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: 9781337694193

Author: EPP, Susanna S.

Publisher: Cengage Learning,

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

1. Find the area of the region enclosed between the curves y = x and y = x. Sketch the region.

for the given rectangular coordinates, find two sets of polar coordinates for which 0≤θ<2π, one with r>0 and the other with r<0. (-2sqrt(3),9)

3) Find the general solution to the following differential equation. d²x = +4x sin² (2t) dt²

Chapter 11 Solutions

Discrete Mathematics With Applications

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