Math Problem Statement
Solution
To find using the Gauss forward interpolation formula, let's go step by step. The given table of and values will be used to calculate this.
Table of values:
\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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Numerical Methods
Interpolation
Gauss Forward Interpolation
Formulas
Gauss Forward Interpolation Formula: f(x) = f0 + pΔf0 + (p(p-1)/2!)Δ²f0 + (p(p-1)(p-2)/3!)Δ³f0 + ...
Forward differences: Δf, Δ²f, etc.
Step size h = x₁ - x₀
Theorems
Gauss Interpolation Theorem
Suitable Grade Level
Undergraduate or Higher Secondary Level
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