hwsol_16 - Homework Assignment 16(12.6 Solutions Page 975 8...

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Homework Assignment 16 - (12.6) - Solutions Page 975: 8*, 10*, 14*, 18, 20*, 26*, 28*,36*, 40*, 46*, 50*, 52*, 56*, 62*, 65* 8. f ! x , y " ! e 3 y / x ! x 2 y 3 " f ! x , y " !" ! 3 y x 2 e 3 y / x ! 2 xy 3 , 3 x e 3 y / x ! 3 x 2 y 2 # 10. f ! x , y " ! sin ! 3 xy " $ y 2 " f ! x , y " !" 3 y cos ! 3 xy " , 3 x cos ! 3 xy " $ 2 y # " f ! ! ,1 " !" ! 3, ! 3 ! $ 2 # 14. f ! x , y , z " ! z 2 e 2 x ! y ! 4 xz 2 " f ! x , y , z " !" 2 z 2 e 2 x ! y ! 4 z 2 , ! z 2 e 2 x ! y , 2 ze 2 x ! y ! 8 xz # " f ! 1,2,2 " !" ! 8, ! 4, ! 12 # 18. f ! x , y " ! x 3 y ! 4 y 2 , ! 2, ! 1 " , u % !" 1 2 , 1 2 # " f ! x , y " !" 3 x 2 y , x 3 ! 8 y # , " f ! 2, ! 1 " !" ! 12, 16 # D u % f ! 2, ! 1 " !" ! 12, 16 # # " 1 2 , 1 2 #! 1 2 4 ! 2 2 20. f ! x , y " ! xy ! y 2 , ! 1,4 " , u % !" 1,4 # " f ! x , y " !" 1 2 x y , 1 2 xy x ! 2 y # , " f ! 1,4 " !" 1, 1 4 ! 8 #!" 1, ! 31 4 # D u % f ! 1,4 " ! 1, ! 31 4 # 1 17 1, 4 ! ! 30 17 17 26. f ! x , y " ! y 2 $ 2 ye 4 x , 0, ! 2 , u % in the direction from 0, ! 2 to ! 4, 4 " f ! x , y " !" 8 ye 4 x , 2 y $ 2 e 4 x # , " f 0, ! 2 !" ! 16, ! 2 # u % ! 1 ! ! 4 " 2 $ 6 2 " ! 4, 6 #!" ! 2 13 , 3 13 # D u % f 0, ! 2 !" ! 16, ! 2 # # " ! 2 13 , 3 13 #! 1 13 ! 32 ! 6 " ! 26 13 ! 2 13 28. f ! x , y , z " ! x 2 $ y 2 $ z 2 , 1, ! 4, 8 , u % !" 1,1, ! 2 # " f ! x , y , z " ! g ! x , y , z " ! 1 ! x 2 $ y 2 $ z 2 " x , 1 ! x 2 $ y 2 $ z 2 " y , 1 ! x 2 $ y 2 $ z 2 " z " f ! 1, ! 4,8 " ! 1 9 , ! 4 9 , 8 9 D u % f 1, ! 4, 8 ! 1 9 , ! 4 9 , 8 9 # 1 6 $ 1,1, ! 2 % ! ! 19 54 6 1
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36. f ! x , y " ! x cos ! 3 y " , ! 2, ! " f ! x , y " ! cos3 y , ! 3 x sin3 y , " f ! x , y " ! ! 1, 0 u % max !" ! 1, 0 # , maximum rate is 1 u % min !" 1, 0 # , minimum rate is -1 40. f ! x , y , z " ! x 2 $ y 2 $ z 2 , ! 1,2, ! 2 " " f ! x , y , z " ! g ! x , y , z " ! 1 ! x 2 $ y 2 $ z 2 " x , 1 ! x 2 $ y 2 $ z 2 " y , 1 ! x 2 $ y 2 $ z 2 " z " f ! 1,2, ! 2 " ! 1 3 , 2 3 , ! 2 3 , || " f ! 1,2, ! 2 " || ! 1 9 $ 4 9 $ 4 9 ! 1 u % max !" 1 3 , 2 3 , ! 2 3 # , maximum rate is 1 u % min !" ! 1 3 , ! 2 3 , 2 3 # , minimum rate is -1 46. z ! x 2 $ y 2 , 3, ! 4, 5 Let F ! x , y , z " ! x 2 $ y 2 ! z . " F ! x , y , z " ! 1 ! x 2 $ y 2 " x , 1 ! x 2 $ y 2 " y , ! 1 , " F 3, ! 4, 5 ! 3 5 , ! 4 5 , ! 1 Tangent plane: 3 5 ! x ! 3 " ! 4 5 ! y $ 4 " ! ! z ! 5 " ! 0 Normal line: r % ! t " ! 3 $ 3 5 t , ! 4 ! 4 5 t , 5 ! t 50. z ! sin x cos y , F ! x , y , z " ! sin x cos y ! z " F ! x , y , z " ! cos x cos y , ! sin x sin y , ! 1 Tangent plane parallel to the xy ! plane if " F ! x , y , z " ! c 0, 0, 1 . cos x cos y , ! sin x sin y , ! 1 ! 0, 0, c , c ! ! 1, cos x cos y ! 0 ! sin x sin y ! 0 cos x ! 0, or cos y ! 0 & x ! n ! 2 , or y ! m ! 2 , n , m ! & 1, & 3, ... and sin x ! 0 or sin y ! 0 & x ! n ! or y ! m ! , m , n ! 0, & 1, & 2, ... x , y , & 1 | x ! n ! 2 , n ! & 1, & 3, ..., y ! m ! , m , n ! 0, & 1, & 2, ... x , y , 0 | x ! n ! , n ! 0, & 1, & 2, ..., y ! m ! 2 , n , m ! & 1, & 3, ... 52. Use the sketch in the text. 2
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56. Use the sketch in the text. 62. f ! x , y " ! 200 ! y 2 ! 4 x 2 . When f !
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