Principles of Mathematics in Operations Research

172 12 Differentiation

12.3 Higher Order Derivatives

Definition 12.3.1 // / has a derivative /' on an interval and if f is itself

differentiate, we denote derivative of f as /", and call the second derivative

of f. Higher order derivatives are denoted by /',/",/^^3 ',... ,f^n\ each of

which is the derivative of the previous one.

Theorem 12.3.2 (Taylor's Theorem) Let f : [a,b] i-> K, n G N, f(n~l)

be continuous on [a,b], and f^n\t) exists Vi G [a,b]. Let a / (3 G [a,b] and

define

fc=0

Then, 3x G (a, (3) 3 /(/3) = p(/3) + ^fsi (/3 - a)".

Remark 12.3.3 Forn = 1, the above theorem is just the mean value theorem.

Proof. Let M 3 /(/3) = p(f3) + M(0 - a)n.

Let g(t) = f(t)-p(t)-M(t-a)n, a < t < b, the error function. We will show

that n\M = f{n){x) for some x G (a, 6). We have g^(t) = fn\t)-n\ M, a <

t < b. If 3x G (a, 6) 9 ^*n^(x) = 0, we are done.

p(k\a) = f{k)(a), k = 0,...,n-l =>

g{a) = g'(a) = g»(a) = • • • = ^""^(a) = 0.

Our choice of M yields g{(i) = 0, thus g'(x\) = 0 for some x\ G (a,/3) by

the Mean Value Theorem. This is for <?"(•), one may continue in this manner.

Thus, g(n\xn) = 0, for some xn G (a, x„_i) C (a,fi). •

Definition 12.3.4 A function is said to be of class Cr if the first r derivatives

exist and continuous. A function is said to be smooth or of class C°° if it is

of class Cr, VrGN.

Theorem 12.3.5 (Taylor's Theorem) Let f : A H-> E, be of class Cr for

A C M", an open set. Let x,y G A and suppose that the segment joining x

and y lies in A. Then, 3c in that segment 3

r_1 1 1

f{y)-f(x) = ^-fW(y-x,...,y-x) + -(c)fV(y-x,...,y-x)

fc=i ' r'

where fW(y-x,...,y-x) = Y;iu...,ik (e,f•*•&,,, ) fan ~ xh) •'' (!/<„ ~ xiJ •
Setting y = x + h, we can write Taylor's formula as

f(X + h) = f{x) + f'(x) • h + • • • + ^J—^f^)ix) .(h,...,h) + Rr-^X, h),