Basic Technical Mathematics
11th Edition
ISBN: 9780134437705
Author: Washington
Publisher: PEARSON
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Chapter 19.2, Problem 43E
To determine
The percent of signal that remains after 100 km.
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7. [10 marks]
Let G
=
(V,E) be a 3-connected graph. We prove that for every x, y, z Є V, there is a
cycle in G on which x, y, and z all lie.
(a) First prove that there are two internally disjoint xy-paths Po and P₁.
(b) If z is on either Po or P₁, then combining Po and P₁ produces a cycle on which
x, y, and z all lie. So assume that z is not on Po and not on P₁. Now prove that
there are three paths Qo, Q1, and Q2 such that:
⚫each Qi starts at z;
• each Qi ends at a vertex w; that is on Po or on P₁, where wo, w₁, and w₂ are
distinct;
the paths Qo, Q1, Q2 are disjoint from each other (except at the start vertex
2) and are disjoint from the paths Po and P₁ (except at the end vertices wo,
W1, and w₂).
(c) Use paths Po, P₁, Qo, Q1, and Q2 to prove that there is a cycle on which x, y, and
z all lie. (To do this, notice that two of the w; must be on the same Pj.)
Chapter 19 Solutions
Basic Technical Mathematics
Ch. 19.1 - Find the 20th term of the arithmetic sequence 2,...Ch. 19.1 - Prob. 2PECh. 19.1 - Prob. 3PECh. 19.1 - Prob. 1ECh. 19.1 - Prob. 2ECh. 19.1 - Prob. 3ECh. 19.1 - Prob. 4ECh. 19.1 - In Exercises 3–6, write the first five terms of...Ch. 19.1 - Prob. 6ECh. 19.1 - Prob. 7E
Ch. 19.1 - Prob. 8ECh. 19.1 - Prob. 9ECh. 19.1 - In Exercises 7–14, find the nth term of the...Ch. 19.1 - Prob. 11ECh. 19.1 - Prob. 12ECh. 19.1 - Prob. 13ECh. 19.1 - Prob. 14ECh. 19.1 - In Exercises 15–18, find the sum of the n terms of...Ch. 19.1 - Prob. 16ECh. 19.1 - Prob. 17ECh. 19.1 - Prob. 18ECh. 19.1 - Prob. 19ECh. 19.1 - Prob. 20ECh. 19.1 - Prob. 21ECh. 19.1 - Prob. 22ECh. 19.1 - Prob. 23ECh. 19.1 - Prob. 24ECh. 19.1 - Prob. 25ECh. 19.1 - Prob. 26ECh. 19.1 - In Exercises 27–56, find the indicated quantities...Ch. 19.1 - Prob. 28ECh. 19.1 - Prob. 29ECh. 19.1 - Prob. 30ECh. 19.1 - In Exercises 27–56, find the indicated quantities...Ch. 19.1 - Prob. 32ECh. 19.1 - Prob. 33ECh. 19.1 - Prob. 34ECh. 19.1 - Prob. 35ECh. 19.1 - Prob. 36ECh. 19.1 - Prob. 37ECh. 19.1 - Prob. 38ECh. 19.1 - Prob. 39ECh. 19.1 - Prob. 40ECh. 19.1 - Prob. 41ECh. 19.1 - In Exercises 27–56, find the indicated quantities...Ch. 19.1 - Prob. 43ECh. 19.1 - In Exercises 27–56, find the indicated quantities...Ch. 19.1 - Prob. 45ECh. 19.1 - Prob. 46ECh. 19.1 - Prob. 47ECh. 19.1 - Prob. 48ECh. 19.1 - In Exercises 27–56, find the indicated quantities...Ch. 19.1 - Prob. 50ECh. 19.1 - Prob. 51ECh. 19.1 -
In Exercises 27–56, find the indicated quantities...Ch. 19.1 - Prob. 53ECh. 19.1 - Prob. 54ECh. 19.1 - Prob. 55ECh. 19.1 - Prob. 56ECh. 19.2 -
Find the sixth term of the geometric sequence 8,...Ch. 19.2 - Prob. 2PECh. 19.2 - Prob. 3PECh. 19.2 - Prob. 1ECh. 19.2 - Prob. 2ECh. 19.2 - Prob. 3ECh. 19.2 - Prob. 4ECh. 19.2 - Prob. 5ECh. 19.2 - Prob. 6ECh. 19.2 - Prob. 7ECh. 19.2 - Prob. 8ECh. 19.2 - Prob. 9ECh. 19.2 - Prob. 10ECh. 19.2 - Prob. 11ECh. 19.2 - Prob. 12ECh. 19.2 - Prob. 13ECh. 19.2 - Prob. 14ECh. 19.2 - In Exercises 15–20, find the sum of the first n...Ch. 19.2 - Prob. 16ECh. 19.2 - Prob. 17ECh. 19.2 - Prob. 18ECh. 19.2 - Prob. 19ECh. 19.2 - Prob. 20ECh. 19.2 - Prob. 21ECh. 19.2 - Prob. 22ECh. 19.2 -
In Exercises 21–28, find any of the values of a1,...Ch. 19.2 - Prob. 24ECh. 19.2 -
In Exercises 21–28, find any of the values of a1,...Ch. 19.2 - Prob. 26ECh. 19.2 - Prob. 27ECh. 19.2 - Prob. 28ECh. 19.2 - Prob. 29ECh. 19.2 - Prob. 30ECh. 19.2 - Prob. 31ECh. 19.2 - Prob. 32ECh. 19.2 - Prob. 33ECh. 19.2 - Prob. 34ECh. 19.2 - Prob. 35ECh. 19.2 - Prob. 36ECh. 19.2 - Prob. 37ECh. 19.2 - Prob. 38ECh. 19.2 - Prob. 39ECh. 19.2 - Prob. 40ECh. 19.2 - Prob. 41ECh. 19.2 - Prob. 42ECh. 19.2 - Prob. 43ECh. 19.2 - Prob. 44ECh. 19.2 - Prob. 45ECh. 19.2 - Prob. 46ECh. 19.2 - Prob. 47ECh. 19.2 - Prob. 48ECh. 19.2 - Prob. 49ECh. 19.2 - Prob. 50ECh. 19.2 -
In Exercises 29–56, find the indicated...Ch. 19.2 - Prob. 52ECh. 19.2 -
In Exercises 29–56, find the indicated...Ch. 19.2 - Prob. 54ECh. 19.2 -
In Exercises 29–56, find the indicated...Ch. 19.2 - Prob. 56ECh. 19.3 - Prob. 1PECh. 19.3 - Prob. 2PECh. 19.3 - Prob. 3PECh. 19.3 - Prob. 1ECh. 19.3 - Prob. 2ECh. 19.3 - Prob. 3ECh. 19.3 - Prob. 4ECh. 19.3 - Prob. 5ECh. 19.3 - Prob. 6ECh. 19.3 - Prob. 7ECh. 19.3 - Prob. 8ECh. 19.3 - Prob. 9ECh. 19.3 - Prob. 10ECh. 19.3 - Prob. 11ECh. 19.3 - Prob. 12ECh. 19.3 - Prob. 13ECh. 19.3 - Prob. 14ECh. 19.3 - Prob. 15ECh. 19.3 - Prob. 16ECh. 19.3 - Prob. 17ECh. 19.3 - Prob. 18ECh. 19.3 - In Exercises 15–24, find the fractions equal to...Ch. 19.3 - In Exercises 15–24, find the fractions equal to...Ch. 19.3 - Prob. 21ECh. 19.3 - Prob. 22ECh. 19.3 - Prob. 23ECh. 19.3 - Prob. 24ECh. 19.3 - Prob. 25ECh. 19.3 - Prob. 26ECh. 19.3 - Prob. 27ECh. 19.3 - In Exercises 25–36, solve the given problems by...Ch. 19.3 - Prob. 29ECh. 19.3 - Prob. 30ECh. 19.3 - Prob. 31ECh. 19.3 - Prob. 32ECh. 19.3 - Prob. 33ECh. 19.3 - Prob. 34ECh. 19.3 - Prob. 35ECh. 19.3 - Prob. 36ECh. 19.4 - Prob. 1PECh. 19.4 - Prob. 2PECh. 19.4 - Prob. 3PECh. 19.4 - Prob. 4PECh. 19.4 - Prob. 1ECh. 19.4 - Prob. 2ECh. 19.4 - Prob. 3ECh. 19.4 - Prob. 4ECh. 19.4 - Prob. 5ECh. 19.4 - Prob. 6ECh. 19.4 - Prob. 7ECh. 19.4 - Prob. 8ECh. 19.4 - Prob. 9ECh. 19.4 - Prob. 10ECh. 19.4 - Prob. 11ECh. 19.4 - Prob. 12ECh. 19.4 - Prob. 13ECh. 19.4 - Prob. 14ECh. 19.4 - Prob. 15ECh. 19.4 - Prob. 16ECh. 19.4 - Prob. 17ECh. 19.4 - Prob. 18ECh. 19.4 - Prob. 19ECh. 19.4 - Prob. 20ECh. 19.4 - Prob. 21ECh. 19.4 - Prob. 22ECh. 19.4 - Prob. 23ECh. 19.4 - Prob. 24ECh. 19.4 - Prob. 25ECh. 19.4 - Prob. 26ECh. 19.4 - Prob. 27ECh. 19.4 - Prob. 28ECh. 19.4 - Prob. 29ECh. 19.4 - Prob. 30ECh. 19.4 - Prob. 31ECh. 19.4 - Prob. 32ECh. 19.4 - Prob. 33ECh. 19.4 - Prob. 34ECh. 19.4 - Prob. 35ECh. 19.4 - Prob. 36ECh. 19.4 - Prob. 37ECh. 19.4 - Prob. 38ECh. 19.4 - Prob. 39ECh. 19.4 - Prob. 40ECh. 19.4 - Prob. 41ECh. 19.4 - Prob. 42ECh. 19.4 - Prob. 43ECh. 19.4 - Prob. 44ECh. 19.4 - Prob. 45ECh. 19.4 - Prob. 46ECh. 19.4 - Prob. 47ECh. 19.4 - Prob. 48ECh. 19.4 - Prob. 49ECh. 19.4 - Prob. 50ECh. 19.4 - Prob. 51ECh. 19.4 - Prob. 52ECh. 19.4 - Prob. 53ECh. 19.4 - Prob. 54ECh. 19.4 - Prob. 55ECh. 19.4 - Prob. 56ECh. 19.4 - In Exercises 45–58, solve the given problems.
57....Ch. 19.4 - Prob. 58ECh. 19 - Prob. 1RECh. 19 - Prob. 2RECh. 19 - Prob. 3RECh. 19 - Prob. 4RECh. 19 - Prob. 5RECh. 19 - Prob. 6RECh. 19 - Prob. 7RECh. 19 - Prob. 8RECh. 19 - Prob. 9RECh. 19 - Prob. 10RECh. 19 - Prob. 11RECh. 19 - Prob. 12RECh. 19 - Prob. 13RECh. 19 - Prob. 14RECh. 19 - Prob. 15RECh. 19 - Prob. 16RECh. 19 - Prob. 17RECh. 19 - Prob. 18RECh. 19 - Prob. 19RECh. 19 - Prob. 20RECh. 19 - Prob. 21RECh. 19 - Prob. 22RECh. 19 - Prob. 23RECh. 19 - Prob. 24RECh. 19 - Prob. 25RECh. 19 - Prob. 26RECh. 19 - Prob. 27RECh. 19 - Prob. 28RECh. 19 - In Exercises 27–30, find the sums of the given...Ch. 19 - Prob. 30RECh. 19 - Prob. 31RECh. 19 - Prob. 32RECh. 19 - In Exercises 31–34, find the fractions equal to...Ch. 19 - Prob. 34RECh. 19 - Prob. 35RECh. 19 - Prob. 36RECh. 19 - Prob. 37RECh. 19 - Prob. 38RECh. 19 - Prob. 39RECh. 19 - Prob. 40RECh. 19 - Prob. 41RECh. 19 - Prob. 42RECh. 19 - Prob. 43RECh. 19 - Prob. 44RECh. 19 - Prob. 45RECh. 19 - Prob. 46RECh. 19 - Prob. 47RECh. 19 - Prob. 48RECh. 19 - Prob. 49RECh. 19 - Prob. 50RECh. 19 - Prob. 51RECh. 19 - Prob. 52RECh. 19 - Prob. 53RECh. 19 - Prob. 54RECh. 19 - Prob. 55RECh. 19 - Prob. 56RECh. 19 - Prob. 57RECh. 19 - Prob. 58RECh. 19 - Prob. 59RECh. 19 - Prob. 60RECh. 19 - Prob. 61RECh. 19 - Prob. 62RECh. 19 - Prob. 63RECh. 19 - Prob. 64RECh. 19 - Prob. 65RECh. 19 - Prob. 66RECh. 19 - Prob. 67RECh. 19 - Prob. 68RECh. 19 - In Exercises 51–98, solve the given problems by...Ch. 19 - Prob. 70RECh. 19 - Prob. 71RECh. 19 - Prob. 72RECh. 19 - Prob. 73RECh. 19 - Prob. 74RECh. 19 - Prob. 75RECh. 19 - Prob. 76RECh. 19 - Prob. 77RECh. 19 - Prob. 78RECh. 19 - Prob. 79RECh. 19 - Prob. 80RECh. 19 - In Exercises 51–98, solve the given problems by...Ch. 19 - Prob. 82RECh. 19 - Prob. 83RECh. 19 - Prob. 84RECh. 19 - Prob. 85RECh. 19 - Prob. 86RECh. 19 - Prob. 87RECh. 19 - Prob. 88RECh. 19 - In Exercises 51–98, solve the given problems by...Ch. 19 - Prob. 90RECh. 19 - Prob. 91RECh. 19 - Prob. 92RECh. 19 - Prob. 93RECh. 19 - Prob. 94RECh. 19 - Prob. 95RECh. 19 - Prob. 96RECh. 19 - Prob. 97RECh. 19 - Prob. 98RECh. 19 - Prob. 99RECh. 19 - Prob. 1PTCh. 19 - Prob. 2PTCh. 19 - Prob. 3PTCh. 19 - Prob. 4PTCh. 19 - Prob. 5PTCh. 19 - Prob. 6PTCh. 19 - Prob. 7PTCh. 19 - Prob. 8PTCh. 19 - Prob. 9PT
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- 6. [10 marks] Let T be a tree with n ≥ 2 vertices and leaves. Let BL(T) denote the block graph of T. (a) How many vertices does BL(T) have? (b) How many edges does BL(T) have? Prove that your answers are correct.arrow_forward4. [10 marks] Find both a matching of maximum size and a vertex cover of minimum size in the following bipartite graph. Prove that your answer is correct. ย ພarrow_forward5. [10 marks] Let G = (V,E) be a graph, and let X C V be a set of vertices. Prove that if |S||N(S)\X for every SCX, then G contains a matching M that matches every vertex of X (i.e., such that every x X is an end of an edge in M).arrow_forward
- Q/show that 2" +4 has a removable discontinuity at Z=2i Z(≥2-21)arrow_forwardRefer to page 100 for problems on graph theory and linear algebra. Instructions: • Analyze the adjacency matrix of a given graph to find its eigenvalues and eigenvectors. • Interpret the eigenvalues in the context of graph properties like connectivity or clustering. Discuss applications of spectral graph theory in network analysis. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440 AZF/view?usp=sharing]arrow_forwardRefer to page 110 for problems on optimization. Instructions: Given a loss function, analyze its critical points to identify minima and maxima. • Discuss the role of gradient descent in finding the optimal solution. . Compare convex and non-convex functions and their implications for optimization. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440 AZF/view?usp=sharing]arrow_forward
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- Only 100% sure experts solve it correct complete solutions okarrow_forwardGive an example of a graph with at least 3 vertices that has exactly 2 automorphisms(one of which is necessarily the identity automorphism). Prove that your example iscorrect.arrow_forward3. [10 marks] Let Go (Vo, Eo) and G₁ = (V1, E1) be two graphs that ⚫ have at least 2 vertices each, ⚫are disjoint (i.e., Von V₁ = 0), ⚫ and are both Eulerian. Consider connecting Go and G₁ by adding a set of new edges F, where each new edge has one end in Vo and the other end in V₁. (a) Is it possible to add a set of edges F of the form (x, y) with x € Vo and y = V₁ so that the resulting graph (VUV₁, Eo UE₁ UF) is Eulerian? (b) If so, what is the size of the smallest possible F? Prove that your answers are correct.arrow_forward
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