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exam1'06 - Bryson Spencer Exam 1 Due 11:00 pm Inst Mar...

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Bryson, Spencer – Exam 1 – Due: Feb 20 2007, 11:00 pm – Inst: Mar Gonzalez 1 This print-out should have 16 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points When -2 -1 0 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 2 4 - 2 - 4 is the graph of a function f , use rectangles to estimate the definite integral I = Z 10 0 | f ( x ) | dx by subdividing [0 , 10] into 10 equal subin- tervals and taking right endpoints of these subintervals. 1. I 22 correct 2. I 19 3. I 20 4. I 18 5. I 21 Explanation: The definite integral I = Z 10 0 | f ( x ) | dx is the area between the graph of f and the interval [0 , 10]. The area is estimated using the gray-shaded rectangles in -2 -1 0 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 2 4 - 2 - 4 Each rectangle has base-length 1; and it’s height can be read off from the graph. Thus Area = 5 + 3 + 2 + 1 + 2 + 2 + 3 + 3 + 1 . Consequently, I 22 . keywords: definite integrals, graph, absolute value 002 (part 1 of 1) 10 points For which integral, I , is the expression 1 15 ˆ r 1 15 + r 2 15 + r 3 15 + . . . + r 15 15 ! a Riemann sum approximation. 1. I = 1 15 Z 1 0 r x 15 dx 2. I = 1 15 Z 1 0 x dx 3. I = 1 15 Z 15 0 x dx 4. I = Z 1 0 x dx correct 5. I = Z 1 0 r x 15 dx
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Bryson, Spencer – Exam 1 – Due: Feb 20 2007, 11:00 pm – Inst: Mar Gonzalez 2 Explanation: When the interval [ a, b ] is divided into n equals intervals, then n X i = 1 f a + i ( b - a ) n b - a n is a Riemann sum approximation for the inte- gral Z b a f ( x ) dx of f over [ a, b ] using right endpoints as sample points. Comparing this with 1 15 ˆ r 1 15 + r 2 15 + r 3 15 + . . . + r 15 15 ! we see that [ a, b ] = [0 , 1], n = 15, and [ a, b ] = [0 , 1] , n = 15 , f ( x ) = x . Consequently, the given expression is a Rie- mann sum for I = Z 1 0 x dx . keywords: integral, Riemann sum, square root 003 (part 1 of 1) 10 points If F ( x ) = Z x 0 e 12 sin 2 θ dθ , find the value of F 0 ( π/ 4). 1. F 0 ( π/ 4) = 6 e 6 2. F 0 ( π/ 4) = 6 e 12 3. F 0 ( π/ 4) = 6 e 4. F 0 ( π/ 4) = e 6 correct 5. F 0 ( π/ 4) = e 12 Explanation: By the Fundamental theorem of calculus, F 0 ( x ) = e 12 sin 2 x . At x = π/ 4, therefore, F 0 ( π/ 4) = e 6 since sin( π 4 ) = 1 2 . keywords: integral, FTC 004 (part 1 of 1) 10 points Evaluate the definite integral I = Z π 8 0 (7 cos 4 x - 3 sin 4 x ) dx . 1. I = 0 2. I = 1 4 3. I = 1 2 4. I = 1 correct 5. I = 3 4 Explanation: To reduce the integral to one involving just sin u and cos u set u = 4 x . Then du = 4 dx , and so I = 1 4 Z π 2 0 (7 cos u - 3 sin u ) du = 1 4 h 7 sin u + 3 cos u i π 2 0 .
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