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Interpolation and Extrapolation Calculator

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Example
Created on 2024-06-20Asked by Owen Thomas (Solvelet student)
Given the data points (1,2)(1, 2), (2,3)(2, 3), and (3,5)(3, 5), find the linear interpolation at x=2.5 x = 2.5 .

Solution

To find the linear interpolation at x=2.5 x = 2.5 using the data points (1,2)(1, 2), (2,3)(2, 3), and (3,5)(3, 5): We use the linear interpolation formula between the points (2,3)(2, 3) and (3,5)(3, 5): y=y1+(x−x1)(y2−y1)x2−x1 y = y_1 + \frac{(x - x_1)(y_2 - y_1)}{x_2 - x_1} Substitute (x1,y1)=(2,3)(x_1, y_1) = (2, 3) and (x2,y2)=(3,5)(x_2, y_2) = (3, 5): y=3+(2.5−2)(5−3)3−2 y = 3 + \frac{(2.5 - 2)(5 - 3)}{3 - 2} y=3+0.5⋅21 y = 3 + \frac{0.5 \cdot 2}{1} y=3+1=4 y = 3 + 1 = 4 Therefore, the linear interpolation at x=2.5 x = 2.5 is: y=4 y = 4 Solved on Solvelet with Basic AI Model
Some of the related questions asked by Layla Rivera on Solvelet
1. Given the data points (0,1)(0, 1), (1,3)(1, 3), and (2,5)(2, 5), estimate the value of yy when x=1.5x = 1.5 using linear interpolation,2. Find the matrix product AB\mathbf{AB} for the matrices A=[12−1302]\mathbf{A} = \begin{bmatrix} 1 & 2 & -1 \\ 3 & 0 & 2 \end{bmatrix} and B=[2−14310]\mathbf{B} = \begin{bmatrix} 2 & -1 \\ 4 & 3 \\ 1 & 0 \end{bmatrix}.,
DefinitionIn terms of estimation of unknown values from the data points, interpolation and extrapolation are implemented. When interpolation is the estimation of the values between the data, then extrapolation is the estimation of the data outside the range. They are in high-usage for making predictions, filling in missing data, and in decision making in the realms of data analytics, numerical analysis, and scientific studies. Example: Linear interpolation implements the formula y=y1​+(x−x1​)x2​−x1​y2​−y1​​ for y values between known data points (x1​,y1​) and (x2​,y2​). For example, extrapolation can predict future values based on past trends by applying a fitted curve.
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