Finite Mathematics for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
14th Edition
ISBN: 9780134675985
Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Christopher J. Stocker
Publisher: PEARSON
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Textbook Question
Chapter 6.1, Problem 17E
In Problems 13-20, write the
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Keity
x२
1. (i)
Identify which of the following subsets of R2 are open and which
are not.
(a)
A = (2,4) x (1, 2),
(b)
B = (2,4) x {1,2},
(c)
C = (2,4) x R.
Provide a sketch and a brief explanation to each of your answers.
[6 Marks]
(ii)
Give an example of a bounded set in R2 which is not open.
[2 Marks]
(iii)
Give an example of an open set in R2 which is not bounded.
[2 Marks
2.
(i)
Which of the following statements are true? Construct coun-
terexamples for those that are false.
(a)
sequence.
Every bounded sequence (x(n)) nEN C RN has a convergent sub-
(b)
(c)
(d)
Every sequence (x(n)) nEN C RN has a convergent subsequence.
Every convergent sequence (x(n)) nEN C RN is bounded.
Every bounded sequence (x(n)) EN CRN converges.
nЄN
(e)
If a sequence (xn)nEN C RN has a convergent subsequence, then
(xn)nEN is convergent.
[10 Marks]
(ii)
Give an example of a sequence (x(n))nEN CR2 which is located on
the parabola x2 = x², contains infinitely many different points and converges
to the limit x = (2,4).
[5 Marks]
2.
(i) What does it mean to say that a sequence (x(n)) nEN CR2
converges to the limit x E R²?
[1 Mark]
(ii) Prove that if a set ECR2 is closed then every convergent
sequence (x(n))nen in E has its limit in E, that is
(x(n)) CE and x() x
x = E.
[5 Marks]
(iii)
which is located on the parabola x2 = = x
x4, contains a subsequence that
Give an example of an unbounded sequence (r(n)) nEN CR2
(2, 16) and such that x(i)
converges to the limit x = (2, 16) and such that x(i)
#
x() for any i j.
[4 Marks
Chapter 6 Solutions
Finite Mathematics for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
Ch. 6.1 - The following linear programming problem has only...Ch. 6.1 - Use the table method to solve the following linear...Ch. 6.1 - Use the table method to solve the following linear...Ch. 6.1 - Refer to Example 1. Find the basic solution for...Ch. 6.1 - Construct the table of basic solutions and use it...Ch. 6.1 - Construct the table of basic solutions and use it...Ch. 6.1 - Refer to Table 5. For the basic solution...Ch. 6.1 - In Problems 1-8, evaluate the expression. (If...Ch. 6.1 - In Problems 1-8, evaluate the expression. (If...Ch. 6.1 - In Problems 1-8, evaluate the expression. (If...
Ch. 6.1 - In Problems 1-8, evaluate the expression. (If...Ch. 6.1 - In Problems 1-8, evaluate the expression. (If...Ch. 6.1 - In Problems 1-8, evaluate the expression. (If...Ch. 6.1 - In Problems 1-8, evaluate the expression. (If...Ch. 6.1 - In Problems 1-8, evaluate the expression. (If...Ch. 6.1 - Problems 9-12 refer to the system...Ch. 6.1 - Problems 9-12 refer to the system...Ch. 6.1 - Problems 9-12 refer to the system...Ch. 6.1 - Problems 9-12 refer to the system...Ch. 6.1 - In Problems 13-20, write the e-system obtained via...Ch. 6.1 - In Problems 13-20, write the e-system obtained via...Ch. 6.1 - In Problems 13-20, write the e-system obtained via...Ch. 6.1 - In Problems 13-20, write the e-system obtained via...Ch. 6.1 - In Problems 13-20, write the e-system obtained via...Ch. 6.1 - In Problems 13-20, write the e-system obtained via...Ch. 6.1 - In Problems 13-20, write the e-system obtained via...Ch. 6.1 - In Problems 13-20, write the e-system obtained via...Ch. 6.1 - Problems 21-30 refer to the table below of the six...Ch. 6.1 - Problems 21-30 refer to the table below of the six...Ch. 6.1 - Problems 21-30 refer to the table below of the six...Ch. 6.1 - Problems 21-30 refer to the table below of the six...Ch. 6.1 - Problems 21-30 refer to the table below of the six...Ch. 6.1 - Problems 21-30 refer to the table below of the six...Ch. 6.1 - Problems 21-30 refer to the table below of the six...Ch. 6.1 - Problems 21-30 refer to the table below of the six...Ch. 6.1 - Problems 21-30 refer to the table below of the six...Ch. 6.1 - Problems 21-30 refer to the table below of the six...Ch. 6.1 - Problems 31-40 refer to the partially completed...Ch. 6.1 - Problems 31-40 refer to the partially completed...Ch. 6.1 - Problems 31-40 refer to the partially completed...Ch. 6.1 - Problems 31-40 refer to the partially completed...Ch. 6.1 - Problems 31-40 refer to the partially completed...Ch. 6.1 - Problems 31-40 refer to the partially completed...Ch. 6.1 - Problems 31-40 refer to the partially completed...Ch. 6.1 - Problems 31-40 refer to the partially completed...Ch. 6.1 - Problems 31-40 refer to the partially completed...Ch. 6.1 - Problems 31-40 refer to the partially completed...Ch. 6.1 - In Problems 41-48, convert the given i-system to...Ch. 6.1 - In Problems 41-48, convert the given i-system to...Ch. 6.1 - In Problems 41-48, convert the given i-system to...Ch. 6.1 - In Problems 41-48, convert the given i-system to...Ch. 6.1 - In Problems 41-48, convert the given i-system to...Ch. 6.1 - In Problems 41-48, convert the given i-system to...Ch. 6.1 - In Problems 41-48, convert the given i-system to...Ch. 6.1 - In Problems 41-48, convert the given i-system to...Ch. 6.1 - In Problems 49-54, graph the system of...Ch. 6.1 - In Problems 49-54, graph the system of...Ch. 6.1 - In Problems 49-54, graph the system of...Ch. 6.1 - In Problems 49-54, graph the system of...Ch. 6.1 - In Problems 49-54, graph the system of...Ch. 6.1 - In Problems 49-54, graph the system of...Ch. 6.1 - For a standard maximization problem in standard...Ch. 6.1 - For a standard maximization problem in standard...Ch. 6.1 - If 5x1+4x21,000 is one of the problem constraints...Ch. 6.1 - If a1x1+a2x2b is one of the problem constraints in...Ch. 6.1 - In Problems 59-66, solve the given linear...Ch. 6.1 - In Problems 59-66, solve the given linear...Ch. 6.1 - In Problems 59-66, solve the given linear...Ch. 6.1 - In Problems 59-66, solve the given linear...Ch. 6.1 - In Problems 59-66, solve the given linear...Ch. 6.1 - In Problems 59-66, solve the given linear...Ch. 6.1 - In Problems 59-66, solve the given linear...Ch. 6.1 - In Problems 59-66, solve the given linear...Ch. 6.1 - In Problems 67-70, explain why the linear...Ch. 6.1 - In Problems 67-70, explain why the linear...Ch. 6.1 - In Problems 67-70, explain why the linear...Ch. 6.1 - In Problems 67-70, explain why the linear...Ch. 6.1 - In Problems 71-72, explain why the linear...Ch. 6.1 - In Problems 71-72, explain why the linear...Ch. 6.1 - A linear programming problem has four decision...Ch. 6.1 - A linear programming problem has five decision...Ch. 6.1 - A linear programming problem has 30 decision...Ch. 6.1 - A linear programming problem has 40 decision...Ch. 6.2 - Graph the feasible region for the linear...Ch. 6.2 - Solve the following linear programming problem...Ch. 6.2 - Solve using the simplex method:...Ch. 6.2 - Repeat Example 3 modified as follows:Ch. 6.2 - For the simplex tableau in Problems 1-4, (A)...Ch. 6.2 - For the simplex tableau in Problems 1-4, (A)...Ch. 6.2 - For the simplex tableau in Problems 1-4, (A)...Ch. 6.2 - For the simplex tableau in Problems 1-4, (A)...Ch. 6.2 - In Problems 5-8, find the pivot element, identify...Ch. 6.2 - In Problems 5-8, find the pivot element, identify...Ch. 6.2 - In Problems 5-8, find the pivot element, identify...Ch. 6.2 - In Problems 5-8, find the pivot element, identify...Ch. 6.2 - In Problems 9-12, (A) Using the slack variables,...Ch. 6.2 - In Problems 9-12, (A) Using the slack variables,...Ch. 6.2 - In Problems 9-12, (A) Using the slack variables,...Ch. 6.2 - In Problems 9-12, (A) Using the slack variables,...Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - In Problems 33 and 34, first solve the linear...Ch. 6.2 - In Problems 33 and 34, first solve the linear...Ch. 6.2 - Solve Problems 35 and 36 by the simplex method and...Ch. 6.2 - Solve Problems 35 and 36 by the simplex method and...Ch. 6.2 - In Problems 37-40, there is a tie for the choice...Ch. 6.2 - In Problems 37-40, there is a tie for the choice...Ch. 6.2 - In Problems 37-40, there is a tie for the choice...Ch. 6.2 - In Problems 37-40, there is a tie for the choice...Ch. 6.2 - In Problems 41-56, construct a mathematical model...Ch. 6.2 - In Problems 41-56, construct a mathematical model...Ch. 6.2 - In Problems 41-56, construct a mathematical model...Ch. 6.2 - In Problems 41-56, construct a mathematical model...Ch. 6.2 - In Problems 41-56, construct a mathematical model...Ch. 6.2 - In Problems 41-56, construct a mathematical model...Ch. 6.2 - In Problems 41-56, construct a mathematical model...Ch. 6.2 - In Problems 41-56, construct a mathematical model...Ch. 6.2 - In Problems 41-56, construct a mathematical model...Ch. 6.2 - In Problems 41-56, construct a mathematical model...Ch. 6.2 - In Problems 41-56, construct a mathematical model...Ch. 6.2 - In Problems 41-56, construct a mathematical model...Ch. 6.2 - In Problems 41-56, construct a mathematical model...Ch. 6.2 - In Problems 41-56, construct a mathematical model...Ch. 6.2 - In Problems 41-56, construct a mathematical model...Ch. 6.2 - In Problems 41-56, construct a mathematical model...Ch. 6.3 - Excluding the nonnegative constraints, the...Ch. 6.3 - The simplex method can be used to solve any...Ch. 6.3 - Form the dual problem:...Ch. 6.3 - Solve the following minimization problem by...Ch. 6.3 - Solve the following minimization problem by...Ch. 6.3 - Repeat Example 4 if the shipping charge from plant...Ch. 6.3 - In Problems 1-8, find the transpose of each...Ch. 6.3 - In Problems 1-8, find the transpose of each...Ch. 6.3 - In Problems 1-8, find the transpose of each...Ch. 6.3 - In Problems 1-8, find the transpose of each...Ch. 6.3 - In Problems 1-8, find the transpose of each...Ch. 6.3 - In Problems 1-8, find the transpose of each...Ch. 6.3 - In Problems 1-8, find the transpose of each...Ch. 6.3 - In Problems 1-8, find the transpose of each...Ch. 6.3 - In Problems 9 and 10, (A) Form the dual problem....Ch. 6.3 - In Problems 9 and 10, (A) Form the dual problem....Ch. 6.3 - In Problems 11 and 12, a minimization problem, the...Ch. 6.3 - In Problems 11 and 12, a minimization problem, the...Ch. 6.3 - In Problems 13-20, (A) Form the dual problem. (B)...Ch. 6.3 - In Problems 13-20, (A) Form the dual problem. (B)...Ch. 6.3 - In Problems 13-20, (A) Form the dual problem. (B)...Ch. 6.3 - In Problems 13-20, (A) Form the dual problem. (B)...Ch. 6.3 - In Problems 13-20, (A) Form the dual problem. (B)...Ch. 6.3 - In Problems 13-20, (A) Form the dual problem. (B)...Ch. 6.3 - In Problems 13-20, (A) Form the dual problem. (B)...Ch. 6.3 - In Problems 13-20, (A) Form the dual problem. (B)...Ch. 6.3 - Solve the linear programming problems in Problem...Ch. 6.3 - Solve the linear programming problems in Problem...Ch. 6.3 - Solve the linear programming problems in Problem...Ch. 6.3 - Solve the linear programming problems in Problem...Ch. 6.3 - Solve the linear programming problems in Problem...Ch. 6.3 - Solve the linear programming problems in Problem...Ch. 6.3 - Solve the linear programming problems in Problem...Ch. 6.3 - Solve the linear programming problems in Problem...Ch. 6.3 - Solve the linear programming problems in Problem...Ch. 6.3 - Solve the linear programming problems in Problem...Ch. 6.3 - Solve the linear programming problems in Problem...Ch. 6.3 - Solve the linear programming problems in Problem...Ch. 6.3 - Solve the linear programming problems in Problem...Ch. 6.3 - Solve the linear programming problems in Problem...Ch. 6.3 - Solve the linear programming problems in Problem...Ch. 6.3 - Solve the linear programming problems in Problem...Ch. 6.3 - A minimization problem has 4 variables and 2...Ch. 6.3 - A minimization problem has 3 variables and 5...Ch. 6.3 - If you want to solve a minimization problem by...Ch. 6.3 - If you want to solve a minimization problem by...Ch. 6.3 - In Problems 41 and 42, (A) Form the dual problem....Ch. 6.3 - In Problems 41 and 42, (A) Form the dual problem....Ch. 6.3 - In Problem 43 and 44, (A) Form an equivalent...Ch. 6.3 - In Problem 43 and 44, (A) Form an equivalent...Ch. 6.3 - Solve the linear programming problem in Problems...Ch. 6.3 - Solve the linear programming problem in Problems...Ch. 6.3 - Solve the linear programming problem in Problems...Ch. 6.3 - Solve the linear programming problem in Problems...Ch. 6.3 - In Problems 49-58, construct a mathematical model...Ch. 6.3 - In Problems 49-58, construct a mathematical model...Ch. 6.3 - In Problems 49-58, construct a mathematical model...Ch. 6.3 - In Problems 49-58, construct a mathematical model...Ch. 6.3 - In Problems 49-58, construct a mathematical model...Ch. 6.3 - In Problems 49-58, construct a mathematical model...Ch. 6.3 - In Problems 49-58, construct a mathematical model...Ch. 6.3 - In Problems 49-58, construct a mathematical model...Ch. 6.3 - In Problems 49-58, construct a mathematical model...Ch. 6.3 - In Problems 49-58, construct a mathematical model...Ch. 6.4 - Repeat Example 1 for...Ch. 6.4 - Solve the following linear programming problem...Ch. 6.4 - Solve the following linear programming problem...Ch. 6.4 - Prob. 4MPCh. 6.4 - Suppose that the refinery in Example 5 has 35,000...Ch. 6.4 - In Problems 1-8, (A) Introduce slack, surplus, and...Ch. 6.4 - In Problems 1-8, (A) Introduce slack, surplus, and...Ch. 6.4 - In Problems 1-8, (A) Introduce slack, surplus, and...Ch. 6.4 - In Problems 1-8, (A) Introduce slack, surplus, and...Ch. 6.4 - In Problems 1-8, (A) Introduce slack, surplus, and...Ch. 6.4 - In Problems 1-8, (A) Introduce slack, surplus, and...Ch. 6.4 - In Problems 1-8, (A) Introduce slack, surplus, and...Ch. 6.4 - In Problems 1-8, (A) Introduce slack, surplus, and...Ch. 6.4 - Use the big M method to solve Problems 9-22....Ch. 6.4 - Use the big M method to solve Problems 9-22....Ch. 6.4 - Use the big M method to solve Problems 9-22....Ch. 6.4 - Use the big M method to solve Problems 9-22....Ch. 6.4 - Use the big M method to solve Problems 9-22....Ch. 6.4 - Use the big M method to solve Problems 9-22....Ch. 6.4 - Use the big M method to solve Problems 9-22....Ch. 6.4 - Use the big M method to solve Problems 9-22....Ch. 6.4 - Use the big M method to solve Problems 9-22....Ch. 6.4 - Use the big M method to solve Problems 9-22....Ch. 6.4 - Use the big M method to solve Problems 9-22....Ch. 6.4 - Use the big M method to solve Problems 9-22....Ch. 6.4 - Use the big M method to solve Problems 9-22....Ch. 6.4 - Use the big M method to solve Problems 9-22....Ch. 6.4 - Solve Problems 5 and 7 by graphing (the geometric...Ch. 6.4 - Solve Problems 6 and 8 by graphing (the geometric...Ch. 6.4 - Problems 25-32 are mixed. Some can be solved by...Ch. 6.4 - Problems 25-32 are mixed. Some can be solved by...Ch. 6.4 - Problems 25-32 are mixed. Some can be solved by...Ch. 6.4 - Problems 25-32 are mixed. Some can be solved by...Ch. 6.4 - Problems 25-32 are mixed. Some can be solved by...Ch. 6.4 - Problems 25-32 are mixed. Some can be solved by...Ch. 6.4 - Problems 25-32 are mixed. Some can be solved by...Ch. 6.4 - Problems 25-32 are mixed. Some can be solved by...Ch. 6.4 - In Problems 33-38, construct a mathematical model...Ch. 6.4 - In Problems 33-38, construct a mathematical model...Ch. 6.4 - In Problems 33-38, construct a mathematical model...Ch. 6.4 - In Problems 33-38, construct a mathematical model...Ch. 6.4 - In Problems 33-38, construct a mathematical model...Ch. 6.4 - In Problems 33-38, construct a mathematical model...Ch. 6.4 - In Problems 39-47, construct a mathematical model...Ch. 6.4 - In Problems 39-47, construct a mathematical model...Ch. 6.4 - In Problems 39-47, construct a mathematical model...Ch. 6.4 - In Problems 39-47, construct a mathematical model...Ch. 6.4 - In Problems 39-47, construct a mathematical model...Ch. 6.4 - In Problems 39-47, construct a mathematical model...Ch. 6.4 - In Problems 39-47, construct a mathematical model...Ch. 6.4 - In Problems 39-47, construct a mathematical model...Ch. 6.4 - In Problems 39-47, construct a mathematical model...Ch. 6 - Problems 1-7 refer to the partially completed...Ch. 6 - Problems 1-7 refer to the partially completed...Ch. 6 - Problems 1-7 refer to the partially completed...Ch. 6 - Problems 1-7 refer to the partially completed...Ch. 6 - Problems 1-7 refer to the partially completed...Ch. 6 - Problems 1-7 refer to the partially completed...Ch. 6 - Problems 1-7 refer to the partially completed...Ch. 6 - A linear programming problem has 6 decision...Ch. 6 - Given the linear programming problem...Ch. 6 - How many basic variables and how many nonbasic...Ch. 6 - Find all basic solutions for the system in Problem...Ch. 6 - Write the simplex tableau for Problem 9, and...Ch. 6 - Solve Problem 9 using the simplex method.Ch. 6 - For the simplex tableau below, identify the basic...Ch. 6 - Find the basic solution for each tableau....Ch. 6 - Form the dual problem of...Ch. 6 - Write the initial system for the dual problem in...Ch. 6 - Write the first simplex tableau for the dual...Ch. 6 - Use the simplex method to find the optimal...Ch. 6 - Use the final simplex tableau from Problem 19 to...Ch. 6 - Solve the linear programming problem using the...Ch. 6 - Form the dual problem of the linear programming...Ch. 6 - Solve Problem 22 by applying the simplex method to...Ch. 6 - Solve the linear programming Problems 24 and...Ch. 6 - Solve the linear programming Problems 24 and...Ch. 6 - Solve the linear programming problem using the...Ch. 6 - Refer to Problem 26. How many pivot columns are...Ch. 6 - In problems 28 and 29, (A) Introduce slack,...Ch. 6 - In problems 28 and 29, (A) Introduce slack,...Ch. 6 - Find the modified problem for the following linear...Ch. 6 - Write a brief verbal description of the type of...Ch. 6 - Write a brief verbal description of the type of...Ch. 6 - Write a brief verbal description of the type of...Ch. 6 - Solve the following linear programming problem by...Ch. 6 - Solve by the dual problem method:...Ch. 6 - Solve Problem 35 by the big M method.Ch. 6 - Solve by the dual problem method:...Ch. 6 - In problems 38-41, construct a mathematical model...Ch. 6 - In problems 38-41, construct a mathematical model...Ch. 6 - In problems 38-41, construct a mathematical model...Ch. 6 - In problems 38-41, construct a mathematical model...
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