Think Deeply About Simple ThingsMiami Fall Conference

September 24, 2010Dedicated to the memory of Prof. Bogdan Baishanski

I have dedicated this talk to the memory of Prof. Bogdan Baishanski of The Ohio StateUniversity who died on August 16. Prof. Baishanski was a well-known researcher in ap-proximation theory and a truly inspirational teacher. I took several graduate analysiscourses from Prof. Baishanski, including a year-long sequence in approximation theoryand a course on group representations and special functions. In all that time, there wasonly one occasion when Prof. Baishanski failed to come to class empty-handed. During oneseemingly ordinary class, he paused, apologized to the class, and reached into his breastpocket for a small scrap of paper. There’s this one formula, it seems, that always gave himtrouble. After referring to the scrap of paper, he ceremoniously discarded it, apologizedto us again, and went on.

I asked him about this some years later and learned that this was simply the way hegauged his preparedness for class: If he needed notes, then apparently he didn’t understandthe material well enough. As a student, I took that answer at face value and didn’t give itmuch more thought. As an old-timer who’s been teaching graduate analysis for roughly 25years now, I find it nothing short of miraculous. You will notice, however, that I broughta whole stack of notes with me!

The title of this talk likewise harkens back to my student days at Ohio State. It wasa mantra that the late Arnold Ross, chair of the math department back in the 70s, wasfond of repeating, and one that’s stuck with me for all these years.

Dr. Ross held a summer program in number theory for gifted high school studentsthat new grad students were encouraged to attend—which I did, some 35 years ago. Itremains one of my most vivid memories, in part because the sting of humiliation is sohard to erase! Dr. Ross would assign dozens of problems each day, and each day he wouldask about our progress. Invariably, the high school students could solve nearly all of theproblems while the the grad students were lucky to solve just a few.

The high school students, with very little formal training, routinely outperformed thegrad students, many of whom came from elite private colleges and most of whom were usedto being at the top of their class. Our heads were full of facts, theorems, and vocabulary,but no one had taught us to think. And certainly no one had forced us to do math “inreal life.” The high school students, on the other hand, were participating in real time,

actually thinking about the problems.

Granted, the high school students were the best and the brightest the country had tooffer, but were we really so different?

My guess is that college ruined us: We had been lectured to extensively, but nevertaught to think outside the rigid confines of named theorems and numbered sections in atextbook. It was a rude awakening and an important lesson—although I’ll admit it waslost on me at the time!

Both of my talks will attempt to address the questions: What can we, as faculty, doto encourage our students to think rather than to recall or recite? And what can studentsdo to become better prepared to do math “in real life”?

In this first talk, I’ll take a literal interpretation of Dr. Ross’s mantra: When presentingstudents with a new idea, let’s focus on simple examples, intuitive thinking, and “industrialgrade” versions of theorems (as my friend Joe Diestel is fond of saying), with as little jargonand terminology as possible. In my experience, the finer points and extra details are mucheasier for students to grasp once they have a firm handle on the basics.

My first example—or pet peeve, really—is l’Hopital’s rule. In the book we’re currentlyusing for Calc I, the authors present l’Hopital’s rule quite early and ask the students tocompute such familiar limits as:

limx→∞

x

exand lim

x→∞

log xx

.

Simple enough, right? Unfortunately, while the functions ex and log x are presented early,with the tacit assumption that students have already seen them, they’re not actuallydefined until the second course!

The upshot is that l’Hopital’s rule becomes yet one more mindless procedure that’snot reinforced by elementary examples and straightforward reasoning. It should comeas no surprise, then, that students are programmed to apply it without thinking, evenwhen it doesn’t apply. Worse still, we reinforce the common notion that ex and log x aremysterious beasts that defy human understanding.

While I have no problem with introducing logs and exponentials early, and I haveno problem with assuming their elementary properties, let’s not get too eager. The firstcalculus offers a great opportunity to demystify these functions. Moreover, virtually allthe elementary limits encountered in the first calculus can be computed without recourseto l’Hopital’s rule.

Personally, I avoid any discussion of the growth rates of ex and log x until we’ve had

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a chance to define them and examine them more carefully. Breaking with the dogma ofthe syllabus, I wait until we’ve covered the Fundamental Theorem, about halfway throughthe course, and then give the familiar definition

log x =∫ x

1

1tdt, x > 0, (1)

from which it’s immediate that

d

dxlog x =

1x, x > 0.

From (1), it’s easy to see that

log x ≤ x− 1 ≤ x for x > 1, (2)

and, so, for example,12

log x = log√x ≤√x for x > 1. (3)

It’s now relatively clear that

0 ≤ limx→∞

log xx≤ lim

x→∞

2√x

x= 0.

Again straying just a bit from the recommended order of events, I define ex as the inverseof log x, a notion familiar to most students. Given this, (2) then yields

t ≤ et =⇒ t2 ≤ e2t =⇒ x2/4 ≤ ex for t, x > 0.

And, from this,0 ≤ lim

x→∞

x

ex≤ lim

x→∞

x

x2/4= 0.

The moral to my story is this: Students have no trouble learning mechanical procedures(most have been trained to do this in high school), but they do have trouble buildingintuition and gaining understanding without a little bit of guidance.

A second pet peeve, this time from Calc III, is the fact that very little attention is paidto elementary techniques for graphing surfaces by-hand. Instead, students’ first exposureto surfaces is often by way of extraordinarily complex figures that I’m certain I would havetrouble drawing by-hand. (See the illustration on the page 5.) Worse still, in the bookthat we’re currently using, for example, an entire section is devoted to classifying varioussurfaces by name (hyperbolic paraboloid, and so forth), a skill that is arguably of littlevalue in practice.

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For the most part, all of my examples live in the first octant—and none of them aregiven names! I try very hard to stick to familiar principles: Start with the intercepts,the traces in the coordinate planes, and draw lots of parallel lines (or contours). (See thehandout on page 6.) I also make a point of identifying the “shadow” of the surface in eachof the coordinate planes, as this will be useful when we speak of integration.

And, when it comes to integration, I likewise stress elementary examples and first-octant solids. For example, when discussing the various coordinate systems (rectangular,cylindrical, and spherical), I always ask my students to setup integrals for the most naturalsolids: Cubes, cylinders, cones, and spheres. (See the handouts on pages 7 and 8.)

On the other hand, as I’ll try to illustrate in my second talk, I don’t shy away fromcomplicated examples. I do, however, try to distinguish between introductory examplesand examples that can be saved for tomorrow. It isn’t necessary to tell the studentseverything on the first day.

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MATH 2330 Project 2A: Graphing February 5, 2010

In each case, sketch the portion of the surface that lies in the first octant.

1. z = 4−x2− 4y2

2. 6x+4y+3z = 12

MATH 2330 Project 5 April 19, 2010

Pictured below are portions of three common solids in the first octant only. In each case,SETUP triple integrals that will give the volume of the solid using Rectangular, Cylindrical,and Spherical Coordinates. In each case, choose ONE of these expressions to EVALUATE.(The volume for each solid is given below.) In other words, you are asked to setup NINEintegral expressions and evaluate THREE of them.

1. The sphere of radius 2. (V = 4π/3)

2. The cylinder of radius 2 and height 4. (V = 4π)

3. The cone of radius 2 and height 4. (V = 4π/3)

Math 233 Some Practice with Double and Triple Integrals

An easy way to get some practice with double and triple integrals is to compute the volumesof all the familiar solids. For instance, a rectangular solid or box with sides of length a,b, and c; a right-circular cylinder with height h and radius a; a right-circular cone withheight h and radius a; a sphere of radius a. Try all the different coordinate systems and asmany different orders of integration as you can stand. (There are as many as 18 differenttriple integrals to choose from for any given solid. And not all of them are easy—just tryfinding the volume of a cube using spherical coordinates!)

In each of the following problems, you should SETUP double or triple integrals in rect-angular, cylindrical, and spherical coordinates which will give the desired volume. Youshould also experiment with different orders of integration (dydxdz, drdzdθ, etc.). In eachcase, try to EVALUATE at least one of these expressions.

In every case, we are interested in a portion of the solid in the first octant inside

the sphere x2 + y2 + z2 = 4. Find the volume of that portion which is:

1. Behind the plane y = x.

2. Inside (or above) the cone z2 = 3(x2 + y2).

3. Behind y = x and outside (or under) the cone z2 = 3(x2 + y2).

4. Inside the cylinder x2 + y2 = 1. (Now try using x2 + z2 = 1!)

5. Inside the cylinder r = 2 cos θ.

6. Outside the sphere x2 + y2 + z2 = 1 (that is, the region between two spheres).

7. Under the plane z = 1 (that is, a sphere minus a “cap”).

8. Behind the plane y = x and above the plane y = z. (Hard!)

9. Behind the plane x = 1 and to the left of the plane y = 1. (Hard!)

10. Inside the sphere ρ = 4 cosφ (that is, the region common to both spheres).

Anyone who has taught an upper division or beginning graduate analysis course willtell you they were shocked to find out how little their students knew about the real numbers.It’s my experience, however, that students know lots of things—they’re just not sure whichthings are true!! The seeds for this lack of certainty are planted early, I fear, and nurtureddaily by our insistence on forward progress at all costs.

On the first day of Calc I, I always begin by asking: What is a real number? Whatbegan as simple curiosity, years ago (I just wanted to hear their answers), has now evolvedinto a mechanism for introducing limits (and, to a lesser extent, the completeness of thereal numbers). By steering the conversation toward decimals, I can bring such numbers as√

2 and π into the discussion, at which point I have a perfect excuse to ask what is meantby those damned dots! √

2 = 1.414 . . . π = 3.14159 . . .

I find this tactic equally worthwhile on the first day of a real analysis or advancedcalculus course, in which case I typically include the question:

Why is 0.4999 . . . =12

?

Invariably, someone will remember a variation on the old trick:

If x = 0.49999 . . . , then

100x = 49.9999, . . . and

10x = 4.9999 . . . , so

90x = 45, or x = 1/2.

This leads naturally to a discussion of geometric series (where essentially the same trick isused to find a closed form for the partial sums) and, of course, a more formal discussionof decimal representations of real numbers.

I highly recommend having this conversation with your class—any class where the realnumbers are of concern. The “dots discussion” alone is worth the price of admission! Youwill learn that your students know all sorts of interesting (and partly true!) facts aboutnumbers. Again, this is a wonderful opportunity to demystify something that, at least forsome students, is a source of confusion.

As long as the notion of geometric series has come up, let me take this opportunityto say a few words about numerical series.

I’ve noticed that the series used in our calculus book are much harder than the onesused in our introductory analysis book. Perhaps you need simpler expressions if your goal

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is to give a formal ε–N proof, whereas any old expression will do if your only concern isrote manipulation. Whatever the reason, it’s a bit backward.

I’ve also noticed that even the most elementary introductions to series often includethe ratio and root tests, and often long before they’re needed. For those of you who aren’tfamiliar with these tests, here is a brief overview: Given a series

∑an with positive terms,

we compute

r = limn→∞

an+1

anrespectively, R = lim

n→∞n√an.

If either of these limits is strictly less than one, then our series converges; if either is strictlybigger than one, then our series diverges. The idea behind both tests is a comparison tothe geometric series; indeed,

an ≈ Crn respectively, an ≈ CRn

for some appropriate constant C and all n sufficiently large.

While these tests are invaluable for a thorough discussion of power series, and they’redefinitely useful for a certain class of numerical series, they’re arguably overkill in anintroductory discussion of numerical series. Moreover, the limits involved can be verychallenging to compute (and frequently involve l’Hopital’s rule!). Because both tests relyheavily on an understanding of geometric series and on comparison of series, I would arguethat that’s the place to start. By way of an example, consider the series

∞∑n=1

3n

n2 + 4n.

The ratio and root tests apply, of course, but each requires rather delicate analysis. Directcomparison to the geometric series

∑(3/4)n, on the other hand, is almost effortless.

In an introductory discussion of numerical series, I stick with the comparison test(and its variant, the limit comparison test), and the integral test—which is a form of thecomparison test that’s both easy to present and relatively easy to understand.

My final example is the Weierstrass theorem. But let me quickly add that I’m notabout to ask you to skip this important theorem! I am, however, going to ask you to skipit’s big brother, the Stone-Weierstrass theorem, at least on your first pass.

The approximation of continuous functions is a nearly ideal topic for testing students’understanding of continuity and uniform continuity. Moreover, it offers a number of av-enues for further study: Power series, Fourier series, even as an introduction to function

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spaces (that is, normed linear spaces and normed algebras of functions) and operator the-ory. In fact, I’ve adopted the habit of treating the Weierstrass theorem as one of theprimary goals of my introductory course in analysis.

Now, as I’ll explain further in my second talk, I never keep secrets. I typically statekey theorems long before we’re ready to prove them. I like to “sneak up” on the big ideas;give them time to take root. And I like to ask questions that will confirm our need forthem. In this case, I begin by asking my class:

Are there non-polynomial continuous functions on [ 0, 1 ]? Specifically, can yougive me an example of a continuous function f on [ 0, 1 ] that does not agree withany polynomial on [ 0, 1 ]?

This is a subtle question: I’m not asking you to give me an example of somethingthat doesn’t look like a polynomial. I’m asking for an example of something that we knowcan’t possibly agree with a polynomial restricted to the interval [ 0, 1 ]. It’s usually aninteresting discussion and I encourage you to have it with your class. In a sense, it’s notmuch different from the question: How do we know that there are real numbers that aren’trational? Both discussions ultimately lead to similar conclusions: Although polynomials(resp., rationals) are rare amongst continuous functions (resp., real numbers), there areapparently enough of them to do our work. In short, the folks who take “business calculus”might have the right idea after all: Polynomials are plenty.

My approach to the Weierstrass theorem is heavily influenced by the presentations inseveral classic textbooks and, to a large extent, by Lebesgue’s first published paper, whichis an absolute gold mine of information. (See the illustrations on pages 12 and 13.)

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Lebesgue offers not only an elementary proof of Weierstrass’s polynomial approxima-tion theorem, he also offers an elementary proof of its equivalence with what is sometimescalled Weierstrass’s Second Theorem, which concerns the approximation of 2π-periodiccontinuous functions by trigonometric polynomials. I take something of a middle groundhere, but the upshot is that I get at Weierstrass’s Second Theorem through purely elemen-tary means.

To begin, I present Lebesgue’s first result from the paper: Given a continuous functionf : [ 0, 1 ]→ R and an ε > 0, we can find a positive integer n and a piecewise linear contin-uous function (or polygonal function) g, with “nodes” at the points k/n, k = 0, 1, . . . , n,that is within ε of f . This is a simple consequence of the uniform continuity of f .

y = f(x)

y = g(x)

!

As Lebesgue then points out, by perturbing g slightly, we can find a polygonal function h,also having nodes at the points k/n, k = 0, 1, . . . , n, but taking rational values at each ofthese nodes, that is within 2ε of f .

y = f(x)

y = g(x)

!

y = h(x)

It’s not hard to see that the collection of polygonal functions taking rational values at thenodes k/n, k = 0, . . . , n, for n = 1, 2, 3, . . ., is a countable set. Thus, Lebesgue has givenus an easy proof that the space C[ 0, 1 ] is separable (that is, has a countable dense subset;in this sense the polygonal functions in C[ 0, 1 ] play the same role as the rational numbersin R).

Lebesgue also includes a clever proof that the collection of polygonal functions basedon a fixed set of nodes x0, . . . , xn is a vector space of dimension n + 1; in fact, he offers

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up a simple basis for the space: The n “angles,” ϕ0, . . . , ϕn−1 together with the constantfunction 1.

xk

!(x) = x +|x|

!k(x) = !(x xk )

The proof is an entertaining exercise and offers a great opportunity to show the interplaybetween analysis and (linear) algebra.

Lebesgue goes on to prove the Weierstrass theorem by noting that |x| has a uniformlyconvergent power series expansion, a fact that’s of independent interest. Indeed, the storygoes that this observation (and Lebesgue’s approach in general) influenced Marshall Stone’sapproach to the Weierstrass theorem.

Although I stray from Lebesgue’s presentation at this point, I return to it by givinghis proof that Weierstrass’s First Theorem implies his Second (and conversely). While theproof is entirely elementary, it’s a bit long, so I won’t go into it here.

I do, however, want to mention an observation made by Dunham Jackson, in hisclassic monograph on Fourier Series and Orthogonal Polynomials: As Jackson points out,it’s easy to show that the Fourier series for any 2π-periodic polygonal function is uniformlyconvergent (and, moreover, must converge to the given function). The proof takes littlemore than integration by-parts but, when paired with Lebesgue’s first observation, leadsto another elementary proof of Weierstrass’s Second Theorem.

Because the key step in the proof is both short and elementary, let me show it to you:

Suppose that f is a 2π-periodic polygonal function with nodes at the points

−π = x0 < x1 < · · · < xm = π,

suppose that λi is the slope of the graph of f in the subinterval [xi−1, xi ], and, finally,suppose that

λ = max{ |λ1|, . . . , |λm| }.

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Now consider∫ xi

xi−1

f(x) cos kx dx

=1k

[f(xi) sin(kxi)− f(xi−1) sin(kxi−1)

]− 1

k

∫ xi

xi−1

λi sin kx dx

=1k

[f(xi) sin(kxi)− f(xi−1) sin(kxi−1)

]+

λi

k2

[cos(kxi)− cos(kxi−1)

].

Summing the first set of bracketed terms over all i yields

m∑i=1

[f(xi) sin(kxi)− f(xi−1) sin(kxi−1)

]= f(π) sin(kπ)− f(−π) sin(−kπ) = 0

(the sum is telescoping), while the sum of the remaining terms is bounded above by∣∣∣∣∣m∑

i=1

λi

k2

[cos(kxi)− cos(kxi−1)

] ∣∣∣∣∣ ≤ 2mλk2

=C

k2

(where C is a constant depending only on f). Thus, we’ve shown that the Fourier cosinecoefficients satisfy |bk| ≤ C/k2. An entirely similar calculation will show that Fouriersine coefficients likewise satisfy |ak| ≤ C/k2. Thus, the Fourier series for f is uniformlyconvergent (and, by elementary considerations, must converge to f).

While I do present the Stone-Weierstrass Theorem in our second real analysis course, Ihope I’ve convinced you that there are a number of approaches to its primary application inelementary courses: Weierstrass’s Second Theorem. I also hope I convinced you that eventhe most elementary beginnings—in this case, the collection of polygonal functions—canlead to deep results.

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References

1. N. L. Carothers, Real Analysis, Cambridge University Press, 2000..2. D. Jackson, Fourier Series and Orthogonal Polynomials, MAA, The Carus Mathematical Mono-

graphs, Volume 6, 1941.3. H. Lebesgue, “Sur l’approximation des fonctions,” Bulletin des Sciences Mathematique, 22 (1898),

278–287.4. M. H. Stone, “Applications of the theory of Boolean rings to general topology,” Transactions of the

American Mathematical Society, 41 (1937), 375–481.5. K. Weierstrass, “Uber die analytische Darstellbarkeit sogenannter willkurlicher Functionen einer

reellen Veranderlichen,” Sitzungsberichte der Koniglich Preussischen Akademie der Wissenshcaften zuBerlin, (1885), 633–639, 789–805. See also, Mathematische Werke, Mayer and Muller, 1895,vol. 3, 1–37. A French translation appears as “Sur la possibilite d’une representationanalytique des fonctions dites arbitraires d’une variable reele,” Journal de MathematiquesPures et Appliquees, 2 (1886), 105–138.

Department of Mathematics and StatisticsBowling Green State UniversityBowling Green, OH 43403carother@bgsu.eduhttp://personal.bgsu.edu/∼carother

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