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| <title>Newton's Divided Difference Interpolation</title> | |
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| <p>Given the values:</p> | |
| <div class="table-container"> | |
| <table class="data-table"> | |
| <thead> | |
| <tr> | |
| <th>\(x\)</th> | |
| <th>\(f(x)\)</th> | |
| </tr> | |
| </thead> | |
| <tbody> | |
| <tr> | |
| <td>5</td> | |
| <td>150</td> | |
| </tr> | |
| <tr> | |
| <td>7</td> | |
| <td>392</td> | |
| </tr> | |
| <tr> | |
| <td>11</td> | |
| <td>1452</td> | |
| </tr> | |
| <tr> | |
| <td>13</td> | |
| <td>2366</td> | |
| </tr> | |
| <tr> | |
| <td>17</td> | |
| <td>5202</td> | |
| </tr> | |
| </tbody> | |
| </table> | |
| </div> | |
| <p>We need to evaluate \( f(9) \) using Newton’s divided difference formula.</p> | |
| <p>First, we compute the divided differences:</p> | |
| <ol> | |
| <li> | |
| <p><b>First divided differences:</b></p> | |
| <div class="equation"> | |
| \[ | |
| \begin{aligned} | |
| f[5, 7] &= \frac{392 - 150}{7 - 5} = 121, \\ | |
| f[7, 11] &= \frac{1452 - 392}{11 - 7} = 265, \\ | |
| f[11, 13] &= \frac{2366 - 1452}{13 - 11} = 457, \\ | |
| f[13, 17] &= \frac{5202 - 2366}{17 - 13} = 709. | |
| \end{aligned} | |
| \] | |
| </div> | |
| </li> | |
| <li> | |
| <p><b>Second divided differences:</b></p> | |
| <div class="equation"> | |
| \[ | |
| \begin{aligned} | |
| f[5, 7, 11] &= \frac{265 - 121}{11 - 5} = 24, \\ | |
| f[7, 11, 13] &= \frac{457 - 265}{13 - 7} = 32, \\ | |
| f[11, 13, 17] &= \frac{709 - 457}{17 - 11} = 42. | |
| \end{aligned} | |
| \] | |
| </div> | |
| </li> | |
| <li> | |
| <p><b>Third divided differences:</b></p> | |
| <div class="equation"> | |
| \[ | |
| \begin{aligned} | |
| f[5, 7, 11, 13] &= \frac{32 - 24}{13 - 5} = 1, \\ | |
| f[7, 11, 13, 17] &= \frac{42 - 32}{17 - 7} = 1. | |
| \end{aligned} | |
| \] | |
| </div> | |
| </li> | |
| <li> | |
| <p><b>Fourth divided difference:</b></p> | |
| <div class="equation"> | |
| \[ | |
| f[5, 7, 11, 13, 17] &= \frac{1 - 1}{17 - 5} = 0. | |
| \] | |
| </div> | |
| </li> | |
| </ol> | |
| <p>Using these divided differences, the interpolating polynomial is constructed as:</p> | |
| <div class="equation"> | |
| \[ | |
| P(x) = 150 + 121(x - 5) + 24(x - 5)(x - 7) + 1(x - 5)(x - 7)(x - 11) | |
| \] | |
| </div> | |
| <p>Evaluating this polynomial at \( x = 9 \):</p> | |
| <div class="equation"> | |
| \[ | |
| \begin{aligned} | |
| P(9) &= 150 + 121(9 - 5) + 24(9 - 5)(9 - 7) + 1(9 - 5)(9 - 7)(9 - 11) \\ | |
| &= 150 + 121 \cdot 4 + 24 \cdot 4 \cdot 2 + 1 \cdot 4 \cdot 2 \cdot (-2) \\ | |
| &= 150 + 484 + 192 - 16 \\ | |
| &= 810. | |
| \end{aligned} | |
| \] | |
| </div> | |
| <p>Thus, the value of \( f(9) \) is:</p> | |
| <div class="boxed-answer"> | |
| \[ | |
| \boxed{810} | |
| \] | |
| </div> | |
| </body> | |
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