DIFFERENTIAL CALCULUS.

FIRST PART.

GENERAL PRINCIPLES AND ANALYTICAL APPLICATIONS.

CHAPTER I.

FUNCTIONS.

1. IF any two quantities are so connected that any variation in the magnitude of the one implies a corresponding variation in the magnitude of the other, either of these quantities is said to be a function of the other. Such a connection is expressed algebraically by means of an equation involving the symbols of the two quantities and any other symbols of invariable magnitudes. Thus x and y are functions the one of the other in the equation

where a and b are supposed to represent invariable magnitudes.

2. If y be expressed directly in terms of x, as in (2), it is said to be an explicit function of x; if it be merely connected

B

with x by an unsolved equation, as in (1), it is said to be an implicit function of x.

3. The connection between x and y may be expressed generally by an equation

where F(x, y) denotes any expression whatever involving x, y, and constants. These constants are usually called parameters, a term borrowed from the theory of conic sections, where the word parameter is used to denote a certain fixed line. If we wish to represent in the most general manner that y is an explicit function of x, we may write

f(x) denoting any algebraical expression which involves x and parameters. The equation (4) is easily seen to be a particular form of the equation (3); for we may write it thus,

y − f(x) = 0,

y -f(x) being merely a particular instance of the general form F(x, y).

4. Functions may be termed mathematical or empirical; mathematical, if the functionality is established by definition; empirical, if discovered by observation. As an instance of the latter functionality, let y denote the attraction of the Sun upon the Earth, and x the distance between these two masses; then it is known by observation that

being a constant quantity: y is in this case an empirical function of x.

5. A function f(x) is said to be continuous when, as x increases continuously, f(x) passes continuously from one possible value to another through all intervening values: the function is said to be discontinuous whenever this condition is violated. Take for instance

then, as a keeps increasing continuously from 0 to a value a – h, where h is a positive quantity less than any assignable magnitude, it is plain that y also keeps continuously changing

through every gradation of value from

∞ to + ∞

Thus we

x changes from a h to a + h, y leaps from without passing through the intervening values. see that in this case y is generally a continuous function, but that it experiences a dissolution of continuity when x becomes equal to a. If we take

then, although, when xa, y assumes the value ∞, yet this value of x does not correspond to a discontinuous state of the function, since, as x passes from a-h to a+h, there is no gap in the range of values of y.

6. Suppose y = f(x) to be the equation to a curve; then, if the function f(x) is continuous for a certain range of values of x, every two points of the locus will be joined by a continuous curve: on the other hand, if there is a dissolution of continuity at any point, and if the function be possible before and after x has passed its critical value, there will be a gap between two points of the curve corresponding to consecutive values of x. Thus, in the instance of the curve

the asymptote, of which the equation is x = a, is touched at opposite ends by the curve for two consecutive values of x, one greater and the other less than a by an indefinitely small magnitude.

7. Functions of x, which are expressed by the ordinary signs of algebra and trigonometry, are usually continuous, if we disregard certain dissolutions of continuity corresponding to peculiar and detached values of x. There are however exceptions to this principle. For example, if

y = (-a)",

a being a positive quantity, it is plain that y will be imaginary whenever x is of the form

λ and u being integers. Now between any two values of x, however little they may differ from each other, we may intercalate an infinite number of fractions of the above form. Thus we see that it is impossible to join by a continuous curve two points of the locus of the equation, corresponding to two systems of real values of x and y, however near they may be to each other. These anomalous functions are inapplicable to questions of natural philosophy, and have attracted but little attention even in pure analysis. In this treatise we shall direct our attention entirely to continuous functions.

8. Functions are distinguished also by the names of algebraical and transcendental. If y be connected with x by an equation involving only the ordinary operations of addition and subtraction, multiplication and division, evolution and involution of assigned degrees, y is said to be an algebraical function of x. If the equation of connection does not satisfy this condition, involving for instance exponential, logarithmic, or circular functions of x and y, then y is said to be a transcendental function of x. Thus, for examples, in the equations

y is an algebraical function of x; and a transcendental one in the equations

x sin y + y sin x = sin (xy).

9. It frequently happens that, if

F(x, y) = 0

be the equation to a curve, y will for a certain value of x experience a dissolution of continuity by becoming impossible, although the curve is itself continuous at the point. Thus if

y2 = 4mx,