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Business Calc Homework w answers_Part_25

Business Calc Homework w answers_Part_25 - 121 Chapter 3...

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38. Use implicit differentiation. y 2 5 } x 1 x 1 } } d d x } y 2 5 } d d x } } x 1 x 1 } 2 y } d d y x } 5 } ( x 1 1 ( x )( 1 1) 1 2 ) 2 ( x )(1) } } d d y x } 5 } 2 y ( x 1 1 1) 2 } 39. x 3 1 y 3 5 1 } d d x } ( x 3 ) 1 } d d x } ( y 3 ) 5 } d d x } (1) 3 x 2 1 3 y 2 y 9 5 0 y 9 5 2 } x y 2 2 } y 0 5 } d d x } 1 2 } x y 2 2 } 2 5 2 5 2 5 2 } 2 xy 3 y 1 5 2 x 4 } 5 2 } 2 x ( x 3 y 1 5 y 3 ) } 5 2 } 2 y x 5 } since x 3 1 y 3 5 1 40. y 2 5 1 2 } 2 x } } d d x } ( y 2 ) 5 } d d x } (1) 2 } d d x } 1 } 2 x } 2 2 yy 9 5 } x 2 2 } y 9 5 } x 2 ( 2 2 y ) } 5 } x 1 2 y } y 0 5 } d d x } 1 } x 1 2 y } 2 5 2 } ( x 2 1 y ) 2 } } d d x } ( x 2 y ) 5 2 } ( x 2 1 y ) 2 } [( x 2 )( y 9 ) 1 ( y )(2 x )] 5 2 } ( x 2 1 y ) 2 } 3 ( x 2 ) 1 } x 1 2 y } 2 1 2 xy 4 5 2 } x 4 1 y 2 } 1 } 1 y } 1 2 xy 2 5 2 } 1 1 x 4 y 2 3 xy 2 } 41. y 3 1 y 5 2 cos x } d d x } ( y 3 ) 1 } d d x } ( y ) 5 } d d x } (2 cos x ) 3 y 2 y 9 1 y 9 5 2 2 sin x (3 y 2 1 1) y 9 5 2 2 sin x y 9 5 2 } 3 2 y 2 si 1 n x 1 } y 0 5 } d d x } 1 2 } 3 2 y 2 si 1 n x 1 } 2 5 2 5 2 5 2 2 42. x 1/3 1 y 1/3 5 4 } d d x } ( x 1/3 ) 1 } d d x } ( y 1/3 ) 5 } d d x } (4) } 1 3 } x 2 2/3 1 } 1 3 } y 2 2/3 y 9 5 0 y 9 5 2 } x y 2 2 2 2 / / 3 3 } 5 2 1 } y x } 2 2/3 y 0 5 } d d x } 3 2 1 } y x } 2 2/3 4 5 2 } 2 3 } 1 } y x } 2 2 1/3 1 } xy 9 2 x 2 ( y )(1) } 2 5 2 } 2 3 } 1 } y x } 2 2 1/3 1 2 5 2 } 2 3 } x 1/3 y 2 1/3 ( 2 x 2 5/3 y 2/3 2 x 2 2 y ) 5 } 2 3 } x 2 4/3 y 1/3 1 } 2 3 } x 2 5/3 y 2/3 43. y 9 5 2 x 3 2 3 x 2 1, y 0 5 6 x 2 2 3, y - 5 12 x , y (4) 5 12, and the rest are all zero. 44. y 9 5 } 2 x 4 4 } , y 0 5 } x 6 3 } , y - 5 } x 2 2 } , y (4) 5 x , y (5) 5 1, and the rest are all zero. ( x ) 3 2 1 } y x } 2 2/3 4 2 y }} x 2 (3 y 2 1 1) 2 cos x 1 12 y sin 2 x }}}} (3 y 2 1 1) 3 (3 y 2 1 1)(2 cos x ) 2 (12 y sin x ) 1 2 } 3 2 y 2 si 1 n x 1 } 2 }}}}} (3 y 2 1 1) 2 (3 y 2 1 1)(2 cos x ) 2 (2 sin x )(6 yy 9 ) }}}} (3 y 2 1 1) 2 ( y 2 )(2 x ) 2 ( x 2 )(2 y ) 1 2 } x y 2 2 } 2 }}} y 4 ( y 2 )(2 x ) 2 ( x 2 )(2 y )( y 9 ) }}} y 4 Chapter 3 Review 121
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45. } d d y x } 5 } d d x } ˇ x 2 w 2 w 2 w x w 5 } 2 ˇ x 2 w 1 2 w 2 w x w } (2 x 2 2) 5 } ˇ x x 2 w 2 2 w 1 2 w x w } At x 5 3, we have y 5 ˇ 3 w 2 w 2 w 2 w (3 w ) w 5 ˇ 3 w and } d d y x } 5 } ˇ 3 w 3 2 w 2 2 w 1 2 w (3 w ) w } 5 } ˇ 2 3 w } . (a) Tangent: y 5 } ˇ 2 3 w } ( x 2 3) 1 ˇ 3 w or y 5 } ˇ 2 3 w } x 2 ˇ 3 w (b) Normal: y 5 2 } ˇ 2 3 w } ( x 2 3) 1 ˇ 3 w or y 5 2 } ˇ 2 3 w } x 1 } 5 ˇ 2 3 w } 46. } d d y x } 5 } d d x } (4 1 cot x 2 2 csc x ) 5 2 csc 2 x 1 2 csc x cot x At x 5 } p 2 } , we have y 5 4 1 cot } p 2 } 2 2 csc } p 2 } 5 4 1 0 2 2 5 2 and } d d y x } 5 2 csc 2 } p 2 } 1 2 csc } p 2 } cot } p 2 } 5 2 1 1 2(1)(0) 5 2 1. (a) Tangent: y 5 2 1 1 x 2 } p 2 } 2 1 2 or y 5 2 x 1 } p 2 } 1 2 (b) Normal: y 5 1 1 x 2 } p 2 } 2 1 2 or y 5 x 2 } p 2 } 1 2 47. Use implicit differentiation. x 2 1 2 y 2 5 9 } d d x } ( x 2 ) 1 } d d x } (2 y 2 ) 5 } d d x } (9) 2 x 1 4 y } d d y x } 5 0 } d d y x } 5 2 } 2 4 x y } 5 2 } 2 x y } Slope at (1, 2): 2 } 2( 1 2) } 5 2 } 1 4 } (a) Tangent: y 5 2 } 1 4 } ( x 2 1) 1 2 or y 5 2 } 1 4 } x 1 } 9 4 } (b) Normal: y 5 4( x 2 1) 1 2 or y 5 4 x 2 2 48. Use implicit differentiation.
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