In interpolation we find a polynomial that passes through the points $x_0<x_1<\cdots<x_n$ and estimates $x\in[x_0,x_n]$, so we say the interpolation polynomial interpolates the points. But as far as I know extrapolation doesn't extrapolate but increase the accuracy of a formula that has the error term $$ M = N(h) + K_1 h + K_2 h^2 + K_3 h^3 +\cdots $$ where the $N(h)$ is our formula and $M$ is the exact solution and $K_i$'s are some constants. Why is extrapolation called extrapolation in spite of there seems be no connection between the method and its name?
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2Online etymology dictionary says the word changed meaning. – Karolis Juodelė Jun 18 '14 at 05:50
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Because you aren't familiar with enough uses of "extrapolate"? See
Interpolate: to infer values between given values
Extrapolate: to infer values outside the range of given values
Eric Towers
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