Analysis and Approaches for IBDP Maths Ebook 1 | Page 193

Front Page
66
Paper 1 Section B – Integration
and transformations of graphs
Example
The graph of a function f has a local maximum point at A , where x � 0 . The derivative
2
of f is given by f �( x) � 3x �3x
� 18.
(a) Find the x -coordinate of A .
(b) The y -intercept of the graph is at (0, � 3) . Find an expression for f( x ).
The graph of a function g is obtained by reflecting the graph of f in the x -axis,
� 7 �
followed by a translation of � �.
�11�
[5]
[6]
15
(c) Find the x -coordinate of the local minimum point on the graph of g .
[3]
Solution
(a) f�( x) � 0
(M1) for setting equation
2
3x
�3x�18 � 0
2
x � x�6�
0
( x�3)( x� 2) � 0
M1A1
x �� 3 or x � 2 (Rejected) A1
�x
� � 3
A1 N2
(b) f( x )
2
� � (3 x � 3 x �18)d
x
(M1) for indefinite integral
3 2
3x
3x
� � �18x�
C
3 2
3 3 2
� x � x �18x � C
A3
2
3 3 2
�3 � 0 � (0) �18(0)
� C
C �� 3
2
3
2
(A1) for correct value
3 2
� f ( x) � x � x �18x
� 3
A1 N3
[5]
[6]
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