Two curves in a cut point would have a same tangent

In summary: I don't know what you mean by "proof". But if you mean an example of two intersecting lines that don't have the same tangent at the point of intersection, I can provide you one. Consider the following two lines: y=x^2+1 and y=x+1. At the point where they intersect, the two lines have a slope of 1 but the y-coordinate of the point of intersection is -1.
  • #1
Nemanja989
79
2
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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  • #2


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.
 
  • #3


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.
 
  • #4


"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!
 
  • #5


it doesn't have same tangent always.
 
  • #6


@Little ant

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

thanks!
 

Related to Two curves in a cut point would have a same tangent

1. What is meant by "two curves in a cut point"?

The term "two curves in a cut point" refers to two curves that intersect at a single point, also known as a "cut point". This means that the two curves share a common point on their graphs.

2. What does it mean for two curves to have the same tangent at a cut point?

Having the same tangent at a cut point means that the two curves have the same slope at the point where they intersect. This can also be thought of as the two curves having the same direction at that point.

3. How can two curves have the same tangent at a cut point?

For two curves to have the same tangent at a cut point, their slopes at that point must be equal. This can occur when the curves have a common derivative at that point, or when the two curves have equivalent equations that produce the same slope at that point.

4. Why is it important for two curves to have the same tangent at a cut point?

Having the same tangent at a cut point can help to determine the behavior of the curves at that point, as well as their relationship to each other. It can also be useful in finding the equations of the curves or the coordinates of the cut point.

5. Can two curves in a cut point have multiple tangents?

No, two curves can only have one tangent at a given cut point. This is because the slope of a curve can only have one value at a specific point. However, if the two curves intersect at multiple points, each of those points may have a different tangent.

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