form following table find the value of e^-1.17 using Grayss forward formula in answers

To find the value of \( e^{-1.17} \) using Gregory-Newton Forward Interpolation Formula, we need a table of function values at certain points. The Gregory-Newton Forward formula is generally used for evenly spaced data points. Here's the formula:

\[
f(x) = f(x_0) + u\Delta f(x_0) + \frac{u(u-1)}{2!}\Delta^2 f(x_0) + \frac{u(u-1)(u-2)}{3!}\Delta^3 f(x_0) + \ldots
\]

where:
- \( u = \frac{x - x_0}{h} \)
- \( h \) is the step size between the data points
- \( \Delta f(x_0), \Delta^2 f(x_0), \) etc., are forward differences.

To proceed with finding \( e^{-1.17} \), please provide the table of values (e.g., values of \( x \) and corresponding function values \( f(x) \)), or let me know if you need help creating one.

Would you like me to guide you through the process step-by-step if you provide the necessary data? 

### Related Questions
1. How do you calculate forward differences for interpolation?
2. What is the significance of the Gregory-Newton Forward Interpolation Formula?
3. How do you determine the step size \( h \) in interpolation problems?
4. What are the applications of interpolation formulas in real-life scenarios?
5. Can backward interpolation be used similarly, and how does it differ?

**Tip:** Always ensure that the data points are evenly spaced when applying the Gregory-Newton Forward Interpolation Formula for accurate results.

Learn how to use Gregory-Newton Forward Interpolation to calculate the value of e^-1.17. This solution includes key formulas and detailed steps, covering interpolation methods, forward differences, and exponential functions. Suitable for undergraduate students studying numerical methods.

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