Editing Linear interpolation (section)

==Linear interpolation as an approximation==

Linear interpolation is often used to approximate a value of some [[Function (mathematics)|function]] {{mvar|f}} using two known values of that function at other points. The ''error'' of this approximation is defined as
<math display="block">R_T = f(x) - p(x),</math>
where {{mvar|p}} denotes the linear interpolation [[polynomial]] defined above:
<math display="block">p(x) = f(x_0) + \frac{f(x_1) - f(x_0)}{x_1 - x_0}(x - x_0).</math>

It can be proven using [[Rolle's theorem]] that if {{mvar|f}} has a continuous second derivative, then the error is bounded by
<math display="block">|R_T| \leq \frac{(x_1 - x_0)^2}{8} \max_{x_0 \leq x \leq x_1} \left|f''(x)\right|.</math>

That is, the approximation between two points on a given function gets worse with the second derivative of the function that is approximated. This is intuitively correct as well: the "curvier" the function is, the worse the approximations made with simple linear interpolation become.