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65. 66. 67. 68. X f(x) X -2 0 || f(x) X -2012 224 x -1 0 1 2 f(x) 1 223 off f(x) -1 -1 0 2 2 24 1 0 1 3 223

65. 66. 67. 68. X f(x) X -2 0 || f(x) X -2012 224 x -1 0 1 2 f(x) 1 223 off f(x) -1 -1 0 2 2 24 1 0 1 3 223

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please do parts 65-68 please… Use the photo from problem 6 for reference for the work to find the answer to those parts
**Problem IV**: Find a linear polynomial \( f(x) \) which is the best least squares fit to the following data. | | \( x \) | \(-1\) | 0 | 1 | 2 | | |---|----------------|---|---|---|---|---| | 61. | \( f(x) \) | 0 | 2 | 2 | 4 | | 62. | \( f(x) \) | 0 | 2 | 2 | 4 | | 63. | \( f(x) \) | 0 | 2 | 2 | 4 | | 64. | \( f(x) \) | 0 | 2 | 2 | 4 | | 65. | \( f(x) \) | 0 | 2 | 2 | 4 | | 66. | \( f(x) \) | 0 | 2 | 2 | 4 | | 67. | \( f(x) \) | 1 | 2 | 2 | 3 | | 68. | \( f(x) \) | 1 | 2 | 2 | 3 | | 69. | \( f(x) \) | 1 | 2 | 2 | 3 | | 70. | \( f(x) \) | 1 | 2 | 2 | 3 | | | \( x \) | \(-1\) | 0 | 1 | 2 | | |---|----------------|---|---|---|---|---| | 71. | \( f(x) \) | 4 | 2 | 2 | 0 | | 72. | \( f(x) \) | 4 | 2 | 2 | 0 | | 73. | \( f(x) \) | 4 | 2 | 2 | 0 | | 74. | \( f(x) \) | 4 | 2 | 2 | 0 | | 75. | \( f(x) \) | 4 | 2 | 2 | 0 | | 76. | \( f(x) \) | 4 | 2 | 2 | 0 | | 77. | \( f(x)
Transcribed Image Text:**Problem IV**: Find a linear polynomial \( f(x) \) which is the best least squares fit to the following data. | | \( x \) | \(-1\) | 0 | 1 | 2 | | |---|----------------|---|---|---|---|---| | 61. | \( f(x) \) | 0 | 2 | 2 | 4 | | 62. | \( f(x) \) | 0 | 2 | 2 | 4 | | 63. | \( f(x) \) | 0 | 2 | 2 | 4 | | 64. | \( f(x) \) | 0 | 2 | 2 | 4 | | 65. | \( f(x) \) | 0 | 2 | 2 | 4 | | 66. | \( f(x) \) | 0 | 2 | 2 | 4 | | 67. | \( f(x) \) | 1 | 2 | 2 | 3 | | 68. | \( f(x) \) | 1 | 2 | 2 | 3 | | 69. | \( f(x) \) | 1 | 2 | 2 | 3 | | 70. | \( f(x) \) | 1 | 2 | 2 | 3 | | | \( x \) | \(-1\) | 0 | 1 | 2 | | |---|----------------|---|---|---|---|---| | 71. | \( f(x) \) | 4 | 2 | 2 | 0 | | 72. | \( f(x) \) | 4 | 2 | 2 | 0 | | 73. | \( f(x) \) | 4 | 2 | 2 | 0 | | 74. | \( f(x) \) | 4 | 2 | 2 | 0 | | 75. | \( f(x) \) | 4 | 2 | 2 | 0 | | 76. | \( f(x) \) | 4 | 2 | 2 | 0 | | 77. | \( f(x)
**Problem 6.** Find a linear polynomial which is the best least squares fit to the following data: \[ \begin{array}{c|cccc} x & -2 & -1 & 0 & 1 & 2 \\ \hline f(x) & -3 & -2 & 1 & 2 & 5 \\ \end{array} \] We are looking for a function \( f(x) = c_1 + c_2 x \), where \( c_1, c_2 \) are unknown coefficients. The data of the problem give rise to an overdetermined system of linear equations in variables \( c_1 \) and \( c_2 \): \[ \begin{cases} c_1 - 2c_2 = -3, \\ c_1 - c_2 = -2, \\ c_1 = 1, \\ c_1 + c_2 = 2, \\ c_1 + 2c_2 = 5. \end{cases} \] This system is inconsistent. We can represent the system as a matrix equation \( Ac = y \), where \[ A = \begin{pmatrix} 1 & -2 \\ 1 & -1 \\ 1 & 0 \\ 1 & 1 \\ 1 & 2 \\ \end{pmatrix}, \quad c = \begin{pmatrix} c_1 \\ c_2 \\ \end{pmatrix}, \quad y = \begin{pmatrix} -3 \\ -2 \\ 1 \\ 2 \\ 5 \\ \end{pmatrix}. \] The least squares solution \( c \) of the above system is a solution of the normal system \( A^T Ac = A^Ty \): \[ \begin{pmatrix} 1 & 1 & 1 & 1 & 1 \\ -2 & -1 & 0 & 1 & 2 \\ \end{pmatrix} \begin{pmatrix} 1 & -2 \\ 1 & -1 \\ 1 & 0 \\ 1 & 1 \\ 1 & 2 \\ \end{pmatrix} \begin{pmatrix} c_1 \\ c_2 \\ \end{pmatrix} = \begin{
Transcribed Image Text:**Problem 6.** Find a linear polynomial which is the best least squares fit to the following data: \[ \begin{array}{c|cccc} x & -2 & -1 & 0 & 1 & 2 \\ \hline f(x) & -3 & -2 & 1 & 2 & 5 \\ \end{array} \] We are looking for a function \( f(x) = c_1 + c_2 x \), where \( c_1, c_2 \) are unknown coefficients. The data of the problem give rise to an overdetermined system of linear equations in variables \( c_1 \) and \( c_2 \): \[ \begin{cases} c_1 - 2c_2 = -3, \\ c_1 - c_2 = -2, \\ c_1 = 1, \\ c_1 + c_2 = 2, \\ c_1 + 2c_2 = 5. \end{cases} \] This system is inconsistent. We can represent the system as a matrix equation \( Ac = y \), where \[ A = \begin{pmatrix} 1 & -2 \\ 1 & -1 \\ 1 & 0 \\ 1 & 1 \\ 1 & 2 \\ \end{pmatrix}, \quad c = \begin{pmatrix} c_1 \\ c_2 \\ \end{pmatrix}, \quad y = \begin{pmatrix} -3 \\ -2 \\ 1 \\ 2 \\ 5 \\ \end{pmatrix}. \] The least squares solution \( c \) of the above system is a solution of the normal system \( A^T Ac = A^Ty \): \[ \begin{pmatrix} 1 & 1 & 1 & 1 & 1 \\ -2 & -1 & 0 & 1 & 2 \\ \end{pmatrix} \begin{pmatrix} 1 & -2 \\ 1 & -1 \\ 1 & 0 \\ 1 & 1 \\ 1 & 2 \\ \end{pmatrix} \begin{pmatrix} c_1 \\ c_2 \\ \end{pmatrix} = \begin{
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