Discrete Mathematics with Graph Theory (Classic Version) (3rd Edition) (Pearson Modern Classics for Advanced Mathematics Series)
3rd Edition
ISBN: 9780134689555
Author: Edgar Goodaire, Michael Parmenter
Publisher: PEARSON
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Chapter 4.4, Problem 1TFQ
A 2003 country music hit by Alan Jackson and Jimmy Buffett contains a line that states (roughly) that, while the clock might read only 12:30, it’s 5:00 somewhere (and so it’s OK to start the evening’s entertainment). Unfortunately, it’s impossible to meet the conditions of this song within America.
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A function is defined on the interval (-π/2,π/2) by this multipart rule:
if -π/2 < x < 0
f(x) =
a
if x=0
31-tan x
+31-cot x
if 0 < x < π/2
Here, a and b are constants. Find a and b so that the function f(x) is continuous at x=0.
a=
b= 3
Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a.
f(x) = (x + 4x4) 5,
a = -1
lim f(x)
X--1
=
lim
x+4x
X--1
lim
X-1
4
x+4x
5
))"
5
))
by the power law
by the sum law
lim (x) + lim
X--1
4
4x
X-1
-(0,00+(
Find f(-1).
f(-1)=243
lim (x) +
-1 +4
35
4 ([
)
lim (x4)
5
x-1
Thus, by the definition of continuity, f is continuous at a = -1.
by the multiple constant law
by the direct substitution property
Chapter 4 Solutions
Discrete Mathematics with Graph Theory (Classic Version) (3rd Edition) (Pearson Modern Classics for Advanced Mathematics Series)
Ch. 4.1 - 227Ch. 4.1 - The Well-Ordering Principle states that any...Ch. 4.1 - Prob. 3TFQCh. 4.1 - Prob. 4TFQCh. 4.1 - Prob. 5TFQCh. 4.1 - Prob. 6TFQCh. 4.1 - Prob. 7TFQCh. 4.1 - Prob. 8TFQCh. 4.1 - Prob. 9TFQCh. 4.1 - Prob. 10TFQ
Ch. 4.1 - Prob. 1ECh. 4.1 - 2. True or false? If false, give a counterexample....Ch. 4.1 - Prob. 3ECh. 4.1 - Prob. 4ECh. 4.1 - Prob. 5ECh. 4.1 - Prob. 6ECh. 4.1 - Prob. 7ECh. 4.1 - Prob. 8ECh. 4.1 - 9. Suppose is a natural number and is the...Ch. 4.1 - Prob. 10ECh. 4.1 - Prob. 11ECh. 4.1 - Prob. 12ECh. 4.2 - True/False Questions
Given two consecutive...Ch. 4.2 - Prob. 2TFQCh. 4.2 - Prob. 3TFQCh. 4.2 - Prob. 4TFQCh. 4.2 - Prob. 5TFQCh. 4.2 - Prob. 6TFQCh. 4.2 - Prob. 7TFQCh. 4.2 - Prob. 8TFQCh. 4.2 - Prob. 9TFQCh. 4.2 - Prob. 10TFQCh. 4.2 - Prob. 1ECh. 4.2 - Prob. 2ECh. 4.2 - Draw the Hasse diagrams for each of the following...Ch. 4.2 - List all minimal, minimum, maximal, and maximum...Ch. 4.2 - Let n be a natural number. Given n consecutive...Ch. 4.2 - Prove that n22(n an integer) is never divisible by...Ch. 4.2 - Given that a and x are integers, a1, a|(11x+3),...Ch. 4.2 - Prob. 8ECh. 4.2 - Prob. 9ECh. 4.2 - Prob. 10ECh. 4.2 - Prob. 11ECh. 4.2 - Prob. 12ECh. 4.2 - Prob. 13ECh. 4.2 - If a and b are relatively prime integers, prove...Ch. 4.2 - Prob. 15ECh. 4.2 - Prob. 16ECh. 4.2 - 17. (a) Find a pair of integers and such that ....Ch. 4.2 - Prob. 18ECh. 4.2 - Prob. 19ECh. 4.2 - Prob. 20ECh. 4.2 - Prob. 21ECh. 4.2 - If aN, prove that...Ch. 4.2 - Prob. 23ECh. 4.2 - Prob. 24ECh. 4.2 - Prob. 25ECh. 4.2 - Prob. 26ECh. 4.2 - Prob. 27ECh. 4.2 - Prob. 28ECh. 4.2 - Prob. 29ECh. 4.2 - Prob. 30ECh. 4.2 - Establish formula (2) of this section by proving...Ch. 4.2 - Prob. 32ECh. 4.2 - Suppose a,b,c are nonzero integers. Show that...Ch. 4.2 - 34. Suppose that and are partial orders.
(a)...Ch. 4.2 - Prob. 35ECh. 4.2 - Let g be the greatest common divisor of integers m...Ch. 4.3 - 127 is a prime number.Ch. 4.3 - Prob. 2TFQCh. 4.3 - Prob. 3TFQCh. 4.3 - Prob. 4TFQCh. 4.3 - Prob. 5TFQCh. 4.3 - Prob. 6TFQCh. 4.3 - Prob. 7TFQCh. 4.3 - Prob. 8TFQCh. 4.3 - Prob. 9TFQCh. 4.3 - Prob. 10TFQCh. 4.3 - Determine whether each of the following integers...Ch. 4.3 - Prob. 2ECh. 4.3 - Suppose p is the smallest prime factor of an...Ch. 4.3 - Prob. 4ECh. 4.3 - Prob. 5ECh. 4.3 - Prob. 6ECh. 4.3 - Prob. 7ECh. 4.3 - Prob. 8ECh. 4.3 - Prob. 9ECh. 4.3 - Prob. 10ECh. 4.3 - Prob. 11ECh. 4.3 - Prob. 12ECh. 4.3 - Estimate the number of primes less than 5000, less...Ch. 4.3 - Prob. 14ECh. 4.3 - Prob. 15ECh. 4.3 - Prob. 16ECh. 4.3 - Prob. 17ECh. 4.3 - 18. (a) Is prime? Explain your answer.
(b) Is...Ch. 4.3 - Prob. 19ECh. 4.3 - Prob. 20ECh. 4.3 - 21. Show that the sum of two consecutive primes is...Ch. 4.3 - Prob. 22ECh. 4.3 - Prob. 23ECh. 4.3 - Prob. 24ECh. 4.3 - Prob. 25ECh. 4.3 - Prob. 26ECh. 4.3 - Let p1p2.......pn+1 denote the first n+1 primes...Ch. 4.3 - Prob. 28ECh. 4.3 - Prob. 29ECh. 4.3 - Let a and b be natural numbers. Find the prime...Ch. 4.3 - Prob. 31ECh. 4.3 - Prob. 32ECh. 4.3 - 33. Show that there are infinitely many triples of...Ch. 4.3 - Prob. 34ECh. 4.3 - Prob. 35ECh. 4.3 - 36. Let denote the set of positive rational...Ch. 4.3 - Prob. 37ECh. 4.3 - Prob. 38ECh. 4.4 - A 2003 country music hit by Alan Jackson and Jimmy...Ch. 4.4 - Prob. 2TFQCh. 4.4 - Prob. 3TFQCh. 4.4 - 4.
Ch. 4.4 - Prob. 5TFQCh. 4.4 - There are 72 congruence classes of integers mod...Ch. 4.4 - Prob. 7TFQCh. 4.4 - Prob. 8TFQCh. 4.4 - Prob. 9TFQCh. 4.4 - Prob. 10TFQCh. 4.4 - Prob. 1ECh. 4.4 - Prob. 2ECh. 4.4 - Prob. 3ECh. 4.4 - 4. True or False? Give a reason for each answer....Ch. 4.4 - Prob. 5ECh. 4.4 - Prob. 6ECh. 4.4 - Prob. 7ECh. 4.4 - Prob. 8ECh. 4.4 - Find all integers x, 0xn, satisfying each of the...Ch. 4.4 - 10. (a) Given integers and a prime , suppose ....Ch. 4.4 - 11. Find all integers and, , that satisfy each of...Ch. 4.4 - Prob. 12ECh. 4.4 - Prob. 13ECh. 4.4 - Prob. 14ECh. 4.4 - Prob. 15ECh. 4.4 - Prob. 16ECh. 4.4 - Prob. 17ECh. 4.4 - Find all integers x, 0xn, that satisfy each of the...Ch. 4.4 - Find all integers x,0xn, that satisfy each of the...Ch. 4.4 - Find all integers x,0xn, that satisfy each of the...Ch. 4.4 - In each case, find the inverse of a(modn) and use...Ch. 4.4 - In each of the following the given integer p is a...Ch. 4.4 - 23. Show that has no solutions.
Ch. 4.4 - 24. Let A be the set of congruence classes of...Ch. 4.5 - Prob. 1TFQCh. 4.5 - Prob. 2TFQCh. 4.5 - Prob. 3TFQCh. 4.5 - Prob. 4TFQCh. 4.5 - Prob. 5TFQCh. 4.5 - Prob. 6TFQCh. 4.5 - Prob. 7TFQCh. 4.5 - Prob. 8TFQCh. 4.5 - Prob. 9TFQCh. 4.5 - Prob. 10TFQCh. 4.5 - Prob. 1ECh. 4.5 - Prob. 2ECh. 4.5 - Prob. 3ECh. 4.5 - Prob. 4ECh. 4.5 - Prob. 5ECh. 4.5 - Prob. 6ECh. 4.5 - Prob. 7ECh. 4.5 - Prob. 8ECh. 4.5 - Prob. 9ECh. 4.5 - Prob. 10ECh. 4.5 - Prob. 11ECh. 4.5 - Prob. 12ECh. 4.5 - Prob. 13ECh. 4.5 - Prob. 14ECh. 4.5 - Prob. 15ECh. 4.5 - Prob. 16ECh. 4.5 - Prob. 17ECh. 4.5 - 18. In each case, find the smallest nonnegative...Ch. 4.5 - For each of the following, find the smallest...Ch. 4.5 - Prob. 20ECh. 4.5 - In each of the following cases, find a positive...Ch. 4.5 - Prob. 22ECh. 4.5 - Prob. 23ECh. 4.5 - Prob. 24ECh. 4.5 - 25. Suppose. Decode each of the following encoded...Ch. 4 - Find the quotient and remainder when...Ch. 4 - Convert (1100101)2 to base 10. Convert 32,145 to...Ch. 4 - Prob. 3RECh. 4 - An integer n, which has exactly eight factors, is...Ch. 4 - Prob. 5RECh. 4 - Prob. 6RECh. 4 - Prob. 7RECh. 4 - Prob. 8RECh. 4 - Illustrate the Euclidean algorithm by showing that...Ch. 4 - Prob. 10RECh. 4 - Prob. 11RECh. 4 - Prob. 12RECh. 4 - 13. Suppose. Show that is not the sum of three...Ch. 4 - Prob. 14RECh. 4 - Prob. 15RECh. 4 - Prob. 16RECh. 4 - Prob. 17RECh. 4 - 18. What is the last digit of ? [Hint: You want...Ch. 4 - Prob. 19RECh. 4 - Prob. 20RECh. 4 - Compute 380(mod7). Find all integers x such that...Ch. 4 - Prob. 22RECh. 4 - Prob. 23RECh. 4 - Prob. 24RECh. 4 - Prob. 25RECh. 4 - Prob. 26RECh. 4 - Find the smallest positive integer x that...
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