Welcome to our community

Be a part of something great, join today!

Parametric Curve of y=|x| and differentiability

Maidenas

New member
Feb 8, 2020
3
Hello to everyone, this is my first post and a have a question about an exercise.
It says

Show that there is (exists) a parametric curve $$\gamma :\Bbb{R}\rightarrow \Bbb{R}^2 $$ which is differentiable such that $$\gamma ( \Bbb{R})=\left\{(x,|x|),x \in \Bbb{R}\right\}$$

Can be this true? That exersice i noticed from other years notes that he is always putting it without solving it.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,613
Leiden
Hi Maidenas, welcome to MHB!

How about the parametrization
$$\gamma: t\mapsto (t^3,|t^3|)$$
It means the curve is not regular.
That is, its derivative is (0,0) at 0.
 

Maidenas

New member
Feb 8, 2020
3
Indeed !! thank you very much my friend :))
 

Maidenas

New member
Feb 8, 2020
3
if we want the curve to be infinietly differentiable what function we could choose??
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,613
Leiden
if we want the curve to be infinitely differentiable what function we could choose??
I don't think that is possible.
An infinitely differentiable curve is smooth, meaning it cannot have an angle in it.