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hw6 - Homework 6 Section 1.3 1(E 10 If x > 1 then x3 < x4...

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Homework 6 Section 1.3 1. (E) 10. If x > 1 then x 3 < x 4 . So clearly x 3 is O ( x 4 ) for C=1 and k=1. But if x 4 is O ( x 3 ) then x 4 < C.x 3 for some C and k. Dividing both sides by x 3 we get x < C . Clearly this condition will not hold for all large x. 2. (E) ( n 3 + n 2 logn )( logn + 1) + (17 logn + 19)( n 3 + 2) = n 3 logn + n 3 + n 2 ( logn ) 2 + n 2 logn + 17 n 3 logn + 19 n 3 + 34 logn + 38 < n 3 logn + n 3 logn + n 3 logn + n 3 logn + 17 n 3 logn + 19 n 3 logn + 34 n 3 logn + 38 n 3 logn (Using logn < n and that n 3 logn > logn for n > 1 etc.) < 112 n 3 logn . So This is O ( n 3 logn ) for C = 112 and n > 1 . 2. (E) 20b. (2 n + n 2 )( n 3 + 3 n ) = 2 n n 3 + 6 n + n 2 3 n + n 5 < = 6 n + 6 n + 6 n + 6 n (As 6 n is larger than other terms for n > 4 .) = 4 . 6 n Thus it is O (6 n ) for C = 4 and n > 4 . 3. (M) 38. From definitions of Big-Theta we have numbers C1, k1, C2, k2, C3, k3, C4 and k4 such that 1. | f ( x ) | ≤ C 1 | g ( x ) | for x > k 1 2. | f ( x ) | ≥ C 2 | g ( x ) | for x > k 2 3. | g ( x ) | ≤ C 3 | h ( x ) | for x > k 3 4. | g ( x ) | ≥ C 4 | h ( x ) | for x > k 4 Combining 1, 3 nad 2,4 we get 5. | f ( x ) | < = C 1 C 3 | h ( x ) | for x > max { k 1 , k 3 } 6. | f ( x ) | > = C 2 C 4 | h ( x ) | for x > max { k 2 , k 4 } This Shows that f(x) is Θ( h ( x )) Section 3.3 4. (E) Each iteration takes a bounded amount of time, and there are at most n iterations, since each iteration decreases the number of cents remaining. Therefore there are O(n) comparisons.
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