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Spring 2007 - Takeda's Class - Practice Exam 1

# Spring 2007 - Takeda's Class - Practice Exam 1 - Math 20E...

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Math 20E, Practice Midterm 1 April 24, 2007 Name: Section: This exam consists of 7 pages including this front page. Ground Rules 1. No calculator is allowed. 2. Show your work for every problem. A correct answer without any justification will receive no credit. 3. You may use one 4-by-6 index card, both sides. 4. Show your ID on your desk. Score 1 10 2 10 3 10 4 10 5 10 6 10 Total 60 1

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1. Let a = 2 i + k and b = - i + 2 j - 3 k be vectors in R 3 . (a) Compute the orthogonal projection of a on b . Answer : The orthogonal projection p is p = b · a || b || 2 b = - 2 - 3 1 + 4 + 9 b = - 5 14 b = 5 14 i - 5 7 j + 15 14 k . (b) Compute a × b . Answer : a × b = i j k 2 0 1 - 1 2 - 3 = 0 1 2 - 3 i - 2 1 - 1 - 3 j + 2 0 - 1 2 = - 2 i + 5 j + 4 k . 2
2. (a) Let f : R 3 R given by f ( x, y, z ) = x 2 + y 2 + z 2 - 2 z . Describe level surfaces by words. Answer : f ( x, y, z ) = x 2 + y 2 + z 2 - 2 z x 2 + y 2 + ( z - 1) 2 - 1 . For each c , the level surface is x 2 + y 2 + ( z - 1) 2 - 1 = c, i.e. x 2 + y 2 + ( z - 1) 2 = c + 1 . Hence the level surfaces are the spheres with center (0 , 0 , 1). (b) Let c : R R 3 be a path in space defined by c ( t ) = ( e t - 1 , t, t 2 ). Find

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