Explanation Given information: ⌈ log 2 n ⌉ i s Θ ( log 2 n ) Definition used: f is of order g: f ( x ) is Θ ( g ( x ) ) if there exists a positive real numbers, A, B and a nonnegative real number k such that A | g ( x ) | ≤ | f ( x ) | ≤ B | g ( x ) | whenever x > k Ceiling function: ⌈ x ⌉ smallest integer that is less than or equal to x. Proof: Since ⌈ x ⌉ ≤ x for all real numbers of x, we have | ⌈ log 2 n ⌉ | ≤ | log 2 n | whenever n ≥ 0 . Since x − 1 < ⌈ x ⌉ ≤ x , | ⌈ log 2 n ⌉ | ≥ | log 2 n − 1 | ⇒ | ⌈ log 2 n ⌉ | ≥ | log 2 n − log 2 2 | ( log 2 2 = 1 ) = | log 2 n 2 | ( log 2 a b = log 2 a<

5th Edition

ISBN: 9780357097717

Author: EPP

Publisher: CENGAGE L

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Chapter 11.4, Problem 42ES

To determine

To prove:

⌈log2n⌉ is Θ(log2n)

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Chapter 11.4, Problem 41ES

Chapter 11.4, Problem 43ES

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Ch. 11.1 - If f is a real-valued function of a real variable,...Ch. 11.1 - Prob. 2TY Ch. 11.1 - Prob. 3TY Ch. 11.1 - Prob. 4TY Ch. 11.1 - Prob. 5TY Ch. 11.1 - Prob. 6TY Ch. 11.1 - Prob. 1ES Ch. 11.1 - The graph of a function g is shown below. a. Is...Ch. 11.1 - Prob. 3ES Ch. 11.1 - Sketch the graphs of the power functions p3 and p4...

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