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Practice Test 2b

# Practice Test 2b - 1 √ n 1 √ n 1-√ n c lim n →∞...

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Practice Test II B for Calculus I, Math 1501, September 19, 2002 Name: This test is to be taken without calculators and notes of any sorts. The allowed time is 50 minutes. Write answers in boxes where provided. Provide exact answers; not decimal approximations! For example, if you mean 2 do not write 1 . 414 . . . .

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I: (25 points, No partial credit) a) True or false: Consider the function f ( x ) = x sin( 1 x ) if x 6 = 0 0 if x = 0 . The equation f ( x ) = 1 2 has a solution in [ - 1 , 1]. b) True or false: The function f ( x ) = sin( x ) + 1 defined on [ - π, π ] has an inverse function. c) True or false: For all a 0 (1 + a ) 1 /n 1 + 1 n a d) True or false: If lim n →∞ a n = a then lim n →∞ a 1 + a 2 + · · · a n n = a e) True or false: a < b implies e a < e b
II: (25 points) Which of the following limits exist? Compute them if they exist. Otherwise explain why they do not exist. a ) lim n →∞ n + 1 2 n + 1 b ) lim

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Unformatted text preview: 1 √ n ( 1 √ n + 1-√ n ) c ) lim n →∞ 1-cos( n ) n d ) lim n →∞ ± 1-1 n 1 + 1 n ² n e ) lim n →∞ (1-1 √ n ) n III: (25 points) Consider the sequence, given recursively by a n +1 = 1-(1-a n ) 2 , a 1 = 1 2 . Is this sequence convergent? If yes calculate the limit. Proceed along the steps outlined below a) Is this sequence bounded above or below? b) Is this sequence monotone increasing or decreasing? c) Calculate the limit c . d) For which values of n is | c-a n | < 10-6 ? IV: (25 points) Consider the recursively defned sequence a n +1 = 4 + 2 a 3 n 3 a 2 n , a 1 = 4 a) Show that this sequence is monotone decreasing and bounded below. b) Calculate its limit. c) Find a stopping rule N ( ε )....
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