Explanation The given data is as follows. x 1 2 3 4 5 f ( x ) 2.4142 2.6734 2.8974 3.0976 3.2804 It is known that, the most appropriate formula to solve this is the five point midpoint formula which is as follows. f ′ ( x 0 ) = 1 12 h ( ( f ( x 0 − 2 h ) + 8 f ( x 0 − h ) + 8 f ( x 0 + h ) ) − f ( x 0 + 2 h ) ) + h 4 f ( 5 ) ( ξ ) 30 Obtain the value of f ′ ( 3 ) as shown below. f ′ ( 3 ) ≈ 1 12 ( ( f ( 1 ) + 8 f ( 2 ) + 8 f ( 4 ) ) − f ( 5 ) ) ≈ 1 12 ( ( 2

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Chapter 4.1, Problem 14ES

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To approximate: The value of f′(3) accurately and to find the bound for the error using the given data and assuming that the third derivative is bounded on [1,5] by 4.

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Chapter 4.1, Problem 13ES

Chapter 4.1, Problem 15ES

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Chapter 4 Solutions

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 approximate: The value of f ′ ( 3 ) accurately and to find the bound for the error using the given data and assuming that the third derivative is bounded on [ 1 , 5 ] by 4.

Solution Summary: The author explains how to approximate the value of fprime (3) accurately and to find the bound for the error using the given data and assuming that the third derivative is bounded

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To approximate: The value of f′(3) accurately and to find the bound for the error using the given data and assuming that the third derivative is bounded on [1,5] by 4.

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Chapter 4 Solutions