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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please do parts 65-68 please…
Use the photo from problem 6 for reference for the work to find the answer to those parts
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{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b19493d-2e0b-4edb-977e-29215d3bb0dd%2F34862aa7-c4b3-4b82-87d9-f1918eac9b6c%2F5t2bhhn_processed.jpeg&w=3840&q=75)
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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