MTH 201 Homework 8 Solution, Fall 2016

MATH 201. Homework 8 Solution. Section 14.1: 1. (a) f 0, 0 0 (b) f 1, 1 0 (c) f 2, 3 58 a b a b a b œ � œ œ (d) f 3, 2 33 a b � � œ 2. (a) f 2, (b) f 3, (c) f , ˆ ‰ ˆ ‰ ˆ ‰ 1 1 6 2 12 4 3 1 1 1 2 2 œ � œ � œ È È È 1 (d) f , 7 1 ˆ ‰ � � œ � 1 2 3. (a) f 3, 1, 2 (b) f 1, , (c) f 0, , 0 3 a b ˆ ‰ ˆ ‰ � œ � œ � œ 4 1 1 8 1 5 2 4 5 3 (d) f 2, 2, 100 0 a b œ 4. (a) f 0, 0, 0 7 (b) f 2, 3, 6 0 (c) f 1, 2, 3 35 a b a b a b È œ � œ � œ (d) f , , Š ‹ É 4 5 6 21 2 2 2 2 È È È œ 5. Domain: all points x y on or above the line a b ß y x 2 œ � 6. Domain: all points x y outside the circle a b ß x y 4 2 � 2 œ 7. Domain: all points x y not liying on the graph a b ß of y x or y x œ œ 3 8. Domain: all points x y not liying on the graph a b ß of x y 25 2 � 2 œ Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley. 796 Chapter 14 Partial Derivatives 9. Domain: all points x y satisfying a b ß x 1 y x 1 2 2 � Ÿ Ÿ � 10. Domain: all points x y satisfying a b ß x 1 y 1 0 a ba b � � � 11. Domain: all points x y satisfying a b ß x 2 x 2 y 3 y 3 0 a ba ba ba b � � � � 12. Domain: all points x y inside the circle a b ß x y 4 such that x y 3 2 2 2 2 � œ � Á 13. 14. 15. 16. 17. (a) Domain: all points in the xy-plane (b) Range: all real numbers Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley. Section 14.1 Functions of Several Variables 797 (c) level curves are straight lines y x c parallel to the line y x � œ œ (d) no boundary points (e) both open and closed (f) unbounded 18. (a) Domain: set of all x y so that y x 0 y x a ß b � Ê (b) Range: z 0 (c) level curves are straight lines of the form y x c where c 0 � œ (d) boundary is y x 0 y x, a straight line È � œ Ê œ (e) closed (f) unbounded 19. (a) Domain: all points in the xy-plane (b) Range: z 0 (c) level curves: for f(x y) 0, the origin; for f x y c 0, ellipses with center 0 0 and major and minor ß œ ß œ � ß a b a b axes along the x- and y-axes, respectively (d) no boundary points (e) both open and closed (f) unbounded 20. (a) Domain: all points in the xy-plane (b) Range: all real numbers (c) level curves: for f x y 0, the union of the lines y x; for f x y c 0, hyperbolas centered at a b a b ß œ œ „ ß œ Á 0 0 with foci on the x-axis if c 0 and on the y-axis if c 0 a ß b � � (d) no boundary points (e) both open and closed (f) unbounded 21. (a) Domain: all points in the xy-plane (b) Range: all real numbers (c) level curves are hyperbolas with the x- and y-axes as asymptotes when f x y 0, and the x- and y-axes a ß b Á when f(x y) 0 ß œ (d) no boundary points (e) both open and closed (f) unbounded 22. (a) Domain: all x y 0 y a b a b ß Á ß (b) Range: all real numbers (c) level curves: for f x y 0, the x-axis minus the origin; for f x y c 0, the parabolas y c x minus the a b a b ß œ ß œ Á œ # origin (d) boundary is the line x 0 œ (e) open (f) unbounded 23. (a) Domain: all x y satisfying x y 16 a ß b � � # # (b) Range: z   " 4 (c) level curves are circles centered at the origin with radii r 4 � (d) boundary is the circle x y 16 # � # œ Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley. (b) Range: z 0 (c) level curves are straight lines of the form y x c where c 0 � œ (d) boundary is y x 0 y x, a straight line È � œ Ê œ (e) closed (f) unbounded 19. (a) Domain: all points in the xy-plane (b) Range: z 0 (c) level curves: for f(x y) 0, the origin; for f x y c 0, ellipses with center 0 0 and major and minor ß œ ß œ � ß a b a b axes along the x- and y-axes, respectively (d) no boundary points (e) both open and closed (f) unbounded 20. (a) Domain: all points in the xy-plane (b) Range: all real numbers (c) level curves: for f x y 0, the union of the lines y x; for f x y c 0, hyperbolas centered at a b a b ß œ œ „ ß œ Á 0 0 with foci on the x-axis if c 0 and on the y-axis if c 0 a ß b � � (d) no boundary points (e) both open and closed (f) unbounded 21. (a) Domain: all points in the xy-plane (b) Range: all real numbers (c) level curves are hyperbolas with the x- and y-axes as asymptotes when f x y 0, and the x- and y-axes a ß b Á when f(x y) 0 ß œ (d) no boundary points (e) both open and closed (f) unbounded 22. (a) Domain: all x y 0 y a b a b ß Á ß (b) Range: all real numbers (c) level curves: for f x y 0, the x-axis minus the origin; for f x y c 0, the parabolas y c x minus the a b a b ß œ ß œ Á œ # origin (d) boundary is the line x 0 œ (e) open (f) unbounded 23. (a) Domain: all x y satisfying x y 16 a ß b � � # # (b) Range: z   " 4 (c) level curves are circles centered at the origin with radii r 4 � (d) boundary is the circle x y 16 # � # œ Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley. Section 14.2: 798 Chapter 14 Partial Derivatives (e) open (f) bounded 24. (a) Domain: all x y satisfying x y 9 a ß b � Ÿ # # (b) Range: 0 z 3 Ÿ Ÿ (c) level curves are circles centered at the origin with radii r 3 Ÿ (d) boundary is the circle x y 9 # � # œ (e) closed (f) bounded 25. (a) Domain: x y 0 0 a b a b ß Á ß (b) Range: all real numbers (c) level curves are circles with center 0 0 and radii r 0 a ß b � (d) boundary is the single point 0 0 a b ß (e) open (f) unbounded 26. (a) Domain: all points in the xy-plane (b) Range: 0 z 1 � Ÿ (c) level curves are the origin itself and the circles with center 0 0 and radii r 0 a ß b � (d) no boundary points (e) both open and closed (f) unbounded 27. (a) Domain: all x y satisfying 1 y x 1 a ß b � Ÿ � Ÿ (b) Range: z � Ÿ Ÿ 1 1 # # (c) level curves are straight lines of the form y x c where 1 c 1 � œ � Ÿ Ÿ (d) boundary is the two straight lines y 1 x and y 1 x œ � œ � � (e) closed (f) unbounded 28. (a) Domain: all x y , x 0 a ß b Á (b) Range: z � � � 1 1 # # (c) level curves are the straight lines of the form y c x, c any real number and x 0 œ Á (d) boundary is the line x 0 œ (e) open (f) unbounded 29. (a) Domain: all points x y outside the circle x y 1 a ß b � œ # # (b) Range: all reals (c) Circles centered ar the origin with radii r 1 � (d) Boundary: the cricle x y 1 # � # œ (e) open (f) unbounded 30. (a) Domain: all points x y inside the circle x y 9 a ß b � œ # # (b) Range: z ln 9 � (c) Circles centered ar the origin with radii r 9 � (d) Boundary: the cricle x y 9 # � # œ Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley. 46. (a) (b) 47. (a) (b) 48. (a) (b) 49. f x y 16 x y and 2 2 2 z 16 2 2 2 6 6 16 x y x y 10 a b Š ‹ Š ‹ Š ‹ È È È È ß œ � � ß Ê œ � � œ Ê œ � � Ê � œ # # # # # # # # 50. f x y x 1 and 1 0 z 1 1 0 x 1 0 x 1 or x 1 a b a b È È ß œ � ß Ê œ � œ Ê � œ Ê œ œ � # # # Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley. 802 Chapter 14 Partial Derivatives 51. f x y x y 3 and 3, 1 z 3 1 3 1 x y 3 1 x y 4 a b a b a b È É ß œ � � � Ê œ � � � œ Ê � � œ Ê � œ 2 2 2 2 52. f x y and 1 1 z 3 3 y 4x 3 a b a b ß œ � ß Ê œ œ Ê œ Ê œ � � 2y x 2y x x y 1 1 1 + 1 x y 1 1 1 � � � � � � � � # a b � � a b a b 53. 54. 55. 56. 57 58 11. lim Ð ß Ñ Ä Ð ß Î Ñ x y 1 6 1 x sin y x 1 1 1 2 4 1 sin 1 2 1 � � Î œ œ œ ˆ ‰ 6 12. lim 2 Ð ß Ñ Ä ß x y 0 ˆ ‰ 2 cos y 1 y sin x 1 cos 0 0 sin 1 1 � � � � " � � œ œ œ � a b ˆ ‰ 13. lim lim lim (x y) ( 1) 0 Ð ß Ñ Ä Ð ß Ñ Ð ß Ñ Ä Ð ß Ñ Ð ß Ñ Ä Ð ß Ñ Á x y 1 1 x y 1 1 x y 1 1 x y x 2xy y (x y) x y x y � � � � � œ œ � œ " � œ 14. lim lim lim (x y) (1 1) 2 Ð ß Ñ Ä Ð ß Ñ Ð ß Ñ Ä Ð ß Ñ Ð ß Ñ Ä Ð ß Ñ Á x y 1 1 x y 1 1 x y 1 1 x y x y (x y)(x y) x y x y � � � � � œ œ � œ�œ 15. lim lim lim (y 2) (1 2) 1 Ð ß Ñ Ä Ð ß Ñ Ð ß Ñ Ä Ð ß Ñ Ð ß Ñ Ä Ð ß Ñ Á Á x y 1 1 x y 1 1 x y 1 1 x 1 x 1 xy y 2x 2 (x 1)(y 2) x 1 x 1 � � � � � � � œ œ � œ � œ � 16. lim lim lim Ð ß Ñ Ä Ð ß � Ñ Ð ß Ñ Ä Ð ß � Ñ Ð ß Ñ Ä Ð ß � Ñ Á � Á Á � Á Á x y 2 4 x y 2 4 x y 2 4 y 4, x x y 4, x x x x y 4 y 4 x y xy 4x 4x x(x 1)(y 4) x(x 1) 1 � � � � � � � � # " œ œ œ (2 � 1) # " œ 17. lim lim lim x y 2 Ð ß Ñ Ä Ð ß Ñ Ð ß Ñ Ä Ð ß Ñ Ð ß Ñ Ä Ð ß Ñ Á Á x y 0 0 x y 0 0 x y 0 0 x y x y x y 2 x 2 y x y x y 2 x y x y � � � � � � � � È È È È È È È È È È ˆ ‰ ˆ ‰ œ œ � � ˆ ‰ È È 0 0 2 2 œ � � œ Š ‹ È È Note: (x y) must approach (0 0) through the first quadrant only with x y. ß ß Á 18. lim lim lim x y 2 Ð ß Ñ Ä Ð ß Ñ Ð ß Ñ Ä Ð ß Ñ Ð ß Ñ Ä Ð ß Ñ � Á � Á � Á x y 2 2 x y 2 2 x y 2 2 x y 4 x y 4 x y 4 x y 4 x y 2 x y 2 x y 2 x y 2 � � � � � � � � � � È È ˆ ‰ ˆ ‰ È È œ œ � � ˆ ‰ È 2 2 2 2 2 4 œ � � œ � œ Š ‹ È Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley. 806 Chapter 14 Partial Derivatives 19. lim lim lim Ð ß Ñ Ä Ð ß Ñ Ð ß Ñ Ä Ð ß Ñ Ð ß Ñ Ä Ð ß Ñ � Á � Á x y 2 0 x y 2 0 x y 2 0 2x y 4 2x y 4 È È ˆ ‰ ˆ ‰ È È È 2x y 2 2x y 2 2x y 4 2x y 2 2x y 2 2x y � � � � � � � � � � � � # " œ œ œ œ œ " " " � � # � È(2)(2) 0 2 2 4 20. lim lim lim Ð ß Ñ Ä Ð ß Ñ Ð ß Ñ Ä Ð ß Ñ Ð ß Ñ Ä Ð ß Ñ � Á � Á x y 4 3 x y 4 3 x y 4 3 x y 1 x y 1 È È È È ˆ ‰ ˆ ‰ È È È È È È x y 1 x y 1 x y 1 x y 1 x y 1 x y 1 � � � � � � � � � � � � " œ œ œ œ œ " " " � � � È È 4 3 1 2 2 4 21. lim lim lim lim cos r 1 Ð ß Ñ Ä Ð ß Ñ Ä Ä Ä x y 0 0 r 0 r 0 r 0 sin x y sin r 2r cos r x y r 2r a b a b a b � † � # œ œ œ œ a b 22. lim lim lim 0 Ð ß Ñ Ä Ð ß Ñ Ä Ä x y 0 0 u 0 u 0 1 cos xy xy u 1 1 cos u sin u � � a b œ œ œ 23. lim lim lim x xy y 1 1 1 1 3 Ð ß Ñ Ä Ð ß �"Ñ Ð ß Ñ Ä Ð ß �"Ñ Ð ß Ñ Ä Ð ß �"Ñ x y 1 x y 1 x y 1 x y x y x y x y x xy y 2 2 2 2 3 3 2 2 � � � � � � œ œ � � œ � � � � œ a bˆ ‰ a b a ba b a b Š ‹ 24. lim lim lim Ð ß Ñ Ä Ð ß Ñ Ð ß Ñ Ä Ð ß Ñ Ð ß Ñ Ä Ð ß Ñ x y 2 2 x y 2 2 x y 2 2 x y x y x y x y x y x y x y x y 2 2 2 2 32 1 1 1 � � � � � � � � � � 4 4 2 2 2 2 2 2 œ œ œ œ a ba ba b a ba b a ba b 25. lim T Ä Ð ß ß Ñ 1 3 4 Š ‹ " " " " " " � � x y z 1 3 4 12 12 12 4 3 19 � � œ � � œ œ 26. lim T Ä Ð ß � ß � Ñ 1 1 1 2xy yz 2(1)( 1) ( 1)( 1) x z 1 ( 1) 1 1 2 � � � � � � � � � # � � " " œ œ œ � 27. lim sin x cos y sec z sin 3 cos 3 sec 0 1 1 2 T Ä Ð ß ß Ñ 3 3 0 a b a b # # # # # # # � � œ � � œ � œ 28. lim tan (xyz) tan 2 tan T Ä � ß ß ˆ ‰ 1 4 2 2 �" �" �" " # œ � œ � ˆ ‰ ˆ ‰ 4 4 † 1 † 1 29. lim ze cos 2x 3e cos 2 (3)(1)(1) 3 T Ä Ð ß ß Ñ 1 0 3 2y 2 0 œ œ œ 1 30. lim ln x y z ln 2 ( 3) 6 ln 49 ln 7 T Ä Ð ß � ß Ñ 2 3 6 È È È # # # # # # � � œ � � � œ œ 31. (a) All x y (b) All x y except 0 0 a b a b a b ß ß ß 32. (a) All x y so that x y (b) All x y a b a b ß Á ß 33. (a) All x y except where x 0 or y 0 (b) All x y a b a b ß œ œ ß Section 14.3: Section 14.2 Limits and Continuity in Higher Dimensions 809 (b) Since f(x y) sin 2 and since 1 sin 2 1 for every , lim f(x y) varies from 1 to 1 k ß œ � Ÿ Ÿ ß � y mx œ ) ) ) Ð ß Ñ Ä Ð ß Ñ x y 0 0 along y mx. œ 60. xy x y xy x y x y x y x y x y x y x y x y k k k k k k k k k k k k k k k k a b È È È È # # # # # # # # # # # # # # # # � œ � Ÿ � œ � Ÿ � � � x y x y x y x y œ � Ê Ÿ œ � Ê � � Ÿ Ÿ � a b a b a b ¹ ¹ # # # # # # # # # � � � � � � xy x y x y xy x y x y x y x y a b a b a b lim xy 0 by the Sandwich Theorem, since lim x y 0; thus, define f 0 0 0 Ê œ „ � œ ß œ Ð ß Ñ Ä Ð ß Ñ Ð ß Ñ Ä Ð ß Ñ x y 0 0 x y 0 0 Š ‹ a b a b x y x y � � # # 61. lim lim lim 0 Ð ß Ñ Ä Ð ß Ñ Ä Ä x y 0 0 r 0 r 0 x xy x y r cos r sin 1 r cos (r cos ) r sin r cos cos sin � � � � � œ œ œ ) ) ) ) ) ) ) ) a b a b 62. lim cos lim cos lim cos cos 0 1 Ð ß Ñ Ä Ð ß Ñ Ä Ä x y 0 0 r 0 r 0 Š ‹ Š ‹ ’ “ x y x y r cos r sin 1 r cos r sin r cos sin � � � � � œ œ œ œ ) ) ) ) ) ) a b 63. lim lim lim sin sin ; the limit does not exist since sin is between Ð ß Ñ Ä Ð ß Ñ Ä Ä x y 0 0 r 0 r 0 y x y r r sin � # # # œ œ œ ) a )b ) ) 0 and 1 depending on ) 64. lim lim lim ; the limit does not exist for co Ð ß Ñ Ä Ð ß Ñ Ä Ä x y 0 0 r 0 r 0 2x 2r cos 2 cos 2 cos x x y r r cos r cos cos � � � � œ œ œ ) ) ) ) ) ) s 0 ) œ 65. lim tan lim tan lim tan ; Ð ß Ñ Ä Ð ß Ñ Ä Ä x y 0 0 r 0 r 0 �" �" �" � � � � ’ “ ’ “ ’ “ k k k k k k k k k k a b k k k k x y r cos r sin r cos sin x y r r œ œ ) ) ) ) if r 0 , then lim tan lim tan ; if r 0 , then Ä œ œ Ä � �" �" � � � # r r Ä ! Ä ! ’ “ ’ “ k k a b k k k k k k k k r cos sin cos sin r r ) ) ) ) 1 lim tan lim tan the limit is r r Ä ! Ä ! �" �" � � � # # ’ “ Š ‹ k k a b k k k k k k k k r cos sin cos sin r r ) ) ) ) 1 1 œ œ Ê 66. lim lim lim cos sin lim (cos 2 ) which ranges between Ð ß Ñ Ä Ð ß Ñ Ä Ä Ä x y 0 0 r 0 r 0 r 0 x y x y r r cos r sin � � � # # œ œ � œ ) ) a b ) ) ) 1 and 1 depending on the limit does not exist � Ê ) 67. lim ln lim ln Ð ß Ñ Ä Ð ß Ñ Ä x y 0 0 r 0 Š ‹ Š ‹ 3x x y 3y x y r 3r cos r cos sin 3r sin � � � � � œ ) ) ) ) lim ln 3 r cos sin ln 3 define f(0 0) ln 3 œ � œ Ê ß œ r Ä 0 a b # # # ) ) 68. lim lim lim 3r cos sin 0 define f(0 0) 0 Ð ß Ñ Ä Ð ß Ñ Ä Ä x y 0 0 r 0 r 0 3xy x y r (3r cos ) r sin � # œ œ œ Ê ß œ ) ) a b ) ) 69. Let 0.1. Then x y x y 0.1 x y 0.01 x y 0 0.01 $ $ œ � � Ê � � Ê � � Ê � � � È È k k # # # # # # # # f(x y) f( ) 0.01 . Ê ß � !ß ! � œ k k % 70. Let 0.05. Then x and y f x y f 0 0 0 y 0.05 . $ $ $ % œ � � Ê ß � ß œ � œ Ÿ � œ k k k k k k k k a b a b ¸ ¸ ¸ ¸ y y x 1 x 1 � � 71. Let 0.005. Then x and y f x y f 0 0 0 x y x y $ $ $ œ � � Ê ß � ß œ � œ Ÿ � � � k k k k k k k k k k k k a b a b ¸ ¸ ¸ ¸ x y x y x 1 x 1 � � � � 0.005 0.005 0.01 . � � œ œ % 72. Let 0.01. Since 1 cos x 1 1 2 cos x 3 1 x y $ œ � Ÿ Ÿ Ê Ÿ � Ÿ Ê Ÿ Ÿ Ê Ÿ Ÿ � " " #� � � � 3 cos x 3 2 cos x x y x y k k ¸ ¸ k k x y . Then x and y f x y f 0 0 0 x y 0.01 0.01 Ÿ � � � Ê ß � ß œ � œ Ÿ � � � k k k k k k k k k k k k k k a b a b ¸ ¸ ¸ ¸ $ $ x y x y 2 cos x 2 cos x � � � � 0.02 . œ œ % Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley. p 73. Let 0.04. Since y x y 1 x x x y f x y f 0 0 $ $ œ Ÿ � Ê Ÿ Ê Ÿ l l œ Ÿ � � Ê ß � ß 2 2 2 y x y x y x y 2 2 2 2 2 2 2 2 2 � � l l È È k k a b a b 0 0.04 . œ � � œ ¹ ¹ x y x y 2 2 � 2 % 74. Let 0.01. If y 1, then y y y x y , so x x x y x y 2 x y . Since $ œ l l Ÿ Ÿ l l œ Ÿ � l l œ Ÿ � Ê l l � Ÿ � 2 2 2 2 2 2 2 2 2 2 È È È È È x x y 1 and y x y 1. Then x y x y 2 2 2 2 2 2 2 2 2 x x x y x y x y x y x y y y x y Ÿ � Ê Ÿ Ÿ � Ê Ÿ Ÿ l l � Ÿ l l � � 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 4 � � � � � l � l $ f x y f 0 0 0 2 0.01 0.002 . Ê ß � ß œ � � œ œ k k a b a b a b ¹ ¹ x y x y 3 4 2 2 � � % 75. Let 0.015. Then x y z f(x y z) f( 0 0) x y z 0 x y z $ $ œ � � � Ê ß ß � !ß ß œ � � � œ � � È È k k k k k k # # # # # # # # # x t x 0.015 0.015 . œ � � � œ œ Š ‹ Š ‹ È È # # # # # % 76. Let 0.2. Then x , y , and z f(x y z) f( 0 0) xyz 0 xyz x y z (0.2) $ $ $ $ œ � � � Ê ß ß � !ß ß œ � œ œ � k k k k k k k k k k k k k k k k k k $ 0.008 . œ œ % 77. Let 0.005. Then x , y , and z f(x y z) f( 0 0) 0 $ $ $ $ œ � � � Ê ß ß � !ß ß œ � k k k k k k k k ¹ ¹ x y z x y z 1 � � � � � x y z x y z 0.005 0.005 0.005 0.015 . œ Ÿ � � Ÿ � � � � � œ œ ¹ ¹ k k k k k k k k x y z x y z 1 � � � � � % 78. Let tan (0.1). Then x , y , and z f(x y z) f( 0 0) tan x tan y tan z $ $ $ $ œ � � � Ê ß ß � !ß ß œ � � �" # # # k k k k k k k k k k tan x tan y tan z tan x tan y tan z tan tan tan 0.01 0.01 0.01 0.03 . Ÿ � � œ � � � � � œ � � œ œ k k k k k k # # # # # # # # # $ $ $ % 79. lim f(x y z) lim (x y z) x y z f(x y z ) f is continuous at Ð ß ß Ñ Ä Ð ß ß Ñ Ð ß ß Ñ Ä Ð ß ß Ñ x y z x y z x y z x y z ß ß œ � � œ � � œ ß ß Ê ! ! ! ! ! ! every (x y z ) ! ! ! ß ß 80. lim f(x y z) lim x y z x y z f(x y z ) f is continuous at Ð ß ß Ñ Ä Ð ß ß Ñ Ð ß ß Ñ Ä Ð ß ß Ñ x y z x y z x y z x y z ß ß œ � � œ � � œ ß ß Ê a b # # # # # # ! ! ! ! ! ! every point (x y z ) ! ! ! ß ß 14.3 PARTIAL DERIVATIVES 1. 4x, 3 2. 2x y, x 2y ` ` ` ` ` ` ` ` f f f f x y x y œ œ � œ � œ � � 3. 2x(y 2), x 1 4. 5y 14x 3, 5x 2y 6 ` ` ` ` ` ` ` ` # f f f f x y x y œ � œ � œ � � œ � � 5. 2y(xy 1), 2x(xy 1) 6. 6(2x 3y) , 9(2x 3y) ` ` ` ` ` ` ` ` # # f f f f x y x y œ � œ � œ � œ � � 7. , 8. , ` ` ` ` " ` ` ` ` � � � � f x f f 2x f x y x y x y x y y x 3 x œ œ œ œ È È É É ˆ ‰ ˆ ‰ y y 9. (x y) , (x y) ` " ` " ` " ` " ` � ` � ` � ` � f f x (x y) x (x y) y (x y) y (x y) œ � � œ � œ � � œ � † † 10. , ` ` ` ` � � � � � � � � � f f x y x y (1) x(2x) x y (0) x(2y) x y x y x y x y y x 2xy œ œ œ œ � a b a b a b a b a b a b 11. , ` ` � � " ` � � ` � � � � � � � � " � � f f x x (xy 1) (xy 1) y (xy 1) (xy 1) (xy 1)(1) (x y)(y) y 1 (xy )(1) (x y)(x) œ œ œ œ œ Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley. Section 14.3 Partial Derivatives 811 12. , ` " ` ` " ` ` ` � ` ` � � � � � f f 1 x x x x x y y y x x y 1 1 y y y y x 1 x 1 œ œ � œ � œ œ œ ˆ ‰ ˆ ‰ ’ “ ’ “ ˆ ‰ ˆ ‰ y y x x y y x x † † ˆ ‰ ˆ ‰ 13. e (x y 1) e , e (x y 1) e ` ` ` ` ` ` ` ` Ð � � Ñ Ð � � Ñ Ð � � Ñ Ð � � Ñ f f x x y y x y 1 x y 1 x y 1 x y 1 œ � � œ œ � � œ † † 14. e sin (x y) e cos (x y), e cos (x y) ` ` ` ` � � � f f x y x x x œ � � � � œ � 15. (x y) , (x y) ` " ` " ` " ` " ` � ` � ` � ` � f f x x y x x y y x y y x y œ � œ œ � œ † † 16. e (xy) ln y ye ln y, e (xy) ln y e xe ln y ` ` ` ` " ` ` ` ` f f e x x y y y y xy xy xy xy xy œ œ œ � œ � † † † † † xy 17. 2 sin (x 3y) sin (x 3y) 2 sin (x 3y) cos (x 3y) (x 3y) 2 sin (x 3y) cos (x 3y), ` ` ` ` ` ` f x x x œ � � œ � � � œ � � † † 2 sin (x 3y) sin (x 3y) 2 sin (x 3y) cos (x 3y) (x 3y) 6 sin (x 3y) cos (x 3y) ` ` ` ` ` ` f y y y œ � � œ � � � œ � � � † † 18. 2 cos 3x y cos 3x y 2 cos 3x y sin 3x y 3x y ` ` ` ` ` ` # # # # # f x x x œ � � œ � � � � a b a b a b a b a b † † 6 cos 3x y sin 3x y , œ � � � a b a b # # 2 cos 3x y cos 3x y 2 cos 3x y sin 3x y 3x y ` ` ` ` ` ` # # # # # f y y y œ � � œ � � � � a b a b a b a b a b † † 4y cos 3x y sin 3x y œ � � a b a b # # 19. yx , x ln x 20. f(x y) and ` ` ` " ` � ` ` ` ` f f ln x f f ln x x y ln y x x ln y y y(ln y) œ œ ß œ Ê œ œ y 1 y 21. g(x), g(y) ` ` ` ` f f x y œ � œ 22. f(x y) (xy) , xy 1 f(x y) (1 xy) and ß œ � Ê ß œ Ê œ � � œ ! _ œ n 0 n k k " ` " ` � ` � ` � 1 xy x (1 xy) x (1 xy) f y † (1 xy) ` " ` ` � ` � f x y (1 xy) y (1 xy) œ � � œ † 23. f y , f 2xy, f 4z 24. f y z, f x z, f y x x y z x y z œ œ œ � œ � œ � œ � # 25. f 1, f , f x y z y y z y z z œ œ � œ � È È � � 26. f x x y z , f y x y z , f z x y z x y z œ � � � œ � � � œ � � � a b a b a b # # # # # # # # # �$Î# �$Î# �$Î# 27. f , f , f x y z yz xy 1 x y z 1 x y z 1 x y z xz œ œ œ È È È � � � 28. f , f , f x y z x yz (x yz) 1 x yz (x yz) 1 x yz (x yz) 1 z y œ œ œ " � � � � � � � � � k k k k k k È È È 29. f , f , f x y z œ œ œ " � � � � � � x 2y 3z x 2y 3z x 2y 3z 2 3 30. f yz (xy) , f z ln (xy) yz ln (xy) z ln (xy) (xy) z ln (xy) z, x y xy x xy x y xy y (yz)(y) yz yz œ œ œ œ � œ � œ � † † † † " ` ` ` ` ` ` f y ln (xy) yz ln (xy) y ln (xy) z z œ � œ † ` ` 31. f 2xe , f 2ye , f 2ze x y z x y z x y z x y z œ � œ � œ � � � � � � � � � � a b a b a b 32. f yze , f xze , f xye x y z xyz xyz xyz œ � œ � œ � � � � Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley. 38. 1 cos , r sin , 1 ` ` ` ` ` ` g g g r z œ � œ œ � ) ) ) 39. W V, W P , W , W , W p v v g v Vv 2V v V v V v 2g 2g 2g g g œ œ � œ œ œ œ � $ $ $ $ $ # 40. m, , , c, ` ` ` ` ` ` ` # ` ` ` # A A A m A k A km h c h k q m q q q q œ œ œ œ � œ � � 41. 1 y, 1 x, 0, 0, 1 ` ` ` ` ` ` ` ` ` ` ` ` ` ` f f f f f f x y x y y x x y œ � œ � œ œ œ œ 42. y cos xy, x cos xy, y sin xy, x sin xy, cos xy xy sin xy ` ` ` ` ` ` ` ` ` ` ` ` ` ` # # f f f f f f x y x y y x x y œ œ œ � œ � œ œ � 43. 2xy y cos x, x sin y sin x, 2y y sin x, cos y, 2x cos x ` ` ` ` ` ` ` ` ` ` ` ` ` ` # g g g g g g x y x y y x x y œ � œ � � œ � œ � œ œ � 44. e , xe 1, 0, xe , e ` ` ` ` ` ` ` ` ` ` ` ` ` ` h h h h h h x y x y y x x y œ œ � œ œ œ œ y y y y 45. , , , , ` " ` " ` �" ` �" ` ` �" ` � ` � ` � ` � ` ` ` ` � r r r r r r x x y y x y x (x y) y (x y) y x x y (x y) œ œ œ œ œ œ 46. , , ` " ` " ` " ` " ` ` � ` ` � � � � � � s s 1 x x x x x x y y y x x x y 1 1 1 1 y y y y œ œ � œ œ œ œ ” • ” • ” • ” • ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ y y y y x x x x † † , , ` ` ` ` ` ` ` ` ` ` � � � � � � � � � � � s s s s x y y x x y y(2x) 2xy x(2y) 2xy y x x y x y x y x y x y x y x y ( 1) y(2y) œ œ œ œ � œ œ œ a b a b a b a b a b a b a b 47. 2x tan xy x sec xy y 2x tan xy x y sec xy , x sec xy x x sec xy , ` ` ` ` w w x y 2 2 2 2 2 2 3 2 œ � † œ � œ † œ a b a b a b a b a b a b 2tan xy 2x sec xy y 2xy sec xy x y 2sec xy sec xy tan xy y ` ` w x 2 2 2 œ � † � � † a b a b a b a b a b a b a b 2tan xy 4xy sec xy 2x y sec xy tan xy , x 2sec xy sec xy tan xy x 2x sec xy tan xy œ � � œ † œ a b a b a b a b a b a b a b a b a b a b 2 2 2 2 3 4 2 w y ` ` 3x sec xy x 2sec xy sec xy tan xy y 3x sec xy x y sec xy tan xy ` ` ` ` ` ` w w y x x y 2 2 3 2 2 3 2 œ œ � † œ � a b a b a b a b a b a b a b a b 48. ye 2x 2xy e , 1 e ye 1 e 1 y , ` ` ` ` � � � � � w w x y x y x y x y x y x y œ † œ œ � † � œ � 2 2 2 2 2 a b a b a b 2y e 2xy e 2x 2ye 1 2x , e 1 1 y e 1 ` ` ` ` � � � � � w w x y x y x y x y 2 x y x y œ � † œ � œ † � � � � 2 2 2 2 2 Š ‹ Š ‹ a b a b a b a b e y 2 , e 2x 1 y 2x e 1 y œ � œ œ † � œ � x y x y x y w w y x x y 2 2 2 � � � ` ` ` ` ` ` a b a b a b Š ‹ 49. sin x y x cos x y 2xy sin x y 2x ycos x y , x cos x y x x cos x y , ` ` ` ` w w x y 2 2 2 2 2 2 2 3 2 œ � † œ � œ † œ a b a b a b a b a b a b cos x y 2xy 4xy cos x y 2x y sin x y 2xy 6xy cos x y 4x y sin x y , ` ` w x 2 2 2 2 2 3 2 2 œ † � � † œ � a b a b a b a b a b x sin x y x x sin x y , 3x cos x y x sin x y 2xy 3x cos x y 2x y sin x y ` ` ` ` ` ` ` ` w w w y y x x y 3 2 2 5 2 2 2 3 2 2 2 4 2 œ � † œ � œ œ � † œ � a b a b a b a b a b a b Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley. Section 14.3 Partial Derivatives 813 50. , , ` ` � � ` ` � � � � � � � � � � � � � � w w x x x y x y x y 2x x y 1 x y x y x y x y x y x 2xy y œ œ œ œ ˆ ‰ ˆ ‰ a ba b a b a b a b a b a b a b 2 2 2 2 2 2 2 2 2 2 2 2 , ` ` � � � � � � � � � � � � � w x x y 2x 2y x 2xy y 2 x y 2x 2 x 3x y 3 xy y x y x y œ œ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ a b a b ’ “ a b a b 2 2 2 3 2 2 2 2 2 2 3 2 , ` � ` ` ` ` ` ` ` � † � � � � † � � � � � � � † � � � w 2x 2x w w y y x x y x y 0 x x 2 x y 1 x y 2x 1 x 2xy y 2 x y 1 x y x y x y œ œ œ œ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ’ “ ’ “ a b a b a b a b 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 œ 2x 3x 2xy y x y 3 2 2 3 � � � � a b 51. , , , and ` ` ` � ` � ` � ` � ` ` � ` ` � w 2 w 3 w 6 w 6 x 2x 3y y 2x 3y y x (2x 3y) x y (2x 3y) œ œ œ œ 52. e ln y , ln x, , and ` ` ` " " ` " " ` ` ` ` ` ` w w x w w x x y y y x y x x y y x y œ � � œ � œ œ � œ � x 53. y 2xy 3x y , 2xy 3x y 4x y , 2y 6xy 12x y , and 2y 6xy 12x y ` ` ` ` ` ` ` ` ` ` # $ # % # # $ $ # # $ # # $ w w w w x y y x x y œ � � œ � � œ � � œ � � 54. sin y y cos x y, x cos y sin x x, cos y cos x 1, and cos y cos x 1 ` ` ` ` ` ` ` ` ` ` w w w w x y y x x y œ � � œ � � œ � � œ � � 55. (a) x first (b) y first (c) x first (d) x first (e) y first (f) y first 56. (a) y first three times (b) y first three times (c) y first twice (d) x first twice 57. f 1 2 lim lim lim x f(1 h 2) f(1 2) h h h 1 (1 h) 2 6(1 h) (2 6) h 6 1 2h h 6 a ß b œ œ œ h 0 h 0 h 0 Ä Ä Ä � ß � ß � � � � � � � � � � � � c d a b lim lim ( 13 6h) 13, œ œ � � œ � h 0 h 0 Ä Ä � 13h � 6h h f (1 2) lim lim lim y f(1 2 h) f(1 2) (2 6 2h) (2 6) h h h 1 1 (2 h) 3(2 h) (2 6) ß œ œ œ h 0 h 0 h 0 Ä Ä Ä ß � � ß � � � � � � � � � � � c d lim ( 2) 2 œ � œ � h 0 Ä 58. f 2 1 lim lim x f 2 h 1 f 2 1 h h 4 2 2 h 3 2 h 3 2 a � ß b œ œ h 0 h 0 Ä Ä a b a b c d a b a b a b � � ß � � ß � � � � � � � � � � lim lim 1 1, œ œ œ h 0 h 0 a b 2h 1 h 1 h � � � Section 14.3 Partial Derivatives 813 50. , , ` ` � � ` ` � � � � � � � � � � � � � � w w x x x y x y x y 2x x y 1 x y x y x y x y x y x 2xy y œ œ œ œ ˆ ‰ ˆ ‰ a ba b a b a b a b a b a b a b 2 2 2 2 2 2 2 2 2 2 2 2 , ` ` � � � � � � � � � � � � � w x x y 2x 2y x 2xy y 2 x y 2x 2 x 3x y 3 xy y x y x y œ œ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ a b a b ’ “ a b a b 2 2 2 3 2 2 2 2 2 2 3 2 , ` � ` ` ` ` ` ` ` � † � � � � † � � � � � � � † � � � w 2x 2x w w y y x x y x y 0 x x 2 x y 1 x y 2x 1 x 2xy y 2 x y 1 x y x y x y œ œ œ œ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ˆ ‰ ’ “ ’ “ a b a b a b a b 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 œ 2x 3x 2xy y x y 3 2 2 3 � � � � a b 51. , , , and ` ` ` � ` � ` � ` � ` ` � ` ` � w 2 w 3 w 6 w 6 x 2x 3y y 2x 3y y x (2x 3y) x y (2x 3y) œ œ œ œ 52. e ln y , ln x, , and ` ` ` " " ` " " ` ` ` ` ` ` w w x w w x x y y y x y x x y y x y œ � � œ � œ œ � œ � x 53. y 2xy 3x y , 2xy 3x y 4x y , 2y 6xy 12x y , and 2y 6xy 12x y ` ` ` ` ` ` ` ` ` ` # $ # % # # $ $ # # $ # # $ w w w w x y y x x y œ � � œ � � œ � � œ � � 54. sin y y cos x y, x cos y sin x x, cos y cos x 1, and cos y cos x 1 ` ` ` ` ` ` ` ` ` ` w w w w x y y x x y œ � � œ � � œ � � œ � � 55. (a) x first (b) y first (c) x first (d) x first (e) y first (f) y first 56. (a) y first three times (b) y first three times (c) y first twice (d) x first twice 57. f 1 2 lim lim lim x f(1 h 2) f(1 2) h h h 1 (1 h) 2 6(1 h) (2 6) h 6 1 2h h 6 a ß b œ œ œ h 0 h 0 h 0 Ä Ä Ä � ß � ß � � � � � � � � � � � � c d a b lim lim ( 13 6h) 13, œ œ � � œ � h 0 h 0 Ä Ä � 13h � 6h h f (1 2) lim lim lim y f(1 2 h) f(1 2) (2 6 2h) (2 6) h h h 1 1 (2 h) 3(2 h) (2 6) ß œ œ œ h 0 h 0 h 0 Ä Ä Ä ß � � ß � � � � � � � � � � � c d lim ( 2) 2 œ � œ � h 0 Ä 58. f 2 1 lim lim x f 2 h 1 f 2 1 h h 4 2 2 h 3 2 h 3 2 a � ß b œ œ h 0 h 0 Ä Ä a b a b c d a b a b a b � � ß � � ß � � � � � � � � � � lim lim 1 1, œ œ œ h 0 h 0 Ä Ä a b 2h 1 h 1 h � � � f 2 1 lim lim y f 2 1 h f 2 1 h h 4 4 3 1 h 2 1 h 3 2 a � ß b œ œ h 0 h 0 Ä Ä a b a b � ‘ a b a b a b � ß � � � ß � � � � � � � � li li li 1 2h 1 � a b � � � � � � � 3 3h 2 4h 2h 1 h 2h a b Section 15.1: ' ' ' ' 1 1 1 1 ’ “ 2 �" 4. 1 dx dy x dy dy y ' ' ' ' 0 0 0 0 1 1 1 1 Š ‹ ’ “ Š ‹ ’ “ � œ � � œ � œ � œ x y x y y y 2 6 2 6 2 6 6 3 x 5 5 2 � 3 3 " 0 0 1 5. 4 y dy dx 4y dx dx x 16 ' ' ' ' 0 0 0 0 3 2 3 3 a b ’ “ � ‘ � œ � œ œ œ # y 3 3 3 16 16 # ! 3 0 6. x y 2xy dy dx xy dx 4x 2x dx 2x 0 ' ' ' ' 0 2 0 0 3 0 3 3 a b a b ’ “ ’ “ # # # # � œ � œ � œ � œ x y 2 3 2x ! �# ! 3 7. dx dy ln 1 x y dy ln 1 y dy y ln 1 y y ln 1 y 2 ln 2 1 ' ' ' ' 0 0 0 0 1 1 1 1 y 1 x y 0 � " œ l � l œ l � l œ l � l � � l � l œ � c d c d 1 0 8. y dx dy x x y dy 4 4 y dy 4y y ' ' ' ' 1 0 1 1 4 4 4 4 4 ˆ ‰ � ‘ ˆ ‰ � ‘ È È x 1 8 92 2 4 3 3 2 1/2 3/2 � œ � œ � œ � œ 4 1 9. e dy dx e dx 5e e dx e e 5 e ' ' ' ' 0 1 0 0 ln 2 ln 5 ln 2 ln 2 ln 2 2x y 2x y 2x 2x 1 2x 2x 1 5 3 2 2 � � � � " # œ œ � œ � œ � c d a b a b � ‘ ln 5 0 " 10. x y e dy dx x y e dx x e dx x e e ' ' ' ' 0 1 0 0 1 2 1 1 x 2 x x x x 3 3 3 3 2 2 2 2 œ œ œ � œ � ‘ � ‘ " # 2 1 0 " 11. y sin x dx dy y cos x dy y dy y ' ' ' ' 1 0 1 1 2 2 2 2 0 2 1 2 œ � œ œ œ c d � ‘ " # 2 3 2 12. sin x cos y dx dy cos x x cos y dy 2 cos y dy 2y sin y 2 ' ' ' ' 2 2 2 0 0 2 a b c d a b c d � œ � � œ � œ � œ 1 1 1 13. 6 y 2 x dA 6 y 2 x dy dx 2 y 2 x y dx 16 4 x dx 16 x 2 x 14 ' ' R a b a b c d a b c d # # � œ � œ � œ � œ � œ ' ' ' ' 0 0 0 0 1 2 1 1 0 0 2 1 3 2 14. dA dy dx dx x dx x ' ' R È È È x x x y y y 3 3 1 2 3 2 1 8 2 2 œ œ � œ œ œ ' ' ' ' 0 1 0 0 4 2 4 4 1 2 0 4 ’ “ � ‘ " # Î Î 15. x y cos y dA x y cos y dy dx x y sin y x cos y dx 2x dx x 0 ' ' R œ œ � œ � œ � œ ' ' ' ' 1 0 1 1 1 1 1 0 1 1 c d a b c d 2 16. y sin x y dA y sin x y dy dx y cos x y sin x y dx ' ' R a b a b c d a b a b � œ � œ � � � � ' ' ' 0 0 0 0 sin x cos x sin x dx cos x sin x cos x 4 œ � � � � œ � � � � � œ ' 0 0 a b c d a b a b a b a b 1 1 1 1 1 1 Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley. 5. 4 y dy dx 4y dx dx x 16 ' ' ' ' 0 0 0 0 3 2 3 3 a b ’ “ � ‘ � œ � œ œ œ # y 3 3 3 16 16 # ! 3 0 6. x y 2xy dy dx xy dx 4x 2x dx 2x 0 ' ' ' ' 0 2 0 0 3 0 3 3 a b a b ’ “ ’ “ # # # # � œ � œ � œ � œ x y 2 3 2x ! �# ! 3 7. dx dy ln 1 x y dy ln 1 y dy y ln 1 y y ln 1 y 2 ln 2 1 ' ' ' ' 0 0 0 0 1 1 1 1 y 1 x y 0 � " œ l � l œ l � l œ l � l � � l � l œ � c d c d 1 0 8. y dx dy x x y dy 4 4 y dy 4y y ' ' ' ' 1 0 1 1 4 4 4 4 4 ˆ ‰ � ‘ ˆ ‰ � ‘ È È x 1 8 92 2 4 3 3 2 1/2 3/2 � œ � œ � œ � œ 4 1 9. e dy dx e dx 5e e dx e e 5 e ' ' ' ' 0 1 0 0 ln 2 ln 5 ln 2 ln 2 ln 2 2x y 2x y 2x 2x 1 2x 2x 1 5 3 2 2 � � � � " # œ œ � œ � œ � c d a b a b � ‘ ln 5 0 " 10. x y e dy dx x y e dx x e dx x e e ' ' ' ' 0 1 0 0 1 2 1 1 x 2 x x x x 3 3 3 3 2 2 2 2 œ œ œ � œ � ‘ � ‘ " # 2 1 0 " 11. y sin x dx dy y cos x dy y dy y ' ' ' ' 1 0 1 1 2 2 2 2 0 2 1 2 œ � œ œ œ c d � ‘ " # 2 3 2 12. sin x cos y dx dy cos x x cos y dy 2 cos y dy 2y sin y 2 ' ' ' ' 2 2 2 0 0 2 a b c d a b c d � œ � � œ � œ � œ 1 1 1 13. 6 y 2 x dA 6 y 2 x dy dx 2 y 2 x y dx 16 4 x dx 16 x 2 x 14 ' ' R a b a b c d a b c d # # � œ � œ � œ � œ � œ ' ' ' ' 0 0 0 0 1 2 1 1 0 0 2 1 3 2 14. dA dy dx dx x dx x ' ' R È È È x x x y y y 3 3 1 2 3 2 1 8 2 2 œ œ � œ œ œ ' ' ' ' 0 1 0 0 4 2 4 4 1 2 0 4 ’ “ � ‘ " # Î Î 15. x y cos y dA x y cos y dy dx x y sin y x cos y dx 2x dx x 0 ' ' R œ œ � œ � œ � œ ' ' ' ' 1 0 1 1 1 1 1 0 1 1 c d a b c d 2 16. y sin x y dA y sin x y dy dx y cos x y sin x y dx ' ' R a b a b c d a b a b � œ � œ � � � � ' ' ' 0 0 0 0 sin x cos x sin x dx cos x sin x cos x 4 œ � � � � œ � � � � � œ ' 0 0 a b c d a b a b a b a b 1 1 1 1 1 1 Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley. 882 Chapter 15 Multiple Integrals 17. e dA e dy dx e dx e e dx e e ' ' R x y x y x y x x x x � � � � � " # œ œ � œ � � œ � � œ ' ' ' ' 0 0 0 0 ln 2 ln 2 ln 2 ln 2 0 0 ln 2 ln 2 ln 2 ln 2 c d a b c d 18. x y e dA x y e dy dx e dx e dx e x e 3 ' ' R x y x y x y x x 2 2 2 2 œ œ œ � œ � œ � ' ' ' ' 0 0 0 0 2 1 2 2 0 1 0 2 ’ “ ˆ ‰ � ‘ a b " " " " " " # # # # # # 19. dA dy dx dx dx 2 ln x 1 2 ln 2 ' ' R x y x y x y x 1 x 1 4 x 1 x 1 4 x 2 3 3 4 2 2 2 2 � � � � œ œ œ œ l � l œ ' ' ' ' 0 0 0 0 1 2 1 1 0 2 0 1 ’ “ c d a b 20. dA dx dy tan x y dy tan y dy y tan y ln 1 y ln 2 ' ' R y y x y 1 4 x y 1 1 1 1 2 2 2 2 � # # � � � � " " œ œ œ œ � l � l œ � ' ' ' ' 0 0 0 0 1 1 1 1 0 1 0 1 a b c d a b � ‘ 1 21. dy dx (ln 2 ln 1) dx (ln 2) dx (ln 2) ' ' ' ' 1 1 1 1 2 2 2 2 1 xy x x œ � œ œ " " # 22. y cos xy dx dy sin xy dy sin y dy cos y ( 1 1) ' ' ' ' 0 0 0 0 1 1 1 œ œ œ � œ � � � œ c d � ‘ 1 1 1 1 ! " " " ! 1 1 2 23. V f x, y dA x y dy dx x y y dx 2 x dx x x œ œ � œ � œ � œ � œ ' ' R a b a b � ‘ ˆ ‰ � ‘ ' ' ' ' 1 1 1 1 1 1 1 1 2 2 2 3 2 3 1 2 2 2 8 3 3 3 3 3 1 1 " " � � 24. V f x, y dA 16 x y dy dx 16 y x y y dx 2 x dx x x œ œ � � œ � � œ � œ � ' ' R a b a b � ‘ ˆ ‰ � ‘ ' ' ' ' 0 0 0 0 2 2 2 2 0 0 2 2 2 2 2 3 2 3 1 88 88 2 3 3 3 3 œ 160 3 25 V f x,