quiz3 solns

Thomas' Calculus: Early Transcendentals

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Unformatted text preview: 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) X n =1 3 n + 2 n +1 5 n X 1 3 5 n + X 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) Z 1 x 2 ln x dx First use integration by parts to find an antiderivative: Z x 2 ln x dx = x 3 ln x 3- Z x 2 3 dx = x 3 ln x 3- x 3 9 + C So Z 1 x 2 ln x dx = lim t x 3 ln x 3- x 3 9 1 t = 1 ln 1 3- 1 9- lim t x 3 ln x 3 + 9 =- 1 9- 1 3 lim t t 3 ln t =- 1 9 . Here we need to show that the limit is 0. We use LH opitals Rule: lim t t 3 ln t = lim t ln t 1 /t 3 = lim t 1 /t- 3 /t 4 = lim t - t 3 3 = 0 . 2. (15 points) Determine whether the following improper integrals converge or diverge. Justify your answers. (a) Z 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...
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This homework help was uploaded on 02/12/2008 for the course MATH 104 taught by Professor Nelson during the Fall '07 term at Princeton.

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quiz3 solns - MATH 104 QUIZ # 3 Spring 2003 Covers Sections...

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