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# ef11k - MAC 2312 Final Exam and Key Prof S Hudson 1(15pts...

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MAC 2312 Aug 11, 2011 Final Exam and Key Prof. S. Hudson 1) (15pts) Compute and simplify. Use limits on any improper integrals. Z + -∞ 1 1 + x 2 dx Z ln(2) 0 e 2 x dx Z 2 x 2 - 1 dx 2) (short answer; 5pts each) 2a) Find the McLaurin Series for f ( x ) = x sin(2 x ). 2b) Use a derivative of a familiar series to get the McLaurin Series for f ( x ) = (1 - x ) - 2 . 2c) Give the McLaurin series for e x . Use a partial sum p n ( x ) to estimate 1 /e = e - 1 to two decimal places. Choose n as small as you can and explain why your answer is accurate enough. Hints; 5! = 120 and 7! = 5040. 3) (10pts) Find the area of the region inside the cardioid r = 1 + cos( θ ). 4) (10pts) Choose ONE: 4a) A cone-shaped reservoir is 20 ft in diameter across the top and 15 feet deep. If the reservoir is filled to a depth 10 ft, how much work is required to pump all the water to the top of the reservoir? (Assume density of water is 62.4 lb per cubic ft). 4b) A 20 ft chain weighs 10 lbs per foot, so 200 lbs. One end is at the top of a tall building, and the other end hangs in the air 20 ft below that. How much work is required

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