CALCULUS, DIFFERENTIAL AND INTEGRAL. The differential calculus and the integral calculus are the two divisions of a branch of mathematics which treats problems involving variable quantities. Such problems arise regularly in geometry, physics and other branches of science. By a quantity, is meant a distance, a weight, a period of time, in short, anything which can be measured.
The two leading problems of the differential calculus are the construction of tangents to curves and the determination of the rate of change of a quantity. A tangent to a curve is a straight line which grazes the curve at a point. Thus AB, in fig. i, is tangent to the curve in the fig ure. A more careful analysis of the notion of tangent will be given below. In elementary geometry, the construction of a tangent to a circle is considered. The differential calculus fur nishes a method for constructing tangents to curves of any type.
A typical situation connected with a rate of change is as fol lows: suppose that water is flowing into a vessel in the shape of an inverted cone (fig. 4), at the uniform rate of I cu. ft./sec. At first the water will rise rapidly in the cone, and then, because the upper part of the cone is wider than the lower part, the level will go up more and more slowly. The differential calculus per mits the determination of the rate at which the level rises at any stated instant. One of the most interesting special problems in the differential calculus is the determination of the maximum or minimum value a quantity can have. The differential calculus also furnishes methods for calculating tables of logarithms, sines, cosines, etc., which are used in trigonometry and in developing two important formulae, Taylor's formula and Maclaurin's.
The integral calculus treats of two classes of problems. The first class deals with such quantities as the amount of area en closed by a curve, the length of a curve, or the amount of volume enclosed by a surface. The second type of problem is the deter mination of a variable quantity when the law of its change is known. For instance, let a body be dropped from a height, in a vacuum. We know that the speed with which it is falling, at any instant, is proportional to the length of time for which it has been falling. The speed is nothing more than the rate of change of the distance which the body has fallen. The integral calculus permits the determination of the distance through which the body falls in any period of time.
If the quantity which is given is called x, and the quantity which is found is called y, the fact that y is a function of x is expressed by the symbolism y = f (x), that is, "y equals f of x". In each particular problem, f(x) becomes a definite mathematical expression. For instance, if x is the length of the side of a square, and y is the area of the square, then y equals so that f(x) in this case is Other letters than f may be used as functional symbols. For instance, where several functions appear in the same problem, one might call them f(x), F(x), g(x), etc., re spectively.
The tangent at P can certainly be constructed if the angle which it makes with the X-axis is known. We shall call the angle between a line and the X-axis the inclination of the line. In what follows immediately, the inclination of the tangent will be found from the fact that, as the secant approaches the tangent, the inclination of the secant approaches that of the tangent. One point deserves special emphasis. We have not said that if Q comes into coincidence with P, the secant becomes a tangent. Unless Q is distinct from P, we do not have two points through which to draw a line. All that has been said is that, as Q comes closer and closer to P, the inclination of the secant comes closer and closer to that of the tangent. This fact suffices for the deter mination of the slope of the tangent.
Limit.—To express the fact that the inclination of the secant approaches that of the tangent, we shall say that the inclination of the tangent is the limit of the inclination of the secant. In general, when a variable quantity comes closer and closer to a fixed quantity, we shall say that the variable quantity has the fixed quantity as a limit. It is not essential that the variable quantity should move steadily in the direction of the fixed quan tity. For instance, as the vibrations of a pendulum die out, the inclination of the pendulum approaches 9o° as a limit, even though the pendulum never (theoretically) stops its movements away from the perpendicular position. When we speak of the "limit" of a constant quantity, we shall mean the quantity itself.
By the slope of a line is meant the trigonometric tangent of the inclination of the line. Obviously, the slope of the secant PQ has for limit, as Q approaches P, the slope of the tangent at P. We shall first seek to make plausible, by arithmetic calcula tions, that the slope of the secant tends toward a limit as Q ap proaches P. Later, a rigorous treatment of the problem will be given.
As Ox approaches zero, that is, as Q approaches P, Ay also approaches zero. This, of course, does not mean that the quotient of Ay by Ox is small. For instance, in our calculations above, when Ax was o• i and Ay was 0-41, Ay/Ax was 4.1. As a matter of fact, when Ox and Ay are both small, Ay/Ox is very close to the slope of the tangent at P, which slope need not be a small number. The idea contained in the last few lines is the central idea of the differential calculus. We proceed to determine the limit of Ay/Ox as Q approaches P. As Q lies on the parabola, the ordinate of Q equals the square of its abscissa, that is, y+Ay= or y+Ay = 2x • Ox+- (2) Now, as P has the co-ordinates (x , y), and as P lies on the parabola, we have y = Using this fact in (2), we find that We now bring Q closer and closer to P, thus making Ax smaller and smaller. It is plain, from (4),that Ay/ Ax comes closer and closer to 2X.
We have thus solved our problem. The tangent at any point P on the parabola has, for slope, twice the abscissa of P. Of course, this fact permits the immediate construction of the tangent at any point. When x = 2, the slope of the tangent is 4. This is the result made reasonable above by arithmetic calculation. It should be appreciated that we have secured an exact, and not an approximate solution of the tangent problem. As Q ap proaches P, the slope of the secant approaches the slope of the tangent. But we have shown that the slope of the secant ap proaches 2X. Hence the slope of the tangent must be identical with 2X.
The limit of Ay/Ax is called the derivative of f(x) for the chosen value of x. Thus the derivative is nothing more than the slope of the tangent to the graph. We represent the derivative of the function y=f(x), for any value of x, by the symbol dy dx' Thus, when y dy = 2X.
dx As to each value of x there corresponds a definite value of dy/dx, the derivative is a function of x.
It cannot be emphasized too strongly that dy/dx is not a quotient. It is a number which is approached by a certain quotient, namely, Ay/ Ax. The symbols dy and dx, as we are using them, have no meaning by themselves. The entire symbol, dy/dx represents, very expressively, the limit of Ay/Ox. It is true that, in one of the chapters of the calculus, "differentials" are defined, for which the symbols dy and dx are used. But the "differential" symbols are entirely distinct from the dy and dx used here.
The process of finding the derivative of a function is called differentiation. We have shown above how to differentiate In treatises on the calculus, it is shown how to differentiate all types of expressions. For instance, the derivative of x" is proved to be for all values of n. For n a positive whole number, this is a very easy problem, and is handled as the case of was handled above. The derivative of sin x is cos x, provided that x is measured in radians. The derivative of log x is 1/x, when the logarithms are taken to the so-called Naperian base e= 2.718281 • • • • (See LOGARITHMS.) The derivative of the sum of two functions is the sum of their derivatives. There exist simple rules for the differentiation of the product or the quotient of two functions.
If y is a constant, then, for every Ox, y+Ay = y, so that Ay = o. Hence Ay/Ox is always zero, so that, according to the definition of the limit of a constant quantity, dy/dx is also zero. This is brought out geometrically by the fact that the graph of a constant is a horizontal line, so that the tangent to the graph at any point, which is the graph itself, has a zero slope. We see that two functions which differ by a constant, for instance, and have the same derivative. This fact will be funda mental in connection with integration.
It has been said above that the quotient Ay/Ox approaches a limit for "the functions commonly met in mathematics." There exist functions, with more or less complicated definitions, for which no derivatives exist. For such functions, as Ax approaches zero, the quotient Ay/Ox either becomes infinite or oscillates without approaching a limit. The subject of functions which are without their derivatives forms a chapter in the theory of func tions (q.v.).
Let y be the depth of the water in the cone, measured in feet, after the water has been flowing in for t seconds. Then y is a func tion of t. To make this situation concrete, suppose that the angle between the extreme elements of the cone, OA and OB, is Then the radius of the upper surface of the water will always be equal to y. Hence the volume of water in the cone at any instant is 3y • = 3 cu. feet. But, in t seconds, t cubic feet of water have entered the tank. Hence What we shall do, according to the central principle of the theory of rates, is to use dy/dt for the rate of change of y. That is, the speed with which the level rises at any instant t (t seconds after the water begins to flow in) will be taken as the derivative of y at that instant. To justify this principle, let us consider any fixed instant t, and the depth y at that instant. At any later instant, t+6t, the depth will have been increased by an amount Ay. Roughly speaking, one might say that, during the time At, y has increased at the "average rate" of Ay/At ft. per sec., but we have no inherent notion of "average rate." What we are doing is to create a meaning for that term. A quantity y, increasing for At sec. at a uniform rate of Ay/At ft. per sec. would increase Ay ft. That is why it is natural to take Ay/At as an "average" in our case. Furthermore, if At is a very small period of time, the level of the water would appear to an observer to be rising practically uniformly, and practically at the rate Ay/At, during the time At.
Because Ay/At can be used as an average speed for the period At, and because Ay/At approaches dy/dt as At approaches zero, we define the instantaneous rate of change of y, at any instant t, to be dy/dt. Thus the instantaneous rate of change is primarily a mathematical notion.
Although the notion of instantaneous rate of change is a math ematical one, it is of the greatest utility in the applied sciences. For when a problem of applied science is to be subjected to mathematical analysis, what is done, effectively, is to replace the problem by an ideal mathematical problem, formulated in terms of the pure mathematician's concepts. To find dy/dt in the problem of the cone, let a represent (3/7r)l. Then The last step is justified by the equation (u (0+ 00 which can be verified by multiplication. As At approaches zero, t+At approaches 1, so that Ay/At approaches a which, then, is dy/dt. For in stance, when t is 8 sec., the level is rising at a/i 2 = o•o8 ft. per sec ond.
Maxima and Minima.—Con sider a function y--= f (x) which has a derivative for every x (fig. 5). At a point such as A, at which the function is a maxi mum, or at a point such as B, where the function is a minimum, the tangent is horizontal; the slope of the tangent is zero.
Example.— Suppose that we have a square sheet of tin, 6 in. on each side, that we cut a square of side x out of each corner, and fold up the sides so as to form an open box. The altitude of the box will be x, and its base will be a square of side 6 — 2X. The volume of the box, call it y, will be This equation has the roots r and 3; that is, if there is an x for which y is a maximum, x is either r or 3. From the nature of the problem, it is evident that some x makes the volume a maximum. Certainly 3 is an impossible value for x. Hence, for x = I, we get a box of maximum volume.
(We shall explain below the origin of this rather peculiar sym bolism.) Thus, f 2X dx = where c is any constant.
The fundamental question arises as to whether all integrals of a function are found by adding constants to any par ticular integral. The reply is affirmative. The only integrals of 2X, for instance, are the functions with c a constant. To give a geometric proof of this fact, let us observe that if F(x) has 2X for derivative, then the function F(x) has a derivative which is everywhere zero. Now only a constant can have a derivative which is zero for every x, for if a function is not a constant, its graph must have points at which the tangent is not horizontal. Thus, F(x) c, with c constant, so that The process of finding the integral of a function is called integration. While the determination of the derivative of a function is a perfectly straightforward procedure, for which definite rules exist, there is no general method for finding the integral of a mathematical expression. Treatises on the calculus give rules for integrating large classes of functions. When these rules fail, more powerful methods can be used which do not give an expression for the integral, but which permit the value of the integral to be calculated for any value of x in which one may be interested. To find a formula for the distance through which a body falls in a given time, under the influence of gravity, let a body, initially at rest, be allowed to fall. If g is the acceleration of gravity, (32 the body will, in t seconds, acquire a speed of gt ft. per second. Let s be the distance through which the body falls in t seconds. Then so that c = o. Hence, s = for every 1.
Suppose x is increased by an amount Ox. The increase, AA, of A, is the strip PMM'P'. To fix our ideas, we shall assume that, between P and P' the graph rises. It will be seen that all other cases are treated in the same way. The strip AA has a greater area than a rectangle of base Lx and height y, but has a smaller area than a rectangle of base Ax and height y+Ay. Thus AA is greater than yLx and less than (y+Ay)Ax. Hence AA /Ax is greater than y, but less than y+ 0y. But, as Ox approaches o, Ay approaches o, so that y+Ay decreases towards y. Hence AA/Ox, which lies between y+Ay and y, must approach y. This proves (5)• We have thus A= f y dx.
3 To determine c, we observe that A = o when x = o. This gives c = o, so that A The area problem is typical of a whole class of problems in geometry and physics. By quite the same means, we determine the lengths of curves, the volumes of solids, the moments of inertia of solids, and other quantities.
We can now point out the origin of the symbol for integration. Let the base OM in fig. 6 be divided into several segments, and rectangles like PM' be constructed on the segments. If the number of segments is very large, and each segment very small, the sum of the areas of the rectangles will be a very good ap proximation to the area under the curve. Let the segments be come smaller and smaller, their number becoming larger and larger. Then the area under the curve will be the limit of a sum of rectangles, each rectangle having an area y Ox, where y is the altitude and Ax the base of the rectangle.
In the symbol f the f , a mediaeval S, stands for summa (sum). The dx refers to the bases Ox of the rectangles. Thus, the notation for integration arose out of the usefulness of the integral for determining the limits of sums.
The methods used by the Greeks for determining the area of a circle and of a segment of a parabola, and the volumes of the cylinder, cone and sphere, were, to the extent that they involved finding the limits of sums, akin to the method of integration. Dur ing the first half of the 17th century, methods of more or less limited scope began to appear among mathematicians for con structing tangents, determining maxima and minima, and finding areas and volumes. But it remained for Isaac Newton and Gott fried Wilhelm Leibniz, working independently of each other, to create during the latter half of the 17th century the concepts of derivative and integral. Newton regarded the derivative of a function as a speed at which the function changes. His symbol for the derivative of a function y, which he called the "fluxion" of the function, was y. The methods of the calculus permitted Newton to carry out one of his finest investigations that of the mechanics of the solar system. Leibniz used the nota tion dy/dx for the derivative, and also introduced the integration symbol.
During the past two and a half centuries numbers of theories have grown up which are offshoots of the calculus. One might mention differential equations, the calculus of variations, differen tial geometry (qq.v.) ; the theory of functions of complex vari ables, with its special chapters like elliptic functions and abelian integrals; the theory of functions of real variables, continuous groups, the calculus of finite differences and integral equations; see CALCULUS OF DIFFERENCES ; FUNCTIONS ; GROUPS.