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<!DOCTYPE html>
<html>
<head>
<title>Newton's Divided Difference Interpolation</title>
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<body>
<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>
</html> |