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The spring 2006 ma 227-6d test 1 in vector calculus. The test consists of two parts. Part 1 has five problems worth 4 points each, with no partial credit. Part 2 has two problems worth 10 points each, where partial credit is awarded. The solutions must include enough detail to justify any conclusions.
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There are 5 problems in Part 1, each worth 4 points. Place your answer on the line to the right of the question. No partial credit will be given on Part 1 problems, only your answer on the answer line will be graded.
(1) Find the dot product of the vectors ใ 2 , 5 , 2 ใ and ใ 1 , โ 2 , โ 1 ใ.
(2) Find the cross product of the vectors ใ 0 , 4 , โ 3 ใ and ใโ 1 , 0 , 1 ใ.
(3) Find the derivative of the vector function ใ 1 , t, t^2 ใ.
(4) Find the derivative of the vector function ใcos t, cos(2t), t sin tใ.
(5) Compute
0 ใ^2 ,^ 1 +^ t,^ e
tใdt.
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There are 2 problems in Part 2, each worth 10 points. On Part 2 problems partial credit is awarded where appropriate. Your solution must include enough detail to justify any conclusions you reach in answering the question.
(1) A parallelepiped is spanned by the vectors ใ 4 , 2 , 2 ใ, ใ 0 , โ 1 , 1 ใ, and ใ 2 , 1 , โ 1 ใ. Find volume and surface area of the parallelepiped.