Most students come to calculus with an intuitive sense of what a tangent line should be for a curve. It is easy enough to give a definition of a tangent to a circle that is both elementary and rigorous. (A line that intersects a circle exactly once.) Yet, when we talk about a curve, such as a polynomial, I think one must talk about infinity to give a rigorous definition. This is not a bad thing… It can help motivate a lesson on the formal definition of a tangent line at a point… But, what intermediate definition could I use to help those students who have no intuition about the matter get an idea of what we are going for before we talk about secants and such?
I have looked at some textbooks and online and I find saying a tangent line is one that "just touches" problematic– it is the kind of phrase that is quite meaningless… Even in the setting of un-rigorous definitions. Yes, I will give examples, but I would like to do better … I think saying that a tangent line is an arrow that points in the direction that the curve is going at that instant might make sense… Though it makes everything 'directional' and that might confuse them later.
What is the best intermediate definition you have seen?
Best Answer
A tangent is a line that intersects the curve once, at least if it is made short enough, but there's a direction such that if you rotate the line about the intersection point in that direction, no matter how little you rotate, it'll hit the curve again. This definition apparently goes back to Euclid in some form, and it works at inflection points, but fails in dimensions higher than $2$. (I assume you're only talking about plane curves.) I think we need the derivative to be continuous for it to work, and we also need it not to be constant in a neighborhood of the intersection point.
I don't see what's wrong with giving a physical definition though: it's where a particle moving along the curve would go if there were suddenly no forces acting on it (Newton's first law). Or, even more physically, it's the line a ball would begin to travel in if you threw the ball in an arc corresponding to the curve and let it go at the point you're interested in. This definition has the advantage that it does not depend on the background mathematics: it is (to a first approximation) an empirical fact about the universe that this concept is consistent.