   Chapter 11.2, Problem 24ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# a. Use one of the methods of Example 11.2.4 to show that 1 4 n 5 − 50 n 3 + 3 n + 12 is Ω ( n 5 ) . b. Show that 1 4 n 5 − 50 n 3 + 3 n + 12 is O ( n 5 ) c. Justify the conclusion that 1 4 n 5 − 50 n 3 + 3 n + 12 is Θ ( n 5 ) .

To determine

(a)

To show that 14n550n3+3n+12 is Ω(n5).

Explanation

Given:

14n550n3+3n+12 is Ω(n5).

Formula used:

Let f and g be real-valued functions defined on the same set of non-negative integers, with g(n)0 for every integer nr, where r is a positive real number. Then

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 ar such that

Ag(n)f(n)   for every integer na.

General procedure to find big-Omega for P(n) :

Let m be a nonnegative integer, Let P(n) be a polynomial of degree m, and suppose the coefficient am of nm is positive.

Let A=12am, and let a be the number obtained as follows:

1. Find the sum of the absolute values of all the coefficients of P(n) except for am.
2. Multiply the result of step 1 by 2am.
3. Let a be the larger of the number 1 and the result of step 2.

Show that AnmP(n) for every integer na.

Calculation:

The coefficient of highest power of 14n550n3+3n+12 is 14 and the sum of the absolute values of its other coefficients is |50|+|3|+|12|.

Hence,

A=12×14=18 and a=214(|50|+|3|+|12|)=8(65)=520

Which is greater than one

To determine

(b)

To show that 14n550n3+3n+12 is O(n5).

To determine

(c)

To justify the conclusion that 14n550n3+3n+12 is Θ(n5).

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