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@TuringTest , @UnkleRhaukus , @phi , @saifoo.khan , @ash2326
|dw:1346864028370:dw|try drawing as many different tangents on this figure as you can
is it 3 ?
What point could you draw to satisfy that?
I count a whole bunch!
@UnkleRhaukus isnt it 3?
Can't you draw an infinite number through each vertex?
@phi y so?
i count three infinite lines
tangents drawn from the vertices would have no direction the only question I see is if you want to talk about the number of \(different\) tangents, which I do figure to be 3
ya different tangents
you could draw infinite tangents on any line segment technically, but that seems arbitrary
my doubt is why dont we count the tangent at the 3 vertices?
which way would the tangents at the vertices point?
Do we have to consider that the slope of a tangent at a sharp point is undefined?
i am not sure
a triangle is made with three tangents
OK, by definition a tangent results from a limiting process.... so they do not exist at a vertex... (per wikipedia)
so the answer is 3?
@CliffSedge that is basically how I figured it
If it's tangent to the triangle, then they cannot coincide with the sides of the triangle.
@CliffSedge y so?
Doesn't 'tangent' mean touch at only a single point?
@CliffSedge yes , but the tangent to a straight line is the straight line itself!!!
tangent is nothing but a flat approximation of curve!!
hmmm... looks like there can be several definitions depending on if you're using geometry, calculus, or an even more generalized usage.
i see what i need to knw is this :|dw:1346864494173:dw| can we draw tangent at point A?
I was thinking of it as like a line graph|dw:1346864498741:dw|there is only one unique equation of the tangent line to this graph\[y=mx+b\implies y'=m\]so \(y'=m\) is the only tangent line in the figure of the triangle you get the same effect for each line segment, of which there are three
Here's my reasoning: 1. tangent is defined as touching at only a single point. 2. in calculus, the slope of *the* tangent line at a sharp point is undefined because there is no single/unique tangent line that can be drawn there. Therefore: infinite tangent lines are possible.
@Turing but that's not the tangent line, that is the slope of the tangent line which is the line itself. You can't draw a line tangent to another line because it would touch at more than one point.
More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c on the curve if the line passes through the point (c, f(c)) on the curve and has slope f'(c) where f' is the derivative of f. -Wikipedia that does not require the line to only touch the graph at a single point, the slope just has to match that of the graph at that point
i asked this qn in my lecture and answered it as 3 then one of the srtudents asked if its 6 as in fig: |dw:1346864689277:dw| i couldnt justify it was 3
i told the student i will answer it tmrw dats y i asked
consider|dw:1346864772930:dw|the tangent at point P intersects point Q on the graph, but that does not mean the line is not tangent to the graph at P
agree @TuringTest can u answer my query above?
Yes, but the tangent doesn't cross at P. It's free to cross elsewhere. see: http://en.wiktionary.org/wiki/tangent def'n. 1.
Is the slope of the tangent the always the same thing as the derivative?
I'm still not convinced that there are fewer than infinite tangent lines that can be drawn to that triangle.|dw:1346864875620:dw|
That definition seems to hold for the 3 tangents of the triangle; they lines do not cross the sides again there is no defined slope at the vertices, so you can't do that @CliffSedge
I don't think the slope of the tangent matters here, since we're not looking for a single unique tangent with a defined slope, just "how many lines?"
The person who gave me this qn for presenting in lecture was @UnkleRhaukus and he has vanished :O
there can only be one unique tangent at each point, the idea that you can draw multiple tangents from each vertex shows that there is no tangent
unkle said that a triangle was composed of three tangents and therefor, I think, meant to imply that there are three tangents
thats a nic reasoning @TuringTest
Thinking of this in terms of first derivatives is over-thinking it, in my opinion. Ask Euclid or Archimedes what they think of this question. It's fundamentally geometrical.
so can i tell the student that slope is not defined at sharp points ?
as a reason for no:of tangent=3 and not 6?
I would like to see some reference that contradicts that, otherwise I would argue along the calculus approach.
@TuringTest answer me pls
I say "yes" there are 3, but there is some debate, and I'm not perfect...
is see but i kthink u r right at this ,the calculus approach is best at this qn
A really good reference I would say that clarifies that tangents are not defined at edges.
thanks a lot evryone for sharing their ideas :")
nice link @siddhantsharan :)
Thank Google :D
Per wolfram's definition (see the last paragraph) http://www.wolframalpha.com/input/?i=tangent+definition&x=0&y=0 a tangent line touches a curve at a single point.... so one could argue that polygons do not have tangents.
Although they have not defined it precisely there. The precise definition is the calculus one.
Can anyone provide a reason why the slopes of the tangents need to be defined? If the claim is that tangents are not defined (i.e. are not unique) at sharp points, then doesn't that mean that the number of lines is infinite?
@siddhantsharan that was really useful thx
page 136 of the link that @siddhantsharan gives a nice non-calculus way of circumventing the definition, but it still leads to the same conclusions; one tangent line per point, and it's okay to draw a tangent to a line segment
the idea is derivative does not exist at sharp points
Why does the existence of the derivative matter? I can draw a triangle, and I can draw numerous tangents at each vertex. I'm looking right at it, so what's the problem?
You can define tangent any way you like. The question is, what is the recognized mathematical definition? Does it exclude straight lines? or is it only defined for "curves"
This is the definition I'm using: (geometry) A straight line touching a curve at a single point without crossing it there. We can all agree that straight lines count as curves, right?
There exists a triangle, and I may, in principle at least, draw as many tangents to its vertices as I wish. QED. I work with a bunch of math tutors/teachers. I'll bring this up with them this afternoon.
The slopes of the tangents need to be defined because the tangent Is the slope at that Point
@CliffSedge The definition you are using is not the mathematical one.
Geometry is not mathematics?
Triangles are geometrical, lines are geometrical; how many hairs are we going to split here?
That is the definition that is implied from the calculus one and is observed to be helpful in many cases. However it is one DERIEVED from the calculus one.
@CliffSedge the definition you said you are working with would imply that this is not a tangent|dw:1346866463441:dw|of course the term "there" is not well-defined
So why is it not useful here? I see a triangle. I can draw tangent lines to its vertices. Why do I need to have a well-defined slope for each one? That was not a condition set forth in the problem statement.
that definition makes no statement about sharp points, so it doesn't seem to help us here
False. It touches at P, but does not cross at P.
Why would the definition need to make a statement about sharp points? Why not generalize? Why not employ Occam's Razor and not introduce more assumptions than are necessary?
Okay, fair enough, so you want to say that a sharp curve can have multiple tangents is the discrepancy. Occam's Razor is all well and good, unless there is a more formal definition such that we don't have to employ it.
I know mathematics is meant to be precise, but it is also about simplifying problems so they may be solved more easily, not more difficultly.
the vertex is only a single point it can only have one tangent
In your attempt to simplify this problem we get infinite tangent lines at a sharp point, or any geometric figure with sharp points. I hardly find that simple, but I see what you are saying.
hmm now im not sure if it is one or zero
Does the formal definition state that if their is no single defined/unique slope for a tangent line then the tangent line can't exist? If that is true, then why is it so easy for me to draw one on paper? I saw above that there exists a definition that states that there may exist only a single tangent line at any point, but if that is true then the triangle has zero tangent lines because each vertex has two lines touching it.
What I meant by simple is that if you gave this problem to a child and gave a definition of 'tangent line' that a child could understand, then you'd get many lines drawn with no fuss over formal definitions.
\[y=|x|\]has tangents \(y=-x,y=x\) though both cross the vertex, so I don't think it matters that the tangents intersect on the graph
It's not that the tangents intersect each other, it's that there is more than one tangent at a single point. If y=|x| can have two tangents, why can't any point have more than one tangent? By one definition (preferred) I say infinite lines, but another, it is zero (and that's not so bad - less work the pencil, eh?
my point was that they intersect \(on\) the graph, at (0,0)
at (0,0) we have to accept that there are either infinite slopes or none, and mathematics when it ever discusses the matter seems to say none.
I'll concede that for now. Anyway, I'm in a rush to get to work, so I'll have to leave this for later. Interesting stuff, gets the mind all geared up for logical thinking!
indeed, let me know if you find out something :)
its a good question