252 
ADDITIONAL THEOREMS IN THE CALCULUS 
[VII 
If we put b = a + h we obtain the equation 
f(a + h) =/(a) + hf'(a) + \№f"{a + 6h) (2), 
which is the standard form of what may be called the Mean 
Value Theorem of the second order. 
The analogy suggested by (1) and (2) at once leads us to 
formulate the following theorem : 
Taylor’s or the General Mean Value Theorem. If 
f{x) is a function of x which has derivatives of the first n orders 
throughout the whole interval {a, h), then 
f(b) —f (a) + (/>- a)f\a) + 
/»+••• 
where a < % < h: or, if b = a + h, 
f{a + h) =f{a) + hf'(a) + \hf\a) + ... 
,/< n -D (a) + ^/ (w, (a + 0/i) 
where 0 < 6 < 1. 
The proof proceeds on precisely the same lines as were adopted 
before in the special cases in which n= 1, 2. We consider the 
function 
where F n (x) =f(b) -f{x) - (b -x)f (x)~^-.(b- xff'fx) - ... 
21 
This function vanishes for x = a and x = b ; its derivative is 
and there must be some value of x between a and b for which 
the derivative vanishes. This leads at once to the desired result. 
In view of the great importance of this theorem we shall give 
at the end of this chapter another proof, not essentially distinct 
from that given above, but different in form and depending on 
the method of integration by parts.