Mean Value

MEAN VALUE. If n quantities are com mensurable, the sum of their units divided by n, the number of the quantities, is the arith metical mean and is the mean value of the is quantities when their number is finite. For example, if the number of units in three quan tities are respectively 4, 8 and 15, the mean value is 4 + 8 + and in general, if the 2 number of units in n quantities is denoted by Xs, . . . Xn, the mean value is — (Xi+ Xi+ X: . . . . X11).

But if a quantity vary continuously in ac cordance with some law, thereby assuming every possible value between two extremes, the number of different quantities is infinite, and the mean value in such a case requires a new defi nition. For example, assume that at every point in the diameter of a semi-circle a perpen dicular to the diameter is drawn to meet the semi-circle. The lengths of these lines form a continuous series of values represented by every number from zero to r, where r is the length of the radius. In such a case if is values of the perpendiculars are obtained and their sum is divided by n, the ratio should approximate to the mean value. It is easy to see, however, that this ratio will approximate to 2 — of the radius of the circle if the n perpendiculars are evenly spaced on the circle, to of the radius if they are evenly spaced on the diameter. It follows that the mean value of a continuously varying quantity is not definite until the law is known by which the of each portion of the scope of variation is determined.

The several forms of the theorem that pass under the name of the mean value theorem involve the principles of the infinitesimal cal culus. The first theorem of the mean value is the following: Let f(x) denote any finite and continuous function of x in the interval between x. and x =-- X; let dx denote the increment and retain the same sign in this interval; and assume that m and M are respectively the least and the greatest values of f(x) in this interval: then the definite integral (x)dx has a defi x• nite value that is greater than m(X—x.) and

less than M(X — x.) ; that is, dx< f(x)dx

The theorem of mean value is of importance in establishing Taylor's series. It follows im mediately from the theorem of mean value that if F(X)=--F(x0)=0, for some f between X and xe. Fy(f)=0, if F is continuous between X and x.. Let be a function of X with derivatives of every order at every point between x---a and x=b. Construct the function (s) =9(X) — (X —0 0'(5) (xs — at "(s) —— (x — s) 1 • • • .

n! n+1 where X is of the form x• h, and X and xs lie between a and b. ty will be of the form —iV(S) -}-e(1)—(X-44"(s)+(X--11)4"(x)— • • (X—zrp - • n! Tsn (.4.1) Furthermore, IKX) is identically 0, and by properly choosing P, 1P(x.) can be made 0, Therefore, by the theorem just stated *W.." z)n [p--on-i-i)(x)j vanishes for some value n of z between and x. As it can be shown that this value need not be X itself, we have for some value of z between x. and X, and since I4(X)=0, if we let we have0(xol-h)(x.)-l- e(e+1)(x.+1.9h) h , where 0 <4<1. Therefore, (n-f-1)! if Os +1) ! can be made as small as you please by increasing n, we have +h)== which is Taylor's theorem. See o n!