Analysis and Approaches for IBDP Maths Ebook 2 | Page 193

3. The functions f and g are defined as f ( x) � asin�
x and g( x) � sin 2�
x respectively,
a � 0 , 0�x
� 1. Let P( r, f ( r )) be the point of intersection of the graphs of f and g .
(a) Consider the case when a � 1.
(i) Find r .
(ii)
Hence, show that the area of the region bounded by graph of f , the graph
of g and the lines x
� r and x � 1 is
9
4� . [10]
13
Q is the local maximum of the graph of g .
(b) Find the value of a if the graph of f passes through Q .
(c)
[4]
Find the least possible value of a such that the graph of f is above the graph of
g for 0�x
� 1.
4. The functions f and g are defined as f ( y) � acos2�
y and g( y) � cos�
y respectively,
a � 0 , 0� y � 1. Let P( f ( r), r ) be the point of intersection of the graphs of f and g .
[4]
(a) Consider the case when a � 1.
(i)
(ii)
� 3 1�
Q �
,
2 6�
� �
Show that
2
r � .
3
Hence, show that the area of the region bounded by graph of f , the graph
of g and the lines y � 0 and y� r is 3 3
4� . [10]
is a fixed point on the graph of g .
(b) Find the value of a if the graph of f passes through Q .
(c)
2
1 �1�
1�8a
�
Show that r � arccos
.
� � 4a
�
� �
[3]
[7]
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