Numerical  Differentiation

This chapter builds on the concepts introduced in the previous chapter on interpolation. Many of the same analytical tools are used here. In interpolation, the goal was to estimate a function f(x) known only at a few discrete points. In this chapter, the focus shifts to approximating the derivatives of such a function as well as its integral over an interval:

Problems of this type arise, for example, when the position of a particle is known at regular time intervals and one wishes to determine its velocity. The derivative of the position, known only at a few points, must be approximated. Similarly, the particle’s acceleration requires computation of the second derivative.

Conversely, if the velocity of a particle is known at certain time intervals, the distance traveled can be obtained by integrating the velocity over the interval [x0, xn].

As discussed in the previous chapter, the function f(x) can be approximated by a polynomial of degree n, with a certain error term. In concise form:

f(x) = pn(x) + En(x)

where En(x) is the error term of order (n + 1). This decomposition forms the foundation for the numerical differentiation and integration techniques developed in this chapter.