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advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

Show the factorization and solve. x1 - x2 + 3x3 + 2x4 = 15-x1 + 5x2 - 5x3 - 2x4 = -353x1 - 5x2 + 19x3
What is the power method for eigenvalues? What are its advantages and disadvantages?
Compute the matrix norm and the condition number corresponding to the l1-vector norm.
In Prob. 3 set B = A - 3I (as perhaps suggested by the diagonal entries) and see whether you may get a sequence of q’s converging to an eigenvalue of A that is smallest (not largest) in absolute
Apply the Gauss–Seidel iteration (3 steps) to the system in Prob. 5, starting from (a) 0, 0, 0(b) 10, 1,0 10.Compare and comment.Data from Prob. 5Do 5 steps, starting from x0 = [1
Fit a parabola (7) to the points (x, y). Check by sketching.The data in Prob. 3. Plot the points, the line, and the parabola jointly. Compare and comment.Data from Prob. 3Fit a straight line to the
Solve the following linear systems by Gauss elimination, with partial pivoting if necessary (but without scaling). Show the intermediate steps. Check the result by substitution. If no solution or
Show the factorization and solve.4x1 + 2x2 + 4x3 = 202x1 + 2x2 + 3x3 + 2x4 = 364x1 + 3x2 + 6x3 + 3x4 = 602x2
Use (4) to obtain an upper bound for the spectral radius: In Prob. 4 Data from Prob. 4 Find and sketch disks or intervals that contain the eigenvalues. If you have a CAS, find the spectrum and
Derive the formula for the normal equations of a cubic least squares parabola.
Let A, B be n × n and positive definite. Are -A, AT, A + B, A - B positive definite?
What is tridiagonalization and QR? When would you apply it?
Use (4) to obtain an upper bound for the spectral radius: In Prob. 1 Data from Prob. 1 Find and sketch disks or intervals that contain the eigenvalues. If you have a CAS, find the spectrum and
Compute the matrix norm and the condition number corresponding to the l1-vector norm.
(a) Write a program for Gauss–Seidel iteration. (b) Apply the program A(t)x = b, to starting from [0 0 0]T, where For t = 0.2, 0.5, 0.8, 0.9 determine the number of steps to obtain the
Fit curves (2) and (7) and a cubic parabola by least squares to (x, y) = (-2, -30), (-1, -4), (0, 4), (1, 4), (2, 22), (3, 68). Graph these curves and the points on common axes. Comment on the
Solve the following linear systems by Gauss elimination, with partial pivoting if necessary (but without scaling). Show the intermediate steps. Check the result by substitution. If no solution or
(a) Write a program for solving linear systems by Cholesky’s method and apply it to Example 2 in the text, to Probs. 7–9, and to systems of your choice. (b) Apply the factorization part of the
Use (4) to obtain an upper bound for the spectral radius:In Prob. 6Data from Prob. 6Find and sketch disks or intervals that contain the eigenvalues. If you have a CAS, find the spectrum and compare.
Find the inverse by the Gauss–Jordan method, showing the details.In Prob. 1Data from Prob. 1Show the factorization and solve by Doolittle’s method. 4x1 + 5x2 = 1412x1 + 14x2 =
Use (4) to obtain an upper bound for the spectral radius: In Prob. 3 Data from Prob. 3 Find and sketch disks or intervals that contain the eigenvalues. If you have a CAS, find the spectrum and
Compute the matrix norm and the condition number corresponding to the l1-vector norm.
Do 5 steps, starting from x0 = [1 1 1]. Compare with the Gauss–Seidel iteration. Which of the two seems to converge faster? Show the details of your work.The system in
Solve the following linear systems by Gauss elimination, with partial pivoting if necessary (but without scaling). Show the intermediate steps. Check the result by substitution. If no solution or
For the given data and for data of your choice find the interpolation polynomial and the least squares approximations (linear, quadratic, etc.). Compare and comment.(a) (-2, 0), (-1, 0), (0, 1), (1,
Verify that the matrix in Prob. 5 is normal. Data from Prob. 5 Find and sketch disks or intervals that contain the eigenvalues. If you have a CAS, find the spectrum and compare.
Verify (11) for x = [3 15 -4]T taken with the l∞-norm and the matrix in Prob. 13. Data from Prob. 13 Compute the matrix norm and the condition number corresponding to the l1-vector norm.
Write a program for the Gauss elimination with pivoting. Apply it to Probs. 13–16. Experiment with systems whose coefficient determinant is small in absolute value. Also investigate the performance
Do 5 steps, starting from x0 = [1 1 1]. Compare with the Gauss–Seidel iteration. Which of the two seems to converge faster? Show the details of your work.Show convergence
Compute the inverse of:
Find the inverse by the Gauss–Jordan method, showing the details.In Prob. 12Data from Prob. 12Show the factorization and solve.4x1 + 2x2 + 4x3 =
Compute the inverse of:
Solve Ax = b1, Ax = b2. Compare the solutions and comment. Compute the condition number of A.
Compute the norms (9), (10), (11) for the following (square) matrices. Comment on the reasons for greater or smaller differences among the three numbers.The matrix in Prob. 5Data from Prob. 5Do 5
Compute the inverse of:
Do 3 steps without scaling, starting from [1 1 1]T.4x1 - x2 = 22.0 4x2 - x3 = 13.4-x1
For Ax = b1 in Prob. 19 guess what the residual of x∼ = [-10.0 14.1]T, very poorly approximating [-2 4]T, might be. Then calculate and comment. Data from Prob. 19 Solve Ax = b1, Ax =
Do 3 steps without scaling, starting from [1 1 1]T.10x1 + x2 - x3 = 17 2x1 + 20x2 + x3 =
The 3 × 3 Hilbert matrix is The n × n Hilbert matrix is Hn = [hjk], where hjk = 1/(j + k - 1). (Similar matrices occur in curve fitting by least squares.) Compute the condition number κ(Hn) for
Make a list of the most important of the many ideas covered in this section and write a two page report on them.
Compute the l1-, l2-, and l∞-norms of the vectors.[8 -21 13 0]T
Maximize or minimize the given objective function f subject to the given constraints.Maximize f = 30x1 + 10x2 in the region in Prob. 5.Data from Prob. 5Describe and graph the regions in the first
Explain how the following can be regarded as a graph or a digraph: a family tree, air connections between given cities, trade relations between countries, a tennis tournament, and memberships of some
If you answer is yes, find S and T: (1) 3 (2) 4)
Guess how much less the probability in Prob. 10 would be if the sign consisted of 100 bulbs. Then calculate.Data from Prob. 10Let p = 2% be the probability that a certain type of light bulb will fail
Do steepest descent steps when:f(x) = 0.1x21 + x22 - 0.02x1, x0 = (3, 3), 5 steps
From memory: Make a list of the three types of alternatives, each with a typical example of your own.
What is a sample? A population? Why do we sample in statistics?
Find and graph the sample regression line of y on x and the given data as points on the same axes. Show the details of your work.(0, 1.0), (2, 2.1), (4, 2.9), (6, 3.6), (8, 5.2)
Let X be normal with mean 80 and variance 9. Find P(X > 83), P(X < 81), P(X < 80), and P(78 < X < 82).
Let X be normal with mean 14 and variance 4. Determine c such that P(X ≤ C) = 95%, P(X ≤ C) = 5%, P(X ≤ C) = 99.5%
Find the mean and variance of a discrete random variable X having the probability function f(0) = 1/4, f(1) =1/2, f(2) = 1/4.
Find the probability function of X = Number of times of tossing a fair coin until the first head appears.
Using Venn diagrams, graph and check De Morgan’s laws
Construct the simplest possible data withx̅ = 100 but qM = 0. What is the point of this problem?
Suppose that 3% of bolts made by a machine are defective, the defectives occurring at random during production. If the bolts are packaged 50 per box, what is the binomial approximation of the
Let X be discrete with probability function f(0) = f(3) = 1/8, f(1) = f(2) = 3/8. Find the expectation of X3.
Show that the random variables with the densitiesf(x, y) = x + yandg(x, y) = (x + 1/2)(y + 1/2)if 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 and f(x, y) = 0 and g((x, y) = 0 elsewhere, have the same marginal
Make up an example similar to Prob. 16, for instance, in terms of divisibility of numbers.Data from Prob. 16You may wonder whether in (16) the last relation follows from the others, but the answer is
Using a Venn diagram, show that if and only if A ⊆ B if and only if A ∪ B = B.
Calculate s for the data 4 1 3 10 2. Then reduce the data by deleting the outlier and calculate s. Comment.
Suppose a trial can result in precisely one of k mutually exclusive events A1, · · ·,Ak with probabilities p1, · · ·, pk, respectively, where P1 + · · · + Pk = 1. Suppose that n independent
Plot a histogram of the data 8, 2, 4, 10 and guess x̅ and s by inspecting the histogram. Then calculate x̅, s2, and s.
James rolls 2 fair dice, and Harry pays k cents to James, where k is the product of the two faces that show on the dice. How much should James pay to Harry for each game to make the game fair?
Let (X, Y) have the probability functionf(0, 0) = f(1, 1) = 1/8,f(0, 1) = f(1, 0) = 3/8,Are X and Y independent?
Show that, by the definition of complement, for any subset A of a sample space S.
Give a systematic discussion of the use of Tables A7 and A8 for obtaining P(X α), P(α b), P(X c) = k, as well as P(μ - c Table A7 Table A8
Let X be a random variable that can assume every real value. What are the complements of the events X ≤ b, X < b, X ≥ c, X ≥ c, b ≤ X ≤ c, b < X ≤ c?
In connection with a trip to Europe by some students, consider the events P that they see Paris, G that they have a good time, and M that they run out of money, and describe in words the events 1, ·
Find the mean and compare it with the median. Find the standard deviation and compare it with the interquartile range.For the release times in Prob. 7Data from Prob. 7Represent the data by a
Suppose that in the production of 60-ohm radio resistors, nondefective items are those that have a resistance between 58 and 62 ohms and the probability of a resistor’s being defective is 0.1%. The
Show that the mean always lies between the smallest and the largest data value.
A small filling station is supplied with gasoline every Saturday afternoon. Assume that its volume X of sales in ten thousands of gallons has the probability density f(x) = 6x(1 - x) if 0 ≤ x ≤ 1
Give an example of two different discrete distributions that have the same marginal distributions.
What is the probability that in a group of 20 people (that includes no twins) at least two have the same birthday, if we assume that the probability of having birthday on a given day is 1/365 for
What gives the greater probability of hitting at least once:(a) Hitting with probability and firing 1 shot(b) Hitting with probability and firing 2 shots(c) Hitting with probability and firing 4
Find the expectation of g(X) = X2, where X is uniformly distributed on the interval -1 ≤ x ≤ 1.
If the resistance X of certain wires in an electrical network is normal with mean 0.01 Ω and standard deviation 0.001 Ω, how many of 1000 wires will meet the specification that they have resistance
Suppose that in an automatic process of filling oil cans, the content of a can (in gallons) is Y = 100 + X, where X is a random variable with density f(x) = 1 - |x| when |x| ≤ 1 and 0 when |x| >
Find the mean and compare it with the median. Find the standard deviation and compare it with the interquartile range.For the medical data in Prob. 3Data from Prob. 3Represent the data by a
Suppose that a test for extrasensory perception consists of naming (in any order) 3 cards randomly drawn from a deck of 13 cards. Find the probability that by chance alone, the person will correctly
Find the mean, standard deviation, and variance in Prob. 11.Data from Prob. 11Make a stem-and-leaf plot, histogram, and boxplot of the data 110, 113, 109, 118, 110, 115, 104, 111, 116, 113.
What is the expected daily profit if a store sells X air conditioners per day with probability f(10) = 0.1, f(11) = 0.3, f(12) = 0.4, f(13) = 0.2 and the profit per conditioner is $55?
Find P(X > Y) when (X, Y) has the densityf(x, y) = 0.25e0.5(x+y) if x ≥ 0, y ≥ 0and 0 otherwise.
(a) Using (7), compute approximate values of for n! for n = 1, · · ·, 20.(b) Determine the relative error in (a). Find an empirical formula for that relative error.(c) An upper bound for that
A pressure control apparatus contains 3 electronic tubes. The apparatus will not work unless all tubes are operative. If the probability of failure of each tube during some interval of time is 0.04,
If sick-leave time X used by employees of a company in one month is (very roughly) normal with mean 1000 hours and standard deviation 100 hours, how much time t should be budgeted for sick leave
Find the probability that none of three bulbs in a traffic signal will have to be replaced during the first 1500 hours of operation if the lifetime X of a bulb is a random variable with the density
Find the mean and compare it with the median. Find the standard deviation and compare it with the interquartile range.For the data in Prob. 1Data from Prob. 1Represent the data by a stem-and-leaf
Make a stem-and-leaf plot, histogram, and boxplot of the data 110, 113, 109, 118, 110, 115, 104, 111, 116, 113.
For what choice of the maximum possible deviation from 1.00 cm shall we obtain 10% defectives in Probs. 9 and 10?
A 5-gear assembly is put together with spacers between the gears. The mean thickness of the gears is 5.020 cm with a standard deviation of 0.003 cm. The mean thickness of the spacers is 0.040 cm with
How many automobile registrations may the police have to check in a hit-and-run accident if a witness reports KDP7 and cannot remember the last two digits on the license plate but is certain that all
A batch of 200 iron rods consists of 50 over sized rods, 50 undersized rods, and 100 rods of the desired length. If two rods are drawn at random without replacement, what is the probability of
Find the mean and variance of the random variable X with probability function or density f(x).f(x) = Cex/2 (x = 0)
Find a shortest spanning tree by Prim’s algorithm. 10/ (5) 3 14 2 8 6 (4) 1 4 4 16 (2
Find the adjacency matrix of the given graph or digraph.
Why are backward edges not considered in the definition of the capacity of a cut set?
Apply the method suggested in Prob. 8 to the graph in Example 1. Do you get the same tree?Data from Prob. 8To get a minimum spanning tree, instead of adding shortest edges, one could think of
Define bipartite graphs and describe some typical applications of them.

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