The differential and integral calculus, containing differentiation, integration, development, series, differential equations, differences, summation, equations of differences, calculus of variations, definite integrals,--with applications to algebra, plane geometry, solid geometry, and mechanics Also, Elementary illustrations of the differential and integral calculus . the function upon lines perpendicular to thevalues of a?, placing each value of the function on the. line drawn throughthe variable extremity of the linear value of x, and measuring it aboveor below the axis of x, according as i
The differential and integral calculus, containing differentiation, integration, development, series, differential equations, differences, summation, equations of differences, calculus of variations, definite integrals,--with applications to algebra, plane geometry, solid geometry, and mechanics Also, Elementary illustrations of the differential and integral calculus . the function upon lines perpendicular to thevalues of a?, placing each value of the function on the. line drawn throughthe variable extremity of the linear value of x, and measuring it aboveor below the axis of x, according as it is positive or negative, we havethe well-known method of representing a function by means of a curve,which is the foundation of the application of algebra to geometry, asgiven by Des Cartes. We have drawn the representation of a functionbelow, so as to exhibit every variety of singular value, and more than theskill of the most practised algebraist would at present be able to find afunction for. The stars mark the singular values, or rather the placesat which there may possibly be a singular value ; all other values areordinary, however near the singular values they may approach in posi-tion. And we see that, however nearly a, the value of a1, may approach tob the value of x at one of the singular points, it must be possible to takea + h lying between a and •zr—<& Postulate 2.—If 0 a be any finite value of <fix, it is always possibleto take h so small, that 0 (a + A) shall be as near to 0 a as we please,and that 0 x shall remain finite from x = a to x = a + h, and alwayslie between 0 a and 0 (a + A) in magnitude. This again is a part of our experience of algebraical functions. It isgenerally assumed under the name of the laiv of continuity. The latterpart of the postulate may be true of the whole extent of some functions:thus, however great A may be, or perpetually increases between a2 and(a + hf. It is possible to imagine a function which does not observe this
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