Mathematics For Machine Technology
8th Edition
ISBN: 9781337798310
Author: Peterson, John.
Publisher: Cengage Learning,
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Textbook Question
Chapter 12, Problem 9A
Multiply the numbers in Exercises 7 through 9. Where necessary, round the answers to 4 decimal places.
9.a.
b.
c.
d.
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(b) An otherwise fair six-sided die has been tampered with in an attempt to
cheat at a dice game. The effect is that the 1 and 6 faces have a
different probability of occurring than the 2, 3, 4 and 5 faces.
Let θ be the probability of obtaining a 1 on this biased die. Then the
outcomes of rolling the biased die have the following probability mass
function.
Table 1 The p.m.f. of outcomes of rolls of a biased die
Outcome 1 2 3 4 5 6
Probability θ
1
4
(1 − 2θ)
1
4
(1 − 2θ)
1
4
(1 − 2θ)
1
4
(1 − 2θ) θ
(i) By consideration of the p.m.f. in Table 1, explain why it is
necessary for θ to be such that 0 < θ < 1/2. [2]
(ii) The value of θ is unknown. Data from which to estimate the value
of θ were obtained by rolling the biased die 1000 times. The result
of this experiment is shown in Table 2.
Table 2 Outcomes of 1000 independent rolls of a biased die
Outcome 1 2 3 4 5 6
Frequency 205 154 141 165 145 190
Show that the likelihood of θ based on these data is
L(θ) = C θ395 (1 − 2θ)
605
,…
Not use ai please
O State Jordan's lemma.
Calculate the following integrals using complex variable techniques.
2π
do
(i)
1+8 cos²0'
I COS x dx
x²-2x+5°
Leave your answers in terms of real functions of real variables.
Chapter 12 Solutions
Mathematics For Machine Technology
Ch. 12 - Round 0.42538 to 2 decimal places.Ch. 12 - Express the decimal fraction 0.056 as a common...Ch. 12 - Add 0.032+0.23+0.0032 . Use Figure 122 to answer...Ch. 12 - Determine the length of A.Ch. 12 - Determine the length of B.Ch. 12 - Determine the length of C.Ch. 12 - Multiply the numbers in Exercises 7 through 9....Ch. 12 - Multiply the numbers in Exercises 7 through 9....Ch. 12 - Multiply the numbers in Exercises 7 through 9....Ch. 12 - A section of a spur gear is shown in Figure 123....
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- Refer to page 138 of the shared file for the numerical methods problem. Use Newton's method to approximate the solution of the given nonlinear system of equations. Start with an initial guess provided in the problem and iterate for at least two steps. Link: [https://drive.google.com/file/d/1RQ2OZk-LSxpRyejKEMg1t2q15dbpVLCS/view? usp=sharing] Show the Jacobian matrix, iteration formulas, and each step of the computation.arrow_forwardRefer to page 128 for the heat equation problem. Solve the one-dimensional heat equation with the given initial and boundary conditions. Link: [https://drive.google.com/file/d/1RQ2OZK-LSxpRyejKEMg1t2q15dbpVLCS/view? usp=sharing] Use Fourier series or other appropriate methods.arrow_forwardGo to page 137 for the real analysis problem. Determine whether the given infinite series converges or diverges using appropriate convergence tests, such as the ratio test, root test, or comparison test. Justify your choice of test and provide clear steps. Link: [https://drive.google.com/file/d/1RQ2OZk-LSxpRyejKEMg1t2q15dbpVLCS/view? usp=sharing] Explain your reasoning and show all calculations.arrow_forward
- Refer to page 132 of the document for the linear algebra problem. Solve the given nonhomogeneous system of equations using Gaussian elimination or matrix inverses, and express the general solution in parametric form. Link: [https://drive.google.com/file/d/1RQ2OZK-LSxpRyejKEMg1t2q15dbpVLCS/view? usp=sharing] Justify each step in the elimination process and interpret the solution.arrow_forwardTum to page 134 for the vector calculus problem. Verify Stokes' theorem by calculating the surface integral of the curl of a vector field over the given surface S and comparing it to the line integral of the same field over the boundary curve C. Link: [https://drive.google.com/file/d/1RQ2OZk-LSxpRyej KEMg1t2q15dbpVLCS/view? usp=sharing] Provide all detailed steps for calculating the curl, surface integral, and line integral.arrow_forwardThe Laplace equation problem is provided on page 136. Solve the two-dimensional Laplace equation on a rectangular region using the method of separation of variables with the specified boundary conditions. Link: [https://drive.google.com/file/d/1RQ2OZk-LSxpRyejKEMg1t2q15dbpVLCS/view? usp=sharing] Include all steps, separation constants, and final solution in series form.arrow_forward
- The dynamical systems problem is on page 127. Determine the stability of the fixed points of the given nonlinear system using linearization techniques. Link: [https://drive.google.com/file/d/1RQ2OZK-LSxpRyejKEMg1t2q15dbpVLCS/view? usp=sharing] Provide phase plane analysis if necessary.arrow_forwardRefer to page 130 in the shared document for the differential equation problem. Solve the given second-order linear ordinary differential equation with the specified boundary conditions using the method of undetermined coefficients or variation of parameters as appropriate. Link: [https://drive.google.com/file/d/1RQ2OZK-LSxpRyejKEMg1t2q15dbpVLCS/view? usp=sharing] Show all intermediate steps, including how you satisfy the given boundary conditions.arrow_forwardWrite down the nth derivative of the function 1/z, and hence find the Taylor series of this function about the point 2 = 2. State the radius of convergence of the power series.arrow_forward
- The Laplace transform problem is on page 129. Compute the Laplace transform of the given function and find its inverse. Link: [https://drive.google.com/file/d/1RQ2OZK-LSxpRyejKEMg1t2q15dbpVLCS/view? usp=sharing] Present all calculations in detail and verify the result.arrow_forwardThe statistics problem is on page 95. Compute the mean (average) of the given dataset. Link: [https://drive.google.com/file/d/1RQ2OZK-LSxpRyejKEMg 1t2q15dbpVLCS/view? usp=sharing] Show your calculations clearly.arrow_forwardRefer to page 118 for the optimization problem. Solve the constrained optimization problem using the Lagrangian method. Link: [https://drive.google.com/file/d/1RQ2OZk-LSxpRyejKEMg1t2q15dbpVLCS/view? usp=sharing] Provide all derivations and interpretations of your solution.arrow_forward
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