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advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

Find and sketch disks or intervals that contain the eigenvalues. If you have a CAS, find the spectrum and compare. -0.2 10 0.1 0.1 6. -0.2 3
Crout€™s method factorizes where L is lower triangular and U is upper triangular with diagonal entries(a) Formulas. Obtain formulas for Crout€™s method similar to (4).(b)
Tridiagonalize. Show the details. 5 4 5 1 4 2 2. 4.
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
Fit a straight line to the given points (x, y) by least squares. Show the details. Check your result by sketching the points and the line. Judge the goodness of fit.Estimate the spring modulus k from
Do 5 steps, starting from x0 = [1 1 1]T and using 6S in the computation. Make sure that you solve each equation for the variable that has the largest coefficient (why?).
Apply the power method without scaling (3 steps), using x0= [1, 1]Tor [1 1 1]T. Give Rayleigh quotients and error bounds. Show the details of your work. 3.6
Find and sketch disks or intervals that contain the eigenvalues. If you have a CAS, find the spectrum and compare. 1 4 3 3 12
Tridiagonalize. Show the details. 1 1 1 1 1
Solve graphically and explain geometrically. -5.00x1 + 8.40x2 = 010.25x1 - 17.22x2 = 0
Fit a straight line to the given points (x, y) by least squares. Show the details. Check your result by sketching the points and the line. Judge the goodness of fit.How does the line in Prob. 1
Show that for the system in Example 2 the Jacobi iteration diverges. Use eigenvalues.
Apply the power method without scaling (3 steps), using x0= [1, 1]Tor [1 1 1]T. Give Rayleigh quotients and error bounds. Show the details of your work. -3 -3 -1
Compute the norms (5), (6), (7). Compute a corresponding unit vector (vector of norm 1) with respect to the l∞-norm.[4 -1 8]
Find and sketch disks or intervals that contain the eigenvalues. If you have a CAS, find the spectrum and compare. 5 10-2 10-2 10-2 8. 10-2 10-2 10-2 9.
Prove that the series converges uniformly in the indicated region.
Find the Taylor series with center z0 and its radius of convergence.ez(z-2), z0 = 1
Compute sinh 0.4 from sinh 0, sinh 0.5 = 0.521, sinh 1.0 = 1.175 by quadratic interpolation.
Integrate by (11) with n = 5sin (x2) from 0 to 1.25
In Example 5 of the text, write down the difference table as needed for (18), then write (18) with general x and then with x = 0.56 to verify the answer in Example 5.
Compute p2(0.75) from the data in Prob. 9 and Newton’s divided difference formula (10).Data from problem 9Calculate the Lagrange polynomial p2(x) for the 5S-values f(0.25) = 0.27633,f(0.5) =
The following integrals cannot be evaluated by the usual methods of calculus. Evaluate them as indicated. Compare your value with that possibly given by your CAS. Si(x) is the sine integral. S(x) and
Apply Newton’s method (6S-accuracy). First sketch the function(s) to see what is going on.What happens in Prob. 15 for any other x0?Data from Prob. 15Apply Newton’s method (6S-accuracy). First
Compute 1 - cos x with 6S arithmetic for x = 0.02(a) As given(b) By 2 sin2 1/2x (derive!).
Find the cubic spline g(x) for the given data with k0 and kn as given.Can you obtain the answer from that of Prob. 15? Data from in Problem 15 fo=f(0) = 2, f = f(2)= -2, f2 = f(4) = 2,
Evaluate the integralsby Simpson€™s rule with 2m as indicated, and compare with the exact value known from calculus.Integrate e-x from 0 to 2 by (7) with h = 1 and with h = 0.5. Give error
Compute √x2 + 4 - 2 with 6S arithmetic for x = 0.001(a) As given and (b) From x2/√x2 + 4 + 2) (derive!).
Using (14), find f(1.25) by linear, by linear, quadratic, and cubic interpolation of the data (values of (40) in App. A31); 6S value and si(1.25) = 1.14645) f(1.0) = 0.94608, f(1.5) = 1.32468, f(2.0)
Evaluate the integralsby Simpson€™s rule with 2m as indicated, and compare with the exact value known from calculus.J, 2m = 10 -2 dx .0.4 dx B = dx, J = хе 1 + x* х ||
Solve by fixed-point iteration and answer related questions where indicated. Show details.Let f(x) = x3 + 2x2 - 3x - 4 = 0. Write this as x = g(x), for g choosing (1) (x3 - f)1/3, (2) (x2 - 1/2f)1/2,
Find the cubic spline g(x) for the given data with k0 and kn as given.f0 = f(0) = 1, f1 = f(2) = 9, f2 = f(4) = 41, f3 = f(6) = 41, k0 = 0, k3 = 12
Compute f'(0.2) for f(x) = x3(a) h = 0.2(b) h = 0.1. Compare the accuracy.
Compute the integral of cos (x2) from 0 to 1 by Simpson’s rule with 2m = 4.
Find the cubic spline q and the interpolation polynomial p for the data (0, 0), (1, 1), (2, 6), (3, 10), with q'(0) = 0, q'(3) = 0 and graph p and q on common axes.
Derive the formula in Prob. 29 from (14)Data from Prob. 29The derivative f'(x) can also be approximated in terms of first-order and higher order differences.Compute f'(0.4) in Prob. 27 from this
1 + 1/2 + 1/3 + · · · diverges. Is the same true for the corresponding series of computer numbers?
A €œfour-point formula€ for the derivative isApply it to f(x) = x4 with x1, · · ·, x4 as in Prob. 27, determine the error, and compare it with that in the
Solve, using x0 and x1 as indicated:x = cos x, x0 = 0.5, x1 = 1
Integrating by parts, show that In = ∫10 exxn dx = e -nIn-1, I0 = e - 1.(a) Compute In, n = 0, · · ·, using 4S arithmetic, obtaining I8 = -3.906. Why is this nonsense? Why is the error so
This method uses the trapezoidal rule and gains precision step wise by halving h and adding an error estimate. Do this for the integral of f(x) = e-xfrom x = 0 to x = 2 with TOL = 10-3, as
Solve, using x0 and x1 as indicated:e-x - tan x = 0, x0 = 1, x1 = 0.7
Figure 430 shows the idea. We assume that f is continuous. We compute the x-intercept c0 of the line through (α0,f(α0)), (b0, f(b0)). If f(c0) = 0, we are done. If f(α0)f(c0) < 0 (as in
Prove that any binary machine number has a finite decimal representation. Is the converse true?
Solve x2 - 100x + 1 = 0. Use 5S-arithmetic.
Integrate by (11) with n = 5cos x from 0 to 1/2π
Apply Newton’s method (6S-accuracy). First sketch the function(s) to see what is going on.At what time x (4S-accuracy only) will the processes governed by f1(x) = 100 (1 - e-0.2x) and f2(x) =
Convert (0.59375)10to (0.10011)2 by successive multiplication by 2 and dropping (removing) the integer parts, which give the binary digits c1, c2, · · · : 0.59375 · 2
(a) Apply this to p1(9.2) and p2(9.2) for the data x0 = 9.0, x1 = 9.5, x2 = 11.0, f0 = ln x0, f1 = ln x1, f2 = ln x2 (6S-values).(b) Given (xj, f(xj)) = (0.2, 0.9980), (0.4, 0.9686), (0.6, 0.8443),
The following integrals cannot be evaluated by the usual methods of calculus. Evaluate them as indicated. Compare your value with that possibly given by your CAS. Si(x) is the sine integral. S(x) and
Compute e10 with 6S arithmetic in two ways (as in Prob. 19).Data from Prob. 19Calculate 1/e = 0.367879 (6S) from the partial sums of 5–10 terms of the Maclaurin series(a) Of e-x with w = 1(b)
In Hermite interpolation we are looking for a polynomial p(x) (of degree 2n + 1 or less) such that p(x) and its derivative p'(x) have given values at n + 1 nodes. (More generally, p(x), p'(x), p"(x),
Do the same task as in Prob. 19 for the difference 3.2 - 6.29.Data from Prob. 19Let 19.1 and 25.84 be correctly rounded. Find the shortest interval in which the sum s of the true (un rounded) numbers
The following integrals cannot be evaluated by the usual methods of calculus. Evaluate them as indicated. Compare your value with that possibly given by your CAS. Si(x) is the sine integral. S(x) and
Apply Newton’s method (6S-accuracy). First sketch the function(s) to see what is going on.Find the largest root of the Legendre polynomial P5(x) given by P5(x) = 1/8 (63x5 - 70x3 + 15x)(a) By
(a) Compute (1 - cos x)/sin x with 6S arithmetic for x = 0.005.(b) Looking at Prob. 16, find a much better formula.Data from Prob. 16Compute 1 - cos x with 6S arithmetic for x = 0.02(a) As given(b)
Compute 0.38755/(5.6815 - 0.38419) as given and then rounded step wise to 4S, 3S, 2S, 1S. Comment.
Derive an error bound in Prob. 9 from (5).Data from Prob. 9Calculate the Lagrange polynomial p2(x) for the 5S-values f(0.25) = 0.27633, f(0.5) = 0.52050, f(1.0) = 0.84270 and from p2(x) an
Evaluate the integralsby Simpson€™s rule with 2m as indicated, and compare with the exact value known from calculus.B, 2m = 10 -2 dx .0.4 dx B = dx, J = хе 1 + x* х ||
For small |α| the equation (x - k)2 = α has nearly a double root. Why do these roots show instability?
Solve by fixed-point iteration and answer related questions where indicated. Show details.Solve x cosh x = 1.
Evaluate the integralsby Simpson€™s rule with 2m as indicated, and compare with the exact value known from calculus.A, 2m = 10 -2 dx .0.4 dx B = dx, J = хе 1 + x* х ||
Does a sketch of the product of the (x - xj) in (5) for the data in Example 2 indicate that extrapolation is likely to involve larger errors than interpolation does?
Solve x2 - 40x + 2 = 0, using 4S-computation.
Prove that the trapezoidal rule is stable with respect to rounding.
Calculate p2(x) in Example 2. Compute from it approximations of ln 9.4, ln 10, ln 10.5, ln 11.5 and ln 12. Compute the errors by using exact 5S-values and comment.
Solve by fixed-point iteration and answer related questions where indicated. Show details.Find a form x = g(x) of f(x) = 0 in Prob. 5 that yields convergence to the root near x = 1.Data from Prob.
Give the details of the derivation of αj2 and αj3 in (13).
Integrate f(x) = x4 from 0 to 1 by (2) with h = 1, h = 0.5, h = 0.25 and estimate the error for h = 0.5 and h = 0.25 by (5).
Estimate the error for p2(9.2 ) in Example 2 from (5).
Solve by fixed-point iteration and answer related questions where indicated. Show details.f = x - cosec x the zero near x = 1.
Order of terms, in adding with a fixed number of digits, will generally affect the sum. Give an example. Find empirically a rule for the best order.
Derive the basic linear system (9) for k1, · · ·, kn-1.
Derive a formula for lower and upper bounds for the rectangular rule. Apply it to Prob. 1.Data from Prob. 1Derive a formula for lower and upperbounds for the rectangular rule. Apply it to Prob. 1.
Estimate the error in Prob. 1 by (5).Data from Prob. 1Calculate P1(x) in Example 1 and from it 3.
Solve by fixed-point iteration and answer related questions where indicated. Show details.Do the iterations (b) in Example 2. Sketch a figure similar to Fig. 427. Explain what happens. 1.0 5, (x) *2
Write -76.437125, 60100, and -0.00001 in floating point form, rounded to 4S.
Find the temperature distribution T(x, y) and the complex potential F(z) in the given thin metal plate whose faces are insulated and whose edges are kept at the indicated temperatures or are
Using (7), find the potential Î¦(r, Î¸) in the unit disk r < 1 having the given boundary values Î¦(1, Î¸). Using the sum of the first few terms of
What areas of physics did we consider? Could you think of others?
State the theorem on the behavior of harmonic functions under conformal mapping. Verify it for Φ* = eu sin v and w = u + iv = z2.
Describe the streamlines for F(z) = 1/2z2 + z.
Interpret Prob. 18 as an electrostatic problem. What are the lines of electric force?Data from Prob. 18Find the potential in the angular region between the plates Arg z = π/6 kept at 800 V and Arg z
Interpret Prob. 17 in Sec. 18.2 as a heat problem, with boundary temperatures, say, 10°C on the upper part and 200°C on the lower.Data from Prob. 17Find the temperature distribution T(x, y) and the
Find the location (u1, v1) of the maximum of Φ* = eu cos v in R*: |w| ≤ 1, v ≥ 0, where w = u + iv where. Find the region R that is mapped onto R* by w = f(z) = z2. Find the potential in R
(a) Show that F(z) = -(Ki/2Ï€) ln z with positive real K gives a flow circulating counterclockwise around z = 0 (Fig. 421). z = 0 is called a vortex. Each time we travel around the
Find the potential in the angular region between the plates Arg z = π/6 kept at 800 V and Arg z = π/3 kept at 600 V.
Find the temperature distribution T(x, y) and the complex potential F(z) in the given thin metal plate whose faces are insulated and whose edges are kept at the indicated temperatures or are
Using (7), find the potential Î¦(r, Î¸) in the unit disk r < 1 having the given boundary values Î¦(1, Î¸). Using the sum of the first few terms of
Verify the maximum principle for Φ(x, y) = ex sin y and the rectangle α ≤ x ≤ b, 0 ≤ y ≤ 2π.
Show that F(z) = arccosh z gives confocal hyperbolas as streamlines, with foci at z = ±1, and the flow may be interpreted as a flow through an aperture (Fig. 419).
Find the equipotential lines of F(z) = i Ln z.
Find the location and size of the maximum of |F(z)| in the unit disk |z| ≤ 1.F(z) = αz + b (α, b complex, α ≠ 0)
Solve Prob. 15 if the sector is -1/8π < Arg z < 1/8π.Data from Prob. 15By applying a suitable conformal mapping, obtain from FIG. 406 the potential Φ in the sector -1/4π < Arg z <
If plates at x1 = 1 and x2 = 10 are kept at potentials U1 = 220 V, U2 = 2kV, is the potential at x = 5 larger or smaller than the potential at r = 5 ? No calculation. Give reason.Data from Prob.
Find the temperature distribution T(x, y) and the complex potential F(z) in the given thin metal plate whose faces are insulated and whose edges are kept at the indicated temperatures or are
Using (7), find the potential Φ(r, θ) in the unit disk r < 1 having the given boundary values Φ(1, θ). Using the sum of the first few terms of the series, compute some values of Φ and sketch
Find the location and size of the maximum of |F(z)| in the unit disk |z| ≤ 1.F(z) = exp z2
Sketch or graph streamlines and equipotential lines of F(z) = iz3. Find V. Find all points at which V is horizontal.
Show that F(z) in Prob. 13 gives the potentials in Fig. 402.Data from Prob. 13Show that F(z) = arccos z gives the potential of a slit in Fig. 401.Fig. 401.Fig. 402. y -1 1
Figure 405 gives the impression that the potential on the y-axis changes more rapidly near 0 than near ±i. Can you verify this? y P 3 kV 12 1 i-1 -2 -3 P:-3 kV Fig. 405. Example 2: z-plane

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