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