七 - A 1 1 z x 2 y 2 2 x 2 y 2 z 2 2 x 4 y 0 3 y x 2 4 x 2...

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A 1 .在空间直角坐标系中,下列方程表示什么形状的图形 . 1 2 2 y x z 2 0 4 2 2 2 2 y x z y x 3 2 x y 4 1 2 4 2 2 2 z y x 5 0 2 2 2 y z x 6 2 2 2 2 z y x 7 0 ) 1 ( 2 2 2 z y x 8 x z y 2 2 解:( 1 )旋转抛物面 . 2 )以( 1,2,0 )为原点 5 为半径的球面 . 3 )抛物线柱 . 4 )以 2) , 2 , 1 ( 为中心的椭球面 . 5 )旋转抛物面 . 6 )圆锥面 . 7 )一个点 8 )双曲抛物面 . 2 .给定两点 3) 1 - 2 ( 1 P 5 0 3 ( 2 P ,求 1 1 P 2 P 之间距离 2 1 P P 2 )线段 2 1 P P 的垂直平分面的方程; 3 )以 2 P 为中心, 2 1 P P 为半径的球面方程 . 解:( 1 30 ) ( ) ( ) ( 2 2 1 2 2 1 2 2 1 2 1 z z y y x x P P . 2 4) , , 2 1 ( 0 2 1 P 2) , 1 , 1 ( 0 ) 2 ( 2 ) 2 1 ( ) 2 1 ( z y x . 3 30 ) 5 ( ) 3 ( 2 2 2 z y x . 3 .求下列函数定义域,并画出定义域示意图 .
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1 2 2 4 1 y x z 2 ) ln( 2 2 y x z 3 1 arcsin x y z 4 2 2 2 1 ) ln( y x x y z 5 ) 1 ln( 4 2 2 2 y x y x z 6 y x z 2 7 y x y x z 1 1 8 9 4 1 2 2 y x z 解: 1 2 2 1 1 0 1 0 1 2 2 y x y x 定义域 2 0 2 2 y x 定义域: y x 3 2 0 0 1 1 1 x y x x y 4 1 0 1 0 2 2 2 2 2 2 y x x y y x x y
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5 4 1 0 4 1 1 0 1 2 2 2 2 2 2 2 2 2 y x y x y x y x y x 6 4 2 0 0 0 x y y y x 7 y x y x y x 0 0 y x 8 3 3 2 2 36 4 9 9 4 1 2 2 2 2 y x y x y x 4 .设 xy y x y x y x f 2 2 ) , ( ,求 ) , ( y x f 解:令 y x v , y x u 4 3 4 ) ( ) ( 2 ) ( ) ( ) . ( 2 2 2 2 2 2 v u y x y x y x y x v u f 所以 4 3 ( 2 2 y x y x f 5 .设 x x y x y x x x y x f ln ) ln (ln , ln 2 ,求 ) , ( y x f . 解:令 x y u , lnx u u u ve y e x
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所以 u v v e u e ve v e e v u f u u u u u ln ln ) ( 2 所以 y x y e y x f x ln ) ( 6 .计算下列函数在给定点处的偏导数; 1 x y z arctan ,求 ) 1 , 1 ( , ) 1 , 1 ( y x z z 2 y x y x z ,求 ) 2 , 1 ( , ) 2 , 1 ( y x z z 3 2 e y x z
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