Tag Archives: tangent

exposition, math, understanding math

Liberal-artsy people

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I graduated from Oberlin College with a B.A. as a math major and minors in philosophy and English literature, with only three semesters of science courses.  I was and am "liberal-artsy".   As professor of math at Case Western Reserve University,  I had lots of colleagues in both pure and applied math who started out with B.Sc. degrees. We did not always understand each other very well!

Caveat: "Liberal-artsy" and "Narrowly Focused B.Sc. type" (I need a better name) are characteristics that people may have in varying amounts, and many professors in science and math have both characteristics.   I do, myself, although I am more L.A. that B.Sc.  Furthermore, I know nothing about any sociological or cognitive-science research on these characteristics.  I am making it all up as I write.  (This is a blog post, not a tome.)

I recently posted on secants and  tangents.  These articles were deliberately aimed to tickle the interests of L.A.  students.

Liberal-artsy types want to know about connections between concepts.  In each post, I wrote on both common meanings of the words (secant line and function, tangent line and function) and the close connections between them.  Some trig teachers / trig texts tell students about these connections but too many don't.   On the other hand, many B.Sc. types are left cold by such discussions.  B.Sc. types are goal-oriented and want to know a) how do I use it? b) how do I calculate it?  They get impatient when you talk about anything else.  I say point out these connections anyway.

L.A. types want to know about the reason for the name of a concept.  The post on secants refers to the metaphor that "secant" means "cutting". This is based on the etymology of "secant", which is hidden to many students  because it is based on Latin.  The post makes the connection that the "original" definition of "secant" was the length of a certain line segment generated by an angle in the unit circle. The post on tangents makes an analogous connection, and also points out that most tangent lines that students see touch the curve at only a single point, which is not a connotation of the English word "touch".

Many people think they have learned something when they know the etymology of a word.  In fact, the etymology of a word may have little or nothing to do with its current meaning, which may have developed over many centuries of metaphors that become dead, generate new metaphors that become dead, umpteen times, so that the original meaning is lost.  (The word "testimony" cam from a Latin phrase meaning hold your testicles, which is really not related to its meaning in present-day English.)

So I am not convinced that etymologies of names can help much in most cases.  In particular, different mathematical definitions of the same concept can be practically disjoint in terms of the data they use, and there is no one "correct" definition, although there may be only one that motivates the name.  (There often isn't a definition that motivates the name.  Think "group".)  But I do know that when I mention the history of a name of a concept in class, some students are fascinated and ask me questions about it.

L.A. types are often fascinated by ETBell-like stories about the mathematician who came up with a concept, and sometimes the stories illuminate the mathematical idea.  But L. A. types often are interested anyway.  It's funny when you talk about such a thing in class, because some students visibly tune out while others noticeably perk up and start paying attention.

So who should you cater to?  Answer:  Both kinds of students.  (Tell interesting stories, but quickly and in an offhand way.)

The posts on secants and tangents also experimented with using manipulable diagrams to illustrate the ideas.  I expect to write about that more in another post.

For more about the role of definitions, check out the abmath article and also Timothy Gowers' post on definitions (one of a series of excellent posts on working with abstract math).


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exposition, language, math, Uncategorized, understanding math

Tangents

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The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook Tangent Line.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

This is an experiment in exposition of the mathematical concepts of tangent.  It follows the same pattern as my previous post on secant, although that post has explanations of my motivation for this kind of presentation that are not repeated here.

Tangent line

A line is tangent to a curve (in the plane) at a given point if all the following conditions hold (Wikipedia has more detail.):

  1. The line is a straight line through the point.
  2. The curve goes through that point.
  3. The curve is differentiable in a neighborhood of the point.
  4. The slope of the straight line is the same as the derivative of the curve at that point.

In this picture the curve is $ y=x^3-x$ and the tangent is shown in red. You can click on the + signs for additional controls and information.

Etymology and metaphor

The word “tangent” comes from the Latin word for “touching”. (See Note below.) The early scholars who talked about “tangent” all read Latin and knew that the word meant touching, so the metaphor was alive to them.

The mathematical meaning of “tangent” requires that the tangent line have slope equal to the derivative of the curve at the point of contact. All of the red lines in the picture below touch the curve at the point (0, 1.5). None of them are tangent to the curve there because the curve has no derivative at the point:

The curve in this picture is defined by

The mathematical meaning restricts the metaphor. The red lines you can generate in the graph all touch the curve at one point, in fact at exactly at one point (because I made the limits on the slider -1 and 1), but there are not tangent to the curve.

Tangents can hug!

On the other hand, “touching” in English usage includes maintaining contact on an interval (hugging!) as well as just one point, like this:

The blue curve in this graph is given by

The green curve is the derivative dy/dx. Notice that it has corners at the endpoints of the unit interval, so the blue curve has no second derivative there. (See my post Curvature).

Tangent lines in calculus usually touch at the point of tangency and not nearby (although it can cross the curve somewhere else). But the red line above is nevertheless tangent to the curve at every point on the curve defined on the unit interval, according to the definition of tangent. It hugs the curve at the straight part.

The calculus-book behavior of tangent line touching at only one point comes about because functions in calculus books are always analytic, and two analytic curves cannot agree on an open set without being the same curve.

The blue curve above is not analytic; it is not even smooth, because its second derivative is broken at $x=0$ and $x=1$. With bump functions you can get pictures like that with a smooth function, but I am too lazy to do it.

Tangent on the unit circle

In trigonometry, the value of the tangent function at an angle $ \theta$ erected on the x-axis is the length of the segment of the tangent at (1,0) to the unit circle (in the sense defined above) measured from the x-axis to the tangent’s intersection with the secant line given by the angle. The tangent line segment is the red line in this picture:


This defines the tangent function for $ -\frac{\pi}{2} < x < \frac{\pi}{2}$.

The tangent function in calculus

That is not the way the tangent function is usually defined in calculus. It is given by \tan\theta=\frac{\sin\theta}{\cos\theta}, which is easily seen by similar triangles to be the same on -\frac{\pi}{2} < x < \frac{\pi}{2}.

We can now see the relationship between the geometric and the $ \frac{\sin\theta}{\cos\theta}$ definition of the tangent function using this graph:


The red segment and the green segment are always the same length.
It might make sense to extend the geometric definition to $ \frac{\pi}{2} < x < \frac{3\pi}{2}$ by constructing the tangent line to the unit circle at (-1,0), but then the definition would not agree with the $ \frac{\sin\theta}{\cos\theta}$ definition.

References

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