Question

The density of a thin circular plate of radius 2 is given by p(x,y)=4+xy.The edge of the plate is described by the parametric equations x=2cost,y=2sint for t in [0,2pi].Find the rate of change of the density with respect to t on the edge of the plate.Does this mean evaluate dx/dt & dy/dt??? or dx/dy which equals (dx/dt)/(dy/dt)?

Answer 1

Use the Multivariable Chain Rule and find
\[{\partial{p} \over \partial{t}}\]

Answer 2

sorry for my ignorance.... but i always thought density was a type of rate of change (a derivative). so if you want the rate of change of the density, aren't you actually looking for the second derivative?

Answer 3

\[dp/dt=dp/dx * dx/dt + dp/dy* dy/dt?\]

Answer 4

but the equation p(x,y) is defined as the density

Answer 5

hey you almost spelled my name.. :)

Answer 6

oh i see...

Answer 7

thanks.... i didn't read that properly....

Answer 8

i think you got it... dp/dt... because it does say with respect to t... that would be my guess. sorry can't be of much help.

Answer 9

\[\frac{\partial{p}}{\partial{t}}=\frac{\partial{p}}{\partial{x}}\cdot \frac{dx}{dt}+\frac{\partial{p}}{\partial{y}}\cdot\frac{dy}{dt}\]

Answer 10

is my final answer then \[dp/dt=-4\sin^2t+4\cos^2t?\]

Answer 11

Yep. :D

Answer 12

Thank you blockcolder!!! :-)

Answer 13

You're welcome. :D