Analysis and Approaches for IBDP Maths Ebook 2 | Page 188

Your Practice Set – Analysis and Approaches for IBDP Mathematics
2. Consider the function
f( x)
�
a
1
� x
2 2
, x � a.
(a)
Show that f is an even function.
(b) (i) State the largest possible domain of f such that
1
f �
exists.
[2]
(ii)
Under this restriction, find the expression of
Let R be the region enclosed by the curve
3
x�� a and
2
3
x� a.
2
y �
a
1
� x
2 2
1
f � .
, the x -axis, the lines
[5]
(c) Find the area of R , giving the answer in terms of � .
The region bounded by the curve y �
a
1
� x
2 2
is rotated through 2π radians about the x -axis.
and the lines
2
x�� a and
2
[5]
2
x� a,
2
(d) By using a suitable substitution and the formula � sec xdx � ln sec x � tan x � C ,
�
show that the volume of the solid generated is ln(3 � 2 2) .
a
[12]
3. Consider the function f( x) �2� 3 x
a
, x � 0 , A( a, 2� 3 ) is a point on the graph of f( x ).
(a)
(b)
Show that f is a one-to-one function.
Find the expression of
1
f � .
[2]
L is the tangent to the graph of f at A . It is given that L passes through the origin O .
[2]
(c) Find the equation of L .
[6]
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