Let's analyze the table step by step, filling in the First Difference and Ratios calculations.
Given values for \( x \) and \( f(x) \):
\( x: [0, 5, 10, 15, 20, 25] \)
\( f(x): [432, 454, 499, 534, 582, 611] \)
1. First Differences:
The First Difference at each point is the difference between the value of \( f(x) \) at that point and the value of \( f(x) \) at the previous point.
- For \( x = 5 \):
[tex]\[
454 - 432 = 22
\][/tex]
- For \( x = 10 \):
[tex]\[
499 - 454 = 45
\][/tex]
- For \( x = 15 \):
[tex]\[
534 - 499 = 35
\][/tex]
- For \( x = 20 \):
[tex]\[
582 - 534 = 48
\][/tex]
- For \( x = 25 \):
[tex]\[
611 - 582 = 29
\][/tex]
2. Ratios:
The Ratio at each point is the value of \( f(x) \) at that point divided by the value of \( f(x) \) at the previous point.
- For \( x = 5 \):
[tex]\[
\frac{454}{432} \approx 1.0509259259259258
\][/tex]
- For \( x = 10 \):
[tex]\[
\frac{499}{454} \approx 1.0991189427312775
\][/tex]
- For \( x = 15 \):
[tex]\[
\frac{534}{499} \approx 1.0701402805611222
\][/tex]
- For \( x = 20 \):
[tex]\[
\frac{582}{534} \approx 1.0898876404494382
\][/tex]
- For \( x = 25 \):
[tex]\[
\frac{611}{582} \approx 1.0498281786941581
\][/tex]
Here is the fully filled table with the calculations:
\begin{tabular}{|c|c|c|c|}
\hline
[tex]$x$[/tex] & [tex]$f(x)$[/tex] & First Difference & Ratios \\
\hline
0 & 432 & & \\
\hline
5 & 454 & [tex]$22$[/tex] & [tex]$1.0509259259259258$[/tex] \\
\hline
10 & 499 & [tex]$45$[/tex] & [tex]$1.0991189427312775$[/tex] \\
\hline
15 & 534 & [tex]$35$[/tex] & [tex]$1.0701402805611222$[/tex] \\
\hline
20 & 582 & [tex]$48$[/tex] & [tex]$1.0898876404494382$[/tex] \\
\hline
25 & 611 & [tex]$29$[/tex] & [tex]$1.0498281786941581$[/tex] \\
\hline
\end{tabular}
