2. 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 dt
is its “vertical” velocity at that instant?
b)If it travels in such away that dx/dt=3 for all t,then what happens to dy/dt as t→+∞?

c) If, however, it travels in such a way that dy/dt remains constant, then what happens
to dx/dt as t→+∞?
well, since y=3x^2
dy/dt = 6x dx/dt
= 6*2*3 = 36
since dx/dt remains constant, dy/dt gets ever faster as x gets larger.
If, however,
k = 6x dx/dt, then dx/dt gets ever slower as x gets larger
To answer these questions, we need to understand the relationship between velocity and position for an object moving along a curve.
The position of the object is given by the equation y = 3x^2. Differentiating this equation with respect to time, we can find the velocity components dx/dt and dy/dt.
a) To find the vertical velocity at the point (2, 12), we need to calculate dy/dt. Since the object is moving along the curve y = 3x^2, we can differentiate this equation with respect to time:
dy/dt = d(3x^2)/dt
Taking the derivative of 3x^2 with respect to x gives:
dy/dt = 6x * dx/dt
We are given that dx/dt = 3 at this instant, and we know that the point (2, 12) lies on the curve. Substituting these values, we get:
12 = 6(2) * (3)
12 = 36
This is not possible, so there must have been an error in the given information or calculations.
b) If dx/dt = 3 for all t, then the horizontal velocity remains constant. As t approaches positive infinity, the object will continue to move with a constant horizontal velocity, which means it will move with a constant speed along the curve. However, since the curve is a parabola and the object has a constant velocity, its vertical velocity (dy/dt) will change at different points on the curve. As t approaches infinity, the vertical velocity may vary depending on the position of the object on the curve.
c) If dy/dt remains constant, then the vertical velocity remains constant. This means the object is moving with a constant speed vertically. However, the value of dy/dt does not provide any direct information about the horizontal velocity (dx/dt). Therefore, as t approaches positive infinity, the behavior of dx/dt depends on the specific way the object is moving horizontally, and we cannot determine what happens to dx/dt without additional information.
In summary, we can determine the behavior of the vertical velocity (dy/dt) and its relationship to dx/dt under certain conditions, but without further information or constraints, we cannot determine the behavior of dx/dt as t approaches positive infinity.