Apply Gauss forward formula to obtain the value of f(x) at x = 3.5 from the tabler

X

2

3

4

5

f(x)

3.818

2.423

-1.027

-2.794

To apply the Gauss forward formula for interpolation, we need to estimate the value of \( f(x) \) at \( x = 3.5 \) using the given table of values. Here’s the step-by-step application of the Gauss forward formula.

### Step 1: Set up the data

Given the table:

\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
2 & 3.818 \\
3 & 2.423 \\
4 & -1.027 \\
5 & -2.794 \\
\hline
\end{array}
\]

### Step 2: Calculate forward differences

Compute the forward differences \(\Delta f\) up to the third order:

\[
\Delta f(x) = f(x+1) - f(x)
\]

Let’s calculate each forward difference:

\[
\begin{array}{c|cccc}
x & f(x) & \Delta f & \Delta^2 f & \Delta^3 f \\
\hline
2 & 3.818 & -1.395 & -2.055 & -0.712 \\
3 & 2.423 & -3.450 & -2.767 & \\
4 & -1.027 & -1.767 & & \\
5 & -2.794 & & & \\
\end{array}
\]

### Step 3: Gauss forward formula

The Gauss forward formula is given by:

\[
f(x) = f_0 + p \Delta f_0 + \frac{p(p-1)}{2!} \Delta^2 f_0 + \frac{p(p-1)(p-2)}{3!} \Delta^3 f_0 + \ldots
\]

where:
- \( f_0 = f(3) = 2.423 \) (taking the nearest point as the reference)
- \( p = \frac{x - x_0}{h} = \frac{3.5 - 3}{1} = 0.5 \)

Now substitute the values:

\[
f(3.5) = 2.423 + (0.5)(-3.450) + \frac{(0.5)(-0.5)}{2}(-2.767) + \frac{(0.5)(-0.5)(-1.5)}{6}(-0.712)
\]

Calculating each term:

\[
f(3.5) = 2.423 - 1.725 + 0.346 - 0.089
\]

\[
f(3.5) \approx 0.955
\]

### Final Answer

The interpolated value of \( f(3.5) \) using the Gauss forward formula is approximately:

\[
\boxed{0.955}
\]

Would you like to go into details or have any questions? Here are some related questions:

1. How is Gauss backward formula different from Gauss forward formula?
2. Can you apply Newton's forward interpolation method to this data?
3. What are the limitations of using interpolation formulas?
4. How would the result differ if using Lagrange interpolation?
5. How to calculate higher order differences in interpolation?

**Tip:** For more accurate results, ensure the data points are evenly spaced before applying interpolation formulas.

Apply Gauss forward formula to obtain the value of f(x) at x = 3.5 from the table where f(x) is given at x = 2, 3, 4, and 5.

Learn how to apply the Gauss forward formula to estimate f(x) at x = 3.5 using a given set of values at x = 2, 3, 4, and 5. This problem covers interpolation techniques and the calculation of finite differences. Suitable for students in Grades 11-12.

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