If x=0, then U,= 0, U, = 1, U1 = 0, U1 = −2, U,=0, U,= 2.3.4, etc. U1 1= = + + + + + etc. = ⚫785398. 1.3 5.7 9.11 13.15 17.19 This series is not so convergent as the former, and it is therefore not so well adapted for computation. 40. Preparatory to the investigation of this important theorem, it is necessary to establish the following proposition : The differential coefficient of u = f (x+y) is the same, whichever of the parts, x and y, is supposed to vary, and x and y are independent of each other. For lety receive the increment h, then will u = f (x + y), Now if h be continually diminished towards zero, the first member u' - u h du du will tend to become or according as his regarded as the dx dy' increment of x or of y; while the other member tends to the same limit in each case; cousequently the two differential coefficients are identical, du dy (x + y); then supposing x to be variable and y constant, And if we regard y as variable, and x constant, then we get 41. Let u denote a function of x, and u' the new value of u when becomes x+h, then will u' = u + du dx d2 u h2 d u h3 h' + + etc. d2 u + dx2 1.2 da3 1.2.3 dx 1.2.3.4 For suppose that the development of f (x + h) is of the form f(x + h) = fx + A h2 + B h3 + Ch + Eh2 + .... (1), where the quantities A, B, C, E, etc., are functions of x independent of h, and a, b, c, e, etc., are constant indices to be determined. It will be readily seen that none of the indices a, b, c, etc., can be negative, for then the second member of (1) would involve a term of the form Qh−a would become ƒ x. = = 0, would be infinite, while the first member Neither can the first term of (1) be different from fx, for when h O, we have the identical equation fx = fx. Now suppose that the terms of (1) are arranged in the ascending order of their positive indices, and let us differentiate (1), first with respect to x, and then with respect to h; then = a a Aha-1+bBh1+c Ch c - 1+eEhe ~1+··· But by (40) the first members of these equations are identical, and therefore the series in the second members must also be identical; whence equating the indices of the several powers of h, and likewise the coefficients of the corresponding terms, we get 2, c = 3, etc. Also, since ha h = 1, we have, by equating the coefficients of the or u' = u + duh+ h2 d3 u + + + etc. dx dx 1.2 d x 1.2.3 dx 1.2.3.4 This theorem is, perhaps, one of the most important in the whole range of pure mathematics. It was first discovered by the celebrated analyst Dr. BROOK TAYLOR, and is known by the name of TAYLOR'S THEorem. If h be negative, or u' ƒ (x − h), then will d u dx h+ d2 u d3 u h3 etc. Taylor's theorem is applicable to the general development of all functions of xh, while no particular value is assigned to x; but it is evident that if a value be assigned to the variable x, such as to render any of the differential coefficients infinite, the theorem will not apply to such a case, and will not give the particular development, which must therefore be sought for by the ordinary operations of algebra. 42. Maclaurin's theorem may be readily deduced from Taylor's in the du d2 u following manner. Let x = O, then the coefficients are functions of x, will become constants if x = as in Maclaurin's theorem, we have But this equation is true for all values of h, and U, U1, U2, etc., are independent of h; therefore writing x for h, for the sake of uniformity of notation, fx 1.2 which is Maclaurin's theorem, and therefore it is only a particular casc of the theorem of Taylor. = sin x, etc.; hence, by Taylor's theorem, sin (x + h) d3u = dx3 d'u dx+ etc. 2 (secx+3 tan x sec *x) = 2 (1 + 4 tan x + 3 tan *x), = 23 (2 tan x + 5 tan 3x + 3 tan 3x), etc. .'. u' = tan (x + h) = tan x + sec 2x h+tan x sec *x. h2 secx (1+3 tan x). where u = tan-1x. ↓ (a − 1)2 + † (a − 1) 3 — ¦ (a − 1)* + etc. MAXIMA AND MINIMA FUNCTIONS OF ONE VARIABLE. 43. The value of a function of a variable quantity may either increase or diminish with the increase of the variable; and after increasing to a certain limit, it may begin to decrease, while the variable increases; or after decreasing to a certain limit, it may begin to increase with the increase of the variable. The value of a function is therefore said to be a maximum when that value is greater than either its immediately preceding or succeeding values, and a minimum when that value is less than any of the values which immediately precede or follow it. 44. A function which increases continually with the increase of the variable does not admit of a maximum or minimum value; but if, after passing a maximum value, the function decreases to a certain limit, and then begins to increase, it admits of both a maximum and minimum value; and if, after passing this minimum value, the function increases to a certain limit, and then begins to decrease, it will admit of a second maximum value; hence a maximum value of a function is not necessarily the greatest value of which the function may be susceptible, nor a minimum value the least value. If the value of the function increase and diminish alternately several times, the maximum and minimum values may succeed each other to any extent. The distinguishing cha racteristic of a maximum value of a function is that it is greater than any of the immediately preceding or succeeding values; and of a minimum, that it is less than any of the values which immediately precede or follow it. 45. Let APP' P" be a curve referred to the line A B C" as axis, and let the variable = fx; and while the variable abscissa A B, or x, increases, the ordinate BH, or u, increases till it attains its maximum position MP. After passing the maximum position MP, the ordinate decreases till it comes into the position M' P', when it is a minimum; and after passing the minimum position M' P', it increases till it attains a second maximuni position M"P", and so on. 46. The determination of these maximum and minimum values of a function constitutes one of the principal applications of the Differential Calculus; but before proceeding to the investigation of the principles for determining such values, the following lemma must be premised: LEMMA.-If, while a variable quantity increases, a function of it likewise increase, the differential coefficient of the function is positive ; but if the function decrease, its differential coefficient is negative. For the differential coefficient of fx is the limit to which the value of ƒ (x + h) − fx the fraction continually approaches, as h, the increment h of x, approaches zero. Now the preceding fraction, as well as the limit du dx' to which it tends, will be positive or negative according as f (x + h) Ꮖ is greater or less than fx; that is, according as fr is increased or diminished by the addition of h. Hence if u = fx, and x increase continudu ally, then will be positive or negative according as u or fx is indx creasing or decreasing. The same relation will exist between any differential coefficient and the one immediately following it, because the latter is the differential coefficient of the former. Thus d2 u dx2 is the differential coefficient of du dx' which is generally a new function of the variable; and is the 47. From the last Article, and the definitions of maximum and minimum values of functions, it follows that the differential coefficient of a function of a variable quantity changes its sign when that function becomes a maximum or a minimum. In the case of a maximum, the function first increases and then dimi |
