Finite Mathematics for Business, Economics, Life Sciences and Social Sciences
14th Edition
ISBN: 9780134677972
Author: Barnett
Publisher: PEARSON
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Chapter 10.4, Problem 4E
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Q5*) Write down an immediate first integral for the Euler-Lagrange equation for the integral
I =
= F(x, y, y″) dx.
Hence write down a first integral of the Euler-Lagrange equation for the integral
I
1 = √(xy ² + x³y²)
dx.
Find the general solution of this ordinary differential equation, seeking first the complementary function and
then the particular integral. (Hint: the ODE is of homogeneous degree. And, for the particular integral, try
functions proportional to log x.)
You are provided with three 2D data points, p1, p2 and p3. Solving A C = B for C provides youwith the coefficients of a natural cubic spline curve that interpolates these points.Additionally, you have been given A and B, but some elements are missing. Moreover, the last two rowsof A are entirely absent. Your task is to determine and fill in the missing elements. For the last two rows,enforce a zero tangent at the beginning (in p1) and a not-a-knot boundary condition in p2. The matricesA and B are given as follows:Explain how to find the entries of A and B . How would you adapt these matrices if the data pointswere 3D? What if your spline should go through five data points? How many “extra rows” would there thenbe (with “extra” meaning “in addition to securing C2-continuity”)?
Q2*) In question P3 we showed that a minimal surface of revolution is given by revolution (about the x-axis)
of the catenary, with equation y = C cosh ((x – B)/C).
-
(a) Suppose, without loss of generality, that the catenary passes through the initial point
P = (x1,y1) = (0, 1). First deduce an expression for the one-parameter family of catenaries passing
through point P. Next calculate the value of x at which y takes its minimum value. By using the inequality
cosh > √2
(you might like to think about how to prove this), show that there are points Q for which it is impossible to find
a catenary passing through both P and Q. In particular, show that it is impossible to find a catenary joining
the points (0, 1) and (2, 1).
(b) A minimal surface of revolution can be realised experimentally by soap films attached to circular wire
frames (see this link and this link for examples). The physical reason for this is that the surface tension, which
is proportional to the area, is being minimised.…
Chapter 10 Solutions
Finite Mathematics for Business, Economics, Life Sciences and Social Sciences
Ch. 10.1 - (A) Using Figure 3, estimate the median annual...Ch. 10.1 - (A) Using Figure 6 estimate the revenue and costs...Ch. 10.1 - Repeat Example 1 for the following intervals: (A)...Ch. 10.1 - The weights (in pounds) were recorded for 20...Ch. 10.1 - (A) Construct a frequency table and histogram for...Ch. 10.1 - (A) Construct a frequency table and histogram for...Ch. 10.1 - Gross domestic product. Graph the data in the...Ch. 10.1 - Corporation revenues. Graph the data in the...Ch. 10.1 - Gold production. Use the double bar graph on world...Ch. 10.1 - Gasoline prices .Graph the data in the following...
Ch. 10.1 - Postal service. Graph the data in the following...Ch. 10.1 - Postal service. Refer to Problem 9. If the data...Ch. 10.1 - Federal income. Graph the data in the following...Ch. 10.1 - Gasoline prices. In April 2017 the average price...Ch. 10.1 - Starting salaries. The starting salaries (in...Ch. 10.1 - Commute times. Thirty-two people were chosen at...Ch. 10.1 - Common stocks. The following table shows...Ch. 10.1 - Mouse weights. One hundred healthy mice were...Ch. 10.1 - Population growth. Graph the data in the following...Ch. 10.1 - Aims epidemic. One way to gauge the toll of the...Ch. 10.1 - Nutrition. Graph the data in the following table...Ch. 10.1 - Greenhouse gases. The U.S. Environmental...Ch. 10.1 - Nutrition. Graph the nutritional information in...Ch. 10.1 - Nutrition. Refer to Problem 21. Suppose that you...Ch. 10.1 - Education. For statistical studies, U.S. states...Ch. 10.1 - Study abroad. Would a pie graph be more effective...Ch. 10.1 - Median age. Use the broken-line graph shown to...Ch. 10.1 - State prisoners. In 1980 in the United States, 6...Ch. 10.1 - Grade point Averages. One hundred seniors were...Ch. 10.2 - For many sets of measurements the median lies...Ch. 10.2 - Find the mean for the sample measurements...Ch. 10.2 - Compute the mean for the grouped sample data...Ch. 10.2 - Add the salary $100,000 to those in Example 3 and...Ch. 10.2 - Find the median for the grouped data in the...Ch. 10.2 - Compute the mode(s), median, and mean for each...Ch. 10.2 - In Problems 1-4, find the mean of the data set....Ch. 10.2 - In Problems 1-4, find the mean of the data set....Ch. 10.2 - In Problems 1-4, find the mean of the data set....Ch. 10.2 - In Problems 1-4, find the mean of the data set....Ch. 10.2 - In Problems 5-8, find the indicated sum. (If...Ch. 10.2 - In Problems 5-8, find the indicated sum. (If...Ch. 10.2 - In Problems 5-8, find the indicated sum. (If...Ch. 10.2 - In Problems 5-8, find the indicated sum. (If...Ch. 10.2 - Find the mean, median, and mode for the sets of...Ch. 10.2 - Find the mean, median, and mode for the sets of...Ch. 10.2 - Find the mean, median, and/or mode, whichever are...Ch. 10.2 - Find the mean, median, and/or mode, whichever are...Ch. 10.2 - Find the mean for the sets of grouped data in...Ch. 10.2 - Find the mean for the sets of grouped data in...Ch. 10.2 - Which single measure of central tendency-mean,...Ch. 10.2 - Which single measure of central tendency mean,...Ch. 10.2 - A data set is formed by recording the results of...Ch. 10.2 - A data set is formed by recording the sums on 200...Ch. 10.2 - (A) Construct a set of four numbers that has mean...Ch. 10.2 - (A) Construct a set of five numbers that has mean...Ch. 10.2 - Price earnings ratios. Find the mean, median, and...Ch. 10.2 - Gasoline tax. Find the mean, median, and mode for...Ch. 10.2 - Light bulb lifetime. Find the mean and median for...Ch. 10.2 - Price earnings ratios. Find the mean and median...Ch. 10.2 - Student loan debt. Find the mean median, and mode...Ch. 10.2 - Tourism Find the mean, median, and mode for the...Ch. 10.2 - Mouse weights. Find the mean and median for the...Ch. 10.2 - Blood cholesterol levels. Find the mean and median...Ch. 10.2 - Immigration Find the mean, median, and mode for...Ch. 10.2 - Grade point averages. Find the mean and median for...Ch. 10.2 - Entrance examination scores. Compute the median...Ch. 10.2 - Presidents. Find the mean and median for the...Ch. 10.3 - (A) When is the sample standard deviation of a set...Ch. 10.3 - Find the standard deviation for the sample...Ch. 10.3 - Find the standard deviation for the grouped sample...Ch. 10.3 - Prob. 1ECh. 10.3 - In Problems 1-8, find the indicated sum. (If...Ch. 10.3 - In Problems 1-8, find the indicated sum. (If...Ch. 10.3 - In Problems 1-8, find the indicated sum. (If...Ch. 10.3 - In Problems 1-8, find the indicated sum. (If...Ch. 10.3 - In Problems 1-8, find the indicated sum. (If...Ch. 10.3 - In Problems 1-8, find the indicated sum. (If...Ch. 10.3 - In Problems 1-8, find the indicated sum. (If...Ch. 10.3 - (A) Find set of the ungrouped mean and sample...Ch. 10.3 - (A) Find the mean and standard deviation of the...Ch. 10.3 - In Problems 11 and 12, find the standard deviation...Ch. 10.3 - In Problems 11 and 12, find the standard deviation...Ch. 10.3 - In Problems 13-18, discuss the validity of each...Ch. 10.3 - In Problems 13-18, discuss the validity of each...Ch. 10.3 - In Problems 13-18, discuss the validity of each...Ch. 10.3 - In Problems 13-18, discuss the validity of each...Ch. 10.3 - In Problems 13-18, discuss the validity of each...Ch. 10.3 - In Problems 13-18, discuss the validity of each...Ch. 10.3 - A data set is formed by recording the sums in 100...Ch. 10.3 - A data set is formed by recording the results of...Ch. 10.3 - Find the mean and standard deviation for each of...Ch. 10.3 - Find the mean and standard deviation for each of...Ch. 10.3 - Find the mean and standard deviation for each of...Ch. 10.3 - Find the mean and standard deviation for each of...Ch. 10.3 - Find the mean and standard deviation for each of...Ch. 10.3 - Find the mean and standard deviation for each of...Ch. 10.3 - Find the mean and standard deviation for each of...Ch. 10.3 - Find the mean and standard deviation for each of...Ch. 10.4 - Find p and q for a single roll of a fair die,...Ch. 10.4 - In Example 2, find the probability of the outcome...Ch. 10.4 - Using the same die experiment as in Example 3,...Ch. 10.4 - Use the binomial formula to expand q+p4.Ch. 10.4 - Repeat Example 5, where the binomial experiment...Ch. 10.4 - Compute the mean and standard deviation for the...Ch. 10.4 - Repeat Example 7 for four patients. The...Ch. 10.4 - Evaluate nCxpxqnx for the values of n,x, and p...Ch. 10.4 - Evaluate nCxpxqnx for the values of n,x, and p...Ch. 10.4 - Evaluate nCxpxqnx for the values of n,x, and p...Ch. 10.4 - Evaluate nCxpxqnx for the values of n,x, and p...Ch. 10.4 - Evaluate nCxpxqnx for the values of n,x, and p...Ch. 10.4 - Evaluate nCxpxqnx for the values of n,x, and p...Ch. 10.4 - In Problems 7-12, a fair coin is tossed four...Ch. 10.4 - In Problems 7-12, a fair coin is tossed four...Ch. 10.4 - In Problems 7-12, a fair coin is tossed four...Ch. 10.4 - In Problems 7-12, a fair coin is tossed four...Ch. 10.4 - In Problems 7-12, a fair coin is tossed four...Ch. 10.4 - In Problems 7-12, a fair coin is tossed four...Ch. 10.4 - In Problems 13-18 construct a histogram for the...Ch. 10.4 - In Problems 13-18 construct a histogram for the...Ch. 10.4 - In Problems 13-18 construct a histogram for the...Ch. 10.4 - In Problems 13-18 construct a histogram for the...Ch. 10.4 - In Problems 13-18 construct a histogram for the...Ch. 10.4 - In Problems 13-18 construct a histogram for the...Ch. 10.4 - In Problems 19-24, round answers to four decimal...Ch. 10.4 - In Problems 19-24, round answers to four decimal...Ch. 10.4 - In Problems 19-24, round answers to four decimal...Ch. 10.4 - In Problems 19-24, round answers to four decimal...Ch. 10.4 - In Problems 19-24, round answers to four decimal...Ch. 10.4 - In Problems 19-24, round answers to four decimal...Ch. 10.4 - If a baseball player has a batting average of 350...Ch. 10.4 - If a true-false test with 10 questions is given,...Ch. 10.4 - A multiple-choice test consists of 10 questions,...Ch. 10.4 - If 60 of the electorate supports the mayor, what...Ch. 10.4 - Construct a histogram for each of the binomial...Ch. 10.4 - Construct a histogram for each of the binomial...Ch. 10.4 - Construct a histogram for each of the binomial...Ch. 10.4 - Construct a histogram for each of the binomial...Ch. 10.4 - A random variable represents the number of...Ch. 10.4 - A random variable represents the number of...Ch. 10.4 - In Problems 35 and 36 a coin is loaded so that the...Ch. 10.4 - In Problems 35 and 36 a coin is loaded so that the...Ch. 10.4 - Find conditions on p that guarantee the histogram...Ch. 10.4 - Consider two binomial distributions for 1,000...Ch. 10.4 - A random variable represents the number of heads...Ch. 10.4 - A random variable represents the number of times a...Ch. 10.4 - Management training each year a company selects a...Ch. 10.4 - Employee turnover. If the probability of a new...Ch. 10.4 - Quality control A manufacturing process produces,...Ch. 10.4 - Guarantees. A manufacturing process produces, on...Ch. 10.4 - Quality control. A manufacturing process produces,...Ch. 10.4 - Management training. Each year a company selects 5...Ch. 10.4 - Medical diagnosis A tuberculosis patient is given...Ch. 10.4 - Harmful drug side effect. A pharmaceutical...Ch. 10.4 - Genetics. The probability that brown-eyed parents,...Ch. 10.4 - Gene mutation the probability of gene mutation...Ch. 10.4 - Epidemics. If the probability of a person...Ch. 10.4 - Drug side effect the probability that a given drug...Ch. 10.4 - Testing A multiple-choice test is given with 5...Ch. 10.4 - Opinion polls. An opinion poll based on a small...Ch. 10.4 - Testing. A multiple-choice test is given with 5...Ch. 10.4 - Sociology. The probability that a marriage will...Ch. 10.4 - Sociology. If the probability is .55 that a...Ch. 10.5 - What percentage of the light bulbs in Example 1...Ch. 10.5 - Refer to Example 1. What is the probability that a...Ch. 10.5 - In Example 3, Use the normal curve to approximate...Ch. 10.5 - Suppose in Example 4 that the manufacturing...Ch. 10.5 - In Problems 1-6, use Appendix C to find the area...Ch. 10.5 - In Problems 1-6, use Appendix C to find the area...Ch. 10.5 - In Problems 1-6, use Appendix C to find the area...Ch. 10.5 - In Problems 1-6, use Appendix C to find the area...Ch. 10.5 - In Problems 1-6, use Appendix C to find the area...Ch. 10.5 - In Problems 1-6, use Appendix C to find the area...Ch. 10.5 - In Problems 7-14, use Appendix C to find the area...Ch. 10.5 - In Problems 7-14, use Appendix C to find the area...Ch. 10.5 - In Problems 7-14, use Appendix C to find the area...Ch. 10.5 - Prob. 10ECh. 10.5 - In Problems 7-14, use Appendix C to find the area...Ch. 10.5 - In Problems 7-14, use Appendix C to find the area...Ch. 10.5 - In Problems 7-14, use Appendix C to find the area...Ch. 10.5 - In Problems 7-14, use Appendix C to find the area...Ch. 10.5 - In Problems 15-20, given a normal distribution...Ch. 10.5 - In Problems 15-20, given a normal distribution...Ch. 10.5 - In Problems 15-20, given a normal distribution...Ch. 10.5 - In Problems 15-20, given a normal distribution...Ch. 10.5 - In Problems 15-20, given a normal distribution...Ch. 10.5 - In Problems 15-20, given a normal distribution...Ch. 10.5 - In Problems 21.26, given a normal distribution...Ch. 10.5 - In Problems 21.26, given a normal distribution...Ch. 10.5 - In Problems 21.26, given a normal distribution...Ch. 10.5 - In Problems 21.26, given a normal distribution...Ch. 10.5 - In Problems 21.26, given a normal distribution...Ch. 10.5 - In Problems 21.26, given a normal distribution...Ch. 10.5 - In Problems 27-34, consider the normal...Ch. 10.5 - In Problems 27-34, consider the normal...Ch. 10.5 - Consider the normal distribution with mean 60 and...Ch. 10.5 - In Problems 27-34, consider the normal...Ch. 10.5 - In Problems 27-34, consider the normal...Ch. 10.5 - In Problems 27-34, consider the normal...Ch. 10.5 - In Problems 27-34, consider the normal...Ch. 10.5 - In Problems 27-34, consider the normal...Ch. 10.5 - In Problems 35-40, discuss the validity of each...Ch. 10.5 - In Problems 35-40, discuss the validity of each...Ch. 10.5 - In Problems 35-40, discuss the validity of each...Ch. 10.5 - In Problems 35-40, discuss the validity of each...Ch. 10.5 - In Problems 35-40, discuss the validity of each...Ch. 10.5 - In Problems 35-40, discuss the validity of each...Ch. 10.5 - In Problems 41-48, use the rule-of-thumb test to...Ch. 10.5 - In Problems 41-48, use the rule-of-thumb test to...Ch. 10.5 - In Problems 41-48, use the rule-of-thumb test to...Ch. 10.5 - In Problems 41-48, use the rule-of-thumb test to...Ch. 10.5 - In Problems 41-48, use the rule-of-thumb test to...Ch. 10.5 - In Problems 41-48, use the rule-of-thumb test to...Ch. 10.5 - In Problems 41-48, use the rule-of-thumb test to...Ch. 10.5 - In Problems 41-48, use the rule-of-thumb test to...Ch. 10.5 - The probability of success in a Bernoulli trial is...Ch. 10.5 - For a binomial distribution with n=100, explain...Ch. 10.5 - A binomial experiment consists of 500 trials. The...Ch. 10.5 - A binomial experiment consists of 500 trials. The...Ch. 10.5 - A binomial experiment consists of 500 trials. The...Ch. 10.5 - A binomial experiment consists of 500 trials. The...Ch. 10.5 - A binomial experiment consists of 500 trials. The...Ch. 10.5 - A binomial experiment consists of 500 trials. The...Ch. 10.5 - A binomial experiment consists of 500 trials. The...Ch. 10.5 - A binomial experiment consists of 500 trials. The...Ch. 10.5 - To graph Problems 59-62, use a graphing calculator...Ch. 10.5 - To graph Problems 59-62, use a graphing calculator...Ch. 10.5 - To graph Problems 59-62, use a graphing calculator...Ch. 10.5 - To graph Problems 59-62, use a graphing calculator...Ch. 10.5 - (A) If 120 scores are chosen from a normal...Ch. 10.5 - (A) If 250 scores are chosen from a normal...Ch. 10.5 - Sales Salespeople for a solar technology company...Ch. 10.5 - Guarantees. The average lifetime for a car battery...Ch. 10.5 - Quality control. A manufacturing process produces...Ch. 10.5 - Quality control. An automated manufacturing...Ch. 10.5 - Marketing claims. A company claims that 60 of the...Ch. 10.5 - Labor relation A union representative 60 claims of...Ch. 10.5 - Medicine. The average healing time of a certain...Ch. 10.5 - Agriculture. The average height of a hay crop is...Ch. 10.5 - Genetics. In a family with 2 children, the...Ch. 10.5 - Genetics. In Problem 73, what is the approximate...Ch. 10.5 - Testing. Scholastic Aptitude Tests (SATs) are...Ch. 10.5 - Politics. Candidate Harkins claims that she will...Ch. 10.5 - Grading on a curve. An instructor grades on a...Ch. 10.5 - Psychology. A test devised to measure...Ch. 10 - Use a bar graph and a broken-line graph to graph...Ch. 10 - Use a pie graph to graph the data on educational...Ch. 10 - (A) Draw a histogram for the binomial distribution...Ch. 10 - For the set of sample measurements...Ch. 10 - If a normal distribution has a mean of 100 and a...Ch. 10 - Given the sample of 25 quiz scores listed in the...Ch. 10 - For the set of grouped sample data given in the...Ch. 10 - (A) Construct a histogram for the binomial...Ch. 10 - What are the mean and standard deviation for a...Ch. 10 - In Problems 10 and 11, discuss the validity of...Ch. 10 - In Problems 10 and 11, discuss the validity of...Ch. 10 - If the probability of success in a single trial of...Ch. 10 - Given a normal distribution with mean 50 and...Ch. 10 - A data set is formed by recording the sums of 100...Ch. 10 - For the sample quiz scores in Problem 6, find the...Ch. 10 - A fair die is rolled five times. What is the...Ch. 10 - Two dice are rolled three times. What is the...Ch. 10 - Ten students take an exam worth 100 points. (A)...Ch. 10 - In the last presidential election, 39 of a city’s...Ch. 10 - A random variable represents the number of wins in...Ch. 10 - Retail sales. The daily number of bad checks...Ch. 10 - Preference survey. Find the mean, median, and/or...Ch. 10 - Plant safety. The weekly record of reported...Ch. 10 - Personnel screening.The scores on a screening test...Ch. 10 - Market research A newspaper publisher claims that...Ch. 10 - Health care. A small town has three doctors on...
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- Q3*) Consider the integral I Yn, Y₁, Y2, . . ., Y'n) dã, [F(x, Y 1, Y2, · · Yng) = - where y1, 2, ...y are dependent variables, dependent on x. If F is not explicitly dependent on x, deduce the equivalent of the Beltrami identity. Optional: Give an example of a function F(y1, Y2, Y₁, y2), and write down the Euler-Lagrange equations and Beltrami Identity for your example. Does having this Beltrami Identity help solve the problem?arrow_forwardWrite an integral that is approximated by the following Riemann sum. Substitute a into the Riemann sum below where a is the last non-zero digit of your banner ID. You do not need to evaluate the integral. 2000 (10 1 ((10-a) +0.001) (0.001)arrow_forwardSolve the following problem over the interval from x=0 to 1 using a step size of 0.25 where y(0)= 1. dy = dt (1+4t)√√y (a) Euler's method. (b) Heun's methodarrow_forward
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