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numerical analysis

Numerical Analysis 9th edition Richard L. Burden, J. Douglas Faires - Solutions

The number e is defined byWhere n! = n(n ˆ’ 1) · · · 2 · 1 for n ‰ 0 and 0! = 1. Use four-digit chopping arithmetic to compute the following approximations to e, and determine the absolute and relative errors.
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
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
a. How many multiplications and additions are required to determine a sum of the formb. Modify the sum in part (a) to an equivalent form that reduces the number of computations.
Let P(x) = anxn + an−1xn−1 + · · · + a1x + a0 be a polynomial, and let x0 be given. Construct an algorithm to evaluate P(x0) using nested multiplication.
Let f (x) = (x+2)(x+1)2x(x −1)3(x −2). To which zero of f does the Bisection method converge when applied on the following intervals? a. [−1.5, 2.5] b. [−0.5, 2.4] c. [−0.5, 3] d. [−3,−0.5]
Let f (x) = (x − 1)10, p = 1, and pn = 1 + 1/n. Show that |f (pn)| < 10−3 whenever n > 1 but that |p − pn| < 10−3 requires that n > 1000.
Let {pn} be the sequence defined byShow that {pn} diverges even though limn†’ˆž (pnˆ’pnˆ’1) = 0.
The function defined by f (x) = sin πx has zeros at every integer. Show that when −1 < a < 0 and 2 < b < 3, the Bisection method converges to a. 0, if a + b < 2 b. 2, if a + b > 2 c. 1, if a + b = 2
Use the Bisection method to find solutions accurate to within 10−2 for x3 − 7x2 + 14x − 6 = 0 on each interval. a. [0, 1] b. [1, 3.2] c. [3.2, 4]
Use the Bisection method to find solutions accurate to within 10−2 for x4 − 2x3 − 4x2 + 4x + 4 = 0 on each interval. a. [−2,−1] b. [0, 2] c. [2, 3] d. [−1, 0]
Use the Bisection method to find solutions accurate to within 10−5 for the following problems. a. x − 2−x = 0 for 0 ≤ x ≤ 1 b. ex − x2 + 3x − 2 = 0 for 0 ≤ x ≤ 1 c. 2x cos(2x) − (x + 1)2 = 0 for −3 ≤ x ≤ −2 and −1 ≤ x ≤ 0 d. x cos x − 2x2 + 3x − 1 = 0 for 0.2
Use the Bisection method to find solutions, accurate to within 10−5 for the following problems. a. 3x − ex = 0 for 1 ≤ x ≤ 2 b. 2x + 3 cos x − ex = 0 for 0 ≤ x ≤ 1 c. x2 − 4x + 4 − ln x = 0 for 1 ≤ x ≤ 2 and 2 ≤ x ≤ 4 d. x + 1 − 2 sin πx = 0 for 0 ≤ x ≤ 0.5 and 0.5
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.
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.
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).
Use algebraic manipulation to show that each of the following functions has a fixed point at p precisely when f (p) = 0, where f (x) = x4 + 2x2 − x − 3. a. g1(x) = (3 + x − 2x2)1/4 b. g2(x) = (x + 3 − x4/2)1/2 c. g3(x) = (x + 3)/(x2 + 2)1/2 d. g4(x) = (3x4 + 2x2 + 3)/(4x3 + 4x - 1)
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
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.
Find all the zeros of f (x) = x2 +10 cos x by using the fixed-point iteration method for an appropriate iteration function g. Find the zeros accurate to within 10−4.
Let A be a given positive constant and g(x) = 2x − Ax2.a. Show that if fixed-point iteration converges to a nonzero limit, then the limit is p = 1/A, so the inverse of a number can be found using only multiplications and subtractions.b. Find an interval about 1/A for which fixed-point iteration
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.]
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. c. Use the results of parts (a) and (b) to show that the sequence in (a)
a. Perform four iterations, if possible, on each of the functions g defined in Exercise 1. Let p0 = 1 and pn+1 = g(pn), for n = 0, 1, 2, 3.b. Which function do you think gives the best approximation to the solution?
a. Show that if A is any positive number, then the sequence defined byxn = 1/2xn−1 + A/2xn−1, for n ≥ 1,converges to√A whenever x0 > 0.b. What happens if x0 < 0?
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.
Suppose that g is continuously differentiable on some interval (c, d) that contains the fixed point p of g. Show that if |g'( p)| < 1, then there exists a δ > 0 such that if |p0 − p| ≤ δ, then the fixed-point iteration converges.
Suppose that g is continuously differentiable on some interval (c, d) that contains the fixed point p of g. Show that if |g'( p)| 0 such that if |p0 ˆ’ p| ‰¤ Î´, then the fixed-point iteration converges.Where g = 32.17 ft/s2 and k represents the coefficient of air resistance in
Let g ∈ C1 [a, b] and p be in (a, b) with g( p) = p and |g' ( p)| > 1. Show that there exists aδ > 0 such that if 0 < |p0 − p| < δ, then |p0 − p| < |p1 − p| . Thus, no matter how close the initial approximation p0 is to p, the next iterate p1 is farther away, so the fixed-point iteration
Use Theorem 2.3 to show that g(x) = π + 0.5 sin(x/2) has a unique fixed point on [0, 2π]. Use fixed-point iteration to find an approximation to the fixed point that is accurate to within 10−2. Use Corollary 2.5 to estimate the number of iterations required to achieve 10−2 accuracy, and
Use Theorem 2.3 to show that g(x) = 2−x has a unique fixed point on [ 1/3, 1]. Use fixed-point iteration to find an approximation to the fixed point accurate to within 10−4. Use Corollary 2.5 to estimate the number of iterations required to achieve 10−4 accuracy, and compare this theoretical
Repeat Exercise 6 using the method of False Position. a. ex + 2−x + 2 cos x − 6 = 0 for 1 ≤ x ≤ 2 b. ln(x − 1) + cos(x − 1) = 0 for 1.3 ≤ x ≤ 2 c. 2x cos 2x − (x − 2)2 = 0 for 2 ≤ x ≤ 3 and 3 ≤ x ≤ 4 d. (x − 2)2 − ln x = 0 for 1 ≤ x ≤ 2 and e ≤ x ≤ 4 e. ex
Use all three methods in this Section to find solutions to within 10−5 for the following problems.a. 3xex = 0 for 1 ≤ x ≤ 2b. 2x + 3 cos x − ex = 0 for 0 ≤ x ≤ 1
Use all three methods in this Section to find solutions to within 10−7 for the following problems. a. x2 − 4x + 4 − ln x = 0 for 1 ≤ x ≤ 2 and for 2 ≤ x ≤ 4 b. x + 1 − 2 sin πx = 0 for 0 ≤ x ≤ 1/2 and for 1/2 ≤ x ≤ 1
Use Newton's method to approximate, to within 10−4, the value of x that produces the point on the graph of y = x2 that is closest to (1, 0). [Hint: Minimize [d(x)]2, where d(x) represents the distance from (x, x2) to (1, 0).]
The following describes Newton's method graphically: Suppose that f'(x) exists on [a, b] and that f'(x) ≠ 0 on [a, b]. Further, suppose there exists one p ∈ [a, b] such that f (p) = 0, and let p0 ∈ [a, b] be arbitrary. Let p1 be the point at which the tangent line to f at (p0, f (p0)) crosses
Use Newton's method to solve the equation 0 = 1/2 + 1/4 x2 − x sin x - 1/2cos 2x, with p0 = π/2.Iterate using Newton's method until an accuracy of 10−5 is obtained. Explain why the result seems unusual for Newton's method. Also, solve the equation with p0 = 5π and p0 = 10π.
The fourth-degree polynomial f (x) = 230x4 + 18x3 + 9x2 − 221x - 9 has two real zeros, one in [−1, 0] and the other in [0, 1]. Attempt to approximate these zeros to within 10−6 using thea. Method of False Positionb. Secant methodc. Newton's methodUse the endpoints of each interval as the
The function f (x) = tan πx − 6 has a zero at (1/π) arctan 6 ≈ 0.447431543. Let p0 = 0 and p1 = 0.48, and use ten iterations of each of the following methods to approximate this root. Which method is most successful and why? a. Bisection method b. Method of False Position c. Secant method
The equation x2−10 cos x = 0 has two solutions,±1.3793646. Use Newton's method to approximate the solutions to within 10−5 with the following values of p0.a. p0 = −100 b. p0 = −50 c. p0 = −25d. p0 = 25 e. p0 = 50f. p0 = 100
The equation 4x2 − ex − e−x = 0 has two positive solutions x1 and x2. Use Newton's method to approximate the solution to within 10−5 with the following values of p0. a. p0 = −10 b. p0 = −5 c. p0 = −3 d. p0 = −1 e. p0 = 0 f. p0 = 1 g. p0 = 3 h. p0 = 5 i. p0 = 10
The function described by f (x) = ln(x2 + 1) − e0.4x cos πx has an infinite number of zeros.a. Determine, within 10−6, the only negative zero.b. Determine, within 10−6, the four smallest positive zeros.c. Determine a reasonable initial approximation to find the nth smallest positive zero of
A drug administered to a patient produces a concentration in the blood stream given by c(t) = Ate−t/3 milligrams per milliliter, t hours after A units have been injected. The maximum safe concentration is 1 mg/mL. a. What amount should be injected to reach this maximum safe concentration, and
Let f (x) = 33x+1 − 7 · 52x.a. Use the Maple commands solve and f solve to try to find all roots of f .b. Plot f (x) to find initial approximations to roots of f.c. Use Newton’s method to find roots of f to within 10−16.d. Find the exact solutions of f (x) = 0 without using Maple.
Let f (x) = x2 − 6. With p0 = 3 and p1 = 2, find p3. a. Use the Secant method. b. Use the method of False Position. c. Which of a. or b. is closer to √6?
Let f (x) = 33x+1 − 7 . 52x.a. Use the Maple commands solve and f solve to try to find all roots of f .b. Plot f (x) to find initial approximations to roots of f.c. Use Newton’s method to find roots of f to within 10−16.d. Find the exact solutions of f (x) = 0 without using Maple.
The logistic population growth model is described by an equation of the form P(t) = PL/(1 − ce−kt) , Where PL, c, and k > 0 are constants, and P(t) is the population at time t. PL represents the limiting value of the population since limt→∞ P(t) = PL. Use the census data for the years
The Gompertz population growth model is described by P(t) = (PLe−ce)−kt , Where PL, c, and k > 0 are constants, and P(t) is the population at time t. Repeat Exercise 31 using the Gompertz growth model in place of the logistic model. In Exercise 31 P(t) = PL/(1 − ce−kt),
In the design of all-terrain vehicles, it is necessary to consider the failure of the vehicle when attempting to negotiate two types of obstacles. One type of failure is called hang-up failure and occurs when the vehicle attempts to cross an obstacle that causes the bottom of the vehicle to touch
Use Newton's method to find solutions accurate to within 10−4 for the following problems. a. x3 − 2x2 − 5 = 0, [1, 4] b. x3 + 3x2 − 1 = 0, [−3,−2] c. x − cos x = 0, [0, π/2] d. x − 0.8 − 0.2 sin x = 0, [0, π/2]
Use Newton's method to find solutions accurate to within 10−5 for the following problems. a. ex + 2−x + 2 cos x − 6 = 0 for 1 ≤ x ≤ 2 b. ln(x − 1) + cos(x − 1) = 0 for 1.3 ≤ x ≤ 2 c. 2x cos 2x − (x − 2)2 = 0 for 2 ≤ x ≤ 3 and 3 ≤ x ≤ 4 d. (x − 2)2 − ln x = 0 for 1
Repeat Exercise 5 using the Secant method. a. x3 − 2x2 − 5 = 0, [1, 4] b. x3 + 3x2 − 1 = 0, [−3,−2] c. x − cos x = 0, [0, π/2] d. x − 0.8 − 0.2 sin x = 0, [0, π/2]
Repeat Exercise 6 using the Secant method. a. ex + 2−x + 2 cos x − 6 = 0 for 1 ≤ x ≤ 2 b. ln(x − 1) + cos(x − 1) = 0 for 1.3 ≤ x ≤ 2 c. 2x cos 2x − (x − 2)2 = 0 for 2 ≤ x ≤ 3 and 3 ≤ x ≤ 4 d. (x − 2)2 − ln x = 0 for 1 ≤ x ≤ 2 and e ≤ x ≤ 4 e. ex − 3x2 = 0
Repeat Exercise 5 using the method of False Position. a. x3 − 2x2 − 5 = 0, [1, 4] b. x3 + 3x2 − 1 = 0, [−3,−2] c. x − cos x = 0, [0, π/2] d. x − 0.8 − 0.2 sin x = 0,
Use Newton's method to find solutions accurate to within 10−5 to the following problems. a. x2 − 2xe−x + e−2x = 0, for 0 ≤ x ≤ 1 b. cos(x +√2) + x(x/2 + √2) = 0, for −2 ≤ x ≤ −1 c. x3 − 3x2(2−x) + 3x(4−x) − 8−x = 0, for 0 ≤ x ≤ 1 d. e6x + 3(ln 2)2e2x − (ln
Suppose p is a zero of multiplicity m of f, where f (m) is continuous on an open interval containing p. Show that the following fixed-point method has g'( p) = 0: g(x) = x − mf (x)/f'(x).
Show that the Bisection Algorithm 2.1 gives a sequence with an error bound that converges linearly to 0.
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.
The iterative method to solve f (x) = 0, given by the fixed-point method g(x) = x, wherehas g'( p) = g''( p) = 0. This will generally yield cubic (Î± = 3) convergence. Expand the analysis of Example 1 to compare quadratic and cubic convergence.
It can be shown (see, for example, [DaB], pp. 228-229) that if {pn}∞ n=0 are convergent Secant method approximations to p, the solution to f (x) = 0, then a constant C exists with |pn+1 − p| ≈ C |pn − p| |pn−1 − p| for sufficiently large values of n. Assume {pn} converges to p of order
Use Newton's method to find solutions accurate to within 10−5 to the following problems. a. 1 − 4x cos x + 2x2 + cos 2x = 0, for 0 ≤ x ≤ 1 b. x2 + 6x5 + 9x4 − 2x3 − 6x2 + 1 = 0, for −3 ≤ x ≤ −2 c. sin 3x + 3e−2x sin x − 3e−x sin 2x − e−3x = 0, for 3 ≤ x ≤ 4 d. e3x
Repeat Exercise 1 using the modified Newton's method described in Eq. (2.13). Is there an improvement in speed or accuracy over Exercise 1?
Repeat Exercise 2 using the modified Newton's method described in Eq. (2.13). Is there an improvement in speed or accuracy over Exercise 2?
Use Newton's method and the modified Newton's method described in Eq. (2.13) to find a solution accurate to within 10−5 to the problem e6x + 1.441e2x − 2.079e4x − 0.3330 = 0, for − 1 ≤ x ≤ 0 This is the same problem as 1(d) with the coefficients replaced by their four-digit
Show that the following sequences converge linearly to p = 0. How large must n be before |pn− p| ≤ 5 × 10−2? a. pn = 1/n , n ≥ 1 b. pn = 1/n2 , n ≥ 1
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.
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.
The following sequences are linearly convergent. Generate the first five terms of the sequence {n} using Aitken's (2 method. a. p0 = 0.5, pn = (2 − epn−1 + p2n−1)/3, n ≥ 1 b. p0 = 0.75, pn = (epn−1/3)1/2, n ≥ 1 c. p0 = 0.5, pn = 3−pn−1 , n ≥ 1 d. p0 = 0.5, pn = cos pn−1, n ≥ 1
Use Steffensen's method to approximate the solutions of the following equations to within 10−5. a. x = (2 − ex + x2)/3, where g is the function. b. x = 0.5(sin x + cos x), where g is the function. c. x = (ex/3)1/2, where g is the function. d. x = 5−x , where g is the function.
Use Steffensen's method to approximate the solutions of the following equations to within 10−5. a. 2 + sin x − x = 0, where g is the function. b. x3 − 2x − 5 = 0, where g is the function. c. 3x2 − ex = 0, where g is the function. d. x − cos x = 0, where g is the function.
A sequence {pn} is said to be super linearly convergent to p ifa. Show that if pn †’ p of order Î± for Î± > 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 Î± for any Î± > 1.
Suppose that {pn} is super linearly convergent to p. Show that
Let Pn(x) be the nth Taylor polynomial for f (x) = ex expanded about x0 = 0. a. For fixed x, show that pn = Pn(x) satisfies the hypotheses of Theorem 2.14. b. Let x = 1, and use Aitken's (2 method to generate the sequence 0, . . . , 8. c. Does Aitken's method accelerate convergence in this
Use Steffensen's method to find, to an accuracy of 10−4, the root of x3 − x − 1 = 0 that lies in [1, 2], and compare this to the results of Exercise 6 of Section 2.2. In Section 2.2 Exercises 6
Find the approximations to within 10−4 to all the real zeros of the following polynomials using Newton's method. a. f (x) = x3 − 2x2 − 5 b. f (x) = x3 + 3x2 − 1 c. f (x) = x3 − x − 1 d. f (x) = x4 + 2x2 − x − 3 e. f (x) = x3 + 4.001x2 + 4.002x + 1.101 f. f (x) = x5 − x4 + 2x3 −
In 1224, Leonardo of Pisa, better known as Fibonacci, answered a mathematical challenge of John of Palermo in the presence of Emperor Frederick II: find a root of the equation x3 +2x2 +10x = 20. He first showed that the equation had no rational roots and no Euclidean irrational root-that is, no
Find approximations to within 10−5 to all the zeros of each of the following polynomials by first finding the real zeros using Newton's method and then reducing to polynomials of lower degree to find out any complex zeros.i. f (x) = x4 + 5x3 − 9x2 − 85x − 136ii. f (x) = x4 − 2x3 − 12x2
Find the approximations to within 10−4 to all the real zeros of the following polynomials using Newton's method. a. f (x) = x3 − 2x2 − 5 b. f (x) = x3 + 3x2 − 1 c. f (x) = x3 − x − 1 d. f (x) = x4 + 2x2 − x − 3 e. f (x) = x3 + 4.001x2 + 4.002x + 1.101 f. f (x) = x5 − x4 + 2x3 −
Find approximations to within 10−5 to all the zeros of each of the following polynomials by first finding the real zeros using Newton's method and then reducing to polynomials of lower degree to determine any complex zeros. a. f (x) = x4 + 5x3 − 9x2 − 85x − 136 b. f (x) = x4 − 2x3 −
Use Newton's method to find, within 10−3, the zeros and critical points of the following functions. Use this information to sketch the graph of f . a. f (x) = x3 − 9x2 + 12 b. f (x) = x4 − 2x3 − 5x2 + 12x − 5
Use Maple to find a real zero of the polynomial f (x) = x3 + 4x − 4.
Use Maple to find a real zero of the polynomial f (x) = x3 − 2x − 5.
For the given functions f (x), let x0 = 0, x1 = 0.6, and x2 = 0.9. Construct interpolation polynomials of degree at most one and at most two to approximate f (0.45), and find the absolute error. a. f (x) = cos x b. f (x) = √1 + x c. f (x) = ln(x + 1) d. f (x) = tan x
Use the following values and four-digit rounding arithmetic to construct a third Lagrange polynomial approximation to f (1.09). The function being approximated is f (x) = log10(tan x). Use this knowledge to find a bound for the error in the approximation. f (1.00) = 0.1924 f (1.05) = 0.2414 f
Use the Lagrange interpolating polynomial of degree three or less and four-digit chopping arithmetic to approximate cos 0.750 using the following values. Find an error bound for the approximation. cos 0.698 = 0.7661 cos 0.733 = 0.7432 cos 0.768 = 0.7193 cos 0.803 = 0.6946 The actual value of cos
Construct the Lagrange interpolating polynomials for the following functions, and find a bound for the absolute error on the interval [x0, xn]. a. f (x) = e2x cos 3x, x0 = 0, x1 = 0.3, x2 = 0.6, n = 2 b. f (x) = sin(ln x), x0 = 2.0, x1 = 2.4, x2 = 2.6, n = 2 c. f (x) = ln x, x0 = 1, x1 = 1.1, x2 =
Let f (x) = ex, for 0 ≤ x ≤ 2. a. Approximate f (0.25) using linear interpolation with x0 = 0 and x1 = 0.5. b. Approximate f (0.75) using linear interpolation with x0 = 0.5 and x1 = 1. c. Approximate f (0.25) and f (0.75) by using the second interpolating polynomial with x0 = 0, x1 = 1, and x2
Repeat Exercise 11 using Maple with Digits set to 10. In Exercise 11 Use the following values and four-digit rounding arithmetic to construct a third Lagrange polynomial approximation to f (1.09). The function being approximated is f (x) = log10(tan x). Use this knowledge to find a bound for the
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
It is suspected that the high amounts of tannin in mature oak leaves inhibit the growth of the winter moth (Operophtera bromata L., Geometridae) larvae that extensively damage these trees in certain years. The following table lists the average weight of two samples of larvae at times in the first
For the given functions f (x), let x0 = 1, x1 = 1.25, and x2 = 1.6. Construct interpolation polynomials of degree at most one and at most two to approximate f (1.4), and find the absolute error. a. f (x) = sin πx b. f (x) = 3√(x - 1) c. f (x) = log10(3x − 1) d. f (x) = e2x - x
In Exercise 26 of Section 1.1 a Maclaurin series was integrated to approximate erf(1), where erf(x) is the normal distribution error function defined bya. Use the Maclaurin series to construct a table for erf(x) that is accurate to within 10ˆ’4 for erf (xi), where xi = 0.2i, for i = 0, 1, . . .
Prove Taylor's Theorem 1.14 by following the procedure in the proof of Theorem 3.3. [LetWhere P is the nth Taylor polynomial, and uses the Generalized Rolle's Theorem 1.10]
Show that maxxj≤x≤xj+1 |g(x)| = h2/4, where g(x) = (x − jh)(x − (j + 1)h).
The Bernstein polynomial of degree n for f ˆˆ C [0, 1] is given byWhere (n/k) denotes n!/k!(n ˆ’ k)!. These polynomials can be used in a constructive proof of the Weierstrass Approximation Theorem 3.1 (see [Bart]) because limn†’ˆž Bn(x) = f (x), for each x ˆˆ [0, 1].a. Find B3(x) for
Use Theorem 3.3 to find an error bound for the approximations in Exercise 1. In Exercise 1 For the given functions f (x), let x0 = 0, x1 = 0.6, and x2 = 0.9. Construct interpolation polynomials of degree at most one and at most two to approximate f (0.45), and find the absolute error. a. f (x) =
Use Theorem 3.3 to find an error bound for the approximations in Exercise 2. In Exercise 2 For the given functions f (x), let x0 = 1, x1 = 1.25, and x2 = 1.6. Construct interpolation polynomials of degree at most one and at most two to approximate f (1.4), and find the absolute error. a. f (x) =
Use appropriate Lagrange interpolating polynomials of degrees one, two, and three to approximate each of the following: a. f (8.4) if f (8.1) = 16.94410, f (8.3) = 17.56492, f (8.6) = 18.50515, f (8.7) = 18.82091 b. f(−1/3) if f (−0.75) = −0.07181250, f (−0.5) = −0.02475000, f (−0.25) =
Use appropriate Lagrange interpolating polynomials of degrees one, two, and three to approximate each of the following: a. f (0.43) if f (0) = 1, f (0.25) = 1.64872, f (0.5) = 2.71828, f (0.75) = 4.48169 b. f (0) if f (−0.5) = 1.93750, f (−0.25) = 1.33203, f (0.25) = 0.800781, f (0.5) =
The data for Exercise 5 were generated using the following functions. Use the error formula to find a bound for the error, and compare the bound to the actual error for the cases n = 1 and n = 2. a. f (x) = x ln x b. f (x) = x3 + 4.001x2 + 4.002x + 1.101 c. f (x) = x cos x − 2x2 + 3x − 1 d. f

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