M11/5/MATHL/HP3/ENG/TZ0/SE

MATHEMATICSHIGHER LEVELPAPER 3 – SERIES AND DIFFERENTIAL EQUATIONS

Monday 9 May 2011 (morning)

INSTRUCTIONS TO CANDIDATES

y�Do not open this examination paper until instructed to do so.y�Answer all the questions.y�Unless otherwise stated in the question, all numerical answers must be given exactly or correct

to three significant figures.

2211-7208 3 pages

1 hour

© International Baccalaureate Organization 2011

22117208

M11/5/MATHL/HP3/ENG/TZ0/SE

2211-7208

– 2 –

Please start each question on a new page. Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. In particular, solutions found IURP�D�JUDSKLF�GLVSOD\�FDOFXODWRU�VKRXOG�EH�VXSSRUWHG�E\�VXLWDEOH�ZRUNLQJ��H�J��LI�JUDSKV�DUH�XVHG�WR�¿QG� a solution, you should sketch these as part of your answer. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. You are therefore advised to show all working.

1. [Maximum mark: 10]

� �D�� )LQG�WKH�¿UVW�WKUHH�WHUPV�RI�WKH�0DFODXULQ�VHULHV�IRU� ln ( )1+ ex. [6 marks]

� �E�� +HQFH��RU�RWKHUZLVH��GHWHUPLQH�WKH�YDOXH�RI� lim ln ( ) lnx

x xx→

+ − −0 22 1 4e

. [4 marks]

2. [Maximum mark: 8]

� &RQVLGHU�WKH�GLIIHUHQWLDO�HTXDWLRQ� ddyx

x y= +2 2 �ZKHUH� y =1�ZKHQ� x = 0 .

� �D�� 8VH� (XOHU¶V� PHWKRG� ZLWK� VWHS� OHQJWK� ���� WR� ¿QG� DQ� DSSUR[LPDWH� YDOXH� RI� �y ZKHQ� x = 0.4 . [7 marks]

� �E�� :ULWH�GRZQ��JLYLQJ�D�UHDVRQ��ZKHWKHU�\RXU�DSSUR[LPDWH�YDOXH�IRU��y��LV�JUHDWHU�WKDQ�RU�OHVV�WKDQ�WKH�DFWXDO�YDOXH�RI��y . [1 mark]

3. [Maximum mark: 11]

� 6ROYH�WKH�GLIIHUHQWLDO�HTXDWLRQ

x yx

y xy x2 2 23 2dd

= + +

� JLYHQ�WKDW� y = −1�ZKHQ� x =1 ���*LYH�\RXU�DQVZHU�LQ�WKH�IRUP� y f x= ( ) .

M11/5/MATHL/HP3/ENG/TZ0/SE

2211-7208

– 3 –

4. [Maximum mark: 15]

� 7KH�LQWHJUDO� In �LV�GH¿QHG�E\ I x xnx

n

n= −+

∫ e dπ

π( )| sin |

1��IRU� n∈` .

� �D�� 6KRZ�WKDW� I0121= + −( )e π

. [6 marks]

� �E�� %\�OHWWLQJ� y x n= − π ��VKRZ�WKDW� I Inn= −e π

0 . [4 marks]

� �F�� +HQFH�GHWHUPLQH�WKH�H[DFW�YDOXH�RI� e d−∞

∫ x x x0

| sin | . [5 marks]

5. [Maximum mark: 16]

� 7KH�H[SRQHQWLDO�VHULHV�LV�JLYHQ�E\� exn

n

xn

==

∞

∑ !0.

� �D�� )LQG�WKH�VHW�RI�YDOXHV�RI��x��IRU�ZKLFK�WKH�VHULHV�LV�FRQYHUJHQW� [4 marks]

� �E�� �L�� 6KRZ��E\�FRPSDULVRQ�ZLWK�DQ�DSSURSULDWH�JHRPHWULF�VHULHV��WKDW�

ex xx

− <−

1 22

��IRU�0 2< <x .

� � �LL�� +HQFH�VKRZ�WKDW� e < +−

⎛⎝⎜

⎞⎠⎟

2 12 1

nn

n

��IRU� n∈ +] . [6 marks]

� �F�� �L�� :ULWH�GRZQ� WKH�¿UVW� WKUHH� WHUPV�RI� WKH�0DFODXULQ� VHULHV� IRU� 1− −e x and

H[SODLQ�ZK\�\RX�DUH�DEOH�WR�VWDWH�WKDW

12

2

− > −−e x x x ��IRU�0 2< <x .

� � �LL�� 'HGXFH�WKDW� e >− +

⎛⎝⎜

⎞⎠⎟

22 2 1

2

2n

n n

n

��IRU� n∈ +] . [4 marks]

� �G�� /HWWLQJ� n =1000��XVH�WKH�UHVXOWV�LQ�SDUWV��E��DQG��F��WR�FDOFXODWH�WKH�YDOXH�RI�H�FRUUHFW�WR�DV�PDQ\�GHFLPDO�SODFHV�DV�SRVVLEOH� [2 marks]