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Pre-Calc Homework Solutions 108

# Pre-Calc Homework Solutions 108 - 108 Section 3.7 x 2 xy...

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41. Find the two points: The curve crosses the x -axis when y 5 0, so the equation becomes x 2 1 0 x 1 0 5 7, or x 2 5 7. The solutions are x 5 6 ˇ 7 w , so the points are ( 6 ˇ 7 w , 0). Show tangents are parallel: x 2 1 xy 1 y 2 5 7 } d d x } ( x 2 ) 1 } d d x } ( xy ) 1 } d d x } ( y 2 ) 5 } d d x } (7) 2 x 1 x } d d y x } 1 ( y )(1) 1 2 y } d d y x } 5 0 ( x 1 2 y ) } d d y x } 5 2 (2 x 1 y ) } d d y x } 5 2 } 2 x x 1 1 2 y y } Slope at ( ˇ 7 w , 0): 2 } ˇ 2 ˇ 7 w 1 7 w 1 2( 0 0) } 5 2 2 Slope at ( 2 ˇ 7 w , 0): 2 } 2 2 ( 2 ˇ ˇ 7 w 1 7 w ) 2 1 (0 0 ) } 5 2 2 The tangents at these points are parallel because they have the same slope. The common slope is 2 2. 42. x 2 1 xy 1 y 2 5 7 } d d x } ( x 2 ) 1 } d d x } ( xy ) 1 } d d x } ( y 2 ) 5 } d d x } (7) 2 x 1 x } d d y x } 1 ( y )(1) 1 2 y } d d y x } 5 0 ( x 1 2 y ) } d d y x } 5 2 (2 x 1 y ) } d d y x } 5 2 } 2 x x 1 1 2 y y } (a) The tangent is parallel to the x -axis when } d d y x } 5 2 } 2 x x 1 1 2 y y } 5 0, or y 5 2 2 x . Substituting 2 2 x for y in the original equation, we have x 2 1 xy 1 y 2 5 7 x 2 1 ( x )( 2 2 x ) 1 ( 2 2 x ) 2 5 7 x 2 2 2 x 2 1 4 x 2 5 7 3 x 2 5 7 x 5 6 !
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