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Unformatted text preview: Math 20E, Practice Final Exam Solutions June 14, 2007 Name : PID : TA : Sec. No : Sec. Time : This exam consists of 11 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 two 4by6 index cards, both sides. 4. You have two hours for this exam. Score 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 Total 100 1 1. (a) Sketch the solid whose spherical coordinates ( ρ, θ, φ ) satisfy the inequal ities 0 ≤ ρ ≤ 1 , ≤ φ ≤ π/ 2. Answer : This is just the sphere x 2 + y 2 + z 2 ≤ 1 with z ≥ 0. (I omit the graph.) (b) Consider the two planes x + y = 1 and y + z = 1. Find the (acute) angle of intersection between these planes. Answer : Note that the normal vectors for x + y = 1 and y + z = 1 are respectively n 1 = (1 , 1 , 0) and n 2 = (0 , 1 , 1). So if the acute angle of those planes is θ , then cos θ = n 1 · n 2  n 1  n 2  = 1 √ 2 √ 2 = 1 2 . Hence θ = π 3 . 2 2. (a) Consider the cone z 2 = x 2 + y 2 . Find the equation of the plane tangent to the cone at the point (3 , 4 , 5). Answer : Let f ( x, y, z ) = x 2 + y 2 z 2 . Then a normal vector to the cone at (3 , 4 , 5) is ∇ f (3 , 4 , 5). Now ∇ f ( x, y, z ) = 2 x i + 2 y j 2 z k...
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 Spring '07
 Enright
 Math, Vector Calculus, Cos, Trigraph, Vector field, Surface normal

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