High School Does a tangent to a curve touch at 2 identical points?

[grzz]

The graph of y = x - 1 CUTS the x-axis at x = 1 while the graph of y = x²- 1 TOUCHES the x-axis at x = 1.
The point at which the tangent touches the curve is shown mathematically by having two solutions of x, i.e. x = 1 (twice).
Is there some deeper meaning to these two identical solutions for x?

 

[member 587159]

The parabola y = x² -1 cuts the x-axis in x = -1 and x = 1.

For an image, http://www.wolframalpha.com/input/?i=y+=+x^2+-1+and+y+=+0

However, do you mean, for example, that:

y = x^2 touches the x-axis twice in x = 0?

For an image, http://www.wolframalpha.com/input/?i=y+=+x^2+and+y+=+0

 

[fresh_42]

The point at which the tangent touches the curve is shown mathematically by having two solutions of x, i.e. x = 1 (twice).

It can be even more. The tangent of a straight line is the straight line itself and therefore "touches" it on the entire length.

Is there some deeper meaning to these two identical solutions for x?

Yes, there is! A tangent at $x_0$ is the limit you get when you approach $x_0$ by secants, which cut the curve twice - say at $x_1$ and $x_2$. $\; x_1 → x_0 \, ,\, x_2 → x_0$ will then result in the tangent.

 

[grzz]

I am very sorry.
I meant comparing the graphs of y = x - 1 and y = (x - 1)².
Both equations have the solution, i.e. x = 1 but that of the tangent is x = 1 (twice).
I am asking if there is some deeper meaning for that 'twice'.

 

[grzz]

It can be even more. The tangent of a straight line is the straight line itself and therefore "touches" it on the entire length.Yes, there is! A tangent at $x_0$ is the limit you get when you approach $x_0$ by secants, which cut the curve twice - say at $x_1$ and $x_2$. $\; x_1 → x_0 \, ,\, x_2 → x_0$ will then result in the tangent.

Thanks 'fresh_42'.
That's the idea that crossed my mind.
But then then when one talks about 'limits' one may say that limits are approached as far as required ... but not attained.

 

[fresh_42]

Why not? What is the limit of say $(5,4,3,2,1,1,1,1,1,1,...)$?

If you solve it by geometrical means, you will find exactly one solution, the tangent. Why do you think calculation will result in a limit that isn't reached? You may calculate secants through the points $(x_0, (x_0-1)^2)$ and $(x_0+h, (x_0+h-1)^2)$ and see what happens if $h → 0$. But don't forget to cancel $h$ out of fractions when it is possible. It is the same calculation, which has to be made to prove $y'=2(x-1)$ or in general to prove the formulas of differentiation like $(x^n)' = nx^{n-1}$. (The calculation should be easy if you choose $x_0= 1$ as touching point, but I suggest to keep $x_0$ arbitrary as an exercise.)

Do you know the fly which flies between two approaching trains at twice the speed of the locomotives? One way, then turns and flies back, then turns again whenever it reaches one of the trains. The distance between the trains is getting smaller and smaller, the fly has ever smaller distances to fly until infinity. Nevertheless the fly will die after a finite amount of time when the trains crash.

 

[grzz]

Why not? What is the limit of say $(5,4,3,2,1,1,1,1,1,1,...)$?
.

The limit of the above is 1 and IS attained.
But although one knows the limit of 1, 1 + 0.5, 1 + 0.5 + 0.25, ... yet is it ever attained?

 

[fresh_42]

Limits are a way to get mathematically hold on something like arbitrary close. Sometimes arbitrary close is zero, the closest possible way, and often just arbitrary close and the limit isn't attained. Since tangents (if defined) touch their function, they are attained by narrowing secants. But you could still object that the secants (as our sequence) are still straights that cut the function's graph twice, no matter how close these two points are. So we get a sequence of secants that converge to the tangent without being one. However, you asked for the meaning of the double solution at one point. One way to see it is the sequence of secants (with two points) becoming a tangent (with one point as the limit of the two secant points) in the limit.

If I misunderstood it, and you were only referring to the exponent $2$ in $y=(x-1)^2$ then it is more an algebraic than a geometric question.
E.g. $(x-1)^n$ has a solution of multiplicity $n$.
See https://en.wikipedia.org/wiki/Multiplicity_(mathematics)#Multiplicity_of_a_root_of_a_polynomial

 

[grzz]

One way to see it is the sequence of secants (with two points) becoming a tangent (with one point as the limit of the two secant points) in the limit.

The geometrical description is a good answer to me.
Thanks