# Test 2b - e lim n →∞(1 1 √ n n III(25 points Defne...

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Test II for Calculus I, Math 1501E1, September 24, 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: The equation x 18 - 5 x 10 + 1 = 0 has a solution in [0 , 1]. b) True or false: The function f ( x ) = x 3 - x deFned on [ - 2 , 2] has an inverse function. c) True or false: If lim n →∞ ( a n b n ) exists and if lim n →∞ a n exists then lim n →∞ b n exists too. d) Calculate exp(5 ln(2)) e) Calculate ln(exp(2 ln(3))) + ln(exp(2 ln(1 / 3)))
II: (25 points) Which of the following limits exist? Compute them if they exist. Otherwise explain why they do not exist. a ) lim n →∞ 4 n 3 ( n 9 + 1) 1 / 3 b ) lim n →∞ n ( n + 1 - n ) c ) lim n →∞ n sin( 3 n ) d ) lim n →∞ (1 - 1 n - 6 n 2 ) n Hint: One way is to factor the expression inside the parenthesis.

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Unformatted text preview: e ) lim n →∞ (1 + 1 √ n ) n III: (25 points) Defne the sequence a n +1 = n + 1 10 n a n , a 1 = 1 a) Find upper and lower bounds on this sequence. b) Is this sequence increasing or decreasing? c) Explain why the sequence is convergent. d) Calculate the limit c . Hint: Calculate the frst ±ew terms a 2 , a 3 , a 4 in this sequence. e) For which values o± n is | a n-c | < 10-10 ? IV: (25 points) Consider the sequence, given recursively by a n +1 = a n + 20 10 , a 1 = 1 . Is this sequence convergent? If yes caculate the limit. Proceed along with 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 | < 10-8 ?...
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Test 2b - e lim n →∞(1 1 √ n n III(25 points Defne...

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