
Mean Value Theorems 9-21
f '' ' x x f '''
f x n x f ''' n
n n
( ) ( ) ( )
( ) !( ) ( ) !
( )
= − ∴ =
= − ∴ =
−
− +
6 1 0 6
1 0
4
1
Now, Maclaurin’s theorem with Lagrange’s form of remainder is
f x f x f '
x
f ''
x
n
f x
x x
x
n
n
( ) ( ) ( )
!
( )
!
( )
( ) ( )
( )
= + + + +
− = + +
−
0 0
2
0
1 1 1
2
1
q
22 3
1 2 3
2
2
3
6
1 1
1
!
( )
!
( )
!
( !)
( )
+ + + +
− = + + + + + +
∴
−
=
−
x x
n
n
x x x x x
n
n
11
2 3
+ + + + + +x x x x
n
Example 9.20 Expand e
x
by Maclaurin’s theorem with Lagrange’s form of remainder after n terms.
Solution:
f x e f
f ' x e f '
f '' x e f ''
f x e f
x
x
x
n x
( ) ( )
( ) ( )
( ) ( )
( )
( )
= =
= =
= =
=
0 1
0 1
0 1
(( )
( )
n x
x eq
q
=
Now, Maclaurin’s theorem with Lagrange’s form of remainder ...