Two curves in a cut point would have a same tangent

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SUMMARY

The discussion centers on the relationship between two curves intersecting at a point and their tangents. Specifically, it is established that while two curves can intersect, they do not necessarily have the same tangent at that intersection point. The curves y=x^2 and x^2+(y-1)^2=1 are tangent at the origin (0,0), demonstrating that tangency implies shared derivatives at that point. However, this is not a universal rule for all intersecting curves, as illustrated by the example of y=x^2+x and y=x^3+x at (1,2), where they do not share a tangent.

PREREQUISITES
  • Understanding of calculus concepts, specifically derivatives and tangents.
  • Familiarity with curve equations, particularly polynomial functions.
  • Knowledge of the geometric interpretation of tangents and intersections.
  • Basic skills in analyzing points of intersection between curves.
NEXT STEPS
  • Study the concept of tangents in calculus, focusing on the conditions for tangency between curves.
  • Learn about the implications of derivatives at points of intersection in multivariable calculus.
  • Explore examples of polynomial curves and their intersections to identify cases of tangency.
  • Investigate the geometric properties of curves, including how to determine points of intersection and tangency.
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Mathematicians, calculus students, and educators interested in the properties of curves and their intersections will benefit from this discussion.

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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