Analysis and Approaches for IBDP Maths Ebook 1 | Page 321

Exercise 51
51.1 (a)
f �( x) �� 5e �
f ��( x) � 25e �
5x
5x
f ( x) �� 125e �
(3) 5x
(b) f ( x) �( � 5) e �
51.2 (a) f �( x) �� cos x
f ��( x) � sin x
(b)
51.3 (a)
( n ) n 5 x
f (3) ( x) � cos x
f (4) ( x) �� sin x
(2 n) n�1
f ( x) �( � 1) sin x
g�( x)
� nx �
n 1
g��( x) �n( n � 1) x n�
g ( x) � n( n �1)( n � 2) x n�
(3) 3
(4)
g x n n n
4
( n�
3) x n�
(b) 12
51.4 (a) g�( x)
� nx �
( ) � ( �1)( � 2)
1
g��( x)
�� nx �
2
g ( x) � 2nx �
(3) 3
g ( x) �� 6nx �
(b) 36
Exercise 52
(4) 4
52.1 (b)
� 2 �
��2, �
� 9 �
(c)
2
�k
� 2
9
52.2 (b) ( �2, � 6)
(c) k �� 6 and
2
2
k ��
3
52.3 (b) (2,1)
(c) k � 1 and k � 3
52.4 (b)
�5 � �
� , � 2 �
�12
�
(c) 4
Exercise 53
53.1 (a) 10
(b) 16
53.2 (a) 3
(b)
13
36
53.3 � 3
53.4
2
4
e
Exercise 54
54.1
54.2
54.3
54.4
7
640x
516311040x
6
560x
16x �3
Exercise 55
55.1 (a)
x 2
�3x� 2�
e � �
(b) � 4.42 and � 0.763
(c) Refer to solution
55.2 (a) cos x�
sin x
(b) 0.785 and 3.93
(c) Refer to solution
55.3 (a)
x x
e cos( e )
(b) (1.23, � 3.29)
(c) Refer to solution
55.4 (a)
�x
�x
� e sin( e )
(b) ( � 1.59, 4.81)
(c) Refer to solution
11
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