(1 point) Find the maximum rate of change of f(x, y) = ln(x + y) at the point (2,-2) and the direction in which it occurs. Maximum rate of change: 1/2(2)^(1/2) Direction (unit vector) in which it occurs: (1/(4(2))^(1/2) -1/(4(2)^(1/2)) (1 point) The temperature at a point (x, y, z) is given by T(x, y, z) = 1200e-x-212-22 where T is measured in C and x, y, and z in meters. 1. Find the rate of change of the temperature at the point P(2, -3, 2) in the direction toward the point Q(3, -5, 3). Answer: D (2,-3,2) = 4200e^(-0.5)/(6^(1/2)) PQ 2. In what direction does the temperature increase fastest at P? Answer: (1 point) Suppose f (x, y) = , P = (3, 1) and v = 3i + 2j. A. Find the gradient of f. (vf)(x, y) = 2x+y^2 i+ 2y+x^2 j Note: Your answers should be expressions of x and y; e.g. "3x-4y" B. Find the gradient of f at the point P. (V) (P) = 7 i+ 11 j Note: Your answers should be numbers C. Find the directional derivative of f at P in the direction of v. (D)(P)=-19/3^(1/2) Note: Your answer should be a number D. Find the maximum rate of change of f at P. 170^(1/2) Note: Your answer should be a number E. Find the (unit) direction vector w in which the maximum rate of change occurs at P. w = Note: Your answers should be numbers i+ j (1 point) A company operates two plants which manufacture the same item and whose total cost functions are C = 6+0.02q and C = 3.8 +0.03q, where q1 and 2 are the quantities produced by each plant. The total quantity demanded, q = 91 + 92, is related to the price, p, by p = 60 -0.03q. How much should each plant produce in order to maximize the company's profit? 91 = 429 92286 Adapted from M. Rosser, Basic Mathematics for Economists, p. 318 (New York: Routledge, 1993).