12.7 Notes - formula becomes: ( 29 ( 29 ( 29 ( 29 ( 29 , ,...

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12.7—Tangent Planes and Normal Lines Definition of Tangent Plane and Normal Line Let F be differentiable at the point ) , , ( 0 0 0 z y x P on the surface S given by F(x,y,z)=0 such that ) , , ( 0 0 0 z y x F 0 1. The plane through P that is normal to ) , , ( 0 0 0 z y x F is called the tangent plane to S at P 2. The line through P having the direction of ) , , ( 0 0 0 z y x F is called the normal line to S at P Example: #10 Theorem 12.13 Equation of Tangent Plane If F is differentiable at ) , , ( 0 0 0 z y x then an equation of the plane to the surface given by F(x,y,z)=0 at ) , , ( 0 0 0 z y x is ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 = - + - + - z z z y x F y y z y x F x x z y x F z y x if the surface is given by ) , ( y x f z = , define F by z y x f z y x F - = ) , ( ) , , ( and the above
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Unformatted text preview: formula becomes: ( 29 ( 29 ( 29 ( 29 ( 29 , , =---+-z z y y y x f x x y x f y x Example: #32 The angle of inclination of a plane is defined to be the angle , 2 between the given plane and the xy-plane. The angle of inclination of a horizontal plane is defined to be 0. Since k is normal to the xy-plane , the angle of inclination of a plane with normal vector n is given by n k n k n k n = = cos Example: #42...
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This note was uploaded on 01/13/2012 for the course MATH 2574 taught by Professor Pamelasatterfield during the Spring '12 term at NorthWest Arkansas Community College.

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12.7 Notes - formula becomes: ( 29 ( 29 ( 29 ( 29 ( 29 , ,...

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