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free/or 0.5$ |
| m52662 | A projectile of mass m = 0.11 kg shot vertically upward with initial velocity v(0) = 8 m/s is slowed due to the force of gravity, Fg = −mg, and due to air resistance, Fr = −kv|v|, where g = 9.8 m/s2 and k = 0.002 kg/m. The differential equation for the velocity v is given by
mv = −mg − kv|v|.
a. Find the velocity after 0.1, 0.2. . . 1.0s.
b. To the nearest tenth of a second, determine when the projectile reaches its maximum height and begins falling. |
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| m52665 | a. Prove that
||x(k) − x|| ≤ ||T||k ||x(0) − x|| and ||x(k) − x|| ≤ ||T||k /1 − ||T|| ||x(1) − x(0) ||,
Where T is an n × n matrix with ||T|| < 1 and
x(k) = Tx(k−1) + c, k = 1, 2, . . . ,
With x(0) arbitrary, c ∈ Rn, and x = Tx + c.
b. Apply the bounds to Exercise 1, when possible, using the l∞ norm. |
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| m52682 | A rectangular parallelepiped has sides of length 3 cm, 4 cm, and 5 cm, measured to the nearest centimeter. What are the best upper and lower bounds for the volume of this parallelepiped? What are the best upper and lower bounds for the surface area? |
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| m52708 | A sequence {pn} is said to be super linearly convergent to p if
a. Show that if pn → p of order α for α > 1, then {pn} is super linearly convergent to p.
b. Show that pn = 1/nn is super linearly convergent to 0 but does not converge to 0 of order α for any α > 1. |
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| m52715 | a. Show that an A-orthogonal set of nonzero vectors associated with a positive definite matrix is linearly independent.
b. Show that if {v(1), v(2), . . . , v(n)} is a set of A-orthogonal nonzero vectors in R and ztv(i) = 0, for each i = 1, 2, . . . , n, then z = 0. |
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| m52718 | a. Show that for any positive integer k, the sequence defined by pn = 1/nk converges linearly to
p = 0.
b. For each pair of integers k and m, determine a number N for which 1/Nk < 10−m. |
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| m52720 | a. Show that H2n+1(x) is the unique polynomial of least degree agreeing with f and f at x0. . . xn. Assume that P(x) is another such polynomial and consider D = H2n+1 − P and D at x0, x1. . . xn.]
b. Derive the error term in Theorem 3.9.
And using the fact that g (t) has (2n + 2) distinct zeros in [a, b].] |
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| m52721 | a. Show that if A is an n × n matrix, then
where λi , . . . , λn are the eigenvalues of A.
b. Show that A is singular if and only if λ = 0 is an eigenvalue of A. |
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| m52722 | a. Show that if A is any positive number, then the sequence defined by
xn = 1/2xn−1 + A/2xn−1, for n ≥ 1,
converges to√A whenever x0 > 0.
b. What happens if x0 < 0? |
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| m52726 | a. Show that
limh→0 ((2 + h)/(2 - h))1/h = e.
b. Compute approximations to e using the formula N(h) = ((2+h)/(2−h))1/h, for h = 0.04, 0.02, and 0.01.
c. Assume that e = N(h)+K1h+K2h2 +K3h3 +· · · . Use extrapolation, with at least 16 digits of precision, to compute an O (h3) approximation to e with h = 0.04. Do you think the assumption is correct?
d. Show that N (−h) = N (h).
e. Use part (d) to show that K1 = K3 = K5 = · · · = 0 in the formula
e = N(h) + K1h + K2h2 + K3h3K4h4 + K5h5 +· · · ,
so that the formula reduces to
e = N(h) + K2h2 + K4h4 + K6h6 +· · · .
f. Use the results of part (e) and extrapolation to compute an O (h6) approximation to e with h = 0.04. |
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| m52728 | a. Show that the cubic polynomials
P(x) = 3 − 2(x + 1) + 0(x + 1)(x) + (x + 1)(x)(x − 1)
And
Q(x) = −1 + 4(x + 2) − 3(x + 2)(x + 1) + (x + 2)(x + 1)(x)
Both interpolate the data
b. Why does part (a) not violate the uniqueness property of interpolating polynomials? |
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| m52730 | a. Show that the Gauss-Jordan method requires
n3/2 + n2 - n/2 multiplications/divisions
and
n3/2 - n/2 additions/subtractions.
b. Make a table comparing the required operations for the Gauss-Jordan and Gaussian elimination |
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| m52731 | a. Show that the Implicit Trapezoidal method is A-stable.
b. Show that the Backward Euler method described in Exercise 12 is A-stable. |
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| m52732 | a. Show that the LU Factorization Algorithm requires
1/3 n3 - 1/3 n multiplications/divisions and 1/3 n3 - 1/2 n2 + 1/6 n additions/subtractions
b. Show that solving Ly = b, where L is a lower-triangular matrix with lii = 1 for all i, requires
1/2 n2 - 1/2 n multiplications/divisions and 1/2 n2 - 1/2 n additions/subtractions
c. Show that solving Ax = b by first factoring A into A = LU and then solving Ly = b and Ux = y requires the same number of operations as the Gaussian Elimination Algorithm 6.1.
d. Count the number of operations required to solve m linear systems Ax(k) = b(k) for k =1, . . . ,m by first factoring A and then using the method of part (c) m times. |
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| m52734 | a. Show that the polynomial nesting technique described in Example 6 can also be applied to the evaluation of
f (x) = 1.01e4x − 4.62e3x − 3.11e2x + 12.2ex − 1.99.
b. Use three-digit rounding arithmetic, the assumption that e1.53 = 4.62, and the fact that enx = (ex)n to evaluate f (1.53) as given in part (a).
c. Redo the calculation in part (b) by first nesting the calculations.
d. Compare the approximations in parts (b) and (c) to the true three-digit result f (1.53) = −7.61. |
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| m52735 | a. Show that the product of two n × n lower triangular matrices is lowers triangular.
b. Show that the product of two n × n upper triangular matrices is upper triangular.
c. Show that the inverse of a nonsingular n × n lower triangular matrix is lowers triangular. |
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| m52736 | a. Show that the quadratic polynomial
P(α) = g1 + h1α + h3α (α − α2)
interpolates the function h defined in (10.18):
h(α) = g(x(0) − α∇g x(0))
at α = 0, α2, and α3.
b. Show that a critical point of P occurs at
α0 = 1/2 (α2 − h1 / h3). |
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| m52737 | a. Show that the rotation matrix
applied to the vector x = (x1, x2)t has the geometric effect of rotating x through the angle θ without changing its magnitude with respect to the l2 norm.
b. Show that the magnitude of x with respect to the l∞ norm can be changed by a rotation matrix. |
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| m52738 | a. Show that the sequence pn = (10−2)n converges quadratically to 0.
b. Show that the sequence pn = (10−n)k does not converge to 0 quadratically, regardless of the size of the exponent k > 1. |
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| m52741 | a. Show that Theorem 2.2 is true if the inequality |g (x)|≤k is replaced by g (x) ≤ k, for all x∈(a,b).
b. Show that Theorem 2.3 may not hold if inequality |g (x)| ≤ k is replaced by g (x) ≤ k. [Show that g(x) = 1 − x2, for x in [0, 1], provides a counterexample.] |
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