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Thomas' Calculus: Early Transcendentals

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MATH 104 QUIZ # 3 Spring 2003 Covers Sections 8.8, 10.1-10.6 of the textbook 1. (10 points) Determine whether the following converge or diverge. If they converge, evaluate. (a) n =1 3 n + 2 n +1 5 n 1 3 5 n + 1 2 2 5 n = 3 / 5 1 - 3 / 5 + 2(2 / 5) 1 - 2 / 5 = 3 / 5 2 / 5 + 4 / 5 3 / 5 = 3 2 + 4 3 = 17 6 (b) 1 0 x 2 ln x dx First use integration by parts to find an antiderivative: x 2 ln x dx = x 3 ln x 3 - x 2 3 dx = x 3 ln x 3 - x 3 9 + C So 1 0 x 2 ln x dx = lim t 0 x 3 ln x 3 - x 3 9 1 t = 1 ln 1 3 - 1 9 - lim t 0 x 3 ln x 3 + 0 9 = - 1 9 - 1 3 lim t 0 t 3 ln t = - 1 9 . Here we need to show that the limit is 0. We use L’Hˆ opital’s Rule: lim t 0 t 3 ln t = lim t 0 ln t 1 /t 3 = lim t 0 1 /t - 3 /t 4 = lim t 0 - t 3 3 = 0 . 2. (15 points) Determine whether the following improper integrals converge or diverge. Justify your answers. (a) 1 x 7 + 100 x x 5 dx converges. The numerator x 7 + 100 x is dominated by the highest power of x , in other words x 7 + 100 x x 7 / 2 as x goes to . So the quotient will be asymptotic to x 7 / 2 /x 5 = 1 /x 3 / 2 as x goes to . Since 1 dx x 3 / 2
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