MATHEMATICS WITH APPL....-ACCESS
12th Edition
ISBN: 9780135240687
Author: Lial
Publisher: PEARSON
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Chapter 10, Problem 9CE
To determine
The bounds for 68% of future annual returns for the Vanguard fund.
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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.)
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.
4. [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
ພ
Chapter 10 Solutions
MATHEMATICS WITH APPL....-ACCESS
Ch. 10.1 - Checkpoint 1 A restaurant trade group commissioned...Ch. 10.1 - Checkpoint 2
Make a histogram and a frequency...Ch. 10.1 - Checkpoint 3
Make a stem-and-leaf plot for the...Ch. 10.1 - Checkpoint 4
List the original data for the...Ch. 10.1 - Checkpoint 5
Characterize the shape of the...Ch. 10.1 - SP 500 Stocks The data for Exercises 1 -4 consist...Ch. 10.1 - SP 500 Stocks The data for Exercises 1 -4 consist...Ch. 10.1 - SP 500 Stocks The data for Exercises 1 -4 consist...Ch. 10.1 - SP 500 Stocks The data for Exercises 1 -4 consist...Ch. 10.1 - The data for Exercises 5-10 consist of random...
Ch. 10.1 - The data for Exercises 5-10 consist of random...Ch. 10.1 - The data for Exercises 5-10 consist of random...Ch. 10.1 - The data for Exercises 5-10 consist of random...Ch. 10.1 - The data for Exercises 5-10 consist of random...Ch. 10.1 - The data for Exercises 5-10 consist of random...Ch. 10.1 - For Exercises 11-20, construct a stem-and-leaf...Ch. 10.1 - For Exercises 11-20, construct a stem-and-leaf...Ch. 10.1 - For Exercises 11-20, construct a stem-and-leaf...Ch. 10.1 - Prob. 14ECh. 10.1 - For Exercises 11-20, construct a stem-and-leaf...Ch. 10.1 - For Exercises 11-20, construct a stem-and-leaf...Ch. 10.1 - Prob. 17ECh. 10.1 - Prob. 18ECh. 10.1 - For Exercises 11-20, construct a stem-and-leaf...Ch. 10.1 - For Exercises 11-20, construct a stem-and-leaf...Ch. 10.1 - Describe the shape of each of the given...Ch. 10.1 - Describe the shape of each of the given...Ch. 10.1 - Describe the shape of each of the given...Ch. 10.1 - Describe the shape of each of the given...Ch. 10.1 - Student Loan Defaults The following histogram...Ch. 10.1 - Stocks The following histogram shows the stock...Ch. 10.1 - Cat Ownership The stem-and-leaf plot below...Ch. 10.1 - Personal Bankruptcies The stem-and-leaf plot below...Ch. 10.1 - 29. Test Scores The grade distribution for scores...Ch. 10.1 - 30. Test Scores The grade distribution for scores...Ch. 10.2 - Checkpoint 1
Find the mean dollar amount of the...Ch. 10.2 - Checkpoint 2
Find for the following frequency...Ch. 10.2 - Checkpoint 3
Find the mean of the following...Ch. 10.2 - Checkpoint 4
Find the mean for the college tuition...Ch. 10.2 - Checkpoint 5
Find the median for the given heights...Ch. 10.2 - Checkpoint 6
Find the mode for each of the given...Ch. 10.2 - Checkpoint 7
Following is a list of the number of...Ch. 10.2 - Prob. 8CPCh. 10.2 - Prob. 1ECh. 10.2 - Prob. 2ECh. 10.2 - Prob. 3ECh. 10.2 - Prob. 4ECh. 10.2 - Prob. 5ECh. 10.2 - Prob. 6ECh. 10.2 - Prob. 7ECh. 10.2 - Prob. 8ECh. 10.2 - Prob. 9ECh. 10.2 - Prob. 10ECh. 10.2 - Prob. 11ECh. 10.2 - Prob. 12ECh. 10.2 - Prob. 13ECh. 10.2 - Prob. 14ECh. 10.2 - Find the mode or modes for each of the given lists...Ch. 10.2 - Find the mode or modes for each of the given lists...Ch. 10.2 - Find the mode or modes for each of the given lists...Ch. 10.2 - Find the mode or modes for each of the given lists...Ch. 10.2 - Find the mode or modes for each of the given lists...Ch. 10.2 - 20. When is the median the most appropriate...Ch. 10.2 - 21. When would the mode be an appropriate measure...Ch. 10.2 - Pet Ownership For Exercises 22 and 23, the...Ch. 10.2 - Pet Ownership For Exercises 22 and 23, the...Ch. 10.2 - 24. To predict the outcome of the next...Ch. 10.2 - Work each problem. (See Example 6.) MLB Payrolls...Ch. 10.2 - Work each problem. (See Example 6.) NFL Team...Ch. 10.2 - Work each problem. (See Example 6.)
27. Business...Ch. 10.2 - Work each problem. (See Example 6.) Sirius XM...Ch. 10.2 - Work each problem. (See Example 6.) Dr Pepper and...Ch. 10.2 - Natural Science The table gives the average...Ch. 10.2 - Natural Science The table gives the average...Ch. 10.2 - For Exercises 32-33 determine the shape of the...Ch. 10.2 - For Exercises 32-33 determine the shape of the...Ch. 10.2 - Prob. 34ECh. 10.2 - Prob. 35ECh. 10.2 - For Exercises 36-38, calculate the weighted...Ch. 10.2 - For Exercises 36-38, calculate the weighted...Ch. 10.2 - For Exercises 36-38, calculate the weighted...Ch. 10.2 - Prob. 39ECh. 10.2 - For Exercises 36-38, calculate the weighted...Ch. 10.3 - Checkpoint 1
Find the range for this sample of the...Ch. 10.3 - Checkpoint 2
Find the deviations from the mean for...Ch. 10.3 - Checkpoint 3
Find the standard deviation for a...Ch. 10.3 - Prob. 4CPCh. 10.3 - Prob. 1ECh. 10.3 - Prob. 2ECh. 10.3 - Prob. 3ECh. 10.3 - Prob. 4ECh. 10.3 - Prob. 5ECh. 10.3 - Prob. 6ECh. 10.3 - Prob. 7ECh. 10.3 - Finance Use the following table for Exercises...Ch. 10.3 - Finance Use the following table for Exercises...Ch. 10.3 - Prob. 10ECh. 10.3 - Education Find the standard deviation for the...Ch. 10.3 - Prob. 12ECh. 10.3 - Prob. 13ECh. 10.3 - Prob. 14ECh. 10.3 - Prob. 15ECh. 10.3 - Prob. 16ECh. 10.3 - Prob. 17ECh. 10.3 - Prob. 18ECh. 10.3 - Prob. 19ECh. 10.3 - Prob. 20ECh. 10.3 - Prob. 21ECh. 10.3 - Prob. 22ECh. 10.3 - Prob. 23ECh. 10.3 - Movie Studios' Revenue For Exercises 23-28, use...Ch. 10.3 - Prob. 25ECh. 10.3 - Prob. 26ECh. 10.3 - Prob. 27ECh. 10.3 - Prob. 28ECh. 10.3 - Prob. 29ECh. 10.3 - Prob. 30ECh. 10.3 - Prob. 34ECh. 10.3 - Prob. 31ECh. 10.3 - Prob. 32ECh. 10.3 - Prob. 33ECh. 10.3 - IBM and Microsoft Revenue The following table...Ch. 10.3 - Prob. 36ECh. 10.3 - Prob. 37ECh. 10.3 - Prob. 38ECh. 10.3 - IBM and Microsoft Revenue The following table...Ch. 10.3 - Prob. 40ECh. 10.3 - Prob. 41ECh. 10.3 - Prob. 42ECh. 10.3 - Prob. 43ECh. 10.3 - Prob. 44ECh. 10.3 - Prob. 45ECh. 10.3 - Prob. 46ECh. 10.3 - Prob. 47ECh. 10.3 - Prob. 48ECh. 10.4 - Prob. 1CPCh. 10.4 - Prob. 2CPCh. 10.4 - Prob. 3CPCh. 10.4 - Prob. 4CPCh. 10.4 - Prob. 5CPCh. 10.4 - Prob. 6CPCh. 10.4 - Prob. 7CPCh. 10.4 - 1. The peak in a normal curve occurs directly...Ch. 10.4 - Prob. 2ECh. 10.4 - Prob. 3ECh. 10.4 - Prob. 4ECh. 10.4 - Prob. 5ECh. 10.4 - Prob. 6ECh. 10.4 - Prob. 7ECh. 10.4 - Prob. 8ECh. 10.4 - Find the percentage of the total area under the...Ch. 10.4 - Prob. 10ECh. 10.4 - Prob. 11ECh. 10.4 - Prob. 12ECh. 10.4 - Prob. 13ECh. 10.4 - Prob. 14ECh. 10.4 - Prob. 15ECh. 10.4 - Prob. 16ECh. 10.4 - Prob. 17ECh. 10.4 - Prob. 18ECh. 10.4 - Prob. 19ECh. 10.4 - 20. Using Chebyshev’s theorem and the normal...Ch. 10.4 - Prob. 21ECh. 10.4 - Prob. 22ECh. 10.4 - Prob. 23ECh. 10.4 - Prob. 24ECh. 10.4 - Prob. 25ECh. 10.4 - Prob. 26ECh. 10.4 - Prob. 27ECh. 10.4 - Prob. 28ECh. 10.4 - Prob. 29ECh. 10.4 - Prob. 30ECh. 10.4 - Prob. 31ECh. 10.4 - Prob. 32ECh. 10.4 - Prob. 33ECh. 10.4 - Prob. 34ECh. 10.4 - Prob. 35ECh. 10.4 - Prob. 36ECh. 10.4 - Prob. 37ECh. 10.4 - Prob. 38ECh. 10.4 - Prob. 39ECh. 10.4 - Prob. 40ECh. 10.4 - Prob. 41ECh. 10.4 - Prob. 42ECh. 10.4 - Prob. 43ECh. 10.4 - Prob. 44ECh. 10.4 - Prob. 45ECh. 10.4 - Prob. 46ECh. 10.4 - Prob. 47ECh. 10.4 - Prob. 48ECh. 10.4 - Prob. 49ECh. 10.4 - Education The mean performance score of a large...Ch. 10.4 - Prob. 51ECh. 10.4 - Prob. 52ECh. 10.4 - Job Satisfaction According to a 2016 study...Ch. 10.4 - Prob. 54ECh. 10.4 - Prob. 55ECh. 10.4 - Prob. 56ECh. 10.4 - Prob. 57ECh. 10.4 - Prob. 58ECh. 10.4 - Student Loan Debt According to a report from the...Ch. 10.4 - Prob. 60ECh. 10.4 - Prob. 61ECh. 10.4 - Prob. 62ECh. 10 - NASDAQ 100 Stocks For Exercises 1-10, the data...Ch. 10 - Prob. 2RECh. 10 - Prob. 3RECh. 10 - Prob. 4RECh. 10 - Prob. 5RECh. 10 - Prob. 6RECh. 10 - NASDAQ 100 Stocks For Exercises 1-10, the data...Ch. 10 - Prob. 8RECh. 10 - Prob. 9RECh. 10 - Prob. 10RECh. 10 - Prob. 11RECh. 10 - Prob. 12RECh. 10 - Prob. 13RECh. 10 - Prob. 14RECh. 10 - Prob. 15RECh. 10 - Prob. 16RECh. 10 - Prob. 17RECh. 10 - Prob. 18RECh. 10 - Prob. 19RECh. 10 - Prob. 20RECh. 10 - Prob. 21RECh. 10 - Prob. 22RECh. 10 - Prob. 23RECh. 10 - Prob. 24RECh. 10 - Find the standard deviation for each of the given...Ch. 10 - Prob. 26RECh. 10 - Prob. 27RECh. 10 - Prob. 28RECh. 10 - Prob. 29RECh. 10 - Prob. 30RECh. 10 - Prob. 31RECh. 10 - Prob. 32RECh. 10 - Prob. 33RECh. 10 - Prob. 34RECh. 10 - Prob. 35RECh. 10 - Prob. 36RECh. 10 - Arm Circumference Data from a recent National...Ch. 10 - Prob. 38RECh. 10 - Prob. 39RECh. 10 - Prob. 40RECh. 10 - Prob. 41RECh. 10 - Prob. 42RECh. 10 - Prob. 43RECh. 10 - Prob. 44RECh. 10 - Prob. 45RECh. 10 - Prob. 46RECh. 10 - Prob. 47RECh. 10 - College or University Education A recent...Ch. 10 - College or University Education A recent...Ch. 10 - Prob. 50RECh. 10 - Prob. 51RECh. 10 - Prob. 52RECh. 10 - Prob. 53RECh. 10 - Prob. 54RECh. 10 - Prob. 1CECh. 10 - Prob. 2CECh. 10 - Prob. 3CECh. 10 - Prob. 4CECh. 10 - Prob. 5CECh. 10 - Prob. 6CECh. 10 - Prob. 7CECh. 10 - Prob. 8CECh. 10 - Prob. 9CECh. 10 - Prob. 10CE
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- 5. [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_forwardQ/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_forward
- Refer 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_forwardRefer to page 140 for problems on infinite sets. Instructions: • Compare the cardinalities of given sets and classify them as finite, countable, or uncountable. • Prove or disprove the equivalence of two sets using bijections. • Discuss the implications of Cantor's theorem on real-world computation. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing]arrow_forwardRefer to page 120 for problems on numerical computation. Instructions: • Analyze the sources of error in a given numerical method (e.g., round-off, truncation). • Compute the error bounds for approximating the solution of an equation. • Discuss strategies to minimize error in iterative methods like Newton-Raphson. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440 AZF/view?usp=sharing]arrow_forward
- Refer to page 145 for problems on constrained optimization. Instructions: • Solve an optimization problem with constraints using the method of Lagrange multipliers. • • Interpret the significance of the Lagrange multipliers in the given context. Discuss the applications of this method in machine learning or operations research. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440 AZF/view?usp=sharing]arrow_forwardOnly 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_forward
- 3. [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_forwardLet T be a tree. Prove that if T has a vertex of degree k, then T has at least k leaves.arrow_forwardHomework Let X1, X2, Xn be a random sample from f(x;0) where f(x; 0) = (-), 0 < x < ∞,0 € R Using Basu's theorem, show that Y = min{X} and Z =Σ(XY) are indep. -arrow_forward
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