Antiderivatives

Abstract

The derivative of a function y=F(x) describes its local rate of change, i.e., the change Δy of the y-value with respect to the change Δx of the x-value in the limit Δx→0. Conversely, the question about the reconstruction of a function F from its local rate of change f leads to the notion of indefinite integrals, which comprises the totality of all functions that have f as their derivative, the antiderivatives of f. Chapter 10 addresses this notion, its properties, some basic examples and applications.

By multiplying the rate of change f(x) with the change Δx one obtains an approximation to the change of the values of the function of the antiderivative F in the segment of length Δx. Adding up these local changes in an interval, for instance between x=a and x=b in steps of length Δx, gives an approximation to the total change F(b)−F(a). The limit Δx→0 (with an appropriate increase of the number of summands) leads to the notion of the definite integral of f in the interval [a,b], which is the subject of Chap. 11.