Linear Interpolation - Linear Interpolation As Approximation

Linear Interpolation As Approximation

Linear interpolation is often used to approximate a value of some function f using two known values of that function at other points. The error of this approximation is defined as

where p denotes the linear interpolation polynomial defined above

It can be proven using Rolle's theorem that if f has a continuous second derivative, the error is bounded by

As you see, 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.