Analysis and Approaches for IBDP Maths Ebook 2 | Page 207

64
Paper 2 Section A – Approximations in Maclaurin Series
Example
(a)
By successive differentiation, find the Maclaurin series of
far as the term in
3
x .
(b) Hence, use this series to find an approximate value of ln 6 .
� 2 �
f( x) � ln � �
� 2 � 3 x � , as
[6]
[3]
14
Solution
(a)
� 2 �
f (0) �ln � ��0
�2 � 3(0) �
f ( x) � ln 2 �ln(2 � 3 x)
1
f� � �
( x) �0 � � � (3)
�2�
3x
�
3
f�( x)
��
2 � 3x
3 3
f �(0)
� � � �
2 � 3(0) 2
� 1 �
f��( x) � �3( �1) � (3)
2 �
�(2 � 3 x)
�
9
f��( x)
�
2
(2 � 3 x)
9 9
f ��(0)
� �
f
f
(3)
(3)
2
(2 � 3(0)) 4
� 1 �
( x) �9( �2) � (3)
3 �
�(2 � 3 x)
�
54
( x)
��
3
(2 � 3 x)
54 27
(2 � 3(0)) 4
(3)
f (0) � � � �
3
(A1) for correct value
(A1) for correct value
(A1) for correct value
(A1) for correct value
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