Practice Test 2a solutions

Practice Test 2a solutions - Solution to Practice Test IIA...

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Solution to Practice Test IIA I: a) Has a solution since f (0) = 4 and f (8) = - 2 where f ( x ) = x 1 / 3 - x + 4 using Bolzano’s theorem. b) Has an inverse on the interval (0 , 1]. c) True d) True e) For y > 0, y 1 + ln( y ) . II: a) 2 / 3, b) 1 / 2, c) 0, d) 1 /e , e) 1 III: First start with some hypothetical considerations. Solving the equation c = 3 c + 4 the limit will be presumaby 4 (the sequence is always positive. Presumably, the sequence will decrease from 5 towards 1. Therefore we conjecture that the sequence will decrease from 5 towards 4 and the limit will be 4. a) First we show that the sequence is bouncec below by 4, i.e., a n 4. We proceed by induction: a 1 = 5 > 4. Assuming a n 4 for some n we have to show that a n +1 4. To see that write a n +1 - 4 = 3 a n + 4 - 4 = 3( a n - 4) 3 a n + 4 + 4 0 . b) Next we show that the sequence is monotone decreasing: a n - a n +1 = a n - 3 a n + 4 = a 2 n - 3 a n - 4 a n + 3 a n + 4 = ( a n - 4)( a n + 1)
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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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Practice Test 2a solutions - Solution to Practice Test IIA...

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