1. Determine an expression for dy/dt if x^2+2y^2=8 and dx/dt=3.
2. A point is moving along the right branch of a hyperbola defined by 4x^2-y^2=64. What is dy/dt when the point is at (5, -6) and dx/dt=3?
I will start you on the first, using implicit differention.
x^2+2y^2=8
2x dx/dt + 4ydy/dt=0
solve for dy/dt
To solve for dy/dt, we can use implicit differentiation. Starting with the equation:
x^2 + 2y^2 = 8
Differentiating both sides of the equation with respect to t (since we're given dx/dt), we get:
2x(dx/dt) + 4y(dy/dt) = 0
Now we can solve for dy/dt by isolating it on one side of the equation:
4y(dy/dt) = -2x(dx/dt)
Dividing both sides of the equation by 4y, we have:
dy/dt = -(2x(dx/dt)) / (4y)
Simplifying further:
dy/dt = -x(dx/dt) / (2y)
Now we can substitute the values we have:
dx/dt = 3
x = ???
Since we don't have a specific value for x, we cannot calculate dy/dt without knowing x. The expression for dy/dt remains as:
dy/dt = -x(dx/dt) / (2y)