Equation of tangent to curve y=lnx/sqrt(x) @ (1,0)?

Yikes! Are we really consigning the ancient Greeks and everyone without calculus to not be able to understand the definition for a tangent?! Somehow, calculus oughtn't to make us forget geometry.

Don't we have a notion that a line is tangent to a curve if it approaches the curve, intersects with it just barely, and then "glances" off? More formally: Line L is tangent to curve C at point P if both C and L pass through P and there exists a circle B about P such that no pair of points of C within P lie on opposite sides of L (In other words, in a local neighborhood around P, all of C's points lie on the same side of L). "Pass through" ensures that P is not isolated on C.

Something like: C passes through P if for every circle B' around P contained in B, there is a line L' through P such that there are at least two points of C - P in B' on opposite sides of L'. Calling y=0 tangent to y=x³ comes from taking parts of two curves (such as -|x³| and |x³|) whose tangents agree at a given point but where the concavity flips. I have always been unhappy with calling this line tangent when it clearly isn't.

Also, by my definition, lines are tangent to themselves, and any line through (0,0) whose slope was <= 1 would be a tangent to y=|x|.

Actually, it's more like a continuum of lines, given that the "moving" point has a continuous set of positions leading up to the limit. Don't get me wrong...I know what's meant by the definition. I just can't think of a rigorous definition of such a limit, offhand.

I'd like to point out that there is more than one kind of tangent. The original definition only applied to circles: "A tangent line intersects the circle at just one point." That's easily extended to arbitrary convex closed curves by adding ", if and only if it is the only line to do so." to the end; but I'm not so sure the addition is all that desirable.

I see no particular problem with a convex polygon having multiple "tangents" at the vertices. The reason I bring this up: It seems to me that "tangent" is a geometric concept and ought to have a geometric definition. Analytic geometry is fine, but it's tied to a coordinate system.

The tangent line that I imaging when somebody says "tangent line" is independent of any coordinate system and doesn't blow up just because dy/dx may not be defined in the some coordinate system that's in use. Mind you, I'd *use* the calculus version in practice. I certainly wouldn't want to do an "existence and uniqueness" proof every time I solved for a tangent line--which is what the convex-curve tangent above requires.

I just like definitions to be rooted somewhere, without wandering off into possible circularity. "What's a tangent?" "Thats the line with the same derivative." "Okay, but what's a derivative?" "Oh, that's the slope of the tangent line." Edit: siamese_scythe has an acceptable definition for the limit of a secant lines, which will work so long as the curve doesn't intersect itself.

Or, at least not in some neighborhood of the point of tangency. I'm still voting for that answer. Quadrillerator has an undeserved thumbs-down.

There are problems, (like which side of the straight line are the points of the straight line on?) but ones that can be cleared up. Rakesh Dubey also offers a "pure tangent" idea that's interesting. From Merriam-Webster's 11th Collegiate Dictionary, "tangent" derives from the Latin tangere: "to touch".

That fits with the definition from Euclid, Book III, Defintion 2: A straight line is said to touch a circle, when meeting it and being produced does not cut it. Google Books has a digitized copy of a 1765 edit of an early translation. Page 450-451 contain remarks about how this definition is deficient for the idea of a tangent for curves other than conic sections.

http://books.google.com/books?id=kT44AAA... One of the points mentioned there is the same as Rakesh Dubey's "local tangent that is still a secant" idea, that the straight line can, in fact, just touch a curve in one place and yet cut it in another. The tangent to a circle has a few useful properies: 1. It intersects the circle at a single point.

2. The remaining points of the tangent line are all outside the circle. (i.e.

The tangent line does not "cut" the circle.) 3. The direction of the line is the same as the "instantaneous" direction of the circle at the point of tangency. Euclid specifies properties 1 and 2 in his definition for circles, while calculus specifies property 3.

You can't have all three for a general smooth curve. For a calculus definition, I like the vector calculus approach. A curve is defined by parmetrization using the vector R(t) = (x(t), y(t)), where x and y are continuous, piecewise differentiable functions of t, and |R'(t)| is nonzero wherever both x'(t) and y'(t) exist.

Then the tangent at some point R_0 = R(t0) is the line through R_0 with the direction given by R'(t0). If R'(t) is undefined at t0, there is no tangent defined by R at t=t0. That last remark is a nod to Rita the Dog, who pointed out the problem of self intersecting curves.

A simple example is a figure-8 that crosses itself at right angles. One parametrization may go smoothly around the curve, with well-defined tangents everywhere, including two different ones at the intersection point. Another my take right-angled turns at the intersection, and produce no definition of either tangent at the intersection.

It's messy, but it does avoid the problem of vertical tangents.

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