A particle is moving along the graph of x^-(1/3). When x=8, the component of its position is increasing at the rate of 1 cm/sec. How fast is the x-component changing at this moment?
To find how fast the x-component is changing at a specific moment, we need to take the derivative of the given function with respect to time.
The equation of the graph is x^-(1/3). We can rewrite this as (1/x)^(1/3).
Now, let's differentiate both sides of the equation:
d/dt [(1/x)^(1/3)] = d/dt (1/x) * (1/3)
To find d/dt (1/x), we need to use the chain rule:
d/dt (1/x) = -1/x^2 * dx/dt
Here, dx/dt represents the rate of change of x with respect to time.
Substituting this into our equation, we get:
d/dt [(1/x)^(1/3)] = -1/x^2 * dx/dt * (1/3)
We know that dx/dt = 1 cm/sec, and we need to find the value of dx/dt when x = 8. So let's plug in these values:
d/dt [(1/x)^(1/3)] = -1/(8^2) * 1 * (1/3)
Simplifying further:
d/dt [(1/x)^(1/3)] = -1/64 * 1/3
d/dt [(1/x)^(1/3)] = -1/192 cm/sec
So, the x-component is changing at a rate of -1/192 cm/sec when x = 8.