Analysis and Approaches for IBDP Maths Ebook 2 | Page 212

Your Practice Set – Analysis and Approaches for IBDP Mathematics
Exercise 65
1. The function f is given by
tan
f( x) � 2 x .
(a) (i) Find the first two derivatives of f( x ).
(b)
(c)
(ii) Hence, find the Maclaurin series for f( x ) up to and including the
term.
By using the Maclaurin series for
2x
4x
ln(1 7e
12 e )
2x
e and ln(1 � x)
� � up to and including the
Hence, find
lim
x�0
2x
4x
ln(1 �7e
�12 e )
f( x)
.
2
x term.
2
x
, find the Maclaurin series for
[8]
[9]
[3]
1
2. The function f is given by f( x)
� 2
1 � x
.
(a) (i) By using the Maclaurin series for arctan x , find the Maclaurin series for
(b)
f( x ) up to and including the
4
x term.
1
(ii) Hence, find the Maclaurin series for gx ( ) � 3
1 � x
up to and including the
6
x term.
Using (a), show that
gx ( ) �1
(c) Show that lim
3
1
x 0 x �� .
�
e �1
1
0.999 � � 0.999001 .
1.001
[6]
[4]
[6]
3. (a) By considering sin 2x and cos2x , show that the Maclaurin series for
2
4
2 256 4
sin 4x up to and including the x term is 16x
� x � .
3
(b)
Hence, show that
� 2
48 � � � �
�sin
� .
16 3 4 4
[7]
[9]
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