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12.6-13.2solns

# 12.6-13.2solns - MATH 231 Group Work Quiz Related 1...

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MATH 231 Group Work Quiz Related 1. Classify and sketch the following surfaces: (a) z = 2 x 2 + y 2 (b) z 2 = x 2 + 1 3 y 2 (c) 1 + 5 z 2 = 7 x 2 - y 2 (d) 4 + z 2 = x 2 + y 2 Answer: (a) Elliptic paraboloid. (b) Cone. (c) Hyperboloid of two sheets. (d) Hyperboloid of one sheet. 1

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2. What is the domain of the vector function r ( t ) = D 2 - t, ln t, t 2 - 1 t - 1 E ? 3. At what point(s) do the curves r ( t ) = t 2 , t 2 + 1 , 4 t and s ( t ) = t - 4 , 1 2 t, - t 2 - 4 intersect? 4. Find a vector function that represents the curve of intersection of z 2 = x 2 + y 2 and 2 x + 3 y - z = 1. Answer: This one is the really ugly one of the bunch. The concepts involved in solving it are more important here than the specific details. For similar problems, look at 13.1 #36-38. We know that there are two pieces to the curve of intersection between these two things. One for z > 0 and one for z < 0. Substituting z = 2 x + 3 y - 1 into z 2 = x 2 + y 2 and using the quadratic formula, we find that x = 1 3 2 - 6 y ± p 1 - 6 y + 12 y 2 Here the + option represents where z > 0 and the - value represents where z < 0. Thus, letting y = t ,
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12.6-13.2solns - MATH 231 Group Work Quiz Related 1...

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