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Unformatted text preview: a h4 ig6 e k# 5 i b iC a aha 2.2 Problem 7 ( a ): f ( x ) = 2 x 3 + x 2 log x is O ( x 3 ). ( b ): f ( x ) = 3 x 3 + (log x ) 4 is O ( x 3 ). ( c ): f ( x ) = ( x 4 + x 2 + 1) / ( x 3 + 1) is O ( x ). ( d ): f ( x ) = ( x 4 + 5 log x ) / ( x 4 + 1) is O (1). 2.2 Problem 13: We know that 2 n 3 n for all n 0, so we can take C = 1 and k = 0. However, if we have chosen C and want 3 n C 2 n for all n k , note that for n = log 3 / 2 C we have equality, and for all n > log 3 / 2 C we have 3 n > C 2 n . Thus 3 n is not O (2 n ). 2.2 Problem 18: We have 1 k + 2 k + 3 k + + n k n k + n k + n k + + n k bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright n times = n ( n k ) = n k +1 . Thus 1 k + + n k is O ( n k +1 ). 2.2 Problem 20 ( a ): f ( x ) = ( n 3 + n 2 log n )(log n + 1) + (17 log n + 19)( n 3 + 2) is O ( n 3 log n )....
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 Spring '06
 CALLAHAN

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