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The image contains a mathematical exercise involving forward and backward difference formulas to approximate the value of a function.

Here’s a breakdown of what’s visible:

### Problem Statement

1. **Using Backward Formula to find \(\sqrt{12547}\)**
   
   Given dataThe image contains a mathematical exercise involving forward and backward difference formulas to approximate the value of a function.

Here’s a breakdown of what’s visible:

### Problem Statement

1. **Using Backward Formula to find \(\sqrt{12547}\)**
   
   Given 
   \[
   \begin{array}{c|c}
   x & y \\
   \hline
   1 & 10 \\
   4 & 16 \\
   9 & 24 \\
   16 & 36 \\
   25 & 64 \\
   \end{array}
   \]

2. **Using Bessel's Forward Formula to find the value of \(f(0.3)\)**
   \[
   \begin{array}{c|c}
   x & y \\
   \hline
   0.1 & 2.3026 \\
   0.2 & 2.9957 \\
   0.3 & 3.097 \\
   0.4 & 3.1781 \\
   \end{array}
   \]

3. **Using Gauss's Backward Formula for some system (related to square roots)**
   
   It seems there are calculations for different square root values:
   \[
   \sqrt{12500}, \sqrt{12510}, \sqrt{12520}, \sqrt{12530}, \sqrt{12540}
   \]
   
   Calculated values:
   \[
   \sqrt{12500} \approx 111.8034, \\
   \sqrt{12510} \approx 111.8482, \\
   \sqrt{12520} \approx 111.8928, \\
   \sqrt{12530} \approx 111.9374, \\
   \sqrt{12540} \approx 111.9822
   \]

### Analysis

From the problem description, there are three different interpolation formulas being used:
1. **Backward Difference Formula**: Typically used when we are given values of \(f(x)\) at equally spaced points and want to estimate the function at a point closer to the last point.
   
2. **Bessel’s Forward Formula**: This is often used for estimating a function value near the middle of the data set.

3. **Gauss’s Backward Formula**: This is another interpolation formula used for estimating values near the middle but using backward differences.

Would you like a detailed step-by-step solution for any of these specific problems? 

### Related Questions:

1. How do you derive the backward difference formula?
2. What are the steps involved in applying Bessel’s Forward Formula?
3. How is Gauss’s Backward Formula different from Newton’s Backward Formula?
4. Can these interpolation methods be used for extrapolation?
5. How accurate are these interpolation methods compared to direct computation?

**Tip:** When using interpolation formulas, make sure your data points are equally spaced to avoid significant errors in your estimates.

Using Backward Formula to find the square root of 12547. Using Bessel's Forward Formula to find the value of f(0.3). Using Gauss's Backward Formula for square roots.

Explore the application of Backward Difference, Bessel’s Forward, and Gauss’s Backward Formulas to approximate the square roots and function values. This detailed problem covers advanced interpolation concepts, making it suitable for students in Grades 11-12.

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