Question 11.3.4: Consider Fig. 11.3.10 which shows some level curves of a fun......
Consider Fig. 11.3.10 which shows some level curves of a function z = f (x, y). On the basis of this figure, answer the following questions:
(a) What are the signs of f^{\prime}_{x}(x, y) and f^{\prime}_{y}(x, y) at the points P and Q? Estimate also the value of f^{\prime}_{x}(3, 1).
(b) What are the solutions of the equations: (i) f (3, y) = 4; and (ii) f (x, 4) = 6?
(c) What is the largest value that f (x, y) can attain when x = 2, and for which y value does this maximum occur?
(a) If you stand at P, you are on the level curve f (x, y) = 2. If you look in the direction of the positive x-axis, along the line y = 4, then you will see the terrain sloping upwards, because the nearest level curves will correspond to larger z values. Hence, f^{\prime}_{x} > 0. If you stand at P and look in the direction of the positive y-axis, along x = 2, the terrain will slope downwards. Thus, at P, we must have f^{\prime}_{y} < 0. At Q, we find similarly that f^{\prime}_{x} < 0 and f^{\prime}_{y} > 0. To estimate f^{\prime}_{x}(3, 1), we use f^{\prime}_{x}(3, 1) ≈ f (4, 1) − f (3, 1) = 2 − 4 = −2.^{7}
(b) Equation (i) has the solutions y = 1 and y = 4, because the line x = 3 cuts the level curve f (x, y) = 4 at (3, 1) and at (3, 4). Equation (ii) has no solutions, because the line y = 4 does not meet the level curve f (x, y) = 6 at all.
(c) The highest value of c for which the level curve f (x, y) = c has a point in common with the line x = 2 is c = 6. The largest value of f (x, y) when x = 2 is therefore 6, and we see from Fig. 11.3.10 that this maximum value is attained when y ≈ 2.2.
^{7} This approximation is actually far from exact. If we keep y = 1 and decrease x by one unit, then f (2, 1) ≈ 4, which should give the estimate f^{\prime}_{x}(3, 1) ≈ 4 − 4 = 0. The “map” is not sufficiently finely graded around Q.