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Unformatted text preview: MAT 1322E, Winter 2005 MIDTERM TEST 2—Solutions Family NAME: Given NAME: Student Number: MAX=18 • Time: 80 min. (Includes time for distribution of papers.) • Only TI 30type calculators are permitted. Notes or books are not permitted. • There are 7 problems. Problems 15 are multiple choice. Circle the best possible answer. (Numerical answers are rounded to the given decimal.) Problems 6 and 7 require a detailed and clearly presented solutions. • Work all problems in the space provided. Use the backpages for rough work if necessary. Do not use any other paper. Write only in nonerasable ink (ballpoint or pen), not in pencil. Cross out, if necessary, but do not erase or overwrite. Work on multiple choice problems will be examined only in case of suspected fraud. 1. [2 points, 7.2 #19] Use Euler’s method with step size 0.1 to estimate y (0 . 2), where y is the solution of the initial value problem y = x 2 + 2 y,y (0) = 1. A.1 . 2 B.1 . 441 C.1 . 431 D.1 . 686 E.1 . 733 F.1 . 676 Solution: y (0 . 1) = 1 + 0 . 1(0 2 + 2(1)) = 1 . 2 , y (0 . 2) = 1 . 2 + 0 . 1(0 . 1 2 + 2(1 . 2)) = 1 . 441 . 2. [2 points, 7.4 #1] A bacteria culture grows at a constant relative growth rate. After 1 hour there are 50 bacteria and after 2 hours the count is 100. Then the number of bacteria after 4 hours will be: A.150 B.200 C.250 D.300 E.400 F.800 Solution: We measure the time t in hours. Let P ( t ) be the population at t hours, then we have P ( t ) = P (0) e kt , where k is the relative growth rate. Note that P (1) = 50 and P (2) = 100, we obtain 50 = P (0) e k , 100 = P (0) e 2 k . These imply that P (0) = 25 , e k = 2 . We thus have P ( t ) = 25(2 t ) , P (4) = 400 . 3. [2 points, 8.2 #15] Determine whether the following series is convergent or divergent. If it is convergent, find its sum. ∞ X n =1 3 n +1 2 2 n A.divergent B.2 C.3 D.9 E.27 F.3 . 25 Solution: We have ∞ X n =1 3 n +1 2 2 n = ∞ X n =1 3 · 3 n 4 n = 3 ∞ X n =1 3 4 ¶ n This series is a geometric series with first term a = 9 /...
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 Fall '09
 ARIANEMASUDA
 Mathematical Series

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