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About 3307 results. 1294 free access solutions
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| m52742 | A silver plate in the shape of a trapezoid (see the accompanying figure) has heat being uniformly generated at each point at the rate q = 1.5 cal/cm3 · s. The steady-state temperature u(x, y) of the plate satisfies the Poisson equation
∂2u / ∂x2 (x, y) + ∂2u ∂y2 (x, y) = − q k,
where k, the thermal conductivity, is 1.04 cal/cm·deg·s. Assume that the temperature is held at 15◦C on L2, that heat is lost on the slanted edges L1 and L3 according to the boundary condition ∂u/∂n = 4, and that no heat is lost on L4; that is, ∂u/∂n = 0. Approximate the temperature of the plate at (1, 0), (4, 0), and 5/2, √3/2 by using Algorithm 12.5. |
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| m52750 | a. Sketch the graphs of y = ex − 2 and y = cos(ex − 2).
b. Use the Bisection method to find an approximation to within 10−5 to a value in [0.5, 1.5] with ex − 2 = cos(ex − 2). |
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| m52751 | a. Sketch the graphs of y = x and y = 2 sin x.
b. Use the Bisection method to find an approximation to within 10−5 to the first positive value of x with x = 2 sin x. |
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| m52752 | a. Sketch the graphs of y = x and y = tan x.
b. Use the Bisection method to find an approximation to within 10−5 to the first positive value of x with x = tan x. |
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| m52790 | a. Suppose that 0 < q < p and that αn = α + O(n−p). Show that αn = α + O(n−q).
b. Make a table listing 1/n, 1/n2, 1/n3, and 1/n4 for n = 5, 10, 100, and 1000, and discuss the varying rates of convergence of these sequences as n becomes large. |
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| m52791 | a. Suppose that 0 < q < p and that F(h) = L + O(hp). Show that F(h) = L + O(hq).
b. Make a table listing h, h2, h3, and h4 for h = 0.5, 0.1, 0.01, and 0.001, and discuss the varying rates of convergence of these powers of h as h approaches zero. |
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| m52813 | a. The Frobenius norm (which is not a natural norm) is defined for an n × n matrix A by
Show that || · ||F is a matrix norm.
b. Find || · ||F for the matrices in Exercise 4.
c. For any matrix A, show that ||A||2 ≤ ||A||F ≤ n1/2 ||A||2. |
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| m52814 | a. The introduction to this chapter included a table listing the population of the United States from 1950 to 2000. Use Lagrange interpolation to approximate the population in the years 1940, 1975, and 2020.
b. The population in 1940 was approximately 132,165,000. How accurate do you think your 1975 and 2020 figures are? |
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| m52815 | a. The introduction to this chapter included a table listing the population of the United States from 1950 to 2000. Use natural cubic spline interpolation to approximate the population in the years 1940, 1975, and 2020.
b. The population in 1940 was approximately 132,165,000. How accurate do you think your 1975 and 2020 figures are? |
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| m52838 | a. Use Algorithm 3.2 to construct the interpolating polynomial of degree three for the unequally spaced points given in the following table:
x f (x)
−0.1 5.30000
0.0 2.00000
0.2 3.19000
0.3 1.00000
b. Add f (0.35) = 0.97260 to the table, and construct the interpolating polynomial of degree four. |
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| m52839 | a. Use Algorithm 3.2 to construct the interpolating polynomial of degree four for the unequally spaced points given in the following table:
x f (x)
0.0 −6.00000
0.1 −5.89483
0.3 −5.65014
0.6 −5.17788
1.0 −4.28172
b. Add f (1.1) = −3.99583 to the table, and construct the interpolating polynomial of degree five. |
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| m52845 | a. Use the following values and five-digit rounding arithmetic to construct the Hermite interpolating polynomial to approximate sin 0.34.
b. Determine an error bound for the approximation in part (a), and compare it to the actual error.
c. Add sin 0.33 = 0.32404 and cos 0.33 = 0.94604 to the data, and redo the calculations. |
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| m52852 | a. Use the result in Exercise 9 to show that if A−1 exists and x, y ∈ Rn, then (A + xyt)−1 exists if and only if ytA−1x ≠ −1.
b. By multiplying on the right by A + xyt, show that when ytA−1 x = −1 we have |
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| m52855 | a. Use Theorem 2.4 to show that the sequence defined by
xn = 1/2xn−1 + 1/xn−1 , for n ≥ 1,
converges to√2 whenever x0 >√2.
b. Use the fact that 0 < (x0−√2)2 whenever x0 ≠√2 to show that if 0 < x0 <√2, then x1 >√2.
c. Use the results of parts (a) and (b) to show that the sequence in (a) converges to√2 whenever x0 > 0. |
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| m52856 | a. Use three-digit chopping arithmetic to compute the sum
first by 1/1 + 1/4 +· · ·+ 1/100 and then by 1/100 + 1/81+· · ·+ 1/1 . Which method is more accurate, and why?
b. Write an algorithm to sum the finite series
in reverse order. |
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| m52863 | a. Verify that the function ||·||1, defined on Rn by
is a norm on Rn.
b. Find ||x||1 for the vectors given in Exercise 1.
c. Prove that for all x ∈ Rn, ||x||1 ≥ ||x||2. |
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| m52903 | A zero of multiplicity m at p if and only if
0 = f ( p) = f ( p) = · · · = f (m−1)( p), but f (m)( p) ≠ 0. |
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| m52925 | Add another line to the extrapolation table in Exercise 1 to obtain the approximation N4(h).
In Exercise 1
a. f (x) = ln x, x0 = 1.0, h = 0.4
b. f (x) = x + ex , x0 = 0.0, h = 0.4
c. f (x) = 2x sin x, x0 = 1.05, h = 0.4
d. f (x) = x3 cos x, x0 = 2.3, h = 0.4 |
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| m52955 | Although q(x) < 0 in the following boundary-value problems, unique solutions exist and are given. Use the Linear Shooting Algorithm to approximate the solutions to the following problems, and compare the results to the actual solutions.
a. y" + y = 0, 0 ≤ x ≤ π/4, y(0) = 1, y(π/4) = 1; use h = π / 20; actual solution y(x) = cos x + (√2 − 1) sin x.
b. y" + 4y = cos x, 0 ≤ x ≤ π/4, y(0) = 0, y(π/4) = 0; use h = π/20; actual solution y(x) = −1/3 cos 2x − √2/6 sin 2x + 1/3 cos x.
c. y" = −4x−1y − 2x−2y + 2x−2 ln x, 1≤ x ≤ 2, y(1) = 1/2, y(2) = ln 2; use h = 0.05; actual solution y(x) = 4x−1 − 2x−2 + ln x − 3/2.
d. y" = 2y − y + xex − x, 0≤ x ≤ 2, y(0) = 0, y(2) = −4; use h = 0.2; actual solution y(x) = 1/6 x3ex - 5/3 xex + 2ex − x − 2. |
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| m52956 | Although q(x) < 0 in the following boundary-value problems, unique solutions exist and are given. Use the Linear Finite-Difference Algorithm to approximate the solutions, and compare the results to the actual solutions.
a. y" + y = 0, 0 ≤ x ≤ π/4, y(0) = 1, y(π/4) = 1; use h = π/20; actual solution y(x) = cos x + (√ 2 − 1) sin x.
b. y" + 4y = cos x, 0 ≤ x ≤ π/4, y(0) = 0, y(π/4) = 0; use h = π/20; actual solution y(x) = −1/3 cos 2x − √2/6 sin 2x + 1/3 cos x.
c. y" = −4x−1y + 2x−2y − 2x−2 ln x, y(1) = 1/2, y(2) = ln 2; use h = 0.05; actual solution y(x) = 4x−1 − 2x−2 + ln x − 3/2.
d. y" = 2y − y + xex − x, 0 ≤ x ≤ 2, y(0) = 0, y(2) = −4; use h = 0.2; actual solution y(x) = 1/6 x3ex - 5/3 xex + 2ex − x − 2. |
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