Analysis and Approaches for IBDP Maths Ebook 1 | страница 215

Front Page
72
Paper 2 Section B – Areas between curves
Example
The following diagram shows the graphs of f ( x) �ln x � 3 and g( x)
� x , for 0�x
� 6.
16
(a) Let A be the area of the region enclosed by the curves of f and g .
(i) Write down the values of x such that f ( x) � g( x)
.
(ii) Calculate the value of A .
(b) Find the value of x for which the gradient of f is equal to the gradient of g .
(c)
a
It is given that � f�
( x )d x�
2 , find a .
1
[6]
[4]
[4]
Solution
(a) (i) x � 0.0524691 and x � 4.5052415
x � 0.0525 and x � 4.51
A2 N2
(ii)
A
�
4.5052415
� ( f ( x) �g(
x))dx
(M1) for definite integral
0.0524691
�
4.5052415
� (ln x � 3 � x )d x
A2
0.0524691
� 5.694451565
� 5.69
A1 N3
(b) f �( x) � g�( x)
(M1) for equal derivatives
1 0 1
x � � A2
x � 1
A1 N3
[6]
[4]
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