Is Slope Purely a Geometric Concept and is it Defined for a Single Point?

Question

Is “slope” purely a geometric concept? For instance can we talk about the “slope” of a function or only the “slope” of a straight line? Furthermore, can we define “slope” for a single point? We often say the slope of the curve at a point, but how can a single point have a slope?

Answer 1

The slope is for the curve (an infinite collection of points). It just happens to vary from point to point. We can use the slope as a synonym for derivative.

Answer 2

[Note: I'm going to use line and line segment interchangeably here, to save typing. I mean it to have a non-infinite length.]

We can define the slope

• of a curve
• at a point

as the slope of the unique tangent to the curve at that point. If the tangent isn't unique, then the curve doesn't have a defined slope there.

Where it gets conceptually difficult is when we try to say the following, which I imagine is what bothers you:

• a curve is made up of infinitely short line segments, which look the same as points since they have zero length.
• the slope at a point is the slope of one of these line segments.

Well, by accepted definitions, there's no such thing as an infinitely short line segment. For example the Wikipedia one:

In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints.

An infinitely short one wouldn't have two distinct endpoints.

No luck if we go back to Euclid, either: his definition was:

A line is breadthless length.

An infinitely short one wouldn't have length.

In geometric representations of differentiation, calculus sidesteps the problem: instead of having an infinitely short line, it treats the slope at a point as the limit of the slope of a line whose length approaches $0$. You can imagine it as "the slope the infinitely short line would have if it could exist", or maybe "the slope left over after the line has disappeared".

We simply define the slope of the curve at a point as being that left-over slope.

If the curve happens to be the graph of a function drawn using $x$–$y$ coordinates, then its slope at a point represents the function's derivative there. Taken literally, the "slope of the function" at a point would be a geometric metaphor based on treating the curve as the actual function. But it's become a standard alternative name for the derivative itself.