w z F z F w y cos zy w z w y F y F w x sin xy z cos yz z w x F x F w y sin xy z

W z f z f w y cos zy w z w y f y f w x sin xy z cos

This preview shows page 42 - 46 out of 115 pages.

w z F z F w y cos zy w z w y F y F w x sin xy z cos yz z w x F x F w y sin xy z cos xy sin yz wz 20 F x , y , z , w 42. w z F z F w 1 2 y z 1 2 x y 1 2 y z w y F y F w 1 2 x y 1 2 1 2 y z 1 2 w x F x F w 1 2 x y 1 2 1 1 2 x y F x , y , z , w w x y y z 0 43. Degree: 1 xy x 2 y 2 1 f x , y xf x x , y yf y x , y x y 3 x 2 y 2 3 2 y x 3 x 2 y 2 3 2 f tx , ty tx ty tx 2 ty 2 t xy x 2 y 2 tf x , y f x , y xy x 2 y 2 44. Degree: 3 3 x 3 9 xy 2 3 y 3 3 f x , y xf x x , y yf y x , y x 3 x 2 3 y 2 y 6 xy 3 y 2 t 3 x 3 3 xy 2 y 3 t 3 f x , y f tx , ty tx 3 3 tx ty 2 ty 3 f x , y x 3 3 xy 2 y 3 45. Degree: 0 xf x x , y yf y x , y x 1 y e x y y x y 2 e x y 0 f tx , ty e tx ty e x y f x , y f x , y e x y
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Section 13.5 Chain Rules for Functions of Several Variables 201 46. Degree: 1 x 2 x 2 y 2 f x , y x 4 x 2 y 2 x 2 y 2 3 2 x 2 x 2 y 2 x 2 y 2 3 2 xf x x , y yf y x , y x x 3 2 xy 2 x 2 y 2 3 2 y x 2 y x 2 y 2 3 2 f tx , ty tx 2 tx 2 ty 2 t x 2 x 2 y 2 tf x , y f x , y x 2 x 2 y 2 47. (Page 923) dw dt w x dx dt w y dy dt 48. (Page 925) w t w x x t w y y t w s w x x s w y y s 49. is the explicit form of a function of two variables, as in The implicit form is of the form as in z x 2 y 2 0. F x , y , z 0, z x 2 y 2 . w f x , y 50. z y f y x , y , z f z x , y , z z x f x x , y , z f z x , y , z dy dx f x x , y f y x , y 51. dS dt 2 2 r h dr dt r dh dt 2 24 36 6 12 4 624 in. 2 min S 2 r r h dV dt 2 rh dr dt r 2 dh dt r 2 h dr dt r dh dt 12 2 36 6 12 4 4608 in. 3 min V r 2 h 52. (Surface area includes base.) 648 10 144 in. 2 min 36 5 20 9 10 in. 2 min 12 10 12 10 6 144 36 10 4 12 2 36 2 144 12 2 36 2 2 12 6 36 12 12 2 36 2 4 dS dt r 2 h 2 r 2 r 2 h 2 2 r dr dt rh r 2 h 2 dh dt S r r 2 h 2 r 2 dV dt 1 3 2 rh dr dt r 2 dh dt 1 3 2 12 36 6 12 2 4 1536 in. 3 min V 1 3 r 2 h
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202 Chapter 13 Functions of Several Variables 53. 2.566 m 2 hr 6 sin 4 1 2 6 2 2 cos 4 90 3 2 2 2 10 m 2 hr dA dt A x dx dt A d dt x sin dx dt x 2 2 cos d dt x x h θ 2 b 2 A 1 2 bh x sin 2 x cos 2 x 2 2 sin 54. (a) (b) 320 2 cm 2 min 25 15 2 10 2 25 15 25 15 25 15 2 10 2 4 25 15 10 25 15 2 10 2 12 25 15 2 10 2 25 15 25 15 25 15 2 10 2 4 R r h R r 2 h 2 dh dt dS dt R r 2 h 2 R r R r R r 2 h 2 dr dt R r 2 h 2 R r R r R r 2 h 2 dR dt S R r R r 2 h 2 3 19,500 6,500 cm 3 min 3 2 15 25 10 4 15 2 25 10 4 15 2 15 25 25 2 12 dV dt 3 2 r R h dr dt r 2 R h dR dt r 2 rR R 2 dh dt V 3 r 2 rR R 2 h 55. m 6 2 8 2 28 m cm 2 sec dI dt 1 2 m 2 r 1 dr 1 dt 2 r 2 dr 2 dt I 1 2 m r 1 2 r 2 2 56. dT dt 1 mR V dp dt p dV dt T 1 mR pV pV mRT 57. (a) —CONTINUED— x 2 tan 2 x 8 tan 0 x tan 2 4 8 x tan tan 2 x 1 2 x tan 4 x tan tan 1 tan tan 4 x tan 4 x θ φ 4 6 8 x tan 2 x
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Section 13.5 Chain Rules for Functions of Several Variables 203 57. —CONTINUED— (b) (c) Thus, x 2 1 x 2 x 8 1 x 0 8 x x x 2 2 ft. d dx 0 2 cos 2 2 x sin cos cos x sin tan 1 x d dx F x F 2 x tan 2 sec 2 x 2 8 2 cos 2 2 x sin cos x 2 8 F x , x 2 8 tan 2 x 0 58. Let then and Now, let and we have Thus, f x x f y y nf x , y .
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