To complete the table, we need to determine the value of [tex]\( y \)[/tex] for each given [tex]\( x \)[/tex] according to the formula:
[tex]\[ y = \left(1 + \frac{1}{x}\right)^x \][/tex]
We are given the values to fill in as follows:
- For [tex]\( x = 1 \)[/tex], [tex]\( y = 2 \)[/tex] (this is already provided).
Now let's determine the values of [tex]\( y \)[/tex] for other given [tex]\( x \)[/tex] values.
### When [tex]\( x = 10 \)[/tex]:
[tex]\[ y = \left(1 + \frac{1}{10}\right)^{10} \][/tex]
[tex]\[ y \approx 2.594 \][/tex]
So, [tex]\( a \approx 2.594 \)[/tex].
### When [tex]\( x = 100 \)[/tex]:
[tex]\[ y = \left(1 + \frac{1}{100}\right)^{100} \][/tex]
[tex]\[ y \approx 2.705 \][/tex]
So, [tex]\( b \approx 2.705 \)[/tex].
### When [tex]\( x = 10,000 \)[/tex]:
[tex]\[ y = \left(1 + \frac{1}{10,000}\right)^{10,000} \][/tex]
[tex]\[ y \approx 2.718 \][/tex]
So, [tex]\( c \approx 2.718 \)[/tex].
### When [tex]\( x = 100,000 \)[/tex]:
[tex]\[ y = \left(1 + \frac{1}{100,000}\right)^{100,000} \][/tex]
[tex]\[ y \approx 2.718 \][/tex]
So, [tex]\( d \approx 2.718 \)[/tex].
### When [tex]\( x = 1,000,000 \)[/tex]:
[tex]\[ y = \left(1 + \frac{1}{1,000,000}\right)^{1,000,000} \][/tex]
[tex]\[ y \approx 2.718 \][/tex]
So, [tex]\( e \approx 2.718 \)[/tex].
Filling these values into the table, we get:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
1 & 2 \\
\hline
10 & 2.594 \\
\hline
100 & 2.705 \\
\hline
10,000 & 2.718 \\
\hline
100,000 & 2.718 \\
\hline
1,000,000 & 2.718 \\
\hline
\end{tabular}
\][/tex]
Therefore:
- [tex]\( a \approx 2.594 \)[/tex]
- [tex]\( b \approx 2.705 \)[/tex]
- [tex]\( c \approx 2.718 \)[/tex]
- [tex]\( d \approx 2.718 \)[/tex]
- [tex]\( e \approx 2.718 \)[/tex]
