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Unformatted text preview: Math 2263 Spring 2009 Midterm 1, WITH SOLUTIONS February 19, 2009 1. (15 points) Find an equation for the plane passing through the two points h x, y, z i = h 4 , 2 , 2 i and h 1 , , 2 i so that the vector ~ i + ~ j + ~ k is tangent to the plane. SOLUTION: A normal vector ~v is the cross product of the vector h 3 , 2 , 4 i from one given point to the other and h 1 , 1 , 1 i . ~v = ~ i ~ j ~ k 3 2 4 1 1 1 = 6 ~ i + ~ j + 5 ~ k. Since h 1 , , 2 i is in the plane, an equation for the plane is 6( x 1) + ( y 0) + 5( z + 2) = 0, or simplifying: 6 x + y + 5 z = 16 . 2. (15 points) Find an equation for the surface in ( x, y, z )space obtained by rotating the ellipse x 2 + 4 y 2 = 1 of the ( x, y )plane about the xaxis. SOLUTION: The distance from the xaxis is p y 2 + z 2 , which replaces  y  in the given equa tion. So the equation for the surface of revolution is x 2 + 4 y 2 + 4 z 2 = 1....
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 Spring '08
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 Math, Multivariable Calculus

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