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Unformatted text preview: Math 53: Multivariable Calculus September 26, 2007 Quiz 4 Lecturer: Prof. Michael Hutchings GSI: Gary Sivek Name: Answers 1. (10 pts) For each of the given surfaces in three dimensions, identify the traces for x = k , y = k , and z = k , then identify and sketch the surface. (a) x 2 + 4 y 2 z 2 = 4 The trace x = k gives k 2 + 4 = z 2 4 y 2 , which is a hyperbola. The trace y = k similarly gives 4 k 2 + 4 = z 2 x 2 , which is also a hyperbola, so we have some sort of hyperboloid. The trace z = k gives x 2 +4 y 2 = k 2 4, which is an ellipse if k 2 4 0, or | k | 2, but when | k | < 2 then there are no solutions. This then divides the surface in two, so we have a hyperboloid of two sheets. The graph is below on the left. (b) y = x 2 The trace z = k yields the standard parabola y = x 2 . The trace x = k gives y = k 2 ; since x and y are constant but z varies freely, this is a line. For the trace y = k , we have x 2 = k ; if k < 0 this has no solutions, and if...
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