Question
A point P(x,y) moves on the graph of y^2 = 2x^3 such that dy/dt = x. Find dx/dt at the point (2,4)
Answers
GPT 3.5
To find dx/dt at the point (2,4), we first need to find dy/dt at that point using the given information.
Given: dy/dt = x
At the point (2,4), x = 2. So, dy/dt = 2.
Now, we can use the equation y^2 = 2x^3 to find dx/dt.
Differentiate both sides of the equation with respect to t:
2y * dy/dt = 6x^2 * dx/dt
Substitute the values we found:
2(4) * 2 = 6(2^2) * dx/dt
16 = 24 * dx/dt
dx/dt = 16/24 = 2/3
Therefore, dx/dt at the point (2,4) is 2/3.
Given: dy/dt = x
At the point (2,4), x = 2. So, dy/dt = 2.
Now, we can use the equation y^2 = 2x^3 to find dx/dt.
Differentiate both sides of the equation with respect to t:
2y * dy/dt = 6x^2 * dx/dt
Substitute the values we found:
2(4) * 2 = 6(2^2) * dx/dt
16 = 24 * dx/dt
dx/dt = 16/24 = 2/3
Therefore, dx/dt at the point (2,4) is 2/3.