153-t1sol - Math 153 - Spring 2010 NAME Signature Exam 1A 7...

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Math 153 - Spring 2010 NAME Signature Exam 1A 7 April 2010 Answer the following questions. The answers must be clear, intelligible, and you must show your work. Provide explanation for all your steps. Your grade will be determined by adherence to these criteria. Use of books, notes and calculators is strictly forbidden. The point value of each problem is given in the left hand margin. (8 pts.) 1. Determine whether the sequence converges or diverges. If it converges, ±nd the limit. a n = 9 n +1 10 n ; n ± 1 We can rewrite a n = 9 ² ± 9 10 ² n and therefore lim n !1 a n = lim n !1 9 ² ± 9 10 ² n = 9 ² lim n !1 ± 9 10 ² n = 9 ² 0 The sequence converges. 1
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2. You are given the sequence f a n g with: a n = n ± 3 5 n + 1 ; n ² 1 Answer the following questions about this sequence. (4 pts.) a). Is the sequence increasing or decreasing? We consider the function: f ( x ) = x ± 3 5 x + 1 for all x ² 1. The ±rst derivative of f ( x ) is f 0 ( x ) = (5 x + 1) ± 5 ³ ( x ± 3) (5 x + 1) 2 = 16 (5 x + 1) 2 which is always positive. Therefore f ( x ) is increasing and the sequence f a n g is also increasing.
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153-t1sol - Math 153 - Spring 2010 NAME Signature Exam 1A 7...

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