940 CHAPTER 14 Partial Derivatives II $4.00 for a medium box, and $4.50 for a large box. Fixed costs are $8000. (a) Express the cost of making x small boxes, y medium boxes, and z large boxes as a function of three variables: C = f ( x, y, z). (b) Find f(3000, 5000, 4000) and interpret it. (c) What is the domain of f? TV 9. Let g(x, y) = cos(x + 2y). III (a) Evaluate g(2, - 1). (b) Find the domain of g. (c) Find the range of g. 10. Let F(x, y) = 1 + 4 - y. (a) Evaluate F (3, 1). (b) Find and sketch the domain of F. VI (c) Find the range of F. 11. Let f (x, y, z) = Vx + vy + vz+ In(4 -x2 - 2 - z2). (a) Evaluate f (1, 1, 1). (b) Find and describe the domain of f. 12. Let g(x, y, z) = x'y2 z ( 10 - x - y - z. (a) Evaluate g (1, 2, 3). (b) Find and describe the domain of g. 33. A contour map for a function f is shown. Use it to estimate the 13-22 Find and sketch the domain of the function. values of f(-3, 3) and f(3, -2). What can you say about the 13. f ( x, y ) = Vx - 2+ vy- 1 shape of the graph? 14 . f ( x, y ) = Vx - 3y 15. f (x, y) = In(9 - x2 9 y? ) 16. f (x, y) = Vx2+ 12 - 4 17. g(x, y) = 18. g(x, y) = = In(2 - x) xty 1 - x2 - y? 70 60 50 40 19. f (x, y ) = - Vy - x2 20. f (x, y) = sin '(x + y) 30 20 21. f ( x, y, z) = V4 - x2 + 9 - y2 + v1-z? 10 22. f(x, y, z) = In(16 - 4x2 - 42 -z?) 34. Shown is a contour map of atmospheric pressure in North 23-31 Sketch the graph of the function. America on August 12, 2008. On the level curves (called isobars) the pressure is indicated in millibars (mb). 23. f (x, y) = y 24. f (x, y) = x2 (a) Estimate the pressure at C (Chicago), N (Nashville), 25. f (x, y) = 10 - 4x - 5y 26. f(x, y) = cos y S (San Francisco), and V (Vancouver). 28. f (x, y) = 2 - x2 - yz (b) At which of these locations were the winds strongest? 27. f (x, y) = sin x 29. f (x, y) = x2 + 42+ 1 30. f ( x, y ) = V 4x2 + yz 31. f (x, y) = V4 - 4x2 - y2 1016 32. Match the function with its graph (labeled I-VI). Give reasons 1016 for your choices. 1012 (a) f(x, y) = 1 + x 2 + 1 2 (b) f (x, y ) = 7 1+ x2 2 (c) f (x, y) = In(x2 + 2) (d) f (x, y) = cos Vx2 + y? -1004 1012 (e) f ( x, y) = 1xyl (f ) f (x, y) = cos(xy) 1008 N942 CHAPTER 14 Partial Derivatives 56. If V(x, y) is the electric potential at a point (x, y) in the 59. f (x, y ) = e-(x2+y?)/3(sin(x2) + cos(y?)) xy-plane, then the level curves of V are called equipotential curves because at all points on such a curve the electric 60. f(x, y) = cos x cos y potential is the same. Sketch some equipotential curves if V(x, y) = c/vr2 - x2 - y2, where c is a positive constant. 61-66 Match the function (a) with its graph (labeled A-F below) 57-60 Use a computer to graph the function using various and (b) with its contour map (labeled I-VI). Give reasons for your domains and viewpoints. Get a printout of one that, in your choices. opinion, gives a good view. If your software also produces level 61. z = sin(xy) 62. z = e* cos y curves, then plot some contour lines of the same function and compare with the graph. 63. z = sin(x - y) 64. z = sin x - sin y 57. f (x, y ) = xy2 -x3 (monkey saddle) * - y 65. z = (1 -x2)(1 - y2) 66. z = - 58. f(x, y) = xy' - yx' (dog saddle) 1+ x + y2 B D E F II III IV VI