for S_n = X_1 + X_2 + ....X_n (independent, identically distributed random variables of finite variance), assume mean is zero, variance is 1 and divide S_n by \sqrt{n} to obtain the variance of the sum to be 1. Then use the Fourier transform on the whole thing (the normalized version) to turn convolution into products, use the definition of Fourier transform and use the Taylor series for the e^{i 2 \pi x \frac{s}{\sqrt{n}}} terms, discard the high order terms, take the limit as n goes to infinity and obtain a Gaussian, which, of course, inverse Fourier transforms to another Gaussian.

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