Summer (Session 1) 2004 - Eggers' Class - Exam 1 (Version 1)

Summer (Session 1) 2004 - Eggers' Class - Exam 1 (Version...

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Unformatted text preview: Math 20E Midterm Exam 1 (version 1) Solution 1. Consider the scalar field f ( x, y, z ) = 2 z 2- 5 xy . (a) Find an equation for the plane tangent to the isotimic surface f ( x, y, z ) = 17 at the point (3 , 1 , 4). f ( x, y, z ) = (- 5 y,- 5 x, 4 z ) and f (3 , 1 , 4) = (- 5 ,- 15 , 16). The vector equation for the plane is thus (- 5 ,- 15 , 16) ( x, y, z ) = (- 5 ,- 15 , 16) (3 , 1 , 4) and the scalar equation for the plane is- 5 x- 15 y + 16 z = 34. (b) Find a unit vector u for which the directional derivative of f in the direction of u at the point (3 , 1 , 4) is 0. (Note: There are many such unit vectors; you need find only one.) Since D u f (3 , 1 , 4) = f (3 , 1 , 4) u , we require a unit vector u such that (- 5 ,- 15 , 16) u = 0. Since (- 5 ,- 15 , 16) (15 ,- 5 , 0) = 0, one such unit vector is u = 1 5 10 (15 ,- 5 , 0). 2. Consider the vector field F ( x, y, z ) = (- y, x, z ). Is R ( t ) = (2 cos( t ) , 2 sin( t ) , 3 e t ) a flow line of F ?...
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