1. Problem 1: Level curves Suppose that the diagram below shows how the elevation i above sea-level (in m) varies over a landscape, with z- and y-coordinates measured in kilometres relative to a field station located at (0,0). ) /42/ 0! 10,// | [ 14_\\\\ i 16 VNN (a) Approximate the rate at which the elevation changes in the z- and y-directions at the t point (2,0) (that is, find the approximate values of the partial derivatives h, and h,; use the level curves to estimate the elevation h at nearby points). Similarly, estimate the partial derivative h, at the points (3,0) and (2,1). Hence approx- imate the rate of change of h,; in the z- and y-directions at (2,0) (that is, estimate h,, and hyy). Hint: For example, with Az = 1 you can estimate ha(2 4+ Az, 0) hae(2,0) hex(2,0) g 0 7 2(2,0) o Suppose that you walk in a straight line from the field station at (0.6,1) (indicated by the black dot), to the first experiment site at grid coordinates (4, 0), and then to a second experiment site at (0, 3), before returning to the field station. Describe how the elevation changes as you follow this path. 2. Problem 2: Linear approximation A model for the surface area of a human body is given by the function S = f(w, h) below, where w is the weight (in pounds), h is the height (in inches), and S is measured in square feet. S = f(w, h) = 0.1091w 420725, (a) Find f(159,69), and interpret this value. (b) Show that the linearization L(w,h) of f when w = 1591b and h = 69in, is given approx- imately by L(w,h) = 0.054w + 0.213 h 3.039. () Compare the values of f(w,h) and L(w,h) when (w,h) = (158,68) and when (w,h) = (150,95). Comment on the relative accuracy of the approximations