Analysis and Approaches for IBDP Maths Ebook 2 | Page 205

63
Paper 2 Section A – Maclaurin Series
Example
By successive differentiation, find the Maclaurin series of
the term in
3
x .
2 x
f ( x) x e sin x
� � , as far as
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Solution
f
2 0
(0) 0 e sin 0 0
� � � (A1) for correct value
( ) 2 x
x
f � x � x � e sin x � e cos x
x
f �( x) � 2 x � e (sin x � cos x)
f� � � � � (A1) for correct value
0
(0) 2(0) e (sin 0 cos 0) 1
x
x
f ��( x) �2 � e (sin x � cos x) � e (cos x � sin x)
x
f ��( x) �2 � 2e cos x
0
f��(0) � 2 � 2e
cos 0 � 4
(A1) for correct value
(3) ( ) 2 x
x
f x � e cos x � 2e sin x
(3) 0 0
f (0) � 2e cos 0 � 2e
sin 0 � 2
(A1) for correct value
2 3
x x (3)
f ( x) � f (0) � xf �(0) � f ��(0) � f (0) �
2! 3!
2 3
x x
f ( x) � 0 � x(1) � (4) � (2) � M1
2 6
2 1 3
f ( x) � x � 2x � x � A1
3
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Exercise 63
1. By successive differentiation, find the Maclaurin series of
term in
2
x .
2. By successive differentiation, find the Maclaurin series of
term in
3
x .
2x
f ( x) � e tan x
2
, as far as the
f ( x) �(1 � x) 3 , as far as the
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