Explanation The Simpson’s rule is exact for f ( x ) = x n if n = 1 , 2 , 3 . The given integral equation is ∫ x 0 x 1 f ( x ) d x = a 0 f ( x 0 ) + a 1 f ( x 1 ) + a 2 f ( x 2 ) + k f 4 ( ξ ) . The error term of Simpson’s rule is − h 5 90 f 4 ( ξ ) and x n = x 0 + n h . For n = 1 , the integral equation becomes, ∫ x 0 x 2 f ( x ) d x = a 0 ( x 0 ) + a 1 ( x 1 ) + a 2 ( x 2 ) ∫ x 0 x 2 x d x = a 0 ( x 0 ) + a 1 ( x 0 + h ) + a 2 ( x 0 + 2 h ) [ x 2 2 ] x 0 x 2 = a 0 ( x 0 ) + a 1 ( x 0 + h ) + a 2 ( x 0 + 2 h ) x 2 2 − x 0 2 2 = a 0 x 0 + a 1 x 0 + a 1 h + a 2 x 0 + 2 a 2 h On further simplification, ( x 0 + 2 h ) 2 − x 0 2 2 = a 0 x 0 + a 1 x 0 + a 1 h + a 2 x 0 + 2 a 2 h x 0 2 + 4 x 0 h + 4 h 2 − x 0 2 2 = a 0 x 0 + a 1 x 0 + a 1 h + a 2 x 0 + 2 a 2 h 2 x 0 h + 2 h 2 = ( a 0 + a 1 + a 2 ) x 0 + ( a 1 + 2 a 2 ) h For n = 2 , the integral equation becomes, ∫ x 0 x 2 f ( x ) d x = a 0 ( x 0 ) + a 1 ( x 1 2 ) + a 2 ( x 2 2 ) ∫ x 0 x 2 x 2 d x = a 0 ( x 0 ) + a 1 ( x 0 + h ) 2 + a 2 ( x 0 + 2 h ) 2 [ x 3 3 ] x 0 x 2 = a 0 ( x 0 ) + a 1 ( x 0 2 + 2 x 0 h + h 2 ) + a 2 ( x 0 2 + 4 x 0 h + 4 h 2 ) ( x 0 + 2 h ) 3 − x 0 3 3 = ( a 1 + a 2 ) x 0 2 + ( a 0 + 2 a 1 h + 4 a 2 h ) x 0 + a 1 h 2 + 4 a 2 h 2 On further simplification, x 0 3 + 6 x 0 2 h + 12 x 0 h 2 + 8 h 3 − x 0 3 3 = ( a 1 + a 2 ) x 0 2 + ( a 0 + 2 a 1 h + 4 a 2 h ) x 0 + a 1 h 2 + 4 a 2 h 2 2 x 0 2 h + 4 x 0

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Chapter 4.3, Problem 26ES

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To derive: The Simpson’s rule with error by using ∫x0x1f(x)dx=a0f(x0)+a1f(x1)+a2f(x2)+kf4(ξ) and find the constants a0,a1,a2 and k.

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Chapter 4.3, Problem 25ES

Chapter 4.3, Problem 27ES

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Numerical Analysis

Ch. 4.1 - Use the forward-difference formulas and...Ch. 4.1 - The data in Exercise 1 were taken from the...Ch. 4.1 - Use the most accurate three-point formula to...Ch. 4.1 - Use the most accurate three-point formula to...Ch. 4.1 - The data in Exercise 5 were taken from the...Ch. 4.1 - The data in Exercise 6 were taken from the...Ch. 4.1 - Prob. 9ES Ch. 4.1 - Use the formulas given in this section to...Ch. 4.1 - The data in Exercise 9 were taken from the...Ch. 4.1 - Prob. 12ES

Ch. 4.1 - Use the following data and the knowledge that the...Ch. 4.1 - Prob. 14ES Ch. 4.1 - Prob. 15ES Ch. 4.1 - Prob. 16ES Ch. 4.1 - Prob. 17ES Ch. 4.1 - Prob. 18ES Ch. 4.1 - Prob. 19ES Ch. 4.1 - Prob. 20ES Ch. 4.1 - Prob. 21ES Ch. 4.1 - In a circuit with impressed voltage (t) and...Ch. 4.1 - In Exercise 9 of Section 3.4, data were given...Ch. 4.1 - Derive an O(h4) five-point formula to approximate...Ch. 4.1 - Use the formula derived in Exercise 24 and the...Ch. 4.1 - a. Analyze the round-off errors, as in Example 4,...Ch. 4.1 - Derive a method for approximating f (x0) whose...Ch. 4.1 - Consider the function e(h)=h+h26M, where M is a...Ch. 4.1 - Prob. 1DQ Ch. 4.1 - Prob. 2DQ Ch. 4.2 - Apply the extrapolation process described in...Ch. 4.2 - Add another line to the extrapolation table in...Ch. 4.2 - The following data give approximations to the...Ch. 4.2 - Prob. 6ES Ch. 4.2 - Prob. 7ES Ch. 4.2 - The forward-difference formula can be expressed as...Ch. 4.2 - Prob. 9ES Ch. 4.2 - Prob. 10ES Ch. 4.2 - Prob. 11ES Ch. 4.2 - Prob. 12ES Ch. 4.2 - Prob. 13ES Ch. 4.3 - Approximate the following integrals using the...Ch. 4.3 - Approximate the following integrals using the...Ch. 4.3 - Find a bound for the error in Exercise 1 using the...Ch. 4.3 - Prob. 4ES Ch. 4.3 - Repeat Exercise 1 using Simpsons rule. 1....Ch. 4.3 - Prob. 6ES Ch. 4.3 - Prob. 7ES Ch. 4.3 - Prob. 8ES Ch. 4.3 - Prob. 9ES Ch. 4.3 - Prob. 10ES Ch. 4.3 - Prob. 11ES Ch. 4.3 - Prob. 12ES Ch. 4.3 - The Trapezoidal rule applied to 02f(x)dx gives the...Ch. 4.3 - Prob. 14ES Ch. 4.3 - Approximate the following integrals using formulas...Ch. 4.3 - Prob. 17ES Ch. 4.3 - Suppose that the data of Exercise 17 have...Ch. 4.3 - Prob. 19ES Ch. 4.3 - Prob. 20ES Ch. 4.3 - The quadrature formula...Ch. 4.3 - The quadrature formula...Ch. 4.3 - Find the constants c0, c1, and x1 so that the...Ch. 4.3 - Find the constants x0, x1, and c1 so that the...Ch. 4.3 - Prob. 25ES Ch. 4.3 - Prob. 26ES Ch. 4.3 - Prob. 27ES Ch. 4.3 - Derive Simpsons Three-Eighths rule (the closed...Ch. 4.3 - Prob. 1DQ Ch. 4.3 - Prob. 2DQ Ch. 4.4 - Use the Composite Trapezoidal rule with the...Ch. 4.4 - Prob. 2ES Ch. 4.4 - Use the Composite Simpsons rule to approximate the...Ch. 4.4 - Prob. 4ES Ch. 4.4 - Prob. 5ES Ch. 4.4 - Prob. 6ES Ch. 4.4 - Prob. 7ES Ch. 4.4 - Prob. 8ES Ch. 4.4 - Prob. 9ES Ch. 4.4 - Prob. 10ES Ch. 4.4 - Determine the values of n and h required to...Ch. 4.4 - Repeat Exercise 11 for the integral 0x2cosxdx. 11....Ch. 4.4 - Determine the values of n and h required to...Ch. 4.4 - Repeat Exercise 13 for the integral 12xlnxdx. 13....Ch. 4.4 - Prob. 15ES Ch. 4.4 - Prob. 17ES Ch. 4.4 - A car laps a race track in 84 seconds. The speed...Ch. 4.4 - Prob. 19ES Ch. 4.4 - Prob. 20ES Ch. 4.4 - Prob. 21ES Ch. 4.4 - Prob. 23ES Ch. 4.4 - Prob. 24ES Ch. 4.4 - Prob. 25ES Ch. 4.4 - Prob. 26ES Ch. 4.4 - Prob. 1DQ Ch. 4.4 - Prob. 2DQ Ch. 4.5 - Use Romberg integration to compute R3, 3 for the...Ch. 4.5 - Use Romberg integration to compute R3, 3 for the...Ch. 4.5 - Prob. 3ES Ch. 4.5 - Prob. 4ES Ch. 4.5 - Use the following data to approximate 15f(x)dx as...Ch. 4.5 - Prob. 9ES Ch. 4.5 - Prob. 10ES Ch. 4.5 - Prob. 11ES Ch. 4.5 - Romberg integration for approximating 01f(x)dx...Ch. 4.5 - Prob. 15ES Ch. 4.5 - Prob. 18ES Ch. 4.5 - Prob. 19ES Ch. 4.5 - Prob. 1DQ Ch. 4.5 - Prob. 4DQ Ch. 4.6 - Prob. 1ES Ch. 4.6 - Prob. 2ES Ch. 4.6 - Prob. 11ES Ch. 4.6 - Prob. 12ES Ch. 4.6 - Could Romberg integration replace Simpsons rule in...Ch. 4.7 - Approximate the following integrals using Gaussian...Ch. 4.7 - Approximate the following integrals using Gaussian...Ch. 4.7 - Repeat Exercise 1 with n = 3. 1. Approximate the...Ch. 4.7 - Repeat Exercise 2 with n = 3. 2. Approximate the...Ch. 4.7 - Repeat Exercise 1 with n = 4. 1. Approximate the...Ch. 4.7 - Repeat Exercise 2 with n = 4. 2. Approximate the...Ch. 4.7 - Repeat Exercise 1 with n = 5. 1. Approximate the...Ch. 4.7 - Repeat Exercise 2 with n = 5. 2. Approximate the...Ch. 4.7 - Describe the differences and similarities between...Ch. 4.7 - Prob. 2DQ Ch. 4.8 - Prob. 1DQ Ch. 4.8 - Prob. 2DQ Ch. 4.8 - Prob. 3DQ Ch. 4.8 - Prob. 4DQ Ch. 4.9 - Suppose a body of mass m is traveling vertically...Ch. 4.9 - The Laguerre polynomials {L0(x), L1(x) ...} form...Ch. 4.9 - Prob. 7ES Ch. 4.9 - Prob. 8ES Ch. 4.9 - Prob. 9ES Ch. 4.9 - Prob. 1DQ Ch. 4.9 - Prob. 2DQ

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To derive: The Simpson’s rule with error by using ∫ x 0 x 1 f ( x ) d x = a 0 f ( x 0 ) + a 1 f ( x 1 ) + a 2 f ( x 2 ) + k f 4 ( ξ ) and find the constants a 0 , a 1 , a 2 and k .

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To derive: The Simpson’s rule with error by using ∫x0x1f(x)dx=a0f(x0)+a1f(x1)+a2f(x2)+kf4(ξ) and find the constants a0,a1,a2 and k.

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