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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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Chapter B.3, Problem 38E
To determine
To show: The statement
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Solve the following boundary value problem using method of separation of variables:
1 ə
ди
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70% (107) +
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T2.3: Prove that there exists a connected graph with degrees d₁ ≥ d₂ >> dn if and only
if d1, d2,..., dn is graphic, d ≥ 1 and di≥2n2. That is, some graph having degree
sequence with these conditions is connected.
Hint - Do not attempt to directly prove this using Erdos-Gallai conditions. Instead work with a
realization and show that 2-switches can be used to make a connected graph with the same degree
sequence. Facts that can be useful: a component (i.e., connected) with n₁ vertices and at least
n₁ edges has a cycle. Note also that a 2-switch using edges from different components of a forest
will not necessarily reduce the number of components. Make sure that you justify that your proof
has a 2-switch that does decrease the number of components.
T2.2 Prove that a sequence s d₁, d₂,..., dn with n ≥ 3 of integers with 1≤d; ≤ n − 1 is the
degree sequence of a connected unicyclic graph (i.e., with exactly one cycle) of order n if and only
if at most n-3 terms of s are 1 and Σ di = 2n.
(i) Prove it by induction along the lines of the inductive proof for trees. There will be a special
case to handle when no d₂ = 1.
(ii) Prove it by making use of the caterpillar construction. You may use the fact that adding an
edge between 2 non-adjacent vertices of a tree creates a unicylic graph.
Chapter B.3 Solutions
Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
Ch. B.3 - Evaluate. (A)4! (B)7!6! (C)8!5!Ch. B.3 - Find (A)5C2 (B)6C0Ch. B.3 - Use the binomial theorem to expand (x + 2)5.Ch. B.3 - Use the binomial theorem to find the fourth term...Ch. B.3 - In Problems 120, evaluate each expression. 1.6!Ch. B.3 - In Problems 120, evaluate each expression. 2.7!Ch. B.3 - In Problems 120, evaluate each expression. 3.10!9!Ch. B.3 - In Problems 120, evaluate each expression....Ch. B.3 - In Problems 120, evaluate each expression. 5.12!9!Ch. B.3 - In Problems 120, evaluate each expression. 6.10!6!
Ch. B.3 - Prob. 7ECh. B.3 - In Problems 120, evaluate each expression....Ch. B.3 - Prob. 9ECh. B.3 - In Problems 120, evaluate each expression....Ch. B.3 - In Problems 120, evaluate each expression....Ch. B.3 - Prob. 12ECh. B.3 - Prob. 13ECh. B.3 - In Problems 120, evaluate each expression. 14.7C3Ch. B.3 - In Problems 120, evaluate each expression. 15.6C5Ch. B.3 - Prob. 16ECh. B.3 - In Problems 120, evaluate each expression. 17.5C0Ch. B.3 - In Problems 120, evaluate each expression. 18.5C5Ch. B.3 - In Problems 120, evaluate each expression....Ch. B.3 - Prob. 20ECh. B.3 - Expand each expression in Problems 2126 using the...Ch. B.3 - Expand each expression in Problems 2126 using the...Ch. B.3 - Expand each expression in Problems 2126 using the...Ch. B.3 - Expand each expression in Problems 2126 using the...Ch. B.3 - Expand each expression in Problems 2126 using the...Ch. B.3 - Expand each expression in Problems 2126 using the...Ch. B.3 - Find the indicated term in each expansion in...Ch. B.3 - Prob. 28ECh. B.3 - Find the indicated term in each expansion in...Ch. B.3 - Find the indicated term in each expansion in...Ch. B.3 - Find the indicated term in each expansion in...Ch. B.3 - Prob. 32ECh. B.3 - Show that nC0 = nCn for n 0.Ch. B.3 - Prob. 34ECh. B.3 - The triangle shown here is called Pascals...Ch. B.3 - Explain why the sum of the entries in each row of...Ch. B.3 - Explain why the alternating sum of the entries in...Ch. B.3 - Prob. 38ECh. B.3 - Show that nCr1+nCr=n+1Cr for n r 1.
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