Chain rule Review

1. Demonstrate basic uses of the chain rule.

A. Suppose that w(x,y,z) = xy + yz + xz and that x(r,t) = rcos(t), y(r,t) = rsin(t), z(r,t) = rt. Find w/rw/r at the point(r,t)=(2,/2)(r,t)=(2,/2).

B. Suppose that z is implicitly defined as a function of x and y by the equation yz + xln(y) = z². Findz/yz/y.

C. The surface area of a box with side lengths x,y, and z (measured in meters) is given by S(x,y,z) = 2xy + 2yz + 2xz. The side lengths are changing with time. Suppose that at a certain time t, x(t) =2, y(t) = 2, and z(t) = 1, and at that specific time, x and y are increasing at a rate of 2 m/s and z is decreasing at a rate of 3 m/s. What is the rate of change of the surface area at that time?