C express as a denite integral the area of the part

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Unformatted text preview: graph at the point with polar coordinates (r, θ) = (1, π/6). c) Express as a definite integral the area of the part of the region in a) that lies outside the circle of radius 1 centered at the origin. Do not evaluate the integral. 2.17 Find the limits of each of the following sequences as n → ∞ or say that the limit does not exist. 4n2 + 3n a) sn = (1 + n)(1 + 2n) (−1)n b) sn = 3 + √ n n2 c) sn = n 2 MTH 142 Spring 2011 Exam 2 Practice 4 2.18 a) Find the sum of the infinite series 3 − 3/2 + 3/4 − 3/8 + 3/16 − 3/32 + 3/64 − 3/128 + · · · . b) Find an explicit formula in terms of a for the sum 21 3a2n . n=1 MTH 142 Exam 3 Practice Problems Spring 2011 1 No calculators will be permitted at the exam. 3.1 A ping-pong ball is launched straight up, rises to a height of 15 feet, then falls back to the launch point and bounces straight up again. It continues to bounce, each time reaching a height 90% of the height reached on the previous bounce. Find the total distance that the ball travels. Note: Questions about convergence or divergence of series may be posed in various ways – some purely multiple choice, others requiring a coherent (specific or nonspecific) reason, some requiring choice and supporting detail. Read questions carefully! 3.2 Use the integral test to determine the convergence of the following series: b) ∞ 1 n=1 a) n 3/2 ∞ n n n=1 e 3.3 Determine if the following series converge or diverge. Give your reasoning using complete sentences. a) b) ∞ ln n 2 n=1 n ∞ n! n=1 (n + 2)! 3.4 For each of the following items a) and b) choose a correct conclusion and reason from among the choices (R), (C), (I) below and provide supporting computation. For example, if you choose (R) calculate and interpret a suitable ratio. MTH 142 Exam 3 Practice Problems Spring 2011 (R) Converges by the ratio test. 2 (C) Diverges by a p-series comparison. (I) Converges by the integral test. a) b) ∞ n2 3n n n=1 n4 ∞ ln n n=1 n ∞ 3n + 4n and en n=1 gives a valid reason? Circle your answer. No other reas...
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