Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

•• Let h(x, y) describe a mountain as in Problem 1. If the same mountain is given by the contour map in Fig. 1, give estimates for ∂h/∂x and ∂h/∂y at points P, Q, R, and the summit S. The scale for x and y (shown by the ruled line) and contours are given in meters.

FIGURE 1

Problem 1

• A mountain can be described by the function h(x, y), which gives the height above sea level of a point that is x east and y north of the origin O. (a) Describe in words the meaning of ∂h/∂x and ∂h/∂y. (b) What does it mean to a hiker who is walking due north if ∂h/∂y is positive? (c) What if he is walking due north, but ∂h/∂y is zero and ∂h/∂x is positive?