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Exam 4 Solutions Sp10 (2)

# Exam 4 Solutions Sp10 (2) - Math 237 Calculus II Exam 4...

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Math 237. Calculus II Exam 4, Version 2 Fall, 2011 Name: _______________________________________ 1. Find the indefinite integral Substitute u = ln x . This gives du = dx / x , and so 2. Find the convergence set for the series Starting off using the Absolute Ratio Test: So the series will converge whenever |3 x 2| < 1 or, equivalently, whenever 1 < 3 x 2 < 1, which simplifies to . To finish up, we need to know whether the series converges at either endpoint. For x = 1 / 3 , the series is equal to . This series, the harmonic series, is known to diverge. For x = 1, the series is equal to , which is the alternating harmonic series and is known to converge. Answer: . 3. Sketch the graph of the hyperbola whose equation is 9 x 2 16 y 2 + 54 x + 64 y 127 = 0. Include foci, vertices, and asymptotes. Here is the pre-sketch algebraic manipulation. 9( x 2 + 6 x + 9) 16( y 2 4 y + 4) = 127 + 81 64 = 144 So this is a horizontal hyperbola with center ( 3, 2), a = 4, and b = 3. That means it has vertices at ( 7, 2) and (1, 2) and asymptotes .

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