math_172_(fall_2008)_(schumaker)_second_exam_(spring07)

math_172_(fall_2008)_(schumaker)_second_exam_(spring07) -...

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Math 172: Second Semester Calculus Spring Semester, 2007 Second Exam # 1 2 3 4 5 6 7 8 Total worth 10 10 15 15 10 15 15 10 100 score N a m e : S e c t i o n : ID Number: Instructions: Work must be shown to receive credit. No calculators. Question 1. ( 10 points ) Let C be the curve (shown below in bold) defined by y = ln x, 1 x 3. Write down, but do not evaluate , an integral that is equal to the arclength of C. C
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Question 2. ( 10 points ) A spring has a natural length of 0.5 meters. It takes 5 Newtons of force to hold the spring at 1.0 meters. How much work (in Joules) is done to stretch the spring from 1.0 meters to 1.5 meters? 1.0 m 0.5 m natural length 5 Newtons
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Question 3. ( 15 points ) Solve the differential equation 4 2 x y dx dy = , subject to the initial condition . 5 ) 1 ( = y
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Question 4. ( 15 points ) Use the integral test to show that the series = 1 3 / 2 1 n n diverges. You must demonstrate the steps of the test in detail; it is insufficient to write “this series diverges since p = 2/3 < 1.”
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Unformatted text preview: Question 5. ( 10 points ) Consider the series . = ) 01 . ( 3 n n (A) List the first four terms in the sequence of partial sums of this series; it is perfectly valid to use decimal notation. (B) Determine the value of the series, or show that it diverges. (A formula along with a short justification will suffice.) Question 6. ( 15 points ) Determine whether the following series are convergent or divergent. Justify your answers. (a) = 1 2 2 ) ( cos n n n (b) = + 2 1 1 n n n (c) = 1 3 ! n n n e Question 7. ( 15 points ) Find the radius of convergence and interval of convergence of the following series = 1 ) 7 ( n n n x Be sure to specify whether the series converges at each endpoint. Question 8. ( 10 points ) Find power series representations for the following functions and determine their radius of convergence. (a) x x f = 5 1 ) ( (b) ) 1 ln( ) ( 5 x x g =...
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math_172_(fall_2008)_(schumaker)_second_exam_(spring07) -...

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