Function Growth.pdf - MAT 243 Written Homework Function...

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MAT 243 Written Homework Function Growth 1. Find the smallest integer n so that f ( x ) = x 5 + x 3 log x x 2 + 1 is big-O of x n . Show your work. 2. Demonstrate that x 2 ln x is big-O of x 3 . You may use a limit theorem given in the lecture, or make an argument based on the exact definition of the big-O relation ship. 3. True or False? (ln n ) 2 is big-O of n . Justify your answer. 4. Find the smallest positive integer k such that 1 2 + 2 2 + 3 2 + . . . + n 2 is big-O of n k . Show your work. 5. Find the mistake in the following argument: We show that 1 2 + 2 2 + 3 2 + . . . + n 2 is big-O of n 2 . Since i 2 n 2 for i = 1 , 2 . . . n , each term in the sum 1 2 +2 2 +3 2 + . . . + n 2 is less than or equal to n 2 . Thus, the maximum of the big-O estimates of each term is n 2 , which implies that the sum 1 2 + 2 2 + 3 2 + . . . + n 2 is big-O of n 2 . 6. (a) Is there a smallest real number a for which

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