An object is moving along the parabola y = 3x^2. (a) When it passes through the point (2,12), its horizontal velocity is dx/dt = 3. what is its vertical velocity at that instant? (b) If it travels in such a way that dx/dt = 3 for all t, then what happens to dy/dt as t --> +infinity? (c) If, however, it travels in such a way that dy/dt remains constant, then what happens to dy/dt as t --> +infinity?
What was wrong with my previous response? I still make no sense of the last part.
For question a, y = 3x^2, so
dy/dt = 6x dx/dt
dy/dt = 6(2) * 3
dy/dt = 36
For question b, as t approacing infinity, x is also approaching infinity (the object is moving along the track of the function, remember(, then dy/dt is also approaching infinity in vertical sense...
For question c, I think you ask for dx/dt, instead of dy/dt since dy/dt always constant... If you ask for dx/dt, dx/dt will be approaching 0 as dt approaching infinity. This is the reason:
dy/dt = 6x dx/dt
as t approaching infinity, x will also be approaching infinity.. Since we hold dy/dt to be constant, dx/dt will have to be a small number to counter the growth of 6x as it approaches infinity.. Therefore, dx/dt will have to approach 0
Hope it helps
No, nothing was wrong with your last response. That was my mistake for posting it up again. Thank you for all your responses.
To find the solution for each part of the question, we will need to use derivatives since we are given the position function in terms of x (y = 3x^2).
(a) To determine the vertical velocity at the point (2, 12) when the horizontal velocity is dx/dt = 3, we can use the chain rule.
First, let's find the derivative of y with respect to time (dy/dt):
dy/dt = dy/dx * dx/dt
Since y = 3x^2, we can differentiate it with respect to x to find dy/dx:
dy/dx = d/dx(3x^2) = 6x
Now, we can substitute the given dx/dt = 3 and the x-coordinate of the point (2, 12):
dy/dt = (dy/dx) * (dx/dt)
= 6x * (dx/dt)
= 6(2) * 3
= 36
Therefore, the vertical velocity at the point (2, 12) when dx/dt = 3 is 36.
(b) If dx/dt is constant at 3 for all t, it means that the object is moving horizontally with a constant rate. As t approaches infinity, the object will continue moving horizontally at the same rate. Since the position function y = 3x^2 is a parabola symmetric around the y-axis, the vertical velocity dy/dt will approach 0 as t increases because the object is not changing its vertical position.
Therefore, as t approaches +infinity, dy/dt approaches 0.
(c) If dy/dt remains constant, it means that the object is moving vertically at a constant rate. In this case, the object is not affected by the horizontal velocity dx/dt.
As t approaches infinity, the vertical velocity dy/dt will remain constant since it is not changing with time.
Therefore, as t approaches +infinity, dy/dt remains constant.