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Summer (Session 1) 2004 - Eggers' Class - Exam 1 (Version 2)

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

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Unformatted text preview: Math 20E Midterm Exam 1 (version 2) Solution 1. Consider the scalar field f ( x, y, z ) = 3 z 2- 4 xy . (a) Find an equation for the plane tangent to the isotimic surface f ( x, y, z ) = 19 at the point (2 , 1 , 3). ∇ f ( x, y, z ) = (- 4 y,- 4 x, 6 z ) and ∇ f (2 , 1 , 3) = 2(- 2 ,- 4 , 9). The vector equation for the plane is thus (- 2 ,- 4 , 9) · ( x, y, z ) = (- 2 ,- 4 , 9) · (2 , 1 , 3) and the scalar equation for the plane is- 2 x- 4 y + 9 z = 19. (b) Find a unit vector u for which the directional derivative of f in the direction of u at the point (2 , 1 , 3) is 0. (Note: There are many such unit vectors; you need find only one.) Since D u f (2 , 1 , 3) = ∇ f (2 , 1 , 3) · u , we require a unit vector u such that (- 2 ,- 4 , 9) · u = 0. Since (- 2 ,- 4 , 9) · (4 ,- 2 , 0) = 0, one such unit vector is u = 1 2 √ 5 (4 ,- 2 , 0). 2. Consider the vector field F ( x, y, z ) = ( y,- x, z ). Is R ( t ) = (3 sin( t ) , 3 cos( t ) , e t ) a flow line of F ?...
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