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About 3307 results. 1294 free access solutions
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free/or 0.5$ |
| m55431 | Repeat Exercise 9 using m = 8. Compare the values of the approximating polynomials with the values of f at the points ξj = −π + 0.2jπ, for 0 ≤ j ≤ 10. Which approximation is better? |
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| m55432 | Repeat Exercise 9 using three-digit rounding arithmetic.
In Exercise 9
a. 0.03x1 + 58.9x2 = 59.2,
5.31x1 − 6.10x2 = 47.0 |
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| m55433 | Repeat Exercise 9 using x0 = π/6.
a. Use P2 (0.5) to approximate f (0.5). Find an upper bound for error |f (0.5) − P2(0.5)| using the error formula, and compare it to the actual error.
b. Find a bound for the error |f(x)−P2(x)| in using P2(x) to approximate f (x) on the interval [0, 1].
c. Approximate
d. Find an upper bound for the error in (c) using
and compare the bound to the actual error. |
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| m55436 | Replace the assumption in Theorem 2.4 that "a positive number k < 1 exists with |g (x)| ≤ k" with
"g satisfies a Lipschitz condition on the interval [a, b] with Lipschitz constant L < 1. Show that the conclusions of this theorem are still valid. |
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| m55457 | Sagar and Payne [SP] analyze the stress-strain relationships and material properties of a cylinder alternately subjected to heating and cooling and consider the equation
∂2T / ∂r2 + 1 / r ∂T / ∂r = 1 / 4K ∂T / ∂t, 1/2 < r < 1, 0 < T,
where T = T(r, t) is the temperature, r is the radial distance from the center of the cylinder, t is time, and K is a diffusivity coefficient.
a. Find approximations to T(r, 10) for a cylinder with outside radius 1, given the initial and boundary conditions:
T(1, t) = 100 + 40t, T (1/2, t) = t, 0≤ t ≤ 10;
T(r, 0) = 200(r − 0.5), 0.5 ≤ r ≤ 1.
Use a modification of the Backward-Difference method with K = 0.1, k=0.5, and h= r=0.1.
b. Use the temperature distribution of part (a) to calculate the strain I by approximating the integral
where α = 10.7 and t = 10. Use the Composite Trapezoidal method with n = 5. |
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| m55481 | Show by example that || · ||∞, defined by ||A||∞ = max1≤i, j≤n |ai j|, does not define a matrix norm. |
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| m55492 | Show that a function F mapping D ⊂ Rn into Rn is continuous at x0 ∈ D precisely when, given any number ε > 0, a number δ > 0 can be found with property that for any vector norm ||·||,
||F(x) − F(x0)|| < ε,
whenever x ∈ D and || x − x0 || < δ. |
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| m55494 | Show that a set {v1, . . . , vk} of k nonzero orthogonal vectors is linearly independent. |
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| m55496 | Show that c0. . . c2m−1 in Algorithm 8.3 are given by
Where ζ = eπi/m. |
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| m55499 | Show that || · ||①, defined by is a matrix norm. Find || · ||① for the matrices in Exercise 4 |
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| m55501 | Show that each of the following initial-value problems has a unique solution and find the solution. Can Theorem 5.4 be applied in each case?
a. y = et−y, 0≤ t ≤ 1, y(0) = 1.
b. y = t−2(sin 2t − 2ty), 1≤ t ≤ 2, y(1) = 2.
c. y = −y + ty1/2, 2≤ t ≤ 3, y(2) = 2.
d. y = (ty + y)/(ty + t) , 2≤ t ≤ 4, y(2) = 4. |
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| m55502 | Show that each of the following matrices is nonsingular but not diagonalizable.
a.
b.
c.
d. |
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| m55505 | Show that, for any constants a and b, the set D = {(t, y) | a ≤ t ≤ b, −∞ < y < ∞} is convex. |
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| m55506 | Show that for any continuous even function f defined on the interval [−a, a], we have |
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| m55507 | Show that for any continuous odd function f defined on the interval [−a, a], we have |
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| m55508 | Show that, for any k, |
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| m55509 | Show that for any positive integers i and j with i > j, we have Ti(x)Tj(x) = 1/2[Ti+j(x) + Ti−j(x)]. |
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| m55510 | Show that for each Chebyshev polynomial Tn(x), we have |
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| m55511 | Show that for each n, the Chebyshev polynomial Tn(x) has n distinct zeros in (−1, 1). |
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| m55512 | Show that for each n, the derivative of the Chebyshev polynomial Tn(x) has n − 1 distinct zeros in (−1, 1). |
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