CHAPTER III - DIFFERENTIAL CALCULUS

Abstract

This chapter discusses differential calculus. Not every continuous function possesses a derivative, i. e. not every curve possesses a tangent at any point. However, there exist continuous functions which do not even have a one-side derivative. It is easily seen that the function f defined by the conditions. A function is called differentiable in an open interval if it has a finite derivative at any point of the interval. According to the geometrical interpretation of the derivative, the tangent to a curve at a point denotes the coordinates on the tangent. Besides the geometrical interpretation the derivative has also important interpretations in physics. In particular, the velocity of the point moving along a straight line is expressed as the derivative of the distance with respect to the time. As regards the relation between the notion of the extremum of a function and the upper and lower bounds of this function one must note first of all that the notion of the extremum is a local notion and the notion of the bound of a function is an integral notion.