Analysis and Approaches for IBDP Maths Ebook 2 | Page 274

Your Practice Set – Analysis and Approaches for IBDP Mathematics
(e) (1) Express A
n
in terms of n .
(2) Write down the formula for An
� 2
.
( e �1)( e �1)( e � 2)
�
(3) Show that An �An�
2
� , n� .
n�3
e
(f)
Find � if
23
e A
� e � �
2 � 0 .
Part II – Investigating the Quantity 2
Let
B be the area bounded by the graph of g , the line y � g( k)
and the line x�k� 1.
k
(g) (1) By considering the rectangle ABCD , find an inequality for B
1.
(2) On the diagram, sketch and shade the region B
2
.
(3) Write down inequalities for B
2
and B
3
.
(4) Suggest an inequality for B
n
.
Part III – Combining the Two Quantities
(h) (1) Express B
1
in terms of A
1.
e �1
(2) Show that B2 � � A
3 2.
e
e �1
(3) Hence, show that Bn
� � A
n 1 n.
(i) (1) State the concavity of the graph of f for x � 1.
e �
(2) Hence, interpret the relationship between A
n
and
B
n
geometrically.
(j) (1) Write down
1
lim
n��
e n
.
(2) Hence, show that lim A �lim B � 0 .
n��
n
n��
n
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