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In calculus classes it is sometimes said that the tangent line to a curve at a point is the line that we get by "zooming in" on that point with an infinitely powerful microscope. This explanation never really translates into a formal definition - we instead approximate the tangent line by secant lines.

I seem to have found a way to obtain tangent lines (and more) by taking "zooming in" seriously.

Example 1

Take the curve $y = x(x-1)(x+1)$.

I want to find an equation for the tangent line to this curve at the origin. So I zoom in on the origin with a microscope of magnification power $c$ (i.e. I stretch both vertically and horizontally by a factor of $c$) to obtain

$\frac{y}{c} = \frac{x}{c}(\frac{x}{c} - 1)(\frac{x}{c}+1)$.

Multiplying through by $c$ I have

$y = x(\frac{x}{c} - 1)(\frac{x}{c}+1) $

Now letting my magnification power go to infinity I have

$y = -x$

Which is the correct answer.

Example 2

Take the curve $y = x^2$.

I want to find an equation for the tangent line to this curve at the point (3,9). I first rewrite the equation as

$(y-9) + 9= ((x-3) + 3)^2$

so that I am focusing on the appropriate point. To zoom on this point with magnification $c$ I have

$\frac{y-9}{c} + 9 = (\frac{x-3}{c} + 3)^2$.

$\frac{y-9}{c} + 9 = \frac{(x-3)^2}{c^2} + 6\frac{x-3}{c} + 9 $ Multiplying through by $c$ I have

$y - 9 = \frac{(x-3)^2}{c} + 6(x-3) $

Now letting my magnification power $c$ go to infinity I have

$y - 9 = 6(x-3)$

Which is the correct answer.

Example 3

Here is the example which actually motivated me to consider this at all:

Take the curve $y^2 = x^2(1 - x)$.

This is a cubic curve with a singularity at the origin, and so it doesn't really have a well defined tangent line. It sort of looks like it should have two tangent lines (y = x, and y = -x), but it is a little bit tricky to formalize this. Let's see what "zooming in" does:

$\frac{y^2}{c^2} = \frac{x^2}{c^2}(1 - \frac{x}{c})$

$y^2 = x^2(1 - \frac{x}{c})$

Letting $c$ go to infinity I have

$y^2 = x^2$, or $(y-x)(y+x) = 0$, which is the pair of lines I desired.

My Questions

  1. Do any books take this approach when developing the derivative?

  2. I would imagine that algebraic geometers do this kind of thing formally. Is there a more rigorous analogue of the prestidigitation I engage in above? Where would I look to read up on such things?

p.s. It would be nice to illustrate each of these examples with a little movie of the "zooming in" process, but I am not sure how to put such things on MO. Any hints?

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Off the top of my head: on the one hand, this looks to me like "just" being the local linear approximation definition of the derivative (cf. multivariable defns), but I've not seen it presented/motivated this way before. In partial answer to your Q2, I guess you could work in the ring $R[\epsilon]$ with $\epsilon^2=0$. –  Yemon Choi Oct 4 '11 at 21:02
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This sort of homothetic scaling is the standard way to think about tangent objects in geometric measure theory. You are just looking at the tangent cone to the graph of your function. –  Rbega Oct 4 '11 at 21:09
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This is explored seriously in Keisler's nonstandard calculus text Elementary Calculus; his treatment of the microscope methodology is based on work of Keith Stroyan. –  Robert Haraway Oct 4 '11 at 21:11
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I find your approach to derivatives really cool! Have you tried to interpret what you are doing in terms of projective geometry? (the substitution x---> x/c; y ---> y/c really reminds me of the homogenization process) –  Tommaso Centeleghe Oct 4 '11 at 21:23
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Incidentally, the zooming in interpretation of a derivative is item (7) on Thurston's list of ways of thinking about this concept at his well-known essay "On proof and progress in mathematics" at arxiv.org/abs/math/9404236 –  Terry Tao Feb 5 '13 at 18:43

9 Answers 9

up vote 24 down vote accepted

In algebraic geometry, this construction is known as the tangent cone to the graph. More generally, suppose we have the zero set of any polynomial $f(x,y) = 0$, and assume $f(0,0)=0$. Then we can write

$f(x,y) = a_m (x,y) + a_{m+1}(x,y) +a_{m+2}(x,y) +\cdots$

where $a_i(x,y)$ is a homogeneous polynomial of degree $i$ and $a_m$ is nonzero. The zero set of $a_m$ is called the tangent cone to the curve at the origin. It is a product of $m$ linear forms (over $\mathbb{C}$), and $m=1$ exactly when the zero set is smooth at the origin. In this case, the tangent cone coincides with the tangent space.

From your point of view, when we substitute $x\mapsto x/c$ and $y\mapsto y/c$ it is clear that the term left in the limit is $a_m$.

We can of course find tangent cones at other points of the zero set by changing coordinates.

In general, for a smooth function $f$ you should be able to take a multivariate Taylor expansion and read off the tangent cone from the lowest degree part. This is where the difficulty comes in for actually defining the tangent line in terms of the tangent cone in a calculus class, as computing the Taylor expansion demands we already have a notion of derivative. This difficulty is obviously not seen in the case of polynomials, although recentering the Taylor expansion of a polynomial at a different point is perhaps easiest done with the aid of derivatives.

Higher dimensional analogues are also available without any real work, although in the singular case the tangent cone is much more interesting than just a union of hyperplanes: it will be a cone over some variety. The homogeneous polynomial $a_m(x_1,\ldots,x_n)$ typically doesn't factor into a product of linear forms when $n>2$.

Tangent cones are treated in any reasonable introduction to algebraic geometry, such as Harris' "First course" book or Shafarevich.

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An animation of your first example, $y=x(x−1)(x+1)$. The animation is flawed in that it jumps jarringly when the tick labels reach the boundary. I removed the ticks and labels, which gains smoothness at the expense of magnitude feedback. (The previous version is here.) And I limited the number of frames so that the file would not be too huge (it is ~1MB now). Frame rate is browser and processor dependent. At best this gives an idea of what a more professional animation might look like.
           alt text

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Awesome. $ $ –  Steven Gubkin Oct 5 '11 at 16:55

I haven't seen any Calculus books take this approach. Possibly, this is because it is not clear how to proceed in the case of functions that are not polynomials. Consider even the really nice function $y = e^x$ and say you are interested in the tangent line to the graph at (0,1).

Using your scheme and the fact that $e^{a+b} = e^a\cdot e^b$, one writes $\frac{y-1}{c} = e^\frac{x}{c}$. This leaves us with
$$ y = 1 + c \cdot e^\frac{x}{c} $$

While taking the limit as $c \to \infty$ certainly gives the correct answer $y = 1+x$, I am not sure how to actually compute this limit without using either Taylor series or L'Hospital's rule, both of which already require you to know the derivative of $e^x$.

Of course you will run into similar difficulties if you try to compute the derivative of this function from the limit-based definition directly. If I remember correctly, standard textbooks on Calculus get around this annoyance either by declaring the answer without proof or by using implicit differentiation and $\ln(y) = x$. It is not too difficult to obtain the derivative of the natural log function from first principles.

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Well, if you say what $e$ is without attempting to motivate it, then it's easy enough to give a proof. That's the approach of the calculus book I'm using (Stewart). –  Frank Thorne Oct 4 '11 at 23:46
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Small mistake: you should have $\frac{y-1}{c} + 1 = e^{\frac{x}{c}}$ so that $y = 1 + c(e^{\frac{x}{c}} - 1) \to 1+x$. –  Faisal Oct 5 '11 at 1:23
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Nonstandard analysis handles things like $\lim_{c\rightarrow\infty} c\cdot e^{\frac{x}{c}}$ by first defining real powers as standard parts of finite hyperrational powers. My approach is to prove that the standard part of $\frac{a^\delta-1}{\delta}$, where $\delta$ is an arbitrary nonzero rational infinitesimal and $a$ is a standard postive real, equals $\int_1^a\frac{dx}{x}$. Then I define $e$ via $\int_1^e\frac{dx}{x}=1$. The pay-off is that $d(e^x)/dx=e^x$ becomes true by definition. –  David Milovich Oct 5 '11 at 17:50

The tangent space $T_pM$ of a Riemannian manifold $M$ with inner metric $d$ can be realized as the pointed Gromov-Hausdorff limit as $\lambda\rightarrow\infty$ of $(\lambda M, p)$ where $\lambda M$ is $M$ with metric $\lambda d$.

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As Rbega says in the comments, if you are really keen to see this rescaling idea put to use in a more rigorous or advanced way, then you can look at some Geometric Measure Theory. While it will look very technical compared to this (because it is designed for potentially badly-behaved or very weakly-defined geometric objects), this sort of homothetic blowing up is standard for defining tangent objects to things. You get a weak kind of convergence of the rescalings of your original object to the tangent object, which, depending on the circumstances, may well (or perhaps will hopefully) display some sort of rigidity, e.g. it may be have to be a cone. It is rigorous and yes you can indeed end up with things like the union of two lines as your tangent object.

In the special case of the graph of a differentiable function, the tangent object at a point will indeed be the graph of the affine function associated with the derivative at the point.

I don't know of any books which take this approach pedagogically, in the development of calculus though.

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I felt I should add that, one of the things that happens with working with such a weak notion of tangent is that tangent objects need not be unique. In other words instead of scaling by a continuous variable $\epsilon\to 0$ you scale by a sequence of numbers $\epsilon_i\to 0$ then a different sequences of $\epsilon_i'\to 0$ might give you a different tangent object. Showing this can't happen in "good" situations is one of the hardest problems in the field... –  Rbega Oct 4 '11 at 23:47
    
Cool! I will have to check out some books on Geometric Measure Theory. –  Steven Gubkin Oct 5 '11 at 16:53

As AmbroseH commented, Keisler's book takes this pedagogical approach. The catch is that his book is old, out-of-print, and uses the infintesimal approach. He talks of "infinite microscopes" and "infinite telescopes" (the latter for horizontal and vertical asymptotes of curves) and has illustrations (sadly without color or animation). You can download the book from his website (Creative Commons License).

You can find more recent notes (not full textbooks yet) that sketch how to teach calculus using various flavors of infinitesimals (without sacrificing a rigorous mathematical foundation):

  1. Keisler uses Robinson-style nonstandard analysis. A more recent variation on this is relativized nonstandard analysis. Instead of two "levels," standard and nonstandard, it's turtles all the way down. See these slides for an introduction and citations.

  2. A very different flavor is smooth infinitesimal analysis, which uses nilsquare infinitesimals (and may use intuitionistic logic). Here is one exposition.

  3. The idea of nilsquare infinitesimals generalizes to nilpotent infinitesimals. This is discussed here, for example.

Since you tagged your question algebraic geometry, my guess is that you'd prefer option 2 or 3.

Finally, you might find this relevant; there's an even older (but not out of print) calculus book that has wonderful pedagogy but implicitly throws around nilsquares without any pretense of rigor.

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Hugh Thurston explored this point of view (or, rather, the tangent-cone reformulation described by Jack Huizenga) in a number of articles in Amer. Math. Monthly and Math. Mag. in the mid-'60's: http://www.ams.org/mathscinet/search/publications.html?pg1=AUCN&s1=Thurston%2C+H%2A&co1=AND&pg2=ALLF&s2=tangent.

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@L Spice: I tried to edit your post by fixing the second link but the MO editor did not like the "H*" for the first name in the author search. I changed it to "Hugh" but that drops two out of the five found items: the two papers in the Monthly. –  Abdelmalek Abdesselam Feb 5 '13 at 17:45
    
@Abdelmalek Abdesselam, I hadn't noticed that the link was problematic; thanks! I think I've fixed it now by percent-encoding the *. –  L Spice Feb 5 '13 at 18:25

For freshman calculus students, I try to explain we're just putting a "microscope" to the different curves and we usually get one of four pictures.

alt text

But picturing the kinds of pathologies that can occur is much harder as you move beyond calculus. The Cantor function. We can picture the Cantor function as flat if you zoom in outside the Cantor set, but the "devil's staircase" if you zoom in at a Cantor set point. Brownian motion will never ``calm down" no matter how much you zoom in -- it does not have bounded variation.

It would be nice to have a picture book of these pathologies.

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I would say you have not applied the "microscope" to the first of your examples: it would become a horizontal line... Also the last example seems not to be quite right (as a result of a zooming in process). –  Qfwfq Oct 5 '11 at 19:11
    
Also, the four pictures don't seem to be exhaustive of all the possible behaviours of "elementary functions". –  Qfwfq Oct 5 '11 at 19:18
    
no they are not... but freshman students are not even aware of these yet. –  john mangual Oct 5 '11 at 19:40

Excellent question! One of the earliest advocates of the "zooming in" approach is the mathematician and education specialist David Tall of Warwick; see e.g. http://www.tallfamily.co.uk/david/papers/2.embodied-calculus His latest related paper is a reevaluation of Cauchy's legacy here. The basic idea is that ideas of zooming and informal infinitesimals should be introduced before any formalisation takes place. Once the student has a grasp of the key concepts, the course can branch out either into standard or non-standard calculus.

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