Answer to Question #273216 in Calculus for reece

Question

The displacement, 

y(m), of a body in damped oscillation is y=2e^-t sin sin 3t . 

The task is to:

1.  Use the Product Rule to find an equation for the velocity of the object if v=dy/dt.

Answer 1

1.

$$v=\dfrac{dy}{dt}=\dfrac{d}{dt}(2e^{-t} \sin(3t))$$

$$=2\sin(3t)\dfrac{d}{dt}(e^{-t})+2e^{-t}\dfrac{d}{dt}(\sin(3t))$$

$$=2\sin(3t)e^{-t}\dfrac{d}{dt}(-t)+2e^{-t}\cos(3t)\dfrac{d}{dt}(3t)$$

$$=-2\sin(3t)e^{-t}+6e^{-t}\cos(3t)$$

$$=2e^{-t}(-\sin(3t)+3\cos(3t))$$

$$v=\dfrac{dy}{dt}=2e^{-t}(-\sin(3t)+3\cos(3t))$$