Parametric differentiation query

[apple2357]

Is it the case that the y part of a parametric equation only determines the number of stationary points?
This feels strange to me.
So for example: x = t^2 , y= t^3-9t looks like this and has two stationary points at x=3 ( when t= +- sqrt(3))



So no matter what i choose for the x part of the parametric equation?

e.g. x = whatever but as long as y= t^3-9t , there will always be two stationary points??

This feels odd and counter intuitive? Presumably x can't be whatever but needs to exist for that particular value of t?

 

[pka]

Honestly don't know what you are saying.
Do you understand that \(\displaystyle \frac{dx}{dt}=2t~\&~\frac{dy}{dt}=3t^2-9~?\)
So \(\displaystyle \frac{dy}{dx}=\frac{3t^2-9}{2t}\). Hence we have stationary points when \(\displaystyle t=\pm\sqrt3\).
The \(\displaystyle 2\sqrt 3\approx 3.46\) does that "look" right?

 

[apple2357]

I think what i am saying is that the number of stationary points seem to only depend upon dy/dt ? In your above work, the '2t' could have been anything and you would still have found two solutions of t?

And i find it strange that dx/dt doesn't impact on the number of stationary points? Is that correct?

 

[pka]

I think what i am saying is that the number of stationary points seem to only depend upon dy/dt ? In your above work, the '2t' could have been anything and you would still have found two solutions of t?
And i find it strange that dx/dt doesn't impact on the number of stationary points? Is that correct?

Believe me, I mean no insult. I don't think that you really understand stationary points.
Have a close look at this plot. How many places on the plot could there be stationary points?
Places where the derivative is zero: the slope of the tangent line is zero; horizontal tangents.
There are only two places where \(\displaystyle 3t^2-9=0\)

 

[Dr.Peterson]

I've moved some words to make your question say what I think you mean:

Is it the case that the y part of a parametric equation alone determines the number of stationary points?
This feels strange to me.
So for example: x = t^2 , y= t^3-9t looks like this and has two stationary points at x=3 ( when t= +- sqrt(3))

So no matter what i choose for the x part of the parametric equation?

e.g. x = whatever but as long as y= t^3-9t , there will always be two stationary points??

This feels odd and counter intuitive? Presumably x can't be whatever but needs to exist for that particular value of t?

I think I understand what you are saying; but the odd thing is that what you call counter-intuitive seems obvious to me, and yet is not quite true!

Since dy/dt = (dy/dt)/(dx/dt), and you are looking for when this is zero (horizontal tangent), that will happen when the numerator, dy/dt, is zero. Obviously. It shouldn't be a surprise that x(t) doesn't affect what y does (namely, become stationary).

Except that what x does can affect the results in special cases, so you can't really ignore it.

What if dx/dt = 0 at the same time dy/dt = 0? Or what if one or both of them were undefined?

So the reality is that x(t) usually has no effect, but it is necessary to verify that it doesn't in a specific case.

 

[apple2357]

Yes, Dr Peterson thanks for helping me clarify what I mean!

That helps, in the sense that we can't completely ignore x(t) because whilst dy/dt could determine the number of stationary points, we would still need to check what is happening with dx/dt or whether whether x(t) is even defined at that value of t to be sure about the number of stationary points. So would it be correct in saying dy/dt alone determines the maximum number of possible stationary points?

Algebraically, I can see that it is not counter-intuitive but visually when I am thinking about the curve ( the number of stationary points seem to mostly(!) depend on how y changes with respect to t ( with exceptions above only) it does seem counter-intuitive.

 

[Dr.Peterson]

So would it be correct in saying dy/dt alone determines the maximum number of possible stationary points?

Possibly. But I don't like making big general statements without an actual proof. There may be some other factor we are missing.

 

[apple2357]

Possibly. But I don't like making big general statements without an actual proof. There may be some other factor we are missing.

Of course, but it is a starting point. Your statement above that we only need to be concerned with the numerator ( dy/dt) to find possible stationary points suggests this may be true. But you are right, i could be overlooking things as i did when i started out with this conjecture.