CHAPTER I. DEFINITIONS. LIMITS. 1. Primary Object of the Differential Calculus. In Nature we frequently meet with quantities which, if observed for some period of time, are found to undergo increase or decrease; for instance, the distance of a moving particle from a known fixed point in its path, the length of a moving ordinate of a given curve, the force exerted upon a piece of soft iron which is gradually made to approach one of the poles of a magnet. When such quantities are made the subject of mathematical investigation, it often becomes necessary to estimate their rates of growth. This is the primary object of the Differential Calculus. 2. In the first six chapters we shall be concerned with the description of an instrument for the measurement of such rates, and in framing rules for its formation and use, and the student must make himself as proficient as possible in its manipulation. These chapters contain the whole machinery of the Differential Calculus. The remaining chapters simply consist of various applications of the methods and formulae here established. A 3. We commence with an explanation of several technical terms which are of frequent occurrence in this subject, and with the meanings of which the student should be familiar from the outset. 4. Constants and Variables. A CONSTANT is a quantity which, during any set of mathematical operations, retains the same value. A VARIABLE is a quantity which, during any set of mathematical operations, does not retain the same value, but is capable of assuming different values. Ex. The area of any triangle on a given base and between given parallels is a constant quantity; so also the base, the distance between the parallel lines, the sum of the angles of the triangle are constant quantities. But the separate angles, the sides, the position of the vertex are variables. It has become conventional to make use of the letters a, b, c, , a, ß, y, . . ., from the beginning of the alphabet to denote constants; and to retain later letters, such as u, v, w, x, y, z, and the Greek letters §, 7, §, for variables. 5. Dependent and Independent Variables. AN INDEPENDENT VARIABLE is one which take up any arbitrary value that may be assigned to it. may A DEPENDENT VARIABLE is one which assumes its value in consequence of some second variable or system of variables taking up any set of arbitrary values that may be assigned to them. 6. Functions. When one quantity depends upon another or upon a system of others, so that it assumes a definite value |