A particle is moving along the plane curve y^2 − 6x^4 = y.?

Question

At the moment when x = −1 the x-coordinate is increasing at a rate of 5 cm/sec. If the y-coordinate is negative at this moment, is y increasing or decreasing? How fast?

take the derivative and??

Answer 1

The equation is
y² - 6x^4 = y

At the moment when x = -1, the equation is:
y² - 6(-1)^4 = y

We then transpose y
y² - y - 6 = 0

Then
(y - 3)(y + 2) = 0

Then
y = 3 or y = -2

Since the y-coordinate is negative at the moment, then y = -2. If we take the derivative of the equation:
y² - 6x^4 = y

We differentiate implicitly
2y dy/dx - 24x³ = dy/dx

Then solve for dy/dx
dy/dx = 24x³/(2y - 1)

Then, we want to find whether the equation increases or decreases, then we substitute the point (-1,-2) to the derivative
dy/dx = 24(-1)³/[2(-2) - 1]

Thus,
dy/dx = -24/(-5)

The derivative
dy/dx = 24/5

The derivative is positive, which means that y is increasing at the moment. The speed is acquired by differentiating the equation with respect to time t.
2y dy/dt - 24x³ dx/dt = dy/dt

Since we know that dx/dt = 5 cm/sec, then
2y dy/dt - 120x³ cm/sec = dy/dt

Solving for dy/dt,
dy/dt = 120x³/(2y - 1) cm/sec

The speed of the particle when it is at (-1,-2) is:
dy/dt = 120(-1)³/[2(-2) - 1] cm/sec

Which is
dy/dt = 24 cm/sec

Therefore, at the moment when x = -1, the x-coordinate increases at 5 cm/sec, and the y-coordinate is negative at the moment, y is increasing at the rate of 24 cm/sec.

^_^

Answer 2

dy/dt = 120x^3/(2y-1)

therefore the top is negative and the bottom is negative therefore the y coordinate is increasing because dy/dt is positive. and it will be increasing at a rate of -120/(2(-2)-1) =40

implicit diff gives you:
dy/dx = 24x^3/(2y-1)
dy/dt=dy/dx * dx/dt
dx/dt=5
when x=-1 y^2-y-6=0
so, (y-3)(y+2)=0
since y is negative, y=-2

Answer 3

y^2 - 6 = y
y^2 - y - 6 = 0
(y + 2)(y - 3) = 0

If we are dealing with the negative value of y, then y = -2.

We want to find dy/dt.

dy/dt = (dy/dx) * (dx/dt)

We are given dx/dt as 5 cm/sec.

To find dy/dx, we use implicit differentiation..

y^2 - y - 6x^4 = 0
2y dy/dx - dy/dx - 24x^3 = 0
dy/dx (2y - 1) = 24x^3
dy/dx = (24x^3) / (2y - 1)

Now put in the values of x and y.

dy/dx = -24 / -5 = 24 / 5

We are given dx/dt as 5, so

dy/dt = 24

y is increasing.

Answer 4

take the derivative of the equation