Precise definition of tangent line to a curve

[MiddleEast]

Homework Statement

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Relevant Equations

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How do we define tangent line to curve accurately ?
I cannot say it is a straight line who intersect the curve in one point because if we draw y = x^2 & make any vertical line, it will intersect the curve and still not the tangent we know. Moreover, tangent line may intersect the curve at other points other than the tangency point.
What is the precise definition for the tangent line?

 

[fresh_42]

Homework Statement:: NA
Relevant Equations:: NA

How do we define tangent line to curve accurately ?
I cannot say it is a straight line who intersect the curve in one point because if we draw y = x^2 & make any vertical line, it will intersect the curve and still not the tangent we know. Moreover, tangent line may intersect the curve at other points other than the tangency point.
What is the precise definition for the tangent line?

Right. The tangent can intersect the curve multiple times. It is a local property, i.e. in a small neighborhood of a certain point. And this isn't a typical intersection point, it is a touching point. Since you posted this question in the pre-calculus section, we only have a geometrical definition.

Say $x_0$ is the point that we want to consider locally. Then we choose another point $x_1$ nearby and draw a straight $\overline{x_0x_1}.$ In case the tangent exists, which is not always the case, then it will be the limit of the straights $\overline{x_0x_1}$ where $x_1$ is chosen ever closer to $x_0.$ They are called secants and intersect the graph twice (or more far away).

 

[MiddleEast]

Thank you for your detailed reply. Surely, using calculus definition will clear any doubt.
Your way is gentle and great, but am looking for definition for the tangent itself without the idea of "approaching xo".

One idea came to mind, what about this "Tangent line is a line that touches the curve and the curve is on one side of the tangent for small neighborhood of tangency point".

 

[fresh_42]

You cannot define a tangent without a limiting process. E.g. $x\longmapsto |x|$ has no tangent at $x=0$ because the limiting process from the left results in a different straight than the limiting process from the right. Therefore you need the secants approaching a unique final straight called tangent. Any straight $y=\alpha \cdot x$ "touches" the graph of $y=|x|$ at $x=0$ without being a tangent.

 

[PeroK]

The tangent line is the straight line that passes through a point on the curve and whose gradient is equal to the derivative of the function at that point.

 

[fresh_42]

The tangent line is the straight line that passes through a point on the curve and whose gradient is equal to the derivative of the function at that point.

And how do you phrase this pre-calculus?

 

[DaveE]

The tangent line is the straight line that passes through a point on the curve and whose gradient is equal to the derivative of the function at that point.

Yes, but...

Since you posted this question in the pre-calculus section, we only have a geometrical definition.

Which is difficult because concepts like "a small neighborhood", or even "slope" end up sort of defining limits and then calculus. For an arbitrary curve, I don't think it can be described without limits. It is essentially a local phenomenon.

 

[FactChecker]

One idea came to mind, what about this "Tangent line is a line that touches the curve and the curve is on one side of the tangent for small neighborhood of tangency point".

That does not work. Consider the function, $f(x)=x^3$. At the point (x=0,y=0), the X-axis is tangent to the curve, but the curve is on one side for $x \lt 0$ and on the other side for $x \gt 0$. There is no way that I know of to avoid the idea of the limit of secants. This is one of the ideas that led to the idea of calculous.

 

[mathwonk]

middle east, your idea of a tangent line at p as a line that meets the curve at p, and in some disc around p lies entirely on one side, is an excellent one. indeed that is euclid's original definition of a tangent line to a circle (a line that meets but does not "cut" i.e. does not cross the circle. however as fact checker points out, this definition can fail for some curves that are not convex at p, i.e. that have an "inflection point" at p.

one must also be careful if one wants to detect curves which do not have a tangent line. i.e. if a curve is known to have a tangent line at p, then any line that meets the curve at p, and lies entirely on one side of the curve near p, is the tangent line. of course there may not be any such line, as with factchecker's cubic. but there may also be more than one such line, as with the graph of the curve y = |x|. then it seems there is no tangent line. i.e. it seems that if there is a curve through p and lying entirely on one side near p, and if there is only one such line, then it seems the curve does have a tangent line at p, and that unique line is it. so for all convex curves, it seems your idea both determines if a tangent line exists, and when it does, identifies that line.

one could try to modify your idea to contain a more naive version of fresh42's definition by saying that it is a line that meets the curve, in some disc around p, only at p, but such that if we rotate it a tiny amount (in one direction or the other) it then meets the curve in another point near p. I.e. looking only at one half of the line at a time, it lies entirely on one side of the curve, but after a small rotation, no longer does so.

this would cover factchecker's cubic, and essentially any algebraic curve, but it does not work for really crinkly curves like y = x^2.sin(1/x), where the tangent line at (0,0) actually meets the curve many times near there and never lies entirely on one side of the curve in any neighborhood of (0,0). of course you may not care about such non algebraic examples.

but in fact such an example can easily be treated by this definition also as follows: I.e. if there is a unique line that meets the curve infinitely often on every disc around p, then that is the tangent line. (In particular this implies that a line is its own tangent at p.) If more than one line does so, there is no tangent line. If every line through p has some disc around p on which it meets the curve only finitely often , then the previous definition works.

note also that an elementary way to state fresh's limit definition is to mimic euclid again and just say the line makes "an angle of zero" with the curve at p. i.e. in euclid's language, "between the tangent line and the curve, no other line (through p) can be interposed", in any small region around p. I.e. if we form any angle at all from two lines intersecting at p, such that the angle is bisected by the tangent line, then in some disc around p, the curve will lie entirely within the angle formed by the two lines.

notice that the concept of limit does appear in our discussion in the form of the "universal quantifier" word "every". when we say something is true for every line, we are really taking a limit, i.e. an infinitely good approximation. so if the correct use of the word "every" may be considered as not requiring calculus, then neither does the definition of a tangent line. if one wonders whether this "limiting" language precedes calculus, note that euclid's proposition that "no other line may be interposed between the tangent and the circle", is logically equivalent to saying that "every other line fails to so interpose", hence is logically a limit statement. thus i agree with fresh42 that some form of limiting statement is needed, but simply point out that euclid already gave one.

 

[mathwonk]

by the way if all you want is a precalculus definition, with no limits of any kind, then you may assume all your curves are "precalculus" curves, hence are given by polynomial equations say y= f(x). Then a line is tangent to the curve at p, if and only if the equation for the intersection of the curve and the line at p has a double root there. i.e. it is the line which meets the curve "twice" at p. this is relates to the previous discussion by the fact that moving the line a little produces a second intersection, which has sprouted out of the doubled one.

here are some free notes where this is explained in detail:
https://www.math.uga.edu/sites/default/files/inline-files/polynomial_tangents_without_limits.pdf

 

[FactChecker]

by the way if all you want is a precalculus definition, with no limits of any kind, then you may assume all your curves are "precalculus" curves, hence are given by polynomial equations say y= f(x).

Precalculus typically includes trigonometry.

 

[mathwonk]

good point. if you wish to include the trig functions, (as indeed also euler does), you might enjoy his brilliantly creative discussion, deriving even their power series without limits, but admittedly by utilizing his intuition of infinitely small and large numbers, and substituting them into his trig formulas. as in article 134, of Introduction to the analysis of the infinite. p. 107 of the springer verlag edition.

 

[mathwonk]

maybe we could try to strengthen middle east's idea to cover all curves? first we have the case where there is some line that meets the curve infinitely often near p. then that is the tangent line if there is only one such line, and there is no tangent line if there is more than one.

now we may assume every line meets the curve only finitely often in some nbhd of p, hence only once in some smaller nbhd. if that line lies entirely on one side near p, and is the only such line, then it is the tangent line. if there is more than one such, there is no tangent line.

now suppose every line is like the tangent to the cubic, namely it lies on one side of the curve in one direction, but lies on the other side of the curve in the other direction. then reflect one half of the curve perpendicularly across the line, so that now the line lies entirely on one side near p. (In the cubic case we get y = |x|^3. Then the point is that the condition of lying on one side works for y = |x|^3, but this curve has the same tangent line as y = x^3 !)

I.e. if now, after reflection, that line is the only line through p and lying entirely on one side of the partially reflected curve near p, then it is the tangent line not only to the reflected curve but to the original curve as well.

now for this type of curve, any line through p lies entirely on one side, after reflection. so there are two cases: after reflection, either for every line, there is always also another line through p and lying entirely on one side, or there is some line such that after reflection it is the only such line. In the first case there is no tangent line, and in the second case the given line is the tangent line.

how about that? does that do it? would you say this uses "no limits"? of course the condition that only one line lies on one side near p, if made precise, involves some language like: for every other line, there is some neighborhood of p in which the line meets the curve again. and this is really equivalent to a statement saying that the line is a limit of secants. but it is euclid's version of limits rather than weierstrass's.

 

[mathwonk]

well that doesn't quite work. if we take y = x^2.sin(1/x) for x ≥ 0, and y = x^3 + x, for x < 0, we seem to have no tangent line at (0,0), and yet the x-axis is the only line that meets the curve infinitely often near (0,0). so I get involved in one sided tangents, something I want to avoid. so it is harder to both recognize a tangent when one exists, and also decide when no tangent exists. perhaps these ideas do work to recognize the tangent when one knows somehow there is a tangent in the first place.

wait a minute, that example is ruled out by the condition that there exists more than one line through p and lying entirely on one side of the curve. anyway. more precision is called for.

so i have to figure out how to distinguish this example from the curve which equals y = x^2.sin(1/x) for x 0, and equals y = x^3 for x < 0. interesting. of course the x-axis here is the only line that meets the curve infinitely often near p, and also every other line through p crosses the curve at p. so maybe the fact that no line through p lies entirely on one side is the key here.

i.e. maybe the bad cases are: more than one line lies on one side, or more than one line meets the curve infinitely often near p; and the good cases are: only one line lies on one side, or no line lies on one side but exactly one line meets infinitely often near p.

or maybe we SHOULD go with one sided tangents. i.e. a necessary condition for a line to be the tangent is that near p, the perpendicular to the line crosses the curve. i.e. removing p separates the curve into two parts, one part on each side of the perpendicular.

now we try to recognize the one sided tangents on each side of the perpendicular. i.e. the ray pointing to one side is the one sided tangent if either it is the only ray meeting curve infinitely often near p, or there is no such ray, this line lies entirely on one side of the curve near p, but every ray obtained by rotating the line towards the curve crosses the curve.

having determined the one sided tangents, they should point in opposite directions in order to have a two sided tangent.

sorry for this mess. eventually one begins to appreciate the usual definition via limits of secants. of course hidden in that definition is that one is really defining one sided tangents and forcing them to be equal all at the same time, by the use of the universal quantifier.

thanks for the fun question!

notice that it is tricky to detect the fact that the x-axis is tangent to the "cuspidal" curve y^2 = x^3 at (0,0) by even this one sided tangent approach, but the limit of secants works fine. i.e. in this case the perpendicular (the y axis) does not separate the curve, and there is only a one sided tangent on one side, not the other, but it is the limit of all secants. usual calculus approaches avoid this case since it is only a graph if rotated upwards, and then the slope of the tangent line is infinite. e.g. y = x^(2/3) does have the y-axis as limit of secants at (0,0), but the derivative is not defined since it is infinite there. oh yes, also the left and right limits give the same line, but the derivative "changes sign" from + to - infinity. in algebraic geometry the tangent at this point is considered as a "doubled line". think of a figure eight, which has two tangent lines at the cross point, and let one loop of the figure eight shrink down to a cusp, and the two distinct tangent lines become one doubled line at the cusp.

 

[FactChecker]

eventually one begins to appreciate the usual definition via limits of secants.

Yes. And an alternative definition that is not much more complicated is hard to imagine. Furthermore, there are a lot of benefits to getting used to the definition using limits of secants because the concept of limits is used everywhere.

 

[MiddleEast]

As I expected, this is not an easy question to ask & will go far.
Am really enjoying & appreciating these long replies even though I need to re-read it again & a little bit above my level.
Keep the discussion up gentlemen, one day I will have deep math knowledge & maybe I can join this discussion.

 

[mathwonk]

couldn't resist: this may get it, i.e. the alternate complicated version, just to show it can be done:

tangents a la euclid

euclid defined a tangent line to a circle as a line meeting it but not crossing it. we want to generalize this characterization of tangent line. Let p be a point of the continuous curve C.

1. If more than one line through p meets C again on every disc around p, then C has no tangent line at p.

2. If more than one line through p lies entirely on one side of C in some disc around p, then C has no tangent line at p.

3. If L is a line through p, and there is no disc around p in which the perpendicular to L at p separates C into two arcs, one on each side of the perpendicular, then L is not tangent to C at p. Hence if no line through p does this, then C has no tangent at p.

Hence assume our curve C has the property that at most one line though p meets C again on every disc around p, and that at most one line through p lies entirely on one side of C in some disc around p, and that we consider only lines through p whose perpendicular at p separates C in some disc into two sides.

4. Given such a line L, assume there is some disc in which L meets C only at p. Then either L lies entirely on one side of C in that disc, in which case L is the tangent line to C at p, or else L crosses C at p. In that case, reflect one side of the curve across L, after which L lies entirely on one side of C. If L is the only such line through p, then L is the tangent to C at p.

If no line through p lies entirely on one side of C near p, and if no line which crosses p becomes the unique such line after reflection, then C has no tangent at p.5. Again given a line L through p whose perpendicular at p separates C, suppose L meets C away from p in every disc around p. By hypothesis no other such line exists, hence either L is the tangent line or there is none. (L is at least a one sided tangent.)

The perpendicular to L separates C at p into two sides, and if L meets C away from p on both sides, in every disc around p, then L is the tangent to C at p. If L meets C away from p in every disc about p but only on one side, there is some disc in which L does not meet C away from p on the other side. L is the only possible tangent line to C and we must decide if it is tangent to C on that other side. It is so, if and only if every other line through p obtained by rotating L slightly toward C, meets C again away from p in that disc.

 

[mathwonk]

of course the limit of secants definition can be given in elementary geometric language too, as suggested earlier. given a line L through p, assume that every angle formed by two lines through p for which L is the angle bisector, has the property that on some disc around p, the entire curve C lies within the angle. Then, and only then, L is the tangent to C at p.

This is not euclid's definition, but it is the key property he proves using his definition, in Proposition 16, Book 3, of the Elements, that says the line makes an angle of zero with the curve at p. Apparently Newton took this property as his own definition, realizing it was much more general. I.e. as long as the curve always gets inside the angle, it does not matter which side of the angle bisector it lies on, or whether it lies entirely on one side of it or not. Of course for simpler curves, the simpler definition in terms of lying on one side is very nice.

 

[WWGD]

IiRC, once youve defined the secant line between x and xo, the tangent line is the limiting line, if it exists, between those two points.