Sure, let's complete the table of values step by step:
Given the function [tex]\( f(x) = 2 e^{-x / 3} \)[/tex], we need to compute the values at specific points [tex]\( x \)[/tex]. We've calculated these values and rounded them to two decimal places.
### Calculating [tex]\( f(x) \)[/tex]:
- For [tex]\( x = -3 \)[/tex]:
[tex]\[
f(-3) = 2 e^{-(-3) / 3} = 2 e^{1} = 5.44
\][/tex]
- For [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = 2 e^{-(-2) / 3} = 2 e^{2/3} = 3.90
\][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) = 2 e^{-(-1) / 3} = 2 e^{1/3} = 2.79
\][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 2 e^{-0 / 3} = 2 e^{0} = 2.00
\][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 2 e^{-1 / 3} = 2 e^{-1/3} = 1.43
\][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 2 e^{-2 / 3} = 2 e^{-2/3} = 1.03
\][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[
f(3) = 2 e^{-3 / 3} = 2 e^{-1} = 0.74
\][/tex]
### Completing the Table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x)=2 e^{-x / 3} \\
\hline
-3 & 5.44 \\
\hline
-2 & 3.90 \\
\hline
-1 & 2.79 \\
\hline
0 & 2.00 \\
\hline
1 & 1.43 \\
\hline
2 & 1.03 \\
\hline
3 & 0.74 \\
\hline
\end{array}
\][/tex]
So, the complete table is:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x)=2 e^{-x / 3} \\
\hline
-3 & 5.44 \\
\hline
-2 & 3.90 \\
\hline
-1 & 2.79 \\
\hline
0 & 2.00 \\
\hline
1 & 1.43 \\
\hline
2 & 1.03 \\
\hline
3 & 0.74 \\
\hline
\end{array}
\][/tex]
