Intuition Behind Particular Related Rates Question

[S.R]

Homework Statement

A particle is moving along the graph of y=sqrt(x). At what point on the curve are the x-coordinate and the y-coordinate of the particle changing at the same rate?

Homework Equations

y = sqrt(x)
y' = 1/(2sqrt(x))

The Attempt at a Solution

The solution to the problem is to differentiate both sides of the eqn with respect to time and then determine when dy/dt = dx/dt.

But I don't understand how the x-coordinate has a rate of change since it's an independent variable? Can someone explain how to comprehensively understand the solution and I suppose, wording of this problem?

Thank-you.

 

[Nick O]

Edit: As shown in later posts, this is wrong and I need sleep.

I think we need to make the assumption that the particle is moving at a constant speed, and that the independent variable is time (not x).

That changes the equation for y:

y = sqrt(t)

and gives us an equation for x:

x = t^2

I admit, though, the question is very vague and I can't be sure that this is what it means.

 

[S.R]

If x = t^2, then sqrt(x) = t, which simplifies the function to:

y = sqrt(sqrt(x))

Not sure I understand.

 

[Nick O]

You are thinking of y as a function of x. Think of them as two independent functions:

x(t) = t^2

y(t) = sqrt(t)

This allows you to differentiate both x and y with respect to t, rather than with respect to one another. Once you have both derivatives, you can find where they have equal rates of change.

Note that t^2 is just a reflection of sqrt(t). It would have been equivalent to say t = sqrt(x), which would make the x and y plots identical, but with different axis labels.

Does that make sense? I'm not sure that I am explaining how I got the equation for x very well.

 

[Dick]

Homework Statement

A particle is moving along the graph of y=sqrt(x). At what point on the curve are the x-coordinate and the y-coordinate of the particle changing at the same rate?

Homework Equations

y = sqrt(x)
y' = 1/(2sqrt(x))

The Attempt at a Solution

The solution to the problem is to differentiate both sides of the eqn with respect to time and then determine when dy/dt = dx/dt.

But I don't understand how the x-coordinate has a rate of change since it's an independent variable? Can someone explain how to comprehensively understand the solution and I suppose, wording of this problem?

Thank-you.

Use the chain rule. dy/dt=(dy/dx)*(dx/dt). If dy/dt=dx/dt then what is dy/dx?

 

[S.R]

Well, dy/dx must equal 1. So I guess we have to assume y and x are functions of another variable (time I suppose)? It's rather unintuitive to do so since x is generally the independent variable.

 

[jackarms]

I don't think we can assume constant speed -- I think all that's implied by the problem is that the independent variable is t, and then velocity is going to depend on dx/dt and dy/dt.

If x = t^2, then sqrt(x) = t, which simplifies the function to:

y = sqrt(sqrt(x))

Not sure I understand.

There's an error in your substitution here. And I think you have the same error too Nick -- eliminating t would also give you sqrt(sqrt(x)) instead of just sqrt(x).

For these parameterization problems, it's usually easiest to assign t to x, and then find what y as a function for t would be. For this problem, you would get:

$x = t$ (by arbitrary assignment) and
$y = \sqrt(t)$

Then you want to find when $\frac{dx}{dt}$ and $\frac{dy}{dt}$ are equal, and then what point that time corresponds to.

Hope this was clear enough.

 

[Nick O]

Gah, you're right. In a post I made an hour or so ago I said I should get some sleep... I really should do that before trying to do any more math.

 

[Dick]

Well, dy/dx must equal 1. So I guess we have to assume y and x are functions of another variable (time I suppose)? It's rather unintuitive to do so since x is generally the independent variable.

Yes, time is the independent variable but dy/dx must equal 1. So at what (x,y) point is dy/dx=1? That doesn't depend on the time.

 

[S.R]

Thanks jackarms, your explanation was great. If dy/dx = 1 then 1/(2sqrt(x)) = 1 and thus x = 1/4.

 

[Ray Vickson]

If x = t^2, then sqrt(x) = t, which simplifies the function to:

y = sqrt(sqrt(x))

Not sure I understand.

To answer the question there is no need to assume a constant speed; it will work perfectly well if you assume the position is <x(t),y(t)>, with y(t) = sqrt(x(t)) and x(t), y(t) differentiable functions of t. That's all you need.