Analysis and Approaches for IBDP Maths Ebook 2 | Page 211

f ( x) � f (0) �xf
�(0)
(ii) 2 3
x x (3)
� f��(0) � f (0) �
2! 3!
� sin 0 �
f ( x) � ln(2 � cos 0) � x � � �
� 2 � cos0 �
2
x � 2cos0 �1
�
� � �
2 �
M2
2 � (2 � cos 0) �
3
x � 2sin 0(1 � cos 0) �
� �
3 ��
6 � (2 � cos 0) �
2 3
x � 1 � x
f ( x) � ln3 � x(0) � �� �� (0) � A1
2 � 3 � 6
1 2
f ( x) � ln3 � x � A1
6
14
(b)
(c)
[10]
2 3
arctan (arctan ) (arctan )
e
x �1� arctan x� x � x � (M1) for valid approach
2! 3!
e
arctan x
3
� x
�1� �x� �
� 3
�
�
�
3
2
3
3
� x � � x �
� x� � � � x� � �
3 3
�
� �
�
� �
�
2! 3!
(A1) for substitution
3 2 3
arctan
1
x x � x �
e
x � � x� � � � (A1) for correct approach
3 2 6
arctan x 1 2 1 3
e �1� x � x � x � A1
2 6
[4]
1 2 1 3
arctan x 1� x � x � x �
e
lim � lim 2 6
A1
x�0 f( x)
x�0 1 2
ln 3�
x � 6
e
1� 0 � 0 � 0 �
�
arctan x
lim �
x�0
f ( x ) ln 3 � 0
e
arctan x
lim
x�0
f ( x ) ln 3
M1
1
� A1
[3]
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