Knuth, big-O calculus, implicit definitions (difficulty of)

Don Knuth says we should teach calculus without limits.

I would define the derivative by first defining what might be called a “strong derivative”: The function $f$ has a strong derivative $f'(x)$ at point $x$ if

$f(x+\epsilon)=f(x)+f'(x)\epsilon+O(\epsilon^2)$

I think this underestimates the difficulty for novices of implicit definitions. We’re quite used to them: “f'(x) is the number such that bla bla, if such a number exists, and, by the way, if such a number exists it is unique.” Students are used to definitions that say, simply, “f'(x) is bla.”

Now I will admit that the usual limit definition has hidden within it an implicit definition of the above kind; but I think the notion of limit is “physical” enough that the implicitness is hidden from the eyes of the student who is willing to understand the derivative as “the number the slope of the chord approaches as the chord gets shorter and shorter.”

Another view — for many if not most calculus students, the definition of the derivative is a collection of formal rules, one for each type of “primitive” function (polynomials, trigonometric, exponential) together with a collection of combination rules (product rule, chain rule) which allow differentiation of arbitrary closed-form functions. For these students, there is perhaps little difference between setting up “h goes to 0” foundations and “O(eps)” foundations. Either set of foundations will be quickly forgotten.

The fact that implicit definitions are hard doesn’t mean we shouldn’t teach them to first-year college students, of course! Knuth is right that the Landau notation is more likely to mesh with other things a calculus student is likely to encounter, simultaneously with calculus or in later years. But Knuth seems to say that big-O calculus would be self-evidently easier and more intuitive, and I don’t think that’s evident at all.

Maybe we could get students over the hump of implicit definitions by means of Frost:

Home is the place where, when you have to go there,

They have to take you in.

(Though it’s not clear the implied uniqueness in this definition is fully justified.)

If I were going to change one thing about the standard calculus sequence, by the way, it would be to do much more Fourier series and much less Taylor series.

8 thoughts on “Knuth, big-O calculus, implicit definitions (difficulty of)”

Jason Starr says:

May 30, 2012 at 6:41 am

Has anybody (presumably a researcher in math education) made a serious attempt to interview freshman calculus students to try to find out how they think about math concepts? After more than a decade teaching calculus, honestly I no longer pretend to understand how my students approach the subject. Of course I know the topics which draw the most complaints (limits without L’Hospital’s rule, derivatives as limits of difference quotients, integrals as limits of Riemann sums, graphing). On the other hand, I feel that I am now quite effective at helping my students learn how to solve certain “standard” types of calculus problems.

majordomo says:

May 30, 2012 at 7:38 am

Yuck Jordan, Fourier series is confusing and very counter-intuitive. It took me at least two separate classes to finally “get” Fourier series. Trust me, no standard calculus sequence should include heavy levels of Fourier series, understanding it requires a level of sophistication that is uncommon in students taking calculus for the first time.

Jeff says:

May 31, 2012 at 9:11 am

Given how little students already understand Taylor series, Fourier series would be an absolute mess. I wish math departments would double down (or triple down) on Taylor series, especially in classes geared towards engineers/scientists.

JSE says:

May 31, 2012 at 9:14 am

But it sounds like you think Fourier series are harder than Taylor series — is that true? I suppose I’ll concede that they’re harder to compute, because second-year calc students have often gotten pretty adept at mechanically computing derivatives, not so much at integration. But Fourier series seems much more physical and intuitive to me than Taylor series do. When I’ve taught both haven’t found that students have more trouble in the Fourier part than the Taylor part.

Jeff says:

May 31, 2012 at 9:46 am

I’ll admit I’m biased towards the applications I work on. Taylor series (and integration by parts) is the bedrock of numerical analysis. Plus, so many models in science/engineering basically boil down to approximating things by a few terms of a Taylor series.

One guess at why students could have an easier time with Fourier (independent of the integration problem mentioned above) is that the general formula is what they remember. My impression with Taylor series the main thing they remember is Taylor series of a small set of functions (e.g. sin, cos, 1/(1-x), etc.) instead of the general formula.

Sam says:

June 2, 2012 at 12:13 am

If we’re going to go as far as Knuth suggests, we might as well go all the way and explicitly use infinitesimals.

David Speyer says:

June 3, 2012 at 8:20 am

Would the vague and incorrect understanding students have of big O be any worse than the vague and incorrect understanding they have of limits? I’ve dreamed about teaching calculus using big O and the appeal was not that I actually expected most students to understand the hidden quantifier, but that I thought that the sloppy impression they would get would be closer to the truth.

An example: An astronomy professor explains how to compute stellar distances via parallex: The diameter of the earth’s orbit is $150 \times 10^9 m$ , the parallex of such and such star is $0.1$ seconds, or $4.8 \times 10^{-7}$ radians. Draw an isosceles triangle with vertices at the star and at the two ends of the Earth’s orbit to conclude that the distance to the star is $(150 \times 10^9/2) / \sin^{-1} (4.8 \times 10^{-7}/2) = (150 \times 10^9) / (4.8 \times 10^{-7})$ . An obnoxious student points out that $\sin^{-1}(x)$ does not actually equal $x$ . The professor replies “Yeah, but the error is $(10^{-7})^3$ , which is completely negligible.” Is the mathematical formalization of that statement more like a big O or more like a limit?

I took a reasonable number of physics courses, and one engineering course, in college, and professors said things like that all the time.

David Speyer says:

June 3, 2012 at 8:35 am

Or, similarly, here is a computation I recall from the engineering course. An I-beam supporting a building has length $L$ and thickness $t$ . It is supported at the ends, while the center is deflected downwards distance $d$ . How much does the bottom stretch by, and how much does the top shrink?

I definitely recall a solution along the following lines: Let’s approximate the curved beam as an arc of a circle with radius $R$ and angle $theta$ , so $R*\theta=L$ . Then $d=R-R \cos (\theta/2) = R \theta^2/8$ . Solving these two equations, $\theta = 8 d/L$ and $R = 8 L^2/d$ . So the top and bottom of the beam are arcs with angle $\theta$ and radii $R \pm t$ , and they have length $L \pm 8 d t/L$ .