Is "locally linear" an appropriate description of a differentiable function?

Question

In this answer on meta, Pete L. Clark said:

I think the question concerns the idea that a differentiable curve becomes more and more like a straight line segment the closer one zooms in on its graph. (And I must say that I regard part of this confusion as an artifact of badly written recent calculus books who describe this phenomenon as "local linearity". Ugh!)

So, what's wrong with calling it "local linearity"? (Examples of the specific language from some relatively recent books follow.)

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From Finney, Demana, Waits, and Kennedy's Calculus: Graphical, Numerical, Algebraic, 1st ed, p107:

A good way to think of differentiable functions is that they are locally linear; that is, a function that is differentiable at a closely resembles its own tangent line very close to a.

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From Hughes-Hallett, Gleason, et al's Calculus: Single Variable, 2nd ed, pp138-9:

When we zoom in on the graph of a differentiable function, it looks like a straight line. In fact, the graph is not exactly a straight line when we zoom in; however, its deviation from straightness is so small that it can't be detected by the naked eye.

Following that, there is discussion of the tangent line approximation, then a theorem titled "Differentiability and Local Linearity" (the first time "local linearity"/"locally linear" appears) stating that if a function f is differentiable at a, then the limit as x goes to a of the quotient of the error in the tangent line approximation and the difference between x and a goes to 0.

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Ostebee and Zorn's Calculus from Graphical, Numerical, and Symbolic Points of View, 1st ed, p110:

Remarkably, the just-illustrated strategy of zooming in to estimate slope almost always works. Zooming in on the graph of almost any calculus function $f$, at almost any point $(a,f(a))$, eventually produces what looks like a straight line with slope $f'(a)$. A function with this property is sometimes called locally linear (or locally straight) at $x=a$. [Margin note: These aren't formal definitions, just descriptive phrases.] Local linearity says, in effect, that $f$ "looks like a line" near $x=a$ and therefore has a well-defined slope at $x=a$.

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(I did not find the term "local linearity" or "locally linear" at a quick glance in Stewart's Calculus: Concepts and Contexts, 2nd ed, or Leithold's The Calculus 7; the rest of the calculus books I have on hand predate the inclusion of graphing calculators/software in textbooks, so are not suitable for comparison.)

Answer 1

None of the books you quoted actually called differentiability "local linearity"; they just used as a good analogy. It is a good analogy, but it is not a good definition. A lot of mathematical terminology, especially from topology, uses the word "local," but it is almost always used with the same purpose. A locally compact set is one where points have arbitrarily small compact neighborhoods. A locally connected set is one where points have arbitrarily small connected neighborhoods. On the other hand, a non-linear differentiable function is not linear on any neighborhood of a point about which it is differentiable. It just looks linear.

Of course, such functions do have tangent lines, which is an equivalent definition of differentiability. I don't think that a tangent counts as local linearity, though.

If you're asking whether people learning calculus should be taught the words "local linearity" instead of "differentiability," I mean, that isn't really much more helpful than using the analogy but keeping the terminology. If you're asking whether we should use the analogy at all, I don't see why not, as long as it's clear that linearity is just an approximation.

EDIT: "Locally linear" only describes differentiable functions $\mathbb{R}\rightarrow\mathbb{R}$. "Differentiable" can be extended to functions between arbitrary differential manifolds. I think it's better to keep the extensible definition.

Answer 2

The "microscope" or "zooming in" explanation of differentiation is basically nonsense. It contradicts the notion of zooming in actually used in calculus (asymptotic expansion of a function near a point, revealing finer and finer information). Magnification matches the mathematics only at order 0 and 1, that is, it captures linear information but also destroys the higher order picture.

Suppose that $f(0)=0$. The $N$-fold magnification of the graph near 0 is the plot of $Nf(x/N)$. This is the straight line $xf'(0)$ to within an error that shrinks to an invisible size as the picture is "zoomed" by increasing $N$. This will, literally, linearize any differentiable function. The higher order approximations -- whose effect is beautifully visible in a fixed scale picture of the first several Taylor polynomials for $f(x)$ near 0 all superimposed on the same coordinate system -- are made indistinguishable from a straight line by flattening everything with the magnification. This is a step backward considering that the difference between a tangent line and an osculating circle is easily illustrated with any curve, and many people see this picture in some form before learning calculus.

What one actually wants to display are the effects away from the point, with the higher approximations being visibly closer to graph over a longer range. The "zooming" picture may be defensible if presented together with these more informative images, but by itself as an explanation of the derivative as a local linear approximation, it is using diagrams that tell a story opposite to the mathematical concepts involved.

Answer 3

I see that I never answered or commented on this question, probably because I was curious to see if others would agree and be able to divine my objections without my explicit input.

This indeed happened: my main objection was that this use of "locally" is at odds with any other nearby use of the term in mathematics. Especially, a topological space $X$ should be locally P if every point admits a base of neighborhoods each having property P. Under this definition, a locally linear function on an interval must actually be linear!

I didn't mean to imply that there was no rigorous mathematical concept behind this terminology. Indeed the "infinite zooming in" can be made precise, as T.. did in his answer above. This really does give a (not completely trivial) characterization of differentiable functions.

However, is this an important or useful intuition for beginning calculus students? I don't think so. I don't think it is meant to be taken very seriously, because if you start taking it seriously you'll find yourself asking what else can happen to a continuous curve upon "infinite zooming in", and then you're well on your way to self-similarity and fractal geometry. This is fascinating mathematics, but it is of course not part of the story of differential calculus.

In fact, as you can see from the above quotes, most of the texts which use the term "local linearity" take some care to emphasize its informality: it is a way of thinking about differentiable functions. However, what they don't explain -- and what is not apparent, even to many people who are both research mathematicians and veteran calculus teachers -- is why we are introducing this (mathematically valid, but not directly mathematically relevant) analogy at the very beginning of the story of differential calculus. Speculating from what I've seen, this is part of a relatively recent wave of calculus texts (I believe the trend started after I myself took calculus, in the early 1990's) which (i) want to start talking about derivatives right at the beginning of the text, but (ii) don't want to get bogged down in any of the attendant technicalities or look like they are giving incomplete explanations. The text that I used as a graduate student teaching calculus in the late 1990's did something similar to this: they talked about "slope-predictors" in Chapter 1 and saved "tangent lines" for Chapter 2.

Perhaps this pedagogical choice is defensible, but it certainly does require defending. Purely as it lies in the calculus text, I do not agree with it at all. Differential calculus has a rich enough cast of characters and ideas; it does not seem wise to make up more terminology and introduce other themes and ideas which will not be followed up on later in the course. When I teach freshman calculus (please note: despite the fact that I have taught freshman calculus more than a few times, if you trust the student evaluations, I am less good at it than the average graduate student in my department: caveat emptor!) I like to get the main ideas out into the open as soon as possible, so I spend the entire first lecture talking about tangent lines and instantaneous velocity. I start here because students have been trained in both of these concepts in their precalculus mathematics (and physics), so generally do have some intuition about these things. However, their intuition falls rather short of an acceptable general definition of either of these important concepts (if things seem to be going well, I may ask for various definitions of tangent lines and then draw counterexamples to them!). What one soon sees is that there is in each case a closely related, but technically much simpler definition -- secant lines and average velocity -- and the matter of it is to explain e.g. how we get from a bunch of average velocities to the instantaneous velocity.

In this way, on the first day I try to set up about the first third to half of the course: we want to compute tangent lines (I try to say a little bit about why, e.g. minimizing and maximizing functions and graphing), and for that we need to learn a little bit about limits. These concepts will get reinforced again and again throughout the rest of the course. Infinite zooming in will never come up again, so why bring it up at all, and especially right at the beginning?

Answer 4

@Paul and @Issac's comments on a wrong answer I had initially posted made me see why "linear on zooming in" is the right picture of differentiability. Suppose $f$ is some function and

$$g(x)=f(a)+m(x-a)$$

is a proposed linear approximation to it at a point $a$.

Suppose I zoom in on the graph of $f$ and $g$ by a multiple $\lambda>0$ and look at a point $h$ distance apart from $a$ in my zoomed coordinates. Then the vertical error in the approximation in my zoomed coordinates is

$$e(\lambda)=\lambda[f(a+h/\lambda)-g(a+h/\lambda)] =\lambda[f(a+h/\lambda)-f(a)]-mh$$

If I keep zooming the approximation will look better and better only if $\lim_{\lambda \to \infty} e(\lambda)=0$. This will be the case only if $f$ is differentiable at $a$ and $f'(a)=m$.