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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?

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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.

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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.

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I like the physical interpretation you give here. –  Mike Spivey Jan 31 '11 at 3:49
    
(i) When you say "move it by a little bit", do you mean "rotate it by a little bit about the point of intersection"? Otherwise I can't see how it works. (ii) Even then, it doesn't work for points of inflexion. You can rotate a tangent in one direction about a point of inflexion without intersecting the curve more than once. –  TonyK May 31 '11 at 16:33
    
@TonyK: yes. I guess I should be more precise: this should work at an inflection point if we rotate in both directions, right? –  Qiaochu Yuan May 31 '11 at 16:39
    
Maybe, but I don't like to think what might happen if the second derivative is not continuous. And, as you say, if the first derivative is not continuous, then the definition fails completely. –  TonyK May 31 '11 at 18:16

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?").

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according to me tangent is a line perpendicular to the radious of curvature of the curve at a given point.

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You have exchanged the problem of finding the line closest to the curve with the problem of finding the circle closest to the curve. This is a step backwards. –  TonyK May 31 '11 at 16:36
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This seems like a silly definition. It is harder to explain what the "radius of curvature" (if this means what I think it means) is than to explain what a tangent line is. –  Qiaochu Yuan May 31 '11 at 16:37

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