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## Fun with obfuscated definitions

I am teaching College Algebra and Pre-Calculus this year out of two college textbooks, Ratti & McWaters for the former and Blitzer for the latter. I happened to be teaching conic sections in both classes on the same day. Here’s the Pre-Calculus take on “parabola”:

A parabola is the set of all points in a plane that are equidistant from a fixed line, the directrix, and a fixed point, the focus, that is not on the line.

Simple, clear, exact.

And here is College Algebra (mind you, the “easier” course meant for those not necessarily taking any higher math):

Let l be a line and F a point not on the line l. Let P be the plane determined by F and l. Then the set of all points P in the plane that are the same distance from F as they are from the line l is called a parabola. Thus, a parabola is the set of points P for which d(F,P) = d(P,l), where d(P,l) denotes the distance between P and l.

Er, what?

### 8 Responses

1. A wonderful question for advanced students pops up in comparing the two… If the plane in question is NOT the Plane determined by F and l, Is the locus of points still a parabola if:
A) The plane is parallel to the plane F,l
B) The plane is skew to plane F,l

2. I actually don’t mind the second definition so much, except for the weirdly asymmetric “d(F,P) = d(P,l)”.

Replacing the commas by parentheses in the first definition would be a big improvement for me. As it is, it’s easy to read it as “equidistant from 1) a fixed line, 2) the directrix, and 3) a fixed point”, and then have to back up. Or the names for the components could be left for a second sentence.

• I suppose I am mostly upset by the inclusion of letters; there might be a point to that if it continued with a proof of some sort, but the definition is it. It’s like defining variables in programming that don’t get used.

And surely the prose structure is non-optimal.

3. I have the impression that many low-level textbooks have a somewhat premodern viewpoint (for example, emphasizing geometric definitions over analytical ones). In a college algebra course and a first calculus course essentially everything will be in the $xy$ plane, so what is the purpose of looking at parabolas in other planes in an algebra course? Another example: a calculus book that I like appears to claim the derivative of $\sin(x)$ would be different if $x$ was measured in degrees instead of radians. There is certainly something that is different there when we want to use degrees instead of radians, but nothing is different enough for a single function to have multiple derivatives. When I inquired about this, it was explained that the authors view sine as a function of an angle rather than a function of a real variable.

• Which calculus book?

• The author is correct. $\frac{d}{dx} sin(x^\circ) = \frac{\pi}{180} \cos(x^\circ)$.

• Sorry, my equation didn’t render. I meant that the functions f(x) = sin(x) and g(x) = sin(x°) clearly have different derivatives, since sin(x°) = sin(x*pi/180), and the author is correct.

4. In the second definition, I would like to see the algebraic condition handled (demonstrated) separately from the locus definition; as is, it rambles a bit. Although as suggested, this definition does have the virtue of specifying that the parabola is in the plane determined by focus and directrix.

As to Carl’s remark about the sine function, we overload sin() to mean two different things: sin(x) and sin((pi/180) x). It doesn’t make mathematical sense to imply that a function has more than one derivative, but sin() is really two different functions, and those different functions have different derivatives. For me, it’s easier to think about it having a “different derivative” than trying to untangle those two functions everywhere