Page:EB1911 - Volume 14.djvu/567

application is the 6th proposition of Archimedes’ treatise On the Sphere and Cylinder, in which it is proved that the area contained between a regular polygon inscribed in a circle and a similar polygon circumscribed to the same circle can be made less than any assigned area by increasing the number of sides of the polygon. The methods of Euclid and Archimedes were specimens of rigorous limiting processes (see Function). The new problems presented by the analytical geometry and natural philosophy of the 17th century led to new limiting processes.

4. In the problem of tangents the new process may be described as follows. Let P, P′ be two points of a curve (see fig. 2). Let x, y be the coordinates of P, and x+Δx, y+Δy those of P′. The symbol Δx means “the difference of two x’s” and there is a like meaning for the symbol Δy.Differentiation.

Fig. 2.

The fraction Δy/Δx is the trigonometrical tangent of the angle which the secant PP′ makes with the axis of x. Now let Δx be continually diminished towards zero, so that P′ continually approaches P. If the curve has a tangent at P the secant PP′ approaches a limiting position (see § 33 below). When this is the case the fraction Δy/Δx tends to a limit, and this limit is the trigonometrical tangent of the angle which the tangent at P to the curve makes with the axis of x. The limit is denoted by

${\frac {dy}{dx}}$.

If the equation of the curve is of the form y=ƒ(x) where ƒ is a functional symbol (see Function), then

${\frac {\Delta y}{\Delta x}}={\frac {f(x+\Delta x)-f(x)}{\Delta x}}$.

and

${\frac {dy}{dx}}=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}$.

The limit expressed by the right-hand member of this defining equation is often written

$f^{\prime }(x)\,$

and is called the “derived function” of ƒ(x), sometimes the “derivative” or “derivate” of ƒ(x). When the function ƒ(x) is a rational integral function, the division by Δx can be performed, and the limit is found by substituting zero for Δx in the quotient. For example, if ƒ(x) = x², we have

${\frac {f(x+\Delta x)-f(x)}{\Delta x}}={\frac {f(x+\Delta x)^{2}-x^{2}}{\Delta x}}={\frac {2x\Delta x+(\Delta x)^{2}}{\Delta x}}=2x+\Delta x$

and

$f^{\prime }(x)=2x\,$

The process of forming the derived function of a given function is called differentiation. The fraction Δy/Δx is called the “quotient of differences,” and its limit dy/dx is called the “differential coefficient of y with respect to x.” The rules for forming differential coefficients constitute the differential calculus.

The problem of tangents is solved at one stroke by the formation of the differential coefficient; and the problem of maxima and minima is solved, apart from the discrimination of maxima from minima and some further refinements, by equating the differential coefficient to zero (see Maxima and Minima).

5. The problem of quadratures leads to a type of limiting process which may be described as follows: Let y=ƒx be the equation of a curve, and let AC and BD be the ordinates of the points C and D (see fig. 3). Let a, b be the abscissae of these points. Let the segment AB be divided into a numberIntegration. of segments by means of intermediate points such as M, and let MN be one such segment. Let PM and QN be those ordinates of the curve which have M and N as their feet. On MN as base describe

Fig. 3.

two rectangles, of which the heights are the greatest and least values of y which correspond to points on the arc PQ of the curve. In fig. 3 these are the rectangles RM, SN. Let the sum of the areas of such rectangles as RM be formed, and likewise the sum of the areas of such rectangles as SN. When the number of the points such as M is increased without limit, and the lengths of all the segments such as MN are diminished without limit, these two sums of areas tend to limits. When they tend to the same limit the curvilinear figure ACDB has an area, and the limit is the measure of this area (see § 33 below). The limit in question is the same whatever law may be adopted for inserting the points such as M between A and B, and for diminishing the lengths of the segments such as MN. Further, if P′ is any point on the arc PQ, and P′M′ is the ordinate of P′, we may construct a rectangle of which the height is P′M′ and the base is MN, and the limit of the sum of the areas of all such rectangles is the area of the figure as before. If x is the abscissa of P, x+Δx that of Q, x′ that of P′, the limit in question might be written

$lim.{\begin{matrix}\sum _{a}^{b}f(x^{\prime })\Delta x\end{matrix}}$

where the letters a, b written below and above the sign of summation Σ indicate the extreme values of x. This limit is called “the definite integral of ƒ(x) between the limits a and b,” and the notation for it is

$\int _{a}^{b}f(x)\,dx$

The germs of this method of formulating the problem of quadratures are found in the writings of Archimedes. The method leads to a definition of a definite integral, but the direct application of it to the evaluation of integrals is in general difficult. Any process for evaluating a definite integral is a process of integration, and the rules for evaluating integrals constitute the integral calculus.

6. The chief of these rules is obtained by regarding the extreme ordinate BD as variable. Let ξ now denote the abscissa of B. The area A of the figure ACDB is represented by the integral $\int _{a}^{\xi }f(x)\,dx$, and it is a function of ξ. Let BD be displaced to B′D′ so that becomes $\xi +\Delta \xi$ (seeTheorem of Inversion. fig. 4). The area of the figure ACD′B′ is represented by the integral $\int _{a}^{\xi +\Delta \xi }f(x)\,dx$ and the increment ΔA is given by the formula:

$\Delta A=\int _{\xi }^{\xi +\Delta \xi }f(x)\,dx$

which represents the area BDD′B′.

Fig. 4.

This area is intermediate between those of two rectangles, having as a common base the segment BB′, and as heights the greatest and least ordinates of points on the arc DD′ of the curve. Let these heights be H and h. Then ΔA is intermediate between HΔξ and hΔξ, and the quotient of differences ΔA/Δξ is intermediate between H and h. If the function ƒ(x) is continuous at B (see Function), then, as Δξ is diminished without limit, H and h tend to BD, or ƒ(ξ), as a limit, and we have:

${\frac {d{\text{A}}}{d\xi }}=f(\xi )$

The introduction of the process of differentiation, together with the theorem here proved, placed the solution of the problem of quadratures on a new basis. It appears that we can always find the area A if we know a function F(x) which has ƒ(x) as its differential coefficient. If ƒ(x) is continuous between a and b, we can prove that

$A=\int _{a}^{b}f(x)\,dx=F(b)-F(a)$

When we recognize a function F(x) which has the property expressed by the equation

${\frac {dF(x)}{dx}}=f(x)$,

we are said to integrate the function ƒ(x), and F(x) is called the indefinite integral of ƒ(x) with respect to x, and is written

$\int f(x)\,dx$

7. In the process of § 4 the increment Δy is not in general equal to the product of the increment Δx and the derived function ƒ′(x). In general we can write down an equation of the formDifferentials.

$\Delta y=f^{\prime }(x)\Delta x+{\text{R}}$,

in which R is different from zero when Δx is different from zero; and then we have not only

$\lim _{\Delta x\to 0}{\text{R}}=0$,

but also

$\lim _{\Delta x\to 0}{\frac {\text{R}}{\Delta x}}=0$

We may separate Δy into two parts: the part ƒ′(x)Δx and the part R. The part ƒ′(x)Δx alone is useful for forming the differential coefficient, and it is convenient to give it a name. It is called the differential of ƒ(x), and is written dƒ(x), or dy when y is written for ƒ(x). When this notation is adopted dx is written instead of Δx, and is called the “differential of x,” so that we have

$df(x)=f^{\prime }(x)dx.$

Thus the differential of an independent variable such as x is a finite difference; in other words it is any number we please. The differential of a dependent variable such as y, or of a function of the independent variable x, is the product of the differential of x and the differential coefficient or derived function. It is important to observe that the differential coefficient is not to be defined as the ratio of differentials, but the ratio of differentials is to be defined as the previously introduced differential coefficient. The differentials