Explanation Given information: 5 n 3 + 65 n + 30. Concept used: Let f and g be real-valued functions defined on the same set of nonnegative integers, with g ( n ) ≥ 0 for every integer n ≥ r , where r is a positive real number. Then f is of order g, written f ( n ) is Θ ( g ( n )) ( f of n is big-Theta of g of n ), if and only if, there exist positive real numbers A, B and k ≥ r such that f is of order at least g, written f (n) is Ω ( g (n)) ( f of n is big-Omega of g of n ), if and only if, there exist positive real numbers A and a ≥ r such that A g ( n ) ≤ f ( n ) for every integer n ≥ a . f is of order at most g, written f (n) is O ( g ( n )) ( f of n is big- O of g of n ), if and only if, there exist positive real numbers B and b ≥ r such that 0 ≤ f ( n ) ≤ B g ( n ) for every integer n ≥ b . Proof: Firstly, to show that 5 n 3 + 65 n + 30 is Ω ( n 3 ) , we need to show that 5 n 3 + 65 n + 30 is greater than or equal to a positive multiple of n 3 for all values of n that are sufficiently large. Now 5 n 3 ≤ 5 n 3 + 65 n + 30 for every integer n ≥ 1 Because when n ≥ 1 , 65 n + 30 is positive. Thus we can let A = 5 and a = 1 to obtain A n 3 ≤ 5 n 3 + 65 n + 30 for every integer n ≥ a , Concluding by definition of Ω -notation, that 5 n 3 + 65 n + 30 is Ω ( n 3 ) . Secondly, to show that 5 n 3 + 65 n + 30 is O ( n 3 ) , we need to show that 5 n 3 + 65 n + 30 is greater than or equal to 0 and less than or equal to some positive multiple of n 3 for all values of n that are sufficiently large. First note that because all terms of 5 n 3 + 65 n + 30 are positive, 0 ≤ 5 n 3 + 65 n + 30 for every integer n ≥ 1

5th Edition

ISBN: 9780357540244

Author: EPP

Publisher: CENGAGE L

See similar textbooks

Concept explainers

Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.

Videos

Question

Chapter 11.2, Problem 13ES

To determine

To show that 5n3+65n+30 is Θ(n3).

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11.2, Problem 12ES

Chapter 11.2, Problem 14ES

Students have asked these similar questions

DO these math problems without ai, show the solutions as well. and how you solved it. and could you do it with in the time spand

The Cartesian coordinates of a point are given. (a) (-8, 8) (i) Find polar coordinates (r, 0) of the point, where r > 0 and 0 ≤ 0 0 and 0 ≤ 0 < 2π. (1, 0) = (r. = ([ (ii) Find polar coordinates (r, 8) of the point, where r < 0 and 0 ≤ 0 < 2π. (5, 6) = =([

The Cartesian coordinates of a point are given. (a) (4,-4) (i) Find polar coordinates (r, e) of the point, where r > 0 and 0 0 and 0 < 0 < 2π. (r, 6) = X 7 (ii) Find polar coordinates (r, 8) of the point, where r < 0 and 0 0 < 2π. (r, 0) = X

Chapter 11 Solutions

WEBASSIGN F/EPPS DISCRETE MATHEMATICS

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...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

To show that 5 n 3 + 65 n + 30 is Θ ( n 3 ) .

Solution Summary: The author explains how to show that 5n3+65n+30 is Theta

Concept explainers

Videos

To show that 5n3+65n+30 is Θ(n3).

Want to see the full answer?

Chapter 11 Solutions