Two curves in a cut point would have a same tangent

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Discussion Overview

The discussion revolves around the conditions under which two curves that intersect at a point have the same tangent at that point. Participants explore examples and clarify terminology related to the intersection and tangency of curves.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions whether two curves that intersect at a point necessarily have the same derivative at that point.
  • Another participant asserts that the two curves in the original example are tangent at the origin, implying they share the same tangent there.
  • A participant seeks clarification on terminology, suggesting alternatives to "cut" for describing the intersection of curves.
  • It is noted that not all intersecting curves have the same tangent; for instance, intersecting straight lines do not share a tangent at their intersection.
  • A participant emphasizes that if two curves are tangent to one another at a point of intersection, they will have the same tangent at that point.
  • One participant expresses uncertainty, stating that it is not always the case that curves have the same tangent at their intersection.
  • A request for proof or an example is made to support the claim that two curves do not always share the same tangent at their intersection.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether two curves that intersect always have the same tangent at the intersection point. Multiple competing views remain regarding the conditions under which tangency occurs.

Contextual Notes

Participants have not fully defined the conditions under which curves are considered to be "like" in terms of their intersection and tangency. There is also ambiguity in the terminology used to describe the intersection of curves.

Nemanja989
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Ola,

If we have two curves, and they cut them selves in a way like these two: y=x^2 and
x^2+(y-1)^2=1. Does it always mean that those two curves in a cut point would have a same tangent, in other words do they need to have the same derivative in that spot?

Thanks!
 
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Your question is confusing - what has "cut" to do with anything? However the two curves are tangent to each other at the origin, so the tangent is the same for both at this point.
 


Well it is little confusing. But I do not know what other word to use instead "cut" maybe
carve, or bisection or contact. And instead "them selves" use one another.

Yes, these two are tangent in point (0,0), but I want to ask is that case with every two curves that contact in the way like in the example that I wrote.

Another example of contact that I am talking about is: y=x^2+x, and y=x^3+x in (0,0) point.
And the contact that I am not talking about is: y=x^2+x, and y=x^3+x in (1,2) point.
 


"itersect" would be better than "cut". And, yes, they do not intersect themselves, they intersect each other. No, the fact that two curves intersect does not mean they have the same tangent there (which was what you asked). Two intersecting straight lines specifically do NOT have the same tangent at the point of intersection. That's so obvious, it's probably not what you were asking.

If you meant to ask "if two curves intersect like these curves do at (0, 0) do they have the same tangent" then, (1) you did not tell us in your original post that you only talking about the intersection at (0, 0), (2) you will need to specify what you mean by "like". The curves you give are tangent to one another at (0,0) so, again obviously, if two curves are tangent to one another at a point where they intersect then, yes, they have the same tangent there. That is precisely what "tangent to one another" means!
 


it doesn't have same tangent always.
 


@Little ant

Could you please give me some proof or an example of that fact?

thanks!
 

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