h11 - moore (jwm2685) HW11 gilbert (55485) 1 This print-out...

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Unformatted text preview: moore (jwm2685) HW11 gilbert (55485) 1 This print-out should have 10 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 10.0 points Find an equation for the plane passing through the origin that is parallel to the tan- gent plane to the graph of z = f ( x, y ) = 4 x 2 2 y 2 + 2 x + 3 y at the point (1 , 1 , f (1 , 1)). 1. z 10 x + 7 y + 16 = 0 2. z 10 x 7 y = 0 correct 3. z 10 x + 7 y = 0 4. z + 10 x 7 y = 0 5. z + 10 x + 7 y 4 = 0 6. z + 10 x 7 y 18 = 0 Explanation: Parallel planes have parallel normals. On the other hand, the tangent plane to the graph of z = f ( x, y ) at the point ( a, b, f ( a, b )) has normal n = ( f x ( a, b ) , f y ( a, b ) , 1 ) . But when f ( x, y ) = 4 x 2 2 y 2 + 2 x + 3 y we see that f x = 8 x + 2 , f y = 4 y + 3 , and so when a = 1 , b = 1, n = ( 10 , 7 , 1 ) . Thus an equation for the plane through the origin with normal parallel to n is ( x, y, z ) n = ( x, y, z )( 10 , 7 , 1 ) = 0 , which after evaluation becomes z 10 x 7 y = 0 . keywords: 002 10.0 points Find an equation for the tangent plane to the graph of f ( x, y ) = radicalbig 8 2 x 2 + y 2 at the point P (2 , 1 , f (2 , 1)). 1. x 4 y + z 6 = 0 2. x + 4 y z 8 = 0 3. 4 x y + z 8 = 0 correct 4. 4 x y z 6 = 0 5. 4 x + y + z 10 = 0 6. x + 4 y z 10 = 0 Explanation: The equation of the tangent plane to the graph of z = f ( x, y ) at the point P ( a, b, f ( a, b )) is given by z = f ( a, b ) + f x vextendsingle vextendsingle vextendsingle ( a, b ) ( x a ) + f y vextendsingle vextendsingle vextendsingle ( a, b ) ( y b ) . Now when f ( x, y ) = radicalbig 8 2 x 2 + y 2 , we see that f x = 2 x radicalbig 8 2 x 2 + y 2 , while f y = y radicalbig 8 2 x 2 + y 2 . moore (jwm2685) HW11 gilbert (55485) 2 Thus at P , f (2 , 1) = 1 , while f x vextendsingle vextendsingle vextendsingle (2 , 1) = 4 , f y vextendsingle vextendsingle vextendsingle (2 , 1) = 1 ....
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This note was uploaded on 05/02/2011 for the course M 408 D taught by Professor Textbookanswers during the Spring '07 term at University of Texas at Austin.

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h11 - moore (jwm2685) HW11 gilbert (55485) 1 This print-out...

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