A slightly more general theorem was given by Cauchy (1823) to the effect that, if ƒ′(x) and F′(x) are continuous between x = a and x = b, then there is a number θ between 0 and 1 which has the property expressed by the equation
| F(b) − F(a) | = | F′ {a + θ(b − a) } | . |
| ƒ(b) − ƒ(a) | ƒ′ {a + θ(b − a) } |
The theorem expressed by the relation (i.) was first noted by Rolle (1690) for the case where ƒ(x) is a rational integral function which vanishes when x = a and also when x = b. The general theorem was given by Lagrange (1797). Its fundamental importance was first recognized by Cauchy (1823). It may be observed here that the theorem of integral calculus expressed by the equation
F(b) − F(a) = ∫ba F′(x) dx
follows at once from the definition of an integral and the theorem of intermediate value.
The theorem of intermediate value may be generalized in the statement that, if ƒ(x) and all its differential coefficients up to the nth inclusive are continuous in the interval between x = a and x = b, then there is a number θ between 0 and 1 which has the property expressed by the equation
| ƒ(b) = ƒ(a) + (b − a) ƒ′(a) + | (b − a)2 | ƒ″(a) + ... + | (b − a)n−1 | ƒ(n−1)(a) |
| 2! | (n − 1)! |
| + | (b − a)n | ƒ(n) {a + θ (b − a) }. |
| n! |
(ii.)
37. This theorem provides a means for computing the values of a function at points near to an assigned point when the value of the function and its differential coefficients at the assigned point are known. The function is expressed by a terminated Taylor’s Theorem. series, and, when the remainder tends to zero as n increases, it may be transformed into an infinite series. The theorem was first given by Brook Taylor in his Methodus Incrementorum (1717) as a corollary to a theorem concerning finite differences. Taylor gave the expression for ƒ(x + z) in terms of ƒ(x), ƒ′(x), ... as an infinite series proceeding by powers of z. His notation was that appropriate to the method of fluxions which he used. This rule for expressing a function as an infinite series is known as Taylor’s theorem. The relation (i.), in which the remainder after n terms is put in evidence, was first obtained by Lagrange (1797). Another form of the remainder was given by Cauchy (1823) viz.,