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, B, 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 §, n, §, for variables. 5. Dependent and Independent Variables. AN INDEPENDENT VARIABLE is one which may take up An any arbitrary value that may be assigned to it. 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 when a system of definite values is given to the others, it is called a FUNCTION of those others. The function itself is a dependent variable, and the variables to which values are given are independent variables. The usual notation to express that one variable y is a function of another x is y=f(x); or y=F(x), or y=p(x); the letters f), F( ), p( ), x( ), .. being generally retained to represent functions of arbitrary or unknown form. If u be an arbitrary or unknown function of several variables x, y, z, we may express the fact by the equation u = f(x, y, z). Ex. In any triangle, two of whose sides are x and y and the included angle 0, we have A=xy sin 0 to express the area. Here A is the dependent variable, and is a function of known form-of x, y, and 0, which are the independent variables. 7. It will be seen that we could write the same equation in other forms, which may be regarded as an expression for sin in terms of the area and two sides; so that now sin may be regarded as the dependent variable, while ▲, x, y, are independent variables. And it is clear that if there be one equation between four variables, as above, it is sufficient to determine one in terms of the other three, so that any one variable may be regarded as dependent and the others as independent. This may be extended. For, if there be one equation between n variables, it will suffice to find one of them in terms of the remaining (n-1), so that any one variable can be considered dependent and the remaining (n−1) independent. And, further, if there be equations connecting n variables (n being greater than ) they will be enough to determiner of the variables in terms of the other n Դ variables, so that any r of the variables can be considered dependent, while the remaining (n-r) are independent. 8. Explicit and Implicit Functions. A function is said to be EXPLICIT when expressed directly in terms of the independent variable or variables. For example, if z=x2, or z=r sin 0, or 2=x2y, 2 is expressed directly in terms of the independent variables, and is therefore in each of the above cases said to be an explicit function of those variables. But, if the function be not expressed directly in terms of the independent variable (or variables), the function is said to be IMPLICIT. or or or Y ax2+2bxy+cy2+2dx+2ey+ƒ=0; in each case is said to be an implicit function of x. Sometimes, however, we can first equation we can write as y to be an explicit function of x. solve the equation for y: e.g., the b-ax2 and in this form is said y X It appears then that if the equation connecting the variables be solved for the dependent variable, that variable is reduced from being an implicit to being an explicit function of the remaining variable or variables. Such solution is not, however, always possible or convenient. 9. Species of Known Functions. Functions which are made up of powers of variables and constants connected by the signs + Xare classed as algebraic functions. If radical signs or fractional indices occur in the function, it is said to be irrational; if not, rational. All other functions are classed as transcendental functions. Of transcendental functions, sines, cosines, tangents, etc., are called trigonometrical or circular functions. Functions such as sinx, tan-1x, etc., are called inverse. trigonometrical functions. 1 Functions such as e, a2, in which the variable occurs in the index, are called exponential functions. While if logarithms are involved, as for instance in loge or log(a+b), etc., the function is called log arithmic. Besides the above we have the hyperbolic functions, sinh x, cosh x, etc., of which a short description follows in Art. 25. 10. Limit of a function. DEF. When a function can be made to approach continually to equality with some fixed value or condition so as to differ from it by less than any assignable quantity, however small, by making the independent variable or variables approach some assigned value or values, that fixed value or condition is called the LIMIT of the function for the value or values of the variable or variables referred to. |