Analysis and Approaches for IBDP Maths Ebook 2 | Page 210

Your Practice Set – Analysis and Approaches for IBDP Mathematics
65
Paper 2 Section B – Miscellaneous Problems
Example
The function f is given by f ( x) �ln(2 � cos x)
.
(a) (i) Find the first three derivatives of f( x ).
(b)
(c)
(ii) Hence, find the Maclaurin series for f( x ) up to and including the
By using the Maclaurin series for arctan x and
arctan x
e up to and including the
arctan x
e
Hence, find lim .
x�0
f ( x )
3
x term.
3
x term.
x
e , find the Maclaurin series for
[10]
[4]
[3]
Solution
(a)
(i)
� 1 �
f �( x) ��
�( �sin x)
�2 � cos x �
(M1) for valid approach
sin x
f�( x)
��
2 � cos x
A1
(2 � cos x)(cos x) � (sin x)( �sin x)
f��( x)
��
2
(2 � cos x)
(M1) for valid approach
2cos cos sin
f��( x)
��
2
(2 � cos x)
2 2
x � x � x
2cos x �1
f��( x)
��
A1
2
(2 � cos x)
f
f
f
(3)
(3)
(3)
2
(2 cos x) ( 2sin x)
� � �
(2 cos x � 1)(2)(2 � cos x)( �sin x)
( x)
��
(M1) for valid approach
4
(2 � cos x)
( �2sin x)(2 � cos x � 2cos x �1)
( x)
��
3
(2 � cos x)
2sin x(1 � cos x)
( x)
�
3
(2 � cos x)
A1
200
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