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Given an exponentially decaying function $f(x) = e^{-kx}$ that

  • passes through the points $\vec a_0= (x_0, y_0)$ and $\vec a_2 = (x_2, y_2)$
  • such that $x_0 < x_2$ and $y_0 > y_2$,

what is the point $\vec a_1 = (x_1, y_1)$ (where $x_0 < x_1 < x_2$) such that the piecewise linear function passing through points $a$ line segments $\vec a_0 \rightarrow \vec a_1 \rightarrow \vec a_2$ are as close to the curve as possible, that is, that minimizes the integral of the absolute vertical difference between the line segments and $f(x)$?

(At first, I assumed that the $\vec a_1$ would be on the curve, but I don't think it is…)

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