Test 2a - e ) lim n →∞ (1 + 1 n ) n 2 III: (25 points)...

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Test II for Calculus I, Math 1501C1, 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 sin( x ) + x - 1 2 = 0 has a solution in [0 , π/ 2]. b) True or false: The function f ( x ) = 1 1+ x deFned on [ - 1 / 2 , 1 / 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(2 ln(5)) e) Calculate ln(exp( - 2 ln(2))) + ln(exp(ln(4)))
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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 2 n 2 + 1 b ) lim n →∞ n ( n + 1 - n ) c ) lim n →∞ n 2 (sin( 1 n )) 2 d ) lim n →∞ (1 - 2 n - 15 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 2 III: (25 points) Defne the sequence a n +1 = n n + 1 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 its limit c . Hint: Calculate the frst ±ew terms a 2 , a 3 , a 4 in this sequence. e) Find a stopping rule, i.e., ±or any ε > fnd N ( ε ) so that | a n-c | < ε ±or all n > N ( ε ) . IV: (25 points) Consider the sequence, given recursively by a n +1 = a n + 90 10 , a 1 = 5 . 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 | < 10-6 ?...
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This note was uploaded on 02/07/2011 for the course MATH 1501 taught by Professor N/a during the Fall '08 term at Georgia Institute of Technology.

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Test 2a - e ) lim n →∞ (1 + 1 n ) n 2 III: (25 points)...

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