Chapter 12.5, Problem 54E

Walking on a surface Consider the following surfaces specified in the form z = f(x, y) and the oriented curve C in the xy-plane.

1. a. In each case, find z’ (t).
2. b. Imagine that you are walking on the surface directly above the curve C in the direction of positive orientation. Find the values of t for which you are walking uphill (that is, z is increasing).

54.     $z=4x^2-y^2+1,C:x=\cos t,y=\sin t;0\le t\le 2\text{π}$