A point is moving along the graph of y = x2 so that dx/dt is 2 centimeters per minute. Find dy/dt when (a)x = 0 and (b)x = 3.
y=x^2
dy/dt = 2xdx/dt (dx/dt=2)
now you just need to put the number into that function above
To find dy/dt, we need to differentiate the equation y = x^2 with respect to time t.
Given:
dx/dt = 2 cm/min (the rate at which x is changing)
Differentiating both sides of the equation y = x^2 with respect to t, we get:
d/dt(y) = d/dt(x^2)
Now, let's find dy/dt:
d/dt(y) = d/dt(x^2)
dy/dt = 2x * dx/dt
(a) When x = 0:
dy/dt = 2(0) * dx/dt
dy/dt = 0 cm/min
Therefore, when x = 0, dy/dt is 0 cm/min.
(b) When x = 3:
dy/dt = 2(3) * dx/dt
dy/dt = 6 * 2
dy/dt = 12 cm/min
Therefore, when x = 3, dy/dt is 12 cm/min.
To find dy/dt, we can use the chain rule, which states that:
dy/dt = dy/dx * dx/dt
In this case, we are given that dx/dt = 2 cm/min. To find dy/dt at a specific point, we need to find dy/dx, the derivative of y = x^2.
(a) When x = 0:
To find dy/dx, differentiate y = x^2 with respect to x:
dy/dx = 2x
Substitute x = 0:
dy/dx = 2(0) = 0 cm
Now we can use the chain rule to find dy/dt:
dy/dt = dy/dx * dx/dt
dy/dt = 0 cm * 2 cm/min
dy/dt = 0 cm/min
So when x = 0, dy/dt = 0 cm/min.
(b) When x = 3:
To find dy/dx, differentiate y = x^2 with respect to x:
dy/dx = 2x
Substitute x = 3:
dy/dx = 2(3) = 6 cm
Now we can use the chain rule to find dy/dt:
dy/dt = dy/dx * dx/dt
dy/dt = 6 cm * 2 cm/min
dy/dt = 12 cm/min
So when x = 3, dy/dt = 12 cm/min.