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Practice Test 2a

# Practice Test 2a - a n 1 = √ 3 a n 4 a 1 = 5 Is this...

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Practice Test II A 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: The equation x 1 / 3 - x + 4 = 0 has a solution in [0 , 8]. b) True or false: The function f ( x ) = 1 1+ x 2 defined on [0 , ) 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) Tue or false: For every positive integer p lim n →∞ log( n ) n 1 /p = 0 e) Using e a 1 + a , valid for all a , derive an anlogous inequality for the log function.
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 2 3 n + 5 b ) lim n →∞ ( p n 2 + n - n ) c ) lim n →∞ n (sin( 1 n )) 2 d ) lim n →∞ (1 - sin( 1 n )) n e ) lim n →∞ (cos( 1 n )) n

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III: (25 points) Consider the sequence, given recursively by
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Unformatted text preview: a n +1 = √ 3 a n + 4 , a 1 = 5 . 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) Suppose you buy a car and you get a loan over P = \$5000 .-at an interest rate of 5%, monthly compounded. Suppose that you want to pay back the loan within 3 years with Fxed monthly payments. How big is your monthly payment? ±ill in the following steps: a) Denote the monthly payment by M and set a = 5 / 100. How much do you owe at the end of the Frst month (i.e. after your Frst payment)? b) How much do you owe after the n-th month? c) Calculate your monthly payment from the answer in b) (Here you may use a calculator)....
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