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How would you define tangent line to a curve at a given point on the curve?

I think in secondary education it is not sufficient to say a tangent line is a straight line touching a curve at a given particular point and you can not say it is a straight line which lies only in one side of the curve when it meets the curve or which have only two coincident intersecting points because those are not applicable in the case of point of inflection , as an example you can consider tangent at the origin to the curve $y = x^3$.

If you say tangent line at the point $(a , f(a))$ to the curve is the straight line going through the point $(a , f(a))$ with gradient $f'(a)$, that is the derivative of $f(x)$ at $x = a$ (as described in the Wolfram mathworld) what happens when the function is not differentiable at $x = a$?

Sometimes a tangent line can exist even without existence of derivative as an example tangent at the origin to the parabola $y^2 = 4ax$, but in some cases tangent line does not exist when the derivative can not be defined as an example there is no tangent line at origin to the curve $y = |x|$.

Therefore what should be the most appropriate way of explaining this to the students?

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    $\begingroup$ Your example of a tangent line existing without existence of the derivative is fixed by replacing "finitely differentiable" with "finitely or infinitely differentiable" (where infinitely differentiable means a 2-sided derivative equal to $+\infty$ or a 2-sided derivative equal to $-\infty).$ One can also ask what it means to be a tangent line to a curve that is not the graph of a function, or more generally, what it means to be a tangent line to an arbitrary subset of the plane -- see On tangents to general sets of points by Besicovitch (1934). $\endgroup$ Jun 28, 2022 at 8:37
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    $\begingroup$ Incidentally, when talking about curves in a fairly general way (which requires one to define exactly what is meant by "a curve"), one also needs to distinguish between the geometrical object (i.e. set of points in the plane) corresponding to the curve and the formulas and parametrizations used to describe the curve -- straight lines can have non-differentiable parametrizations, the graph of the Peano curve is the entire plane, etc. $\endgroup$ Jun 28, 2022 at 8:49
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    $\begingroup$ The entire post was written in italics, which is very distracting. I got rid of that and typeset the mathematical expressions in LaTeX. $\endgroup$
    – KCd
    Jul 9, 2022 at 20:17

2 Answers 2

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Part of our role as mathematics educators is to welcome students into the cultural practices of mathematicians.

Too often we present definitions as "set in stone". It is more honest to acknowledge that definitions in mathematics are carefully crafted to balance several competing aesthetic and practical needs. We want our definitions to capture the intuition we gain from working with the most important specific examples we have grappled with, we want them to be formulated in a way that they "play nicely" with other definitions we have already made, we want them to only admit pathology if that pathology is easily controlled (or is unavoidable in some sense). I am sure there are many other considerations that we all make. A definition is a work of art, not a commandment set down by our mathematical grandparents.

I think that you are showcasing these tensions in your original post.

If we are concerned with zero sets of polynomials, then our intuition about tangents is related to multiplicity of intersection. Following this path seriously might lead you to rediscover the theory of schemes! If we are concerned with more general curves, it is not so easy to capture tangency using algebra in this way, and we end up needing differential calculus. We would also need an "official" definition of "curve" as a precursor to making a definition of "tangent line to a curve". Whether your definition of curve is based on level sets or parameterizations will influence the definition you make for tangent lines.

As an instructor you have a choice to make with every new definition you want to introduce:

  1. Present the definition "straight from the textbook" and let the students figure it out.
  2. Present the definition "straight from the textbook" and explain, using examples, why you think the definition was chosen this way.
  3. Mimic the definition creation process. Give a few examples, create a "rough definition #1" pick holes in this definition by showing some more examples where the rough definition is inconvenient, iterate until you settle on an "official definition".
  4. Show the students examples and guide them toward creating their own definitions. Get them to pick holes in each others definitions and patch them up. If they miss a hole, pick it yourself. Continuing picking and patching until you have a workable definition. If this is different from the "official definition" you want to work with for the course, compare and contrast the two and indicate why you will be using the "official definition".

Personally I go with option #2 and #3 most often, and only use #4 if I want to make a whole class (or two!) where the focus is the definition creation process.

Here is an example of how option #4 might go down in (maybe in a Real Analysis class?):

Teacher: Here are $20$ graphs. Each graph has a curve and a line. What do you notice?

Students: [many interesting categorizations]

Teacher: Those are all interesting! What about the difference between graph #7 and graph #11? Anything interesting you can say there?

Student: In graph #7 the line is tangent to the curve, while in graph #11 it is not.

Teacher: Exactly. This is what I want to focus on today: creating a rigorous definition for what it means for a line to be tangent to a curve at a point.

I want to stress something: the point of class today is to learn what it is like to make a good definition. We will probably not end up with the same definition as the textbook (which is the definition we will use for the rest of the course). That is okay. Again, the point of the exercise is to learn how to make good definitions.

Okay, so everyone get into pairs and take a stab at creating a definition.

Students: [Create various definitions which are all flawed in some way].

Teacher: Okay now pair up with another pair to make groups of 4. Critique each others defintions.

Students: [Poke holes in and try to repair each others definitions].

Iterate this process combined with whole class discussions until a final definition is reached:

Teacher: Okay, after an hour of work we have finally come to a definition the class feels satisfied with.

Definition A line $Ax + By = C$ is said to be tangent to the curve $F(x,y) = 0$ at the point $p$ if

  1. $p$ is a point of intersection between the line and the curve.
  2. For every $\epsilon_1, \epsilon_2 > 0$ there exist $0 \leq \delta_A, \delta_B, \delta_C \leq \epsilon_1$ so that the perturbed line $(A+\delta_A)x + (B + \delta_B)y = C + \delta_C$ has at least two points of intersections with the curve $F(x,y) = 0$ in the disk centered at $p$ of radius $\epsilon_2$.

This is an amazing definition! I am really surprised and pleased that we were able to come up with this definition together. I hope that the process of creating this definition has been illuminating about the whole process of how definitions are created, and how it involves a lot of choices to be made.

In our discussion, we talked about how this definition means that (technically) we are accepting that the "curve" $0=0$ is "tangent" to every line. We were okay with that. We discussed that the curve $ y = |x|$ has multiple tangent lines through $(0,0)$. We were also okay with that!

The authors of our textbook have made different aesthetic choices. They would prefer to say that $y = |x|$ doesn't have any tangent lines passing through $(0,0)$. I hope we can all agree that this is valid. Here is the definition our textbook is using. We will need to digest this definition and also compare/contrast with the definition we developed.

[Teacher puts "official definition" on the board, and discussion continues].

To answer your question more directly though: it would depend on the course.

In an Algebra 2 course I might define a tangent line to a conic section as a line which intersects the conic section only once. I would say that this definition doesn't work for more general curves, but it works for these curves we are studying.

In a calc 1 course, I would define define the tangent line to the graph of a function $f$ only for points where $f$ is differentiable. In that case I would define it as the line $y - f(a) = f'(a)(x-a)$ where $(a,f(a))$ is the point of tangency. I might make a distinct definition for the definition of vertical tangent line, or for tangent lines to curves which are locally the graph of a function (such as $x^2 + y^2 = 1$).

In a multivariable course, I would define a smooth curve $\mathbb{C}$ as the image of an injective differentiable function $\gamma: [0,1] \to \mathbb{R}^n$ with non-vanishing derivative. I would define the tangent line to a smooth curve at the point $p$ as the line passing through the point $p$ with vector $\gamma'(t)$, where $\gamma$ is any such parameterization. This would require the proof (via the chain rule) that re-parameterization does not change the tangent line.

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  • $\begingroup$ thank you very much for giving much interesting explanation. As you know we can see this important clarifications because of the question posted but due to some other reasons question still not received upvotes. $\endgroup$ Jun 28, 2022 at 16:19
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There is no universally appropriate definition that will suit all pedagogical purposes for students at all levels. However, a pretty nice definition is given by Marsden and Weinstein, in their book Calculus Unlimited. Paraphrased:

A line L through a point P on the graph of a function f is said to be a tangent line if all lines through P with slopes less than that of L cut through the graph in one direction, while all lines with slopes greater than L's cut through it in the opposite direction.

(The notion of "cutting through" is defined by saying that in a small enough region around the given point, we're above or below the graph depending on whether we're to the left or right of the point.)

In reality, you want to give your students multiple ways to look at this, multiple ways of experiencing it.

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  • $\begingroup$ This definition would seem to say that $f(x) = x^2\sin(\frac{1}{x})$ (with $f(0) = 0$) does not have a tangent line at $x=0$. $\endgroup$ Jun 30, 2022 at 0:04
  • $\begingroup$ @StevenGubkin I believe $y=0$ satisfies that definition of the tangent line to your example. Every other line through the origin lies outside the region between the parabolas $y=\pm x^2$, except at the origin of course, in a small enough disk about the origin. But the graph of $y=f(x)$ lies inside the region. $\endgroup$
    – user1815
    Jul 11, 2022 at 3:17

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