| № |
Condition |
free/or 0.5$ |
| m53997 | Find the next largest and smallest machine numbers in decimal form for the numbers
a. 0 10000001010 1001001100000000000000000000000000000000000000000000
b. 1 10000001010 1001001100000000000000000000000000000000000000000000
c. 0 01111111111 0101001100000000000000000000000000000000000000000000
d. 0 01111111111 0101001100000000000000000000000000000000000000000001 |
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| m53998 | Find the nth Maclaurin polynomial Pn(x) for f (x) = arctan x. |
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| m54016 | Find the rates of convergence of the following functions as h → 0.
a. limh→0 (sin h)/h = 1
b. limh→0 (1 − cos h)/h = 0
c. limh→0 (sin h − h cos h)h = 0
d. limh→0 (1 - eh)/h = −1 |
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| m54017 | Find the rates of convergence of the following sequences as n→∞.
a. limn→∞ sin1/n = 0
b. limn→∞ sin 1/n2 = 0
c. limn→∞ (sin 1/n)2 = 0
d. limn→∞ [ln(n + 1) − ln(n)] = 0 |
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| m54018 | Find the row interchanges that are required to solve the following linear systems using Algorithm 6.1.
a. x1 − 5x2 + x3 = 7,
10x1 + 20x3 = 6,
5x1 − x3 = 4
b. x1 + x2 − x3 = 1,
x1 + x2 + 4x3 = 2,
2x1 − x2 + 2x3 = 3
c. 2x1 − 3x2 + 2x3 = 5,
−4x1 + 2x2 − 6x3 = 14,
2x1 + 2x2 + 4x3 = 8.
d. x2 + x3 = 6,
x1 − 2x2 − x3 = 4,
x1 − x2 + x3 = 5. |
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| m54019 | Find the second Taylor polynomial P2(x) for the function f (x) = ex cos x about x0 = 0.
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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| m54020 | Find the sixth Maclaurin polynomial for xex , and use Chebyshev economization to obtain a lesser degree polynomial approximation while keeping the error less than 0.01 on [−1, 1]. |
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| m54024 | Find the smallest values for n = m so that Algorithm 4.4 can be used to approximate the integrals in Exercise 1 to within 10−6 of the actual value.
In Exercise 1 |
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| m54025 | Find the smallest values for n = m so that Algorithm 4.4 can be used to approximate the integrals in Exercise 3 to within 10−6 of the actual value.
In Exercise 3 |
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| m54026 | Find the smallest values of n = m so that Algorithm 4.5 can be used to approximate the integrals in Exercise 1 to within 10−6. Do not continue beyond n = m = 5. Compare the number of functional evaluations required to the number required in Exercise 2.
In Exercise 2 |
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| m54037 | Find the third Taylor polynomial P3(x) for the function f (x) = (x − 1) ln x about x0 = 1.
a. Use P3(0.5) to approximate f (0.5). Find an upper bound for error |f (0.5) − P3(0.5)| using the error formula, and compare it to the actual error.
b. Find a bound for the error |f (x) − P3(x)| in using P3(x) to approximate f (x) on the interval [0.5, 1.5].
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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| m54038 | Find the third Taylor polynomial P3(x) for the function f (x) = √(x + 1) about x0 = 0. Approximate √0.5, √0.75, √1.25, and √1.5 using P3(x), and find the actual errors. |
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| m54080 | Following along the line of Exercise 11 in Section 6.3 and Exercise 15 in Section 7.2, suppose that a species of beetle has a life span of 4 years, and that a female in the first year has a survival rate of 1/2, in the second year a survival rate of 1/4, and in the third year a survival rate of 1/8 . Suppose additionally that a female gives birth, on the average, to two new females in the third year and to four new females in the fourth year. The matrix describing a single female s contribution in 1 year to the female population in the succeeding year is
where again the entry in the ith row and jth column denotes the probabilistic contribution that a female of age j makes on the next year s female population of age i.
a. Use the Geršgorin Circle Theorem to determine a region in the complex plane containing all the eigenvalues of A.
b. Use the Power method to determine the dominant eigenvalue of the matrix and its associated eigenvector.
c. Use Algorithm 9.4 to determine any remaining eigenvalues and eigenvectors of A.
d. Find the eigenvalues of A by using the characteristic polynomial of A and Newton s method. |
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| m54106 | For each choice of f (t, y) given in parts (a)-(d):
i. Does f satisfy a Lipschitz condition on D = {(t, y) | 0 ≤ t ≤ 1, −∞ < y < ∞}?
ii. Can Theorem 5.6 be used to show that the initial-value problem
y = f (t, y), 0≤ t ≤ 1, y(0) = 1,
is well-posed?
a. f (t, y) = t2y + 1
b. f (t, y) = ty
c. f (t, y) = 1 − y
d. f (t, y) = −ty +4t/y |
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| m54107 | For each choice of f (t, y) given in parts (a)-(d):
i. Does f satisfy a Lipschitz condition on D = {(t, y) | 0 ≤ t ≤ 1, −∞ < y < ∞}?
ii. Can Theorem 5.6 be used to show that the initial-value problem
y = f (t, y), 0≤ t ≤ 1, y(0) = 1,
is well-posed?
a. f (t, y) = et−y
b. f (t, y) = (1 + y)/(1 + t)
c. f (t, y) = cos(yt)
d. f (t, y) = y2/(1 + t) |
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| m54112 | For each of the following equations, determine an interval [a, b] on which fixed-point iteration will converge. Estimate the number of iterations necessary to obtain approximations accurate to within 10−5, and perform the calculations.
a. x = 2 − ex + x2/3
b. x = 5/x2 + 2
c. x = (ex/3)1/2
d. x = 5−x
e. x = 6−x
f. x = 0.5(sin x + cos x) |
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| m54113 | For each of the following equations, use the given interval or determine an interval [a, b] on which fixed-point iteration will converge. Estimate the number of iterations necessary to obtain approximations accurate to within 10−5, and perform the calculations.
a. 2 + sin x − x = 0 use [2, 3]
b. x3 − 2x − 5 = 0 use [2, 3]
c. 3x2 − ex = 0
d. x − cos x = 0 |
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| m54118 | For each of the following linear systems, obtain a solution by graphical methods, if possible. Explain the results from a geometrical standpoint.
a. x1 + 2x2 = 3,
x1 − x2 = 0.
b. x1 + 2x2 = 3,
2x1 + 4x2 = 6.
c. x1 + 2x2 = 0,
2x1 + 4x2 = 0.
d. 2x1 + x2 = −1,
4x1 + 2x2 = −2,
x1 − 3x2 = 5. |
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| m54119 | For each of the following linear systems, obtain a solution by graphical methods, if possible. Explain the results from a geometrical standpoint.
a. x1 + 2x2 = 0,
x1 − x2 = 0.
b. x1 + 2x2 = 3,
−2x1 − 4x2 = 6
c. 2x1 + x2 = −1,
x1 + x2 = 2,
x1 − 3x2 = 5.
d. 2x1 + x2 + x3 = 1,
2x1 + 4x2 − x3 = −1. |
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| m54120 | For each of the following matrices determine if it diagonalizable and, if so, find P and D with A = PDP−1.
a.
b.
c.
d. |
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