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- <!doctype html>
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  <html>
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- <head>
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- <meta charset="utf-8" />
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- <meta name="viewport" content="width=device-width" />
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- <title>My static Space</title>
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- <link rel="stylesheet" href="style.css" />
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- </head>
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- <body>
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- <div class="card">
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- <h1>Welcome to your static Space!</h1>
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- <p>You can modify this app directly by editing <i>index.html</i> in the Files and versions tab.</p>
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- <p>
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- Also don't forget to check the
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- <a href="https://huggingface.co/docs/hub/spaces" target="_blank">Spaces documentation</a>.
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- </p>
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- </div>
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- </body>
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- </html>
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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+ <!DOCTYPE html>
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  <html>
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+ <head>
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+ <title>Newton's Divided Difference Interpolation</title>
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+ <script src="https://polyfill.io/v3/polyfill.min.js?presets=full"></script>
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+ <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
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+ <style>
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+ body {
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+ font-family: Arial, sans-serif;
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+ }
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+ .equation {
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+ display: block;
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+ margin-left: auto;
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+ margin-right: auto;
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+ text-align: center;
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+ margin-top: 1em;
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+ margin-bottom: 1em;
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+ }
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+ .table-container {
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+ display: flex;
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+ justify-content: center;
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+ margin-top: 1em;
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+ margin-bottom: 1em;
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+ }
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+ .data-table {
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+ border-collapse: collapse;
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+ }
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+ .data-table th, .data-table td {
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+ border: 1px solid black;
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+ padding: 8px;
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+ text-align: center;
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+ }
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+ .boxed-answer {
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+ border: 1px solid black;
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+ padding: 10px;
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+ display: inline-block;
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+ margin-top: 1em;
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+ }
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+ ol li {
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+ margin-bottom: 0.5em;
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+ }
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+ </style>
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+ </head>
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+ <body>
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+
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+ <p>Given the values:</p>
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+
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+ <div class="table-container">
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+ <table class="data-table">
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+ <thead>
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+ <tr>
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+ <th>\(x\)</th>
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+ <th>\(f(x)\)</th>
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+ </tr>
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+ </thead>
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+ <tbody>
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+ <tr>
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+ <td>5</td>
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+ <td>150</td>
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+ </tr>
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+ <tr>
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+ <td>7</td>
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+ <td>392</td>
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+ </tr>
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+ <tr>
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+ <td>11</td>
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+ <td>1452</td>
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+ </tr>
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+ <tr>
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+ <td>13</td>
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+ <td>2366</td>
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+ </tr>
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+ <tr>
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+ <td>17</td>
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+ <td>5202</td>
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+ </tr>
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+ </tbody>
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+ </table>
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+ </div>
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+
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+ <p>We need to evaluate \( f(9) \) using Newton’s divided difference formula.</p>
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+
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+ <p>First, we compute the divided differences:</p>
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+
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+ <ol>
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+ <li>
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+ <p><b>First divided differences:</b></p>
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+ <div class="equation">
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+ \[
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+ \begin{aligned}
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+ f[5, 7] &= \frac{392 - 150}{7 - 5} = 121, \\
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+ f[7, 11] &= \frac{1452 - 392}{11 - 7} = 265, \\
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+ f[11, 13] &= \frac{2366 - 1452}{13 - 11} = 457, \\
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+ f[13, 17] &= \frac{5202 - 2366}{17 - 13} = 709.
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+ \end{aligned}
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+ \]
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+ </div>
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+ </li>
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+ <li>
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+ <p><b>Second divided differences:</b></p>
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+ <div class="equation">
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+ \[
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+ \begin{aligned}
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+ f[5, 7, 11] &= \frac{265 - 121}{11 - 5} = 24, \\
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+ f[7, 11, 13] &= \frac{457 - 265}{13 - 7} = 32, \\
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+ f[11, 13, 17] &= \frac{709 - 457}{17 - 11} = 42.
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+ \end{aligned}
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+ \]
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+ </div>
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+ </li>
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+ <li>
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+ <p><b>Third divided differences:</b></p>
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+ <div class="equation">
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+ \[
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+ \begin{aligned}
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+ f[5, 7, 11, 13] &= \frac{32 - 24}{13 - 5} = 1, \\
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+ f[7, 11, 13, 17] &= \frac{42 - 32}{17 - 7} = 1.
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+ \end{aligned}
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+ \]
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+ </div>
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+ </li>
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+ <li>
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+ <p><b>Fourth divided difference:</b></p>
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+ <div class="equation">
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+ \[
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+ f[5, 7, 11, 13, 17] &= \frac{1 - 1}{17 - 5} = 0.
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+ \]
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+ </div>
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+ </li>
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+ </ol>
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+
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+ <p>Using these divided differences, the interpolating polynomial is constructed as:</p>
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+ <div class="equation">
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+ \[
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+ P(x) = 150 + 121(x - 5) + 24(x - 5)(x - 7) + 1(x - 5)(x - 7)(x - 11)
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+ \]
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+ </div>
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+
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+ <p>Evaluating this polynomial at \( x = 9 \):</p>
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+ <div class="equation">
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+ \[
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+ \begin{aligned}
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+ P(9) &= 150 + 121(9 - 5) + 24(9 - 5)(9 - 7) + 1(9 - 5)(9 - 7)(9 - 11) \\
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+ &= 150 + 121 \cdot 4 + 24 \cdot 4 \cdot 2 + 1 \cdot 4 \cdot 2 \cdot (-2) \\
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+ &= 150 + 484 + 192 - 16 \\
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+ &= 810.
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+ \end{aligned}
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+ \]
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+ </div>
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+
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+ <p>Thus, the value of \( f(9) \) is:</p>
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+ <div class="boxed-answer">
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+ \[
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+ \boxed{810}
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+ \]
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+ </div>
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+
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+ </body>
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+ </html>