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?
dy/dt = 6x(dx/dt)
so at (2,12) dx/dt = 3
then dy/dt = 6(2)(3) = 36
If dx/dt is always 3, then dy/dt = 18x
so as x --> + infinity , 18x ---> + inf.
and dy/dt --> + inf.
for c) I think you meant to say,
"then what happens to dx/dt as t --> +infinity?"
if dy/dt = c
then c = 18(dx/dt)
dx/dt = c/(18x), now as x becomes larger and larger, what happens to the c/(18x) , and as a result, what happens to dx/dt ?
its not right
To answer these questions, we can use calculus to find the derivatives of the equation y = 3x^2, which will give us the velocities at any given point.
(a) To find the vertical velocity, we need to calculate dy/dt when dx/dt = 3.
The given parabolic equation is y = 3x^2. Taking the derivative of both sides with respect to time (t), we have:
dy/dt = d(3x^2)/dt
Differentiating term by term, we get:
dy/dt = 6x * dx/dt
Substituting dx/dt = 3 and the given point (2, 12) into the equation:
dy/dt = 6(2) * 3
dy/dt = 36
Therefore, the vertical velocity at the instant the object passes through the point (2, 12) is 36.
(b) If dx/dt = 3 for all t, it means the horizontal velocity is constant. Let's find out what happens to dy/dt as t approaches positive infinity.
Using the same derivative dy/dt = 6x * dx/dt, since dx/dt is constant, dy/dt will be the derivative of 6x times a constant:
dy/dt = 6(dx/dt)
As dx/dt = 3, we substitute this into the equation:
dy/dt = 6(3)
dy/dt = 18
Therefore, as t approaches positive infinity, dy/dt becomes a constant value of 18.
(c) If dy/dt remains constant, let's examine what happens when t approaches positive infinity.
Using the derivative formula dy/dt = 6x * dx/dt, we know that dy/dt remains constant.
Since dy/dt = constant, let's denote it as k (a constant):
k = 6x * dx/dt
Dividing both sides by 6 and dx/dt:
k/6 = x * (dx/dt)
Now, we can integrate both sides with respect to t:
∫(k/6) dt = ∫x * (dx/dt) dt
(k/6) t = ∫x * dx
Integrating x with respect to itself:
(k/6) t = (1/2) x^2 + C
As t approaches positive infinity, t increases indefinitely, so (k/6) t becomes infinite. For this equation to hold, the right side of the equation, ((1/2) x^2 + C), must also approach infinity. Thus, dy/dt approaches infinity as t approaches positive infinity.
In conclusion:
(a) The vertical velocity at the point (2, 12) is 36.
(b) As t approaches positive infinity, dy/dt remains constant at 18.
(c) As t approaches positive infinity, dy/dt increases without bound and approaches infinity.