Math Problem Statement
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
Solution
To apply the Gauss forward formula for interpolation, we need to estimate the value of at 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:
\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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Interpolation
Gauss Forward Formula
Finite Differences
Formulas
Gauss forward formula: f(x) = f0 + pΔf0 + (p(p−1)/2!)Δ^2f0 + (p(p−1)(p−2)/3!)Δ^3f0
Forward difference: Δf(x) = f(x+1) − f(x)
Theorems
Interpolation Theorem
Finite Difference Theorem
Suitable Grade Level
Grades 11-12