What is an intermediate definition for a tangent to a curve?

Solution 1:

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.

Solution 2:

One possible way to introduce a tangent to a curve at a point is to do as follows. Take a point on the curve where we want to define a tangent to the curve. Consider all set of lines passing through the point. There will be a unique line such that in a neighborhood of the curve the entire curve in the neighborhood lies on only one side of the line. But of course this way of motivating will fail if we are near a point of inflexion.

Solution 3:

My favorite calculus-level definition (which is fairly rigorous, not hard to motivate, and does not rely on pictures) is that the tangent line to $y=f(x)$ at $x_0$ is the (unique) line that goes through $(x_0,f(x_0))$ and affords the best linear approximation to $y=f(x)$ near $x_0$. That is, if you let $g(x)$ be the point on the line with coordinate $x$, then $f(x)-g(x)$ goes to zero faster than $x-x_0$ goes to $0$; that is, $\frac{f(x)-g(x)}{x-x_0} \to 0$ as $x\to x_0$. This captures the idea that the tangent is the line that "best approaches" the graph when you are near the point $x_0$.

See about halfway through this previous answer (starting in paragraph 7, where it says "Now, does the line that join $A$ and $B$ really have a slope that approaches the slope of the tangent?").