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To find \( e^{1.17} \) using the **Gauss forward interpolation formula**, let's go step by step. The given table of \( x \) and \( e^x \) values will be used to calculate this.

### Table of values:
\[
\begin{array}{|c|c|}
\hline
x & e^x \\
\hline
1.00 & 2.7183 \\
1.05 & 2.8577 \\
1.10 & 3.0042 \\
1.15 & 3.1582 \\
1.20 & 3.3201 \\
1.25 & 3.4903 \\
1.30 & 3.6693 \\
\hline
\end{array}
\]

The **Gauss forward interpolation formula** is:
\[
f(x) = f_0 + p \Delta f_0 + \frac{p(p-1)}{2!} \Delta^2 f_0 + \frac{p(p-1)(p-2)}{3!} \Delta^3 f_0 + \dots
\]
Where:
- \( p = \frac{x - x_0}{h} \)
- \( h \) is the step size, which is \( h = 1.05 - 1.00 = 0.05 \).
- \( f_0 \) corresponds to \( x_0 = 1.15 \) (the nearest value to \( 1.17 \)).

We will calculate the forward differences \( \Delta f \), \( \Delta^2 f \), etc., and then apply the formula.

Let me perform the calculations for this.The value of \( e^{1.17} \) using the Gauss forward interpolation formula is approximately:

\[
e^{1.17} \approx 3.2220
\]

Here is the difference table used in the calculation:

\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & \Delta^0 f & \Delta^1 f & \Delta^2 f & \Delta^3 f & \Delta^4 f & \Delta^5 f \\
\hline
1.00 & 2.7183 & 0.1394 & 0.0071 & 0.0004 & 0.0000 & -0.0000 \\
1.05 & 2.8577 & 0.1465 & 0.0075 & 0.0004 & -0.0000 & 0.0001 \\
1.10 & 3.0042 & 0.1540 & 0.0079 & 0.0004 & 0.0001 & 0.0000 \\
1.15 & 3.1582 & 0.1619 & 0.0083 & 0.0005 & 0.0000 & 0.0000 \\
1.20 & 3.3201 & 0.1702 & 0.0088 & 0.0000 & 0.0000 & 0.0000 \\
1.25 & 3.4903 & 0.1790 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
1.30 & 3.6693 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
\hline
\end{array}
\]

Would you like more details on the interpolation process or further explanation of the differences? 

### Follow-up questions:
1. How does the Gauss forward formula compare to other interpolation methods?
2. What is the significance of the forward differences in interpolation?
3. Can the Gauss backward formula be used here, and what would change?
4. How does changing the step size \( h \) affect the accuracy of the interpolation?
5. What are some applications of interpolation methods in real-life scenarios?

**Tip**: When working with interpolation, choosing the closest reference point helps minimize error and improve accuracy.

Find e^1.17 using Gauss forward formula from the provided table of x and e^x values.

Learn how to apply the Gauss forward interpolation formula to calculate e^1.17 from a table of x and e^x values. This solution explains step-by-step how to use forward differences and the interpolation formula, suitable for students studying numerical methods.

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