Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
14th Edition
ISBN: 9780134668574
Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Christopher J. Stocker
Publisher: PEARSON
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Textbook Question
Chapter A.6, Problem 59E
Rationalize the denominators in Problems 55–60.
59. 
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7.
Define the sequence {b} by
bo = 0
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= 2
8.
bn=4bn-1-4bn-2 for n ≥ 2
(a) Give the first five terms of this sequence.
(b) Prove: For all n = N, bn = 2nn.
Let a Rsuch that a 1, and let nЄ N. We're going to derive a formula for
Σoa without needing to prove it by induction. Tip: it can be helpful to use C1+C2+...+Cn
notation instead of summation notation when working this out on scratch paper.
(a) Take a a² and manipulate it until it is in the form Σ.a.
i=0
(b) Using this, calculate the difference between a Σ0 a² and Σ0 a², simplifying away the
summation notation.
i=0
(c) Now that you know what (a – 1) Σ0 a² equals, divide both sides by a − 1 to derive the
formula for
a².
(d) (Optional, just for induction practice) Prove this formula using induction.
3.
Let A, B, and C be sets and let f: A B and g BC be functions. For
each of the following, draw arrow diagrams that illustrate the situation, and then prove the
proposition.
(a) If ƒ and g are injective, then go f is injective.
(b) If ƒ and g are surjective, then go f is surjective.
(c) If gof is injective then f is injective. Make sure your arrow diagram shows that 9 does
not need to be injective!
(d) If gof is surjective then g is surjective. Make sure your arrow diagram shows that f
does not need to be surjective!
4.
5.
6.
Let X be a set and let f: XX be a function. We say that f is an involution if
fof idx and that f is idempotent if f f = f.
(a) If f is an involution, must it be invertible? Why or why not?2
(b) If f is idempotent, must it be invertible? Why or why not?
(c) If f is idempotent and x E range(f), prove that f(x) = x.
Prove that [log3 536] 5. You proof must be verifiable by someone who does not
have access to a scientific calculator or a logarithm table (you cannot use log3 536≈ 5.7).
Define the sequence {a} by a = 2-i for i≥ 1.
(a) Give the first five terms of the sequence.
(b) Prove that the sequence is increasing.
Chapter A.6 Solutions
Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
Ch. A.6 - Evaluate each of the following: (A)161/2 (B)16...Ch. A.6 - Convert to radical form. (A)u1/5 (B)(6x2y5)2/9...Ch. A.6 - Prob. 3MPCh. A.6 - Prob. 4MPCh. A.6 - Write the following expression in the form axp +...Ch. A.6 - Prob. 6MPCh. A.6 - Prob. 7MPCh. A.6 - Rationalize each numerator. (A)332 (B)2n4n...Ch. A.6 - Change each expression in Problems 16 to radical...Ch. A.6 - Change each expression in Problems 16 to radical...
Ch. A.6 - Change each expression in Problems 16 to radical...Ch. A.6 - Prob. 4ECh. A.6 - Prob. 5ECh. A.6 - Change each expression in Problems 16 to radical...Ch. A.6 - Prob. 7ECh. A.6 - Prob. 8ECh. A.6 - Prob. 9ECh. A.6 - Prob. 10ECh. A.6 - Prob. 11ECh. A.6 - Prob. 12ECh. A.6 - In Problems 1324, find rational number...Ch. A.6 - In Problems 1324, find rational number...Ch. A.6 - In Problems 1324, find rational number...Ch. A.6 - In Problems 1324, find rational number...Ch. A.6 - In Problems 1324, find rational number...Ch. A.6 - In Problems 1324, find rational number...Ch. A.6 - In Problems 1324, find rational number...Ch. A.6 - In Problems 1324, find rational number...Ch. A.6 - In Problems 1324, find rational number...Ch. A.6 - Prob. 22ECh. A.6 - In Problems 1324, find rational number...Ch. A.6 - In Problems 1324, find rational number...Ch. A.6 - Prob. 25ECh. A.6 - Prob. 26ECh. A.6 - Prob. 27ECh. A.6 - Prob. 28ECh. A.6 - Prob. 29ECh. A.6 - Prob. 30ECh. A.6 - Prob. 31ECh. A.6 - Prob. 32ECh. A.6 - Prob. 33ECh. A.6 - Prob. 34ECh. A.6 - Simplify each expression in Problems 3540 using...Ch. A.6 - Simplify each expression in Problems 3540 using...Ch. A.6 - Simplify each expression in Problems 3540 using...Ch. A.6 - Prob. 38ECh. A.6 - Simplify each expression in Problems 3540 using...Ch. A.6 - Simplify each expression in Problems 3540 using...Ch. A.6 - In Problems 4148, multiply, and express answers...Ch. A.6 - Prob. 42ECh. A.6 - In Problems 4148, multiply, and express answers...Ch. A.6 - In Problems 4148, multiply, and express answers...Ch. A.6 - Prob. 45ECh. A.6 - In Problems 4148, multiply, and express answers...Ch. A.6 - Prob. 47ECh. A.6 - In Problems 4148, multiply, and express answers...Ch. A.6 - Write each expression in Problems 4954 in the form...Ch. A.6 - Write each expression in Problems 4954 in the form...Ch. A.6 - Prob. 51ECh. A.6 - Write each expression in Problems 4954 in the form...Ch. A.6 - Write each expression in Problems 4954 in the form...Ch. A.6 - Write each expression in Problems 4954 in the form...Ch. A.6 - Rationalize the denominators in Problems 5560. 55....Ch. A.6 - Rationalize the denominators in Problems 5560. 56....Ch. A.6 - Prob. 57ECh. A.6 - Prob. 58ECh. A.6 - Rationalize the denominators in Problems 5560....Ch. A.6 - Rationalize the denominators in Problems 5560....Ch. A.6 - Rationalize the denominators in Problems 6166....Ch. A.6 - Rationalize the denominators in Problems 6166....Ch. A.6 - Rationalize the denominators in Problems 6166....Ch. A.6 - Rationalize the denominators in Problems 6166....Ch. A.6 - Prob. 65ECh. A.6 - Prob. 66ECh. A.6 - Prob. 67ECh. A.6 - Problems 6770 illustrate common errors involving...Ch. A.6 - Problems 6770 illustrate common errors involving...Ch. A.6 - Problems 6770 illustrate common errors involving...Ch. A.6 - In Problems 7182, discuss the validity of each...Ch. A.6 - In Problems 7182, discuss the validity of each...Ch. A.6 - Prob. 73ECh. A.6 - Prob. 74ECh. A.6 - Prob. 75ECh. A.6 - In Problems 7182, discuss the validity of each...Ch. A.6 - Prob. 77ECh. A.6 - In Problems 7182, discuss the validity of each...Ch. A.6 - In Problems 7182, discuss the validity of each...Ch. A.6 - In Problems 7182, discuss the validity of each...Ch. A.6 - In Problems 7182, discuss the validity of each...Ch. A.6 - In Problems 7182, discuss the validity of each...Ch. A.6 - In Problems 8388, simplify by writing each...Ch. A.6 - Prob. 84ECh. A.6 - In Problems 8388, simplify by writing each...Ch. A.6 - Prob. 86ECh. A.6 - In Problems 8388, simplify by writing each...Ch. A.6 - In Problems 8388, simplify by writing each...Ch. A.6 - In Problems 8994, evaluate using a calculator....Ch. A.6 - In Problems 8994, evaluate using a calculator....Ch. A.6 - Prob. 91ECh. A.6 - In Problems 8994, evaluate using a calculator....Ch. A.6 - In Problems 8994, evaluate using a calculator....Ch. A.6 - In Problems 8994, evaluate using a calculator....Ch. A.6 - In Problems 95 and 96, evaluate each expression on...Ch. A.6 - In Problems 95 and 96, evaluate each expression on...
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