Question

A particle moves on a surface whose equation is
z = f(x, y) = (x − 1)^2+ y^2

Assume that the x and y coordinates of the particle at time t are given by x = 2 cos t and
y = 2 sin t , where 0 ≤ t ≤ 2π. Use the total derivative rule to show that the rate of change of
the height of the particle above the xy plane is given by

dz/dt = 4 sin t.

Hence find the maximum value of the height and the coordinates (x, y,z) of the particle when
the maximum occurs.

Answer 1

\[\frac{dz}{dt} = \frac{dz}{dx}*\frac{dx}{dt} + \frac{dz}{dy}*\frac{dy}{dt}\]
\[\frac{dz}{dx} = 2(x-1)\]
\[\frac{dx}{dt} =-2\sin(t)\]
\[\frac{dz}{dy} = 2y\]
\[\frac{dy}{dt} = 2\cos(t)\]
substitute into top equation to verify dz/dt = 4sin(t)

Answer 2

Oh, it was simpler than I thought. Thanks!