To approximate \( f(x) = 3^{\sqrt{2}} \) to three decimal places, you want to choose a value of \( x \) that will yield \( \sqrt{2} \) closely, as \( f(x) = 3^{\sqrt{2}} \).
The values given for \( x \) are approximations of \( \sqrt{2} \):
- \( 1 \)
- \( 1.4 \)
- \( 1.41 \)
- \( 1.414 \)
- \( 1.4142 \)
- \( 1.41421 \)
- \( 1.414213 \)
Among these, the value \( 1.414213 \) is the most accurate approximation of \( \sqrt{2} \) (which is approximately \( 1.41421356 \ldots \)).
To find \( f(x) = 3^{\sqrt{2}} \), you can use \( x = 1.414213 \). This would give you the closest approximation of \( f(x) \) to three decimal places.
Thus, you should use 1.414213 for your sequence to approximate \( f(x) = 3^{\sqrt{2}} \) to three decimal places.
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