Explanation The cubic spline S is defined as: S ( x ) = { S 0 ( x ) = 1 + 2 x − x 3 0 ≤ x ≤ 1 S 1 ( x ) = 2 + b ( x − 1 ) + c ( x − 1 ) 2 + d ( x − 1 ) 3 1 ≤ x ≤ 2 Differentiate, the above function with respect to x . S ′ ( x ) = { S 0 ′ ( x ) = 2 − 3 x 2 S 1 ′ ( x ) = b + 2 c ( x − 1 ) + 3 d ( x − 1 ) 2 S ″ ( x ) = { S 0 ″ ( x ) = − 6 x S 0

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Chapter 3.5, Problem 11ES

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The value b,c,d for cubic spline S, on [0,2].

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Chapter 3.5, Problem 10ES

Chapter 3.5, Problem 12ES

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Chapter 3 Solutions

EBK NUMERICAL ANALYSIS

Ch. 3.1 - For the given functions f(x), let x0 = 0, x1 =...Ch. 3.1 - Use Theorem 3.3 to find an error bound for the...Ch. 3.1 - Prob. 4ES Ch. 3.1 - The data for Exercise 6 were generated using the...Ch. 3.1 - Prob. 9ES Ch. 3.1 - Prob. 10ES Ch. 3.1 - Prob. 11ES Ch. 3.1 - Prob. 12ES Ch. 3.1 - Prob. 15ES Ch. 3.1 - Prob. 17ES

Ch. 3.1 - It is suspected that the high amounts of tannin in...Ch. 3.1 - Prob. 21ES Ch. 3.1 - Prove Taylors Theorem 1.14 by following the...Ch. 3.1 - Prob. 1DQ Ch. 3.1 - If we decide to increase the degree of the...Ch. 3.2 - Let P3(x) be the interpolating polynomial for the...Ch. 3.2 - Nevilles method is used to approximate f(0.4),...Ch. 3.2 - Nevilles method is used to approximate f(0.5),...Ch. 3.2 - Suppose xj = j, for j = 0, 1, 2, 3, and it is...Ch. 3.2 - Nevilles Algorithm is used to approximate f(0)...Ch. 3.2 - Prob. 11ES Ch. 3.2 - Prob. 13ES Ch. 3.2 - Can Nevilles method be used to obtain the...Ch. 3.3 - Use Eq. (3.10) or Algorithm 3.2 to construct...Ch. 3.3 - Use Eq. (3.10) or Algorithm 3.2 to construct...Ch. 3.3 - Use the Newton forward-difference formula to...Ch. 3.3 - Use the Newton forward-difference formula to...Ch. 3.3 - Use the Newton backward-difference formula to...Ch. 3.3 - Use the Newton backward-difference formula to...Ch. 3.3 - a. Use Algorithm 3.2 to construct the...Ch. 3.3 - a. Use Algorithm 3.2 to construct the...Ch. 3.3 - a. Approximate f(0.05) using the following data...Ch. 3.3 - The following data are given for a polynomial P(x)...Ch. 3.3 - The following data are given for a polynomial P(x)...Ch. 3.3 - The Newton forward-difference formula is used to...Ch. 3.3 - Prob. 14ES Ch. 3.3 - Prob. 16ES Ch. 3.3 - Prob. 17ES Ch. 3.3 - Show that the polynomial interpolating the...Ch. 3.3 - Prob. 20ES Ch. 3.3 - Prob. 21ES Ch. 3.3 - Prob. 22ES Ch. 3.3 - Prob. 23ES Ch. 3.3 - Compare and contrast the various...Ch. 3.3 - Is it easier to add a new data pair using...Ch. 3.3 - Prob. 3DQ Ch. 3.4 - Use Theorem 3.9 or Algorithm 3.3 to construct an...Ch. 3.4 - Use Theorem 3.9 or Algorithm 3.3 to construct an...Ch. 3.4 - The data in Exercise 1 were generated using the...Ch. 3.4 - The data in Exercise 2 were generated using the...Ch. 3.4 - Let f (x) = 3xex e2x. a. Approximate f (1.03) by...Ch. 3.4 - The following table lists data for the function...Ch. 3.4 - a. Show that H2n + 1 (x) is the unique polynomial...Ch. 3.4 - Prob. 1DQ Ch. 3.4 - Prob. 2DQ Ch. 3.4 - Prob. 3DQ Ch. 3.5 - Determine the natural cubic spline S that...Ch. 3.5 - Determine the clamped cubic spline s that...Ch. 3.5 - Construct the natural cubic spline for the...Ch. 3.5 - Construct the natural cubic spline for the...Ch. 3.5 - The data in Exercise 3 were generated using the...Ch. 3.5 - Prob. 6ES Ch. 3.5 - Prob. 8ES Ch. 3.5 - Prob. 11ES Ch. 3.5 - Prob. 12ES Ch. 3.5 - Prob. 13ES Ch. 3.5 - Prob. 14ES Ch. 3.5 - Given the partition x0 = 0, x1 = 0.05, and x2 =...Ch. 3.5 - Prob. 16ES Ch. 3.5 - Prob. 21ES Ch. 3.5 - Prob. 22ES Ch. 3.5 - Prob. 23ES Ch. 3.5 - It is suspected that the high amounts of tannin in...Ch. 3.5 - Prob. 29ES Ch. 3.5 - Prob. 30ES Ch. 3.5 - Prob. 31ES Ch. 3.5 - Prob. 32ES Ch. 3.5 - Let f C2[a, b] and let the nodes a = x0 x1 xn...Ch. 3.5 - Prob. 34ES Ch. 3.5 - Prob. 35ES Ch. 3.6 - Let (x0, y0) = (0,0) and (x1, y1) = (5, 2) be the...Ch. 3.6 - Prob. 2ES Ch. 3.6 - Prob. 5ES Ch. 3.6 - Prob. 1DQ

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The value b , c , d for cubic spline S, on [ 0 , 2 ] .

Solution Summary: The author explains the value of b,c,d for cubic spline S, and the required value for the function.

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The value b,c,d for cubic spline S, on [0,2].

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Chapter 3 Solutions