áuÙ ãdß ê1ß Ý Ù3Ú ÛÍëbÝ5æ êaÜ7Ú Û è?éxåuÝ...

8

Click here to load reader

Upload: lytuyen

Post on 29-Jul-2018

212 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: áUÙ ãDß ê1ß Ý Ù3Ú ÛÍëbÝ5æ êaÜ7Ú Û è?éXåuÝ …rboyer/courses/math533/week06.pdf · ØcÙ?Ú ÛÜ7Ý5Ù*Þsß à Ùȧ§§ §§§ §§§§ §§§ §§§ §RáUâRãDãDÙ

������������ � �� ��������������������������� � !�#"%$&"('*)+,��-/.10325476�8#9�0;:<4#.&0;4=0?>#@%A#BC=CD@�0?25:E9=F1.#6HGJIE0K@�2K.F�0K:�.#L&M%25478N#OPQ@D25@�9�FRIEI�03>#@%SFROK:E9D6@3TU.#:<0;:<4#.#OVF1.#6&25@�O38#I<0;OXWY254#CZI�FRO;0J[\@�@�]�^!_�a`b�c�����d)c�(eDf,@�0%gch&Si@=0?>#@DOKBC=CD@�0?2j:<9kL725478NH47.mlnI<@�0;0K@�25O�op0?>FR0X:<O�qc0?>#@DOK@�0X4RWrF�I<IsS#: t5@�9�0K:E47.#O4RWsFDOK@�0u4�WvOK:Ew�@=lm8.#6@�2xWY8.#9�0;:<4#.FRI�9�4#C=N41O;:<0;:<4#.,^u-�0X:<OyF*TU.#:E0K@*L725478Nz4RWv4#256@�2ul�{�^PQ@D25@�9�FRIEI�03>#@�:|2XSFROK:E9*OK0?2K8#9�0?82j@1^}%�),~�)U���d���d)c�(eJGy.aB�N@�2;CV8#03FR0K:E47.H4RWbF�TU.#:<0;@=OK@�0yg�:EOy@�:E0?>#@�2=�5���u0?>#@D:E6@�.a0K:E0�B1qs�Y�1�JFk9�B#9�I<@1q\���1�9�47C=Ni41OK:E0K:E47.H4RW(6:EO�t�4a:|.a0y9�B#9�I<@�O�^}%�),)��3eDPQ@�8#OK@�:�.#6�8#9�0K:E47.�47.n0?>#@=OK:Ew�@�4RWJgu^H-�WV��gJ�,���1q�0?>#@�25@�O?8#IE0=>#41IE6O�^HPQ@�F�OKO?8CD@_0?>#@25@�O?8#IE0xW�4#2uFRIEI�O;@�0KOu[u:E0?>zOK:Ew�@%IE@�OKOy0?>F1.��,^��F1]R@ ��gJ�����,^��u>#4#41OK@mF�Ni@�2KCD8#0?FR0K:E47.����ZgX�����^��(:E9�]��i ��Zgu^¡f¢@�0��a£Q�������a£�¤, ?��q�W�472¥D���1¦?�i¦�§�§�§�^sf,@�0ylmS@%03>#@%I�F125L1@�O;0J:�.a0K@�L1@�2xOK4=0?>#@y@�IE@�CD@�.a0KO%�i ¨¦?�#©R¦�§�§�§K¦3�7h_F125@%F�I<I�6:EOK0K:�.#9�0�^�\I�FR:�C�e������#h�(���i �^ª�B�FROKO38C=N#0K:E47.,q�0?>#@�@�IE@�CD@�.10;O��   ¦?� © ¦�§�§�§K¦3� h ¦?� hR«�  F125@�.#410D6:EOK0K:�.#9�0�^�¬x@�.#9�@1q\� hR«�  ��� £ W�472OK47C�@u¥U^(-�W¢¥_­¡�1q03>#@�.�� £ �®���Y� £�¤,  �bFa.#6�� £ ������� h �\[X>#:<9�>�:EOu:�C=Ni41OKOK:�S#IE@=OK:�.#9�@*0?>#@%N@�2KCD8#0?F¨¯0K:E47.k��:EOJ47.#@3¯°0;4R¯°47.#@1^\¬u@�.#9�@1q#¥V�±�y472x� b�������#h��^f,@�0X²³�¡g�´¶µ¨� R¦�§�§�§K¦3�7hi·7^¸%¹�º�»?¼�½�¾7¿�À�Á¨Â e\:ÃW(Äu��²%q#0?>#@J����Ä��\��²ÅOK:�.#9�@¶�z:<Ox47.#@3¯°0K4R¯°47.#@���@�Æ1@�25B��_�&Ç��ȵ¨�i ¨¦�§�§�§;¦?�#h·%:EOX0?>#@:�C=FRL1@*4RWsF1._@�IE@�CD@�.10XW°2547CÉÇJ��^ª�B®:�.#6�8#9�0K:E47.,qs��� ²%qs03>#@�25@�OK0325:E9�0K:E47.®4RWu�®0K4n0?>#@&O?8S#O;@�0�²ÊCD8#OK0_S@&03>#@&N25476�8#9�0�4�W¶6:EO�t541:�.109�B#9�I<@�O�^yPQ@VCkF�B�@�Ë#0K@�.#6�Fa.1B&Ni@�2KCD8#0?FR0K:E47.H47.H²�S1B&>F�Æ#:|.#LH:E0¶FR9�0yFROX0?>#@D:E6@�.10K:E05Bm47.&Ç*^J�x>#@25@�O?8#IE0y.#4R[�W�4aI<IE4R[uO�^}%�),~�)U���d���d)c�(e³ÌÍÆ1@�25B³Ni@�2KCD8#0?FR0K:E47.Å47.�F�OK@�0�[u:E0?>ÅFR0�IE@�FROK0�05[\4®@�I<@�C�@�.10KOH:EO�F®N254#6�8#9�0&4RW0?2KFa.#O?N41O;:<0;:<4#.#O�^}%�),)��3eb�r47.#OK:E6@�2u@�:E0?>#@�2X:E6@�.a0K:E05BUe���i �¦?�#©R¦�§�§�§3¦?�#h7�(�Z�Y� R¦?�#©������#©R¦?�aÎÏ�Ч�§�§c���#h#¤, ¨¦3�7h��¦\472���   ¦?� © ¦�§�§�§3¦?� h �(�Z�Y�   ¦?� h �����   ¦3� h#¤,  �y§�§�§U���   ¦3� © ��¦\472!_�a`b�c�����d)c�(es�uF�I<IFuNi@�2KCD8#0?FR0K:E47. »�½R»? :ÃW�:E0c:EO�03>#@bN254#6�8#9�0�4�WiF1.X@�Æa@�.%.#8CVSi@�2�4RW0?2KF1.#O?Ni41OK:E0K:E47.#O�o410?>#@�2j[u:EOK@1q�:E0X:EO Á#ÑÑ ^s�u>#@ÐWYFR9�0x0?>FR0u0?>#:EOX6@3TU.#:E0K:E47.m:EOJ[\@�IEIï°6@3TU.#@�6�:EOySFROK@�6H47._0?>#@yW�41IEIE4R[u:�.#L�^}%�),~�)U���d���d)c�(e%Ì(Æ1@�2jB�N@�2;CV8#03FR0K:E47.m:EO%@�:E0?>#@�2Ð4766n472u@�Æa@�.,oc0?>FR0y:EO�q,:EO*0?>#@=N25476�8#9�0*4RWbF1.H4#66��@�Æ1@�.�X.#8CVSi@�2u4�Wv0?2;F1.#O?N4aOK:E0K:E47.#O�^}%�),)��3esPn@yI<@�0yF=Ni@�2KCD8#0?FR0K:E47.mFR9�0u4#._0?>#@%:�.#6:E9�@�O%4RW(0?>#@%Ni41IEB�.#47C�:|F�I(L1:EÆ1@�.&FROX0?>#@%N254#6�8#9�0�eÒ ?Ó�Ô�Õ£�Ó�h ��Ö�Ôp×�Ö�£Ï�p§

Page 2: áUÙ ãDß ê1ß Ý Ù3Ú ÛÍëbÝ5æ êaÜ7Ú Û è?éXåuÝ …rboyer/courses/math533/week06.pdf · ØcÙ?Ú ÛÜ7Ý5Ù*Þsß à Ùȧ§§ §§§ §§§§ §§§ §§§ §RáUâRãDãDÙ

ØcÙ?Ú�Û�Ü7Ý5Ù*Þsß�à�Ùȧ�§�§�§�§�§�§�§�§�§�§�§�§�§�§�§�§RáUâRãDãDÙ�Û�Ý3ßäÚ�åuÝ5æ1Ü�ç�è3é\áUÙ�ãDß�ê1ß�Ý�Ù3Ú�ÛÍëbÝ5æ�êaÜ7Ú�Û�è?éXåuÝ�æaÜ�çQìJÚ�Û�ßäæaíè¸%¹�º�»?¼�½�¾7¿�À�Á¨Â#º e%�j���\Gy.�l¯°9�B#9�IE@=:EOJ@�Æa@�.�:ÃWblH:<OX4#66co�:E0X:<OX4#66&:ÃWblH:<OX@�Æ1@�.,^���1�\�u>#@yO;@�0b4�WvFRIEIc@�Æ1@�.�Ni@�2KCD8#0?FR0K:E47.#OxW�4#2KCDOuF*O?8S#L725478N,qi9�FRIEI<@�6m0?>#@ ¾�î�¿j»?¼KÂ�¾7¿�À�Âaï�º¨ð¹�ï¼3Á�ð�ñ ÇXh4RWvgph,^sòu4a0K@y0?>FR0uÇXh_:<Ox0?>#@%]R@�2K.#@�Ic4�W�0?>#@%.FR0?82;FRI�>#47CD4#CD472KN>#:EO?CÉL1:EÆ1@�.�SaB�C=FaNN#:|.#L�gch_:�.a0K4�5×y���ÏóÏqp[J>#@�25@�ôD:EO*0?>#@�.#8CVSi@�2y4RW\0?2KF1.#O3N41OK:E0K:E47.#O=8#OK@�6n:|.õ6@�OK9�25:�S#:�.#LQ0?>#@=N@�2KCD8#0?F�0K:E47.Q9�F�I<IE@�60?>#@ ºKÀÃï� 4RWs0?>#@VNi@�2KCD8#0?FR0K:E47.,^yA#:�.#9�@=0?>#@*:�C=FRL1@DL7254#8N&>FROX47256@�2X�q�[b@ÐTU.#6m0?>FR0X0?>#@%472j6@�2u4RWÇ h :<OX@�ËFR9�0KIEB�>FRIÃWs0?>#@y4#256@�2u4�Wv0?>#@XW°8#IEI�O;B�C=C�@�0?25:E9DL725478N�g h ^PQ@D.#4¨[Å9�47.10;:|.#8#@%4#82u6@�OK9�2j:|N#0;:<4#.&4�Wv0?>#@*OKBC=CD@�0?25:E9*L725478N,^}%�),~�)U���d���d)c�(euÌ(Æ1@�25B�@�Æa@�.&N@�2;CV8#03FR0K:E47.�:EOX0?>#@VN2j476�8#9�0X4RWs��¯°9�B79�IE@�O�^}%�),)��3eb�x>#@%N25476�8#9�0y4�WÍF1.aB_0�[\4D0?2KFa.#O?N41O;:<0;:<4#.#Oy:<OyF=��¯°9�B#9�IE@1^u�\47.#OK:E6@�2x0?>#@%05[\4D:E6@�.10K:E0K:E@�O�e

����¦3ö�����öR¦K÷��p�Z����¦?öR¦?÷���¦����¦3ö�����÷R¦3ø7�p�Z��÷R¦?��¦?ø#���Y��¦KöR¦K÷���§}%�),~�)U���d���d)c�(e\GJIEI(�a¯°9�B79�IE@�O%:�.Hgch�F125@y9�47.�tK8#L7FR0;@1^}%�),)��3eb�r47.#OK:E6@�2u03>#@%:E6@�.10;:<05B,e

� ¤,  ���i �¦?�#©R¦�§�§�§K¦3�ù����p¦\[J>#@�25@J�_��ú � � §�§�§ ��  �7© §�§�§û�iùcü §ý%),�),�ä�d���+se\�\[\4�N@�2KCD8#0?F�0K:E47.#OJ:�.mgch�F12j@y9�47.�tj8#L#FR0K@%:ÃW(0?>#@�B&>F�Æ1@y0?>#@*O?F1CD@y9�B#9�IE@VO;0?2K8#9�0?825@1^òx410K@1e(0?>#@y9�4#.�tj8#L7F�9�B�9�I�FROKO;@�OJ4RW(gch�Fa25@%.FR0?82KF�I<IEB&NF12;F1CD@�0?25:Ew�@�6mS1B_0?>#@yNF1250K:E0K:E47.#OX4RWvl�q[J>#@�2j@F�NF1250K:E0K:E47.¡l±�þ�i %ÿ,�����7©Dÿ��� ÿ�ÿ�ÿ����#h&ÿlÅ9�472K25@�O?Ni47.#6O&0;4 F®Ni@�2KCD8#0?FR0K:E47.�[x:<03>³�i Q�3¯9�B#9�I<@�O�qJ�#©���¯°9�B79�IE@�O�qy§�§�§Eq\�7h�.a¯°9�B79�IE@�O�^¡�x>#@�25@&:EO�FmO3N@�9�:�FRI%.#410?F�0K:E47.nW�4#2D0?>#@�9�47.�tK8#L7FR9�B®9�I�FROKO� l ù�� ¦¨��l_×���� ù���� ¦�§�§�§?¦�� ù�� q�[J>#:E9Ï>mCD@�Fa.#Oy0?>FR0J� h 6@�.#410;@�Oy0?>#@V.#8CDS@�2X4RWbli¯°9�B#9�I<@�O�q�� h7¤¢  0?>#@.#8CVSi@�2u4�WÍl�× �y9�B#9�I<@�O�qUF1.#6HOK4=47.,^fcFR0;@�2®[\@¡O?>FRIEIm@�OK0?FaS#I<:EO?> 03>#@¡W�41IEIE4¨[x:|.#L±W�472KCD8#I�F W�472�03>#@�472j6@�2®4RWQ0?>#@¡9�I�FRO;O � l ù�� ¦¨�Yl�×��� ù���� ¦�§�§�§K¦�� ù e

l�{l�{ l ù�� �Yl�×®����{ä�Yl�×®��� ù���� ÿ�ÿ�ÿK�1{�� ù §}%�),~�)U���d���d)c�(e\GJIEI�9�B79�IE@�Oy4RW(47256@�2X��F125@y9�47.�tK8#L7FR0;@%:�.&ÇXh�:ÃWs���³l_×��^}%�),)��3e�PQ@J[x:<IEIcCD4#6:ÃW�B_0?>#@JN254#4RW,W�472(0?>#@xW°8#I<I,OKBC=CD@�0325:E9JL72j478N,^(-�WU�k:<OuF1.�@�Æ1@�.�Ni@�2KCD8#0?FR0K:E47.,q[\@VF12j@y647.#@1^u-�W��cq�0?>#478#L#>�:EOJ4#66cq�[b@DCD4#6: W�B��HS1B�9�47CkN41OK:�.#L�:E0X[u:E0?>&FkOK:�.#L1IE@D0?2KF1.#O?Ni41OK:E0K:E47.��l_×®�1¦3lU��0K4D47S#03FR:�.�0?>#@%Ni@�2KCD8#0?FR0K:E47.��7^��u>#@�.�0?>#@y9�B#9�IE@D@��8FRIEO�� ¤,  �i �¦?�#©R¦�§�§�§3¦?�iùÏ���*OK:�.#9�@D0?>#@0?2KFa.#O?N41O;:<0;:<4#.&:EOX6:EO�t�4a:|.a0xWY254#C 0?>#@%O38NN47250X4RWc�c^

Page 3: áUÙ ãDß ê1ß Ý Ù3Ú ÛÍëbÝ5æ êaÜ7Ú Û è?éXåuÝ …rboyer/courses/math533/week06.pdf · ØcÙ?Ú ÛÜ7Ý5Ù*Þsß à Ùȧ§§ §§§ §§§§ §§§ §§§ §RáUâRãDãDÙ

ØcÙ?Ú�Û�Ü7Ý5Ù*Þsß�à�Ùȧ�§�§�§�§�§�§�§�§�§�§�§�§�§�§�§�§RáUâRãDãDÙ�Û�Ý3ßäÚ�åuÝ5æ1Ü�ç�è3é\áUÙ�ãDß�ê1ß�Ý�Ù3Ú�ÛÍëbÝ5æ�êaÜ7Ú�Û�è?éXåuÝ�æaÜ�çQìJÚ�Û�ßäæaíè� »��_¾R¼���� -�0D25@�C=FR:�.#OD0K4�9�47CkN#I<@�0;@�0?>#@k9�I|F�OKOK:ÃT�9�FR0K:E47.�4RWb@�Æ1@�.�N@�2;CV8#03FR0K:E47.#O=8Nm0K4�9�47.�tK8#L7FR9�B1^-�0�:EOm.#410�0?2K8#@m0?>FR0�05[\4®l¯°9�B#9�IE@�Om:|.ÅÇJh�.#@�@�6�S@Q9�47.�tK8#L7FR0;@�:�.¡ÇXh @�Æ1@�.�0?>#4#8#L7> 0?>#@�B¡F125@9�47.�tK8#L7FR0K@\:�.=gch,^(-/.=NF1250K:E9�8#I�F12�q#0?>#@ugch*9�47.�tK8#L7FR9�BD9�I�FROKO�9�47.a0?FR:�.#:�.#LVFyN@�2;CV8#03FR0K:E47.���C=F�ByO?N#IE:E0:�.10;4&05[\4�9�47.#OK:E6@�25@�6�FRODF�ÇJhm9�4#.�tj8#L7F�9�BQ9�I�FROKOD6@�N@�.#6:�.#L�47.m0?>#@�NFa25:E0�Bm4RWu0?>#@_N@�2KCD8#0?FR0;:<4#.:�C=N#IE@�CD@�.a0K:�.#L�03>#@%9�47.�tK8#L7FR9�B1^��y.#@*9�F1.�O?>#4R[È0?>FR0u03>#:<OX9�FROK@*479�9�82jOy: WbFa.#6&47.#IEB�:ÃW��m64#@�Oy.#4109�47C=CD8#0K@y[x:<03>�F1.=4#66&N@�2;CV8#03FR0K:E47._:ÃWÍFa.#6�47.#IEB�:ÃW(0?>#@¶NFa250K:E0K:E47.�4�Wvl�[J>#:E9�>�9�472;25@�O?N4#.#6OJ0K4��9�47.#OK:EOK0;OJ4RW(6:EOK0K:�.#9�0y4#66&:�.10;@�L1@�25O�^fcFR0;@�2s[\@u[x:<IEI�N2j4¨Æa@u0?>FR0�0?>#@JL#25478N#OuÇJh,q�W�4#2bl��! %F125@XOK:�C=N#IE@yL7254#8N#Ob4�W¢4#256@�2\l�{#"1�^$�y.#@y9�F1.O?>#4R[¡0?>F�0JÇ&%('� �sAf\���¦� 1��[J>#:EI<@=Ç*)+'� �sAf\���¦�,a�}%�),~�)U���d���d)c�(eJ�u>#@*L725478NHÇ�-D>FROJO38S#L725478N#OX4RWs47256@�2x�qU�iq�F1.#6/.�^s�u>#@DO?8S#L7254#8N�4RW(47256@�2*.:EO¶.#4#2KC=FRI�^\Ç�-%>FROy.#4=O?8S#L#25478NH4RW(47256@�2�0^}%�),)��3euÇ - >FROy4#.#@%@�IE@�CD@�.a0%4RWs472j6@�2D�1q0?>25@�@D@�IE@�CD@�.a0KO%4�Ws47256@�2X�qUFa.#621k@�IE@�CD@�.a0KO%4RWs4#256@�2�^%�x>#@�.#47.a¯°0?25:EÆ#:�FRIsO?8S#L#25478N#O%4�WrÇ - F125@DL1:EÆ1@�.QSaB,eu0?>25@�@kO?8S#L725478N#O*4RWb4#256@�2y�q,47.#@=.#472KCkFRIO?8S#L72j478N_4RW�472j6@�23.D[J>#:E9�>_:EOu:EOK47CD472;N>#:<9*0K454s©76�4s©1qF1.#6_@�:EL7>10xO?8S#L725478N#Ox4RW�47256@�2\�^(�u>#@�2j@:EOJò8�ÈO?8S#L725478N_4�W�47256@�2�0i^�GXOJF*9�47.#OK@�8#@�.#9�@1q,0?>#@X9�47.aÆ1@�25OK@y4RW(fcFRL72KFa.#L1@:9�OJ�u>#@�4#25@�C :EO\W°FRIEOK@1^f,@�0;9�O%2K8#IE@V4#8#0y0?>#@D9�FROK@D0?>FR0yÇ�-=>FRO%FkO?8S#L72j478Nm4RWs472j6@�2�0^JA8NNi41OK@_0?>FR0X:<0y6:E6cqp9�FRIEI(:<0=<�^�u>#@�.>< >FROD:�.#6@�Ë®�&:�.�Ç�-�O;4&:E0D:<O_.#472KC=FRI�^m-°.�NF1250K:E9�8#I�F12�q�@�Æ1@�25Bn9�41OK@�0=��< >F�OV4#256@�2*0�[\4:�.QÇ*-?":<�^D-°.m4103>#@�2y[\47256O�qc� © �>< W�472yFRIEIs�Q�®Ç*-1^=ª\8#0%0?>#@DO��8F125@=4�WrF1.aBQ��¯°9�B79�IE@_:�.QÇ�-k:<OF1.#4103>#@�2J��¯°9�B#9�IE@1^bA#4�q�< CV8#O;0u9�47.a0?FR:�.&FRIEI$1=��¯°9�B#9�IE@�O�^u�u>#:EOJ9�4#.10?2;FR6:E9�0KOJ03>#@%OK:Ew�@%4�W@<�^¸%¹�º�»?¼�½�¾7¿�À�Á¨Â#º e%�j���\�u>#@%L72j478N�Ç - 9�F1.�S@*:E6@�.10K:ÃT�@�6Q[x:<03>�0?>#@yL72j478N�4RW(25410?FR0;:<4#.FRIcOKBC=CD@�0?2j:<@�O4RWvFk25@�L78#I�F12\0K@�032KF1>#@�6�2547._[X>#@�25@%0?>#@%�¨¯°9�B79�IE@�OX9�472K25@�O?Ni47.#6&0;4V03>#@%0?>25@�@3¯�W�41IE6&O;B�C=C�@�0?25:E@�OJ[X>#41OK@FRË#:EO%4RWJ254103FR0K:E47.H:EOVLa:<Æa@�.QS1Bm0?>#@kI<:�.#@�0?>F�0%NFROKOK@�OD0?>254#8#L7>QF_Æ1@�250;@�ËõFa.#6Q0?>#@=9�@�.a0?2541:E6�4RWu0?>#@47NNi41OK:E0K@_W°FR9�@1^��x>#@�25@�Fa25@A1zO?8#9�>®25410?FR0;:<4#.#O�^&�u>#@�25@&Fa25@=05[\4R¯�W�4aI<6�OKBC=CD@�0?2j:<@�Ok[J>#41OK@�FRË#:<OD4RW254103FR0K:E47.®:EO�L1:EÆ1@�.�SaB®0?>#@&IE:�.#@Q0?>F�0�NFRO;OK@�O�03>25478#L7> 0?>#@mCD:E6�N41:�.a0KO&4RW%05[\4Q47NNi41OK:�.#L�@�6La@�O�^�u>#@�25@yF125@*.%O?8#9�>�2j410?FR0K:E47.#O�^�P�:E0?>k0?>#@u:E6@�.a0K:E0�B1qi[b@X>F�Æa@u:E6@�.10K:ÃT�@�6H0?>#@J05[\@�IEÆ1@u@�IE@�CD@�.a0KOu4RW�Ç*-1^PQ@DC=F1N�F=254a0?FR0K:E47._:�.a0K4�03>#@%OKBC=CD@�0?2jB�L72j478N�47._03>#@yW�4782\Æ1@�250;:<9�@�OX4RW(0?>#@%0;@�0?2KF1>#@�6�254#.,^���1�¢G®OK:�CD:EI|Fa2(F1.FRIEB7O;:<O(9�Fa.%S@\647.#@�0K4X:<6@�.a0K:ÃW�B%0?>#@ÍWY8#IEI#OKB�CkCD@�0?25:E9\L725478N*g - [x:<03>%03>#@b254103FR0K:E47.FRIOKBC=CD@�0?2j:<@�OX4RW(0?>#@%9�8Si@1^\PQ@%[\478#IE6mS@VFaS#I<@D0K4D9�47.#O;0?2K8#9�0yF=C=F1N�WY254#C 0?>#@V2j410?FR0K:E47.#Ox:|.a0K4_0?>#@OKBC=CD@�0?2jB�L72j478N�47._03>#@yW�4782\6:�FRL147.F�I<Oy4�Wv0?>#@*9�8S@1^���1�*�y.#@%9�F1.zO?>#4R[¡0?>FR0JÇ*- '� ��Afb�Y�¦?�1��^PQ@D.#4¨[ÅOK0?8#6B�0?>#@%FRIE0K@�2;.FR0K:�.#L�L725478N�Ç % ^PQ@D25@�9�47256m0?>#@*9�47.�tj8#L#FR9�B&9�I�FROKO;@�Oy4RWbÇ&%ao�03>#@�B&F125@1e � � % ¦ � � © ¦�� ¦ � �i¦�� © q�Fa.#6 � CB ¦ � ED ^u-/.&4a0?>#@�2[\47256O�qa0?>#@X:<6@�.a0K:E05B�Ni@�2KCD8#0?FR0K:E47.,qFÐ9�I|F�OKO\[u:E0?>=05[\4%6:EO�t541:�.a0r�¨¯°9�B79�IE@�O�q,Fy9�I�FROKO\[u:E0?>k47.#@y��¯°9�B#9�I<@1qF1.#6H0?>#@%05[\4D9�I|F�OKOK@�Oy0?>FR0x9�47CD@yW°2547C 0?>#@%O?N#IE:E0K0K:�.#L�4RWs03>#@y9�47.�tj8#L#FR9�B&9�I�FROKOX4RW3 �¯°9�B#9�I<@�O�^%��òu4a0K@1e0?>#@*9�47.�tj8#L#FR9�B�9�I�FROKOX0?>FR0uO?N#IE:E0KOX:EO¶F=NFa250K:E0K:E47.H4RW3 D[u:E0?>&F�OK:�.#L1IE@V4#66&@�.a0?25B1^Ã��u>#@�:�2*47256@�25ODF125@&�a¦��; ¦3�:F¦����¦����D25@�O?Ni@�9�0K:EÆ1@�IEB1^&Pn@=9�F1.ztj8#OK0;: W�Bm0?>#:EOD9�FRIE9�8#I|F�0K:E47.®[u:E0?> ¾1ÑHG�ÁJI

Page 4: áUÙ ãDß ê1ß Ý Ù3Ú ÛÍëbÝ5æ êaÜ7Ú Û è?éXåuÝ …rboyer/courses/math533/week06.pdf · ØcÙ?Ú ÛÜ7Ý5Ù*Þsß à Ùȧ§§ §§§ §§§§ §§§ §§§ §RáUâRãDãDÙ

ØcÙ?Ú�Û�Ü7Ý5Ù*Þsß�à�Ùȧ�§�§�§�§�§�§�§�§�§�§�§�§�§�§�§�§RáUâRãDãDÙ�Û�Ý3ßäÚ�åuÝ5æ1Ü�ç�è3é\áUÙ�ãDß�ê1ß�Ý�Ù3Ú�ÛÍëbÝ5æ�êaÜ7Ú�Û�è?éXåuÝ�æaÜ�çQìJÚ�Û�ßäæaíè25@�FRO;47.#:�.#L1O¶8#O;:|.#L&S#:�.#47C�:|F�IsF1.#6&CD8#I<0;:|.#4#CD:�FRI(9�47@�K_9�:<@�.a0KO�^Ly@�0?FR:EIEO�e � � © ¦�� 9�47.#OK:EOK0;O%4RWu0?>#@=N254#6�8#9�0D4RWb05[\4�6:EO�t541:�.a0V032KF1.#O?Ni41OK:E0K47.#O�^=�x>#:<O*:EO%L1:EÆ1@�.QSaBm0?>#@CD8#IE0K:�.#47CD:�FRI\9�47@�K_9�:<@�.a0�ú �¦3�cü [J>#:E9Ï>õ9�478.10;Oy0?>#@�.#8CVSi@�2y4RWs[uF�B#OJ4RW\9Ï>#4#41OK:�.#LH0�[\4_O?8S#OK@�0KO4RW\0�[\4=@�IE@�CD@�.a0KO*W°2547C µ7�1¦?�i¦?�¦M.�¦� a·y[u:E0?>#478#0D25@�N#I�FR9�@�CD@�.a0%OK4&F_05BN#:<9�F�I(@�I<@�C�@�.10D>FROy0?>#@*W�4#2KC����¦?ö�����÷R¦?ø#�3^&A#:�.#9�@&0?>#@�OK@_W°FR9�0K4#25OD9�47C=CD8#0K@1q(0?>#@�0325:�.#47CD:�FRIu9�4#@�K�9�:E@�.10_4RÆ1@�259�478.a0KO=03>#@�FR9�038FRI.#8CVSi@�2JSaB&F*W°FR9�0K472r4RWÍ�i^v¬x@�.#9�@1q,0?>#@y472j6@�2u4RW � � © ¦�� ���:FN"1�y���; ^�u>#@�9�I�FROKO � �¦�� © C=F�BQS@�L1:EÆ1@�.�0?>25478#L#>Q0?>#@&S#:�.#47C�:|F�Iu9�47@�K_9�:<@�.a0 ú � ü ^m�u>#:EO�8.#6@�259�478.a0KO0?>#@DFR9�0?8FRI�.#8CDS@�2ySaB�F�W°FR9�0K472\4�WÍ�=Si@�9�F18#OK@&����¦?öR¦3÷��sF1.#6®����¦3÷R¦?ö��ÍL1:EÆ1@D6:<O;0K:�.#9�0%��¯°9�B79�IE@�O�^u�x>#@25@�O?8#IE0X:EO%�:F^(Gy.#410?>#@�2x[uF�B_0K4=9�4#8.10X0?>#:EOJ9�I�FRO;Oy:<OX0K4=9�4#8.10X0?>#@V.#8CDS@�2x4RWs[uF�B7Ou4�W�W�472KCD:�.#L_:�.0?82K.���:|.Q�Y��¦3öR¦?÷���qR0?>#@�.�öXF�W�0K@�2�9Ï>#4#41OK:�.#L���qF1.#6_TU.FRIEIEB�÷XF�W�0K@�2\�=F1.#6&öX>FR6�Si@�@�.�9�>#41OK@�.,o03>FR0:EO�qc03>#@V9�4#8.10y:EOO P6/.�6m�^%ª\8#0y0?>#:EO%4RÆ1@�2j9�478.10;Oy0?>#@=N41OKO;:|S#IE@���¯°9�B#9�I<@�O=SaB�FkW°FR9�0K472x4RWu�=OK:�.#9�@����¦?öR¦3÷��c�Z��öR¦3÷R¦?�#���Z�Y÷R¦?��¦?ö���^�u>#@=9�I�FRO;O � :�.�gchn>FRO%OK:Ew�@Q�Q P6R.S6��1��": �SaBm0?>#@=O?F1C�@�25@�FROK4#.#:|.#LQFROD��¯°9�B#9�I<@�O�^�òx4R[%qT�¢ %��5�1¦3�¦?�i¦�.�¦� 1�xF1.#6U� © �����¦��a¦?�¦M.�¦� 1�uSi@�I<4#.#LQ0K4&6:#Vc@�25@�.a0=9�47.�tK8#L7FR9�BQ9�I�FRO;OK@�OD:|.®Ç % OK:�.#9�@&ô © :<O9�47.�tK8#L7FR0K@*0K4S� © S1B&FD9�4#.�tj8#L7F�0K:E47.�L1:EÆ1@�.HS1B�03>#@yOK:�.#L1IE@=0?2KF1.#O3N41OK:E0K:E47.��5�1¦3�1��^òx4¨[*q�IE@�0OW S@=F�N2547Ni@�2y.#472KC=FRI�O?8S#L72j478Nm4RWbÇ8%1^y�u>#@�.m:E0KOy4#256@�2yCD8#OK0¶Si@�F_N2547N@�2X6:EÆ#:EOK4724RW30:Fq�.FaCD@�IEB1qc�q��qX q�0qp���qc�; iq472u�:Fi^3Y,8250?>#@�2�q�OK:�.#9�@�W :EO¶.#472;C=FRI�q�:ÃW[Zm�/W�q03>#@�.&O;4�CD8#OK0:E0KO\9�47.�tK8#L7FR9�B_9�I|F�OKO�^3�xW�9�4#825OK@1q#0?>#@X:<6@�.a0K:E05B�CD8#OK0uSi@¶F1.k@�IE@�CD@�.a0u4RW�F1.aB=O?8S#L7254#8N,^bòx410K@u03>FR00?>#@=9�4#.�tj8#L7F�9�BQ9�I�FROKOK@�O*4RWJÇ&%=Fa25@=9�IE41OK@�6®8.#6@�2*:|.aÆ1@�25O;:<4#.,^��x>#:<OD:EO%O3N@�9�:�FRI\0K4&03>#@�FRIE0K@�2K.FR0;:|.#LL7254#8N,^¬x@�.#9�@1q�0?>#@*47256@�2x4RW3W CD8#OK0yS@VF�6:<Æ#:EOK472X4RW30:F*[J>#:E9�>&9�Fa.&FRIEOK4�Si@D[J25:E0K0K@�.�FRO�e

�\W®���±�3���; a�&� �:F1ö]�¡���1÷$�����1ø�¦[J>#@�25@=��¦?öR¦3÷Rq�Fa.#6õø_@�8FRI�@�:E0?>#@�2=Fk472*�1^u-�0y:EO8��8#:E9�]1IEBQÆ1@�2j: T�@�6m0?>#@�2j@�Fa25@V.#4�OK4aI|8#0;:<4#.#Oy0K4�03>#:<O@�8FR0K:E47.,^¬x@�.#9�@1qUÇ % :EOX4782\TU2jOK0u@�ËF1C=N#IE@D4RWÍFk.#47.a¯�F1S@�IE:�F1.mO;:|CkN#I<@DL72j478N,^sPn@%25@�9�47256ce}%�),~�)U���d���d)c�(euÇ&%Ð:<OyFDOK:�C=N#IE@=L#25478N,^P�:E0?>&FDIE:E0K0KIE@=CD472j@y[b4#2K]�qi[b@y9�Fa.�O?>#4R[¡0?>F�0JÇJh_:EOyFRIE[uF�B#OuOK:�C=N#IE@*W�472ul��^ ^PQ@\9�F1.=2j@�9�47256=0?>F�0vÇ8%u>FRO3 XO?8S#L#25478N#Os4�WU47256@�2].�q�;FXO?8S#L7254#8N#Os4RW,47256@�2(�q#F1.#6�0XO?8S#L725478N#O4RWs472j6@�2& ^*YU82503>#@�2�q�O38S#L725478N#Oy4RWs03>#@�OK@*T�Ë7@�6m47256@�2jO¶F12j@VFRIEI(9�47.�tK8#L7FR0K@*0K4_@�FR9Ï>H4103>#@�2�^\P�:<03>0?>#@=A#B#I<4R[±0?>#@�4725BH0?>FR0y[\@DO?>FRIEI(OK@�@=I�FR0K@�2X03>#:<O*0K@�2KC�q,[\@D[u:EIEIbSi@�FaS#I<@_0K4�N25@�6:E9�0D0?>#@=.78CDS@�24RW(O?8S#L7254#8N#OJ4RW(0?>#@�OK@*47256@�25O�^Gy.#410?>#@�2k[rF�B�0K4Q0?2j@�FR0�Ç8%�:EO�F�O=0?>#@&L#25478N�4RW%25410?F�0K:E47.FRIuOKBC=CD@�0325:E@�O�4RW¶03>#@&64#6@�9�F1>#@�6�2547.[J>#:E9�>Q:EOVFH25@�L78#I�F12ÐOK41IE:E6®[u:E0?>®����Ni@�.10?F�L147.õWYFR9�@�O�q��:F_Æ1@�2j0K:E9�@�O�q(F1.#6 �NF_@�6L1@�O�^�Y,8250?>#@�2�qp@�FR9�>

.

Page 5: áUÙ ãDß ê1ß Ý Ù3Ú ÛÍëbÝ5æ êaÜ7Ú Û è?éXåuÝ …rboyer/courses/math533/week06.pdf · ØcÙ?Ú ÛÜ7Ý5Ù*Þsß à Ùȧ§§ §§§ §§§§ §§§ §§§ §RáUâRãDãDÙ

ØcÙ?Ú�Û�Ü7Ý5Ù*Þsß�à�Ùȧ�§�§�§�§�§�§�§�§�§�§�§�§�§�§�§�§RáUâRãDãDÙ�Û�Ý3ßäÚ�åuÝ5æ1Ü�ç�è3é\áUÙ�ãDß�ê1ß�Ý�Ù3Ú�ÛÍëbÝ5æ�êaÜ7Ú�Û�è?éXåuÝ�æaÜ�çQìJÚ�Û�ßäæaíèÆ1@�250;@�Ë&:EOy:�.#9�:<6@�.a0D[u:E0?>H@�Ë�F�9�0KIEB��_N@�.a0?FRL147.#O�^���P�:E0?>H0?>#:EOy:|.aW�472KCkFR0K:E47.,q,:E0y:EO%.#410y0K4#4�6:#K�9�8#IE00K4�O?]�@�0K9Ï>n>FRIÃWr4�Wr03>#@=6476@�9�Fa>#@�6�2547.,^���2jB�:E0�{Ã�®fcFR0K@�2Ð[b@k[u:EI<I�6@�OK9�25:�S@�03>#:<O=254a0?FR0K:E47.FRI(L#25478N0?>254#8#L7>HL725478NmFR9�0K:E47.#O�^&Y�472X.#4¨[*q,[b@*6@�OK9�25:�S@_:E0%S1BHIE:<O;0K:�.#L�254103FR0K:E47.#OX:|.n��¯°O?NFR9�@D:�.m0K@�2KC�OJ4RW0?>#@�:�2XFRË7@�OX4RWs25410?F�0K:E47._[u:E0?>#478#0XÆ1@�25:ÃW�B7:�.#L�0?>#@�:�2XFRIEL1@�S2KFR:E9=25@�I�FR0K:E47.#O�^�u>#@�25@DF125@*OK:EËH �¯�W�41IE6�FRË#@�OX4RWb25410?F�0K:E47.Q��0?>FR0x:<O�q,0?>#@V254a0?FR0K:E47._:EOy0?>254#8#L7>&�:0NFJ_�": D6@�L72j@�@�O?�u03>FR0t541:�.�0?>#@*9�@�.10?2j41:E6OJ4RW(47NNi41OK:�.#L_WYF�9�@�O�^s�u>#:EOyB#:<@�IE6O%�`.k@�I<@�C�@�.10KOy4�Wv472j6@�2� ^�u>#@�25@DF125@y0K@�.H��¯�W�41IE6�FRË#@�Ou4RWs25410?F�0K:E47.Q��0?>FR0\:EO�q�0?>#@%254103FR0K:E47.�:EOu03>25478#L7>H�:0:FJ_�"1�*6@�L72j@�@�O?�u03>FR0t541:�.&NFR:�25OX4RW(47NN41O;:<0;@VÆa@�250K:E9�@�O�^\�u>#:EOJB#:E@�I<6O%�NFD@�IE@�CD@�.10;Oy4RWv4#256@�2u�^�u>#@�25@mF125@_TW�0K@�@�.��¨¯�W�41IE6�FRË#@�O_4RW¶254103FR0K:E47.�0?>254#8#L7>®0?>#@&C�:<6�Ni41:�.10;O�4RW¶4#NN41OK:E0K@H@�6L1@�O�^��x>#:<OB#:<@�IE6O=�; �@�I<@�C�@�.10KOy4�Wv472j6@�2u�^�xWs9�47825O;@1q0?>#@�25@*:<OXOK0K:EIEI�0?>#@%:E6@�.a0K:E0�B&CkF1N�[J>#:E9�>HL1:EÆ1@�OJFD0K4a0?FRIc4RW30:FD254103FR0K:E47.FRIcO;B�C=C�@�0?25:E@�O�^� »��_¾R¼���� fcF�0K@�2_47. [\@�O3>FRIEIyO?>#4R[ S1B OK:�CD:EI|Fa2�2j@�FROK47.#:�.#L1OH0?>FR0��sAf\���¦�aa�=:EO&FRIEOK4�F�OK:�C=N#IE@L7254#8N&4�Ws47256@�2*�;0:1^X-b>F�Æ1@D647.#@=Fk9�FRIE9�8#I�FR0K:E47.n0?>FR0y8#OK@�Oy0?>#@%O3F1CD@=CD@�0?>#4#6QFROuW�472XÇ&%DSFRO;@�6@�Ë#9�I|8#O;:<Æa@�I<B®4#.m9�47.�tK8#L7FR9�Bm9�I�FROKOK@�O�^�-Y0%:EO%IE@�.#L7>a05B1^��u>#@�S8#I�]n4RWb03>#@=[\472K]�9�4#.#OK:EOK0KOD4RWbO3>#4¨[x:|.#L0?>FR0%��Afb�Y�¦�a1�x>FROyOK:EËm9�47.�tj8#L#FR9�B&9�I�FROKO;@�O%4RW\47256@�2D�1¦3��1¦�.#�¦� :0i¦K�`.�qF1.#6Q�`.�^yGXL7FR:�.QF_.#472KCkFRIO?8S#L72j478N�W CD8#OK0\9�47.a0?FR:�._9�47C=N#IE@�0K@*9�47.�tK8#L7FR9�B_9�I|F�OKOK@�O�^\�u>#@%N4aOKOK:�S#:EI<:E0K:d@�Oy4RW�:E0KOu472j6@�2bFa25@y0?>#@6:EÆ7:EOK4#25OJ4�Wb�;0:1y�¡� Î ÿ?�\ÿ�aq#0?>FR0s:EO�q�ø��õµ¨�¦?�i¦�.�¦�0¦�a¦�1¦��.�¦K��1¦K�`.�¦K�N1¦b.7�¦� :0i¦�1`.·7^�PQ@y9�F1.k9Ï>#@�9�]SaB�@�Ë>F18#OK0K:E47.H0?>F�0u0?>#@�25@DF125@%.#4_OK41I�8#0K:E47.#OX0K4=0?>#@*I<:�.#@�F12X@�8FR0K:E47.ø*���3� ��;Z=�c.#�:d&�! :01Ö�� �`.7ôe���`.1Ä3¦[J>#@�25@�Zc¦�d�¦3Ö¦?ôa¦KÄsCD8#OK0JSi@%@�:E0?>#@�28F=472X�1^�u>#@*O?F1CD@*L1@�.#@�2KFRI(F1NN2547F�9Ï>z9�478N#IE@�6Q[u:E0?>mA#B7IE4R[È0?>#@�4725B�:EOyCV8#9�>���8#:E9�]R@�2�^u-�[u:EIEIsN25@�OK@�.a0y0?>#@6@�0?FR:EIEOy0?>#@�.z: Ws0;:|C�@VFRIEIE4R[uO�^PQ@*9�I<4aOK@VSaB�F�L1@�.#@�2KFRI(F125L78C�@�.10\W�472uÇXh,e}%�),~�)U���d���d)c�(eu�u>#@VFRIE0K@�2;.FR0K:�.#L�L725478N�ÇXh,q#W�472ul��f �:<OXOK:�C=N#IE@1^}%�),)��3e%f,@�0�WþSi@�F�N254#N@�2D.#472KC=F�I(O?8S#L725478Nõ4RWJÇJh,^DPQ@kO?>FRIEIsO?>#4R[�0?>F�0=W CV8#O;0y9�47.103FR:�.F_��¯°9�B79�IE@1^DA#:�.#9�@AW :EO%.#472;C=FRI�qgW�[x:<IEI(9�47.a0?FR:�.m@�Æa@�25B���¯°9�B#9�IE@1^%ª\8#0y0?>#@=��¯°9�B#9�I<@�O*L1@�.#@�2KFR0;@=0?>#@@�.a0K:�25@%L72j478N&ÇXh,^s¬x@�.#9�@1qhW�9�F1..#410yS@DN2547Ni@�2�^PQ@�Si@�L1:�.�SaB�9Ï>#4#41OK:�.#L^i �jW [J>#@�2j@ki±:<Om.#410�0?>#@Q:E6@�.a0K:E05B1^Ê�u>#@�Fa25L78CD@�.a0&9�47.#O;:<O;0KO&4RW9�47.#OK032K8#9�0K:�.#L�F_.#@�[�@�IE@�CD@�.a0Oi]lp:|.mÇ h [J>#:E9Ï>H[u:EIEI�T�Ë�CD4#25@%@�IE@�CD@�.a0KO%4�Wuµ7�1¦?�i¦�§�§�§5¦?lp·%0?>F1.�i�^-�W[i l :EOy.#410JF=��¯°9�B#9�IE@1q,[\@%F1NN#IEB&0?>#:EOyN25479�@�6�82j@�F�L7FR:�.&8.a0K:EI�F=��¯°9�B#9�IE@D:<OX47S#0?F�:|.#@�6c^-�W$im:EOJ.#410bFD��¯°9�B#9�IE@1q0?>#@�._:E0KOuN254#6�8#9�0\W�472KCZF�Ob6:EO�t541:�.a0u9�B79�IE@�Ou[x:<IEIc9�4#.10?F�:|.�FÐ9�B79�IE@*4RWcI<@�.#La0?>��472ÍL725@�FR0K@�2�472Í:E0b:EOJF*N254#6�8#9�0u4RWc0?2KF1.#O3N41OK:E0K:E47.#O�^b-/._410?>#@�2�[\47256O�qa@�:<03>#@�2�i ©Pm�¡-Y6�472\.#410�^v-/._0?>#@OK@�9�47.#6H9�FROK@1q�0?>#@%N254#6�8#9�0XW�4#2KC 4RW3i�CD8#OK0x9�47.103FR:�.�47.#IEB&6:EO�t541:�.a0J0?2;F1.#O?N4aOK:E0K:E47.#O�^\PQ@*[J25:E0K@1e

im�Z�Qn   ¦on © ¦�n Î ¦c§�§�§ �pÿ�ÿ�ÿK¦\472im�Z�Qnj �¦�n°©����Qn�ÎR¦�n�-��y§�§�§

Page 6: áUÙ ãDß ê1ß Ý Ù3Ú ÛÍëbÝ5æ êaÜ7Ú Û è?éXåuÝ …rboyer/courses/math533/week06.pdf · ØcÙ?Ú ÛÜ7Ý5Ù*Þsß à Ùȧ§§ §§§ §§§§ §§§ §§§ §RáUâRãDãDÙ

ØcÙ?Ú�Û�Ü7Ý5Ù*Þsß�à�Ùȧ�§�§�§�§�§�§�§�§�§�§�§�§�§�§�§�§RáUâRãDãDÙ�Û�Ý3ßäÚ�åuÝ5æ1Ü�ç�è3é\áUÙ�ãDß�ê1ß�Ý�Ù3Ú�ÛÍëbÝ5æ�êaÜ7Ú�Û�è?éXåuÝ�æaÜ�çQìJÚ�Û�ßäæaíè-°.�03>#@_TU25OK0=9�FRO;@1qs[\@_9Ï>#4#41OK@&F1.�@�IE@�CD@�.a05p��qW OK4�p © m��-Y6�^��x>#@�.®0?>#@�25@&CD8#OK0D@�Ë#:EOK0��õOK4pc�rp¢�Y�7�K� m�¡��^sA#@�0yöb�ts?uov���7�XF1.#6&÷u�wp¢��ö���^ª�B&FROKO?8CkN#0K:E47.,qU��¦?öRqF1.#6&÷yFa25@%FRIEI�6:EOK0;:|.#9�0�^uòu@�Ë#0�q9�>#474aOK@%05[\4=CD4725@y6:EOK0K:�.#9�0y@�IE@�CD@�.a0KO�q�O?F�B�øF1.#6�x1^òx4¨[*qcIE@�0=pyl�����÷R¦?ø�¦�x��bp¢�Y÷R¦Kø�¦�x�� ¤,  ^=A#:�.#9�@/W :<O=.#472;C=FRI�qc[\@DTU.#6cpzl��>W�^�YU8250?>#@�2�qgpyl°���#�J� öF1.#6�p l ��ö��(�¡ø�^A#:�.#9�@�p l �Yö�� m�tp¢�Yö���q10?>#@DN@�2KCD8#0?F�0K:E47.#O&p�F1.#6�p l Fa25@y6:EOK0K:�.#9�0�^Y(:�.FRIEIEB1q,[b@yIE@�0��&�^p ¤,  p l qOK45�s���#�(�¡��F1.#6/� m�Å-�6�^�u>#@*OK@�9�47.#6&9�F�OK@%:EOuO?>#4R[J.zOK:�CD:EI�F125IEB1^uª�@�9�F18#OK@D4RW(0?>#@XTU25OK0X9�FROK@1q[\@%C=F�B�FROKO?8C�@y0?>FR0�p © �¡-Y6W�472uFRIEIgp��/W�^PQ@_9Ï>#4#41OK@�p®�^W [u:E0?>cp m�Ê-Y6�^�Pn@&N#:<9�]Q�mO;4Hpc���#� m� ��^mf,@�0=ö=�{p¢�Y�7��^HA#:|.#9�@�pm:<Ok@�Æ1@�.,qpQ9�F1..#4a0V@�8FRI\0?>#@_0?2KF1.#O3N41OK:E0K:E47.È����¦?ö���^&¬x@�.#9�@1q�0?>#@�25@_@�Ë7:EOK0KO_÷=OK4m0?>F�0V÷ m�Ê��¦�� m��öD[x:<03>pc��÷�� m�¡÷R^òx4¨[*qIE@�0¶ø*�^p¢�Y÷���^�GJL7F�:|.,q�[\@XTU.#6m0?>FR0J��¦?öR¦3÷Rq1F1.#6�ø_F125@DFRIEIc6:EOK0K:�.#9�0yOK:�.#9�@�p�>FROx47256@�2X�^A#:�.#9�@=l��| q#[\@y9�F1.�TU.#62xÐ0?>FR0u:EOX6:EOK0K:�.#9�0XWY2j47C ��¦?öR¦?÷Rq#F1.#6�ø�^f,@�0�p l � ��÷R¦3ø�¦�x��bpc��÷R¦3ø¦�x�� ¤,  ^XªsB�9�4#.#OK0?2K8#9�0K:E47.,q}p l ���#�\��ö%F1.#6�p l ��ø7���~x1^XA#:|.#9�@5p l �Yø7� m�tp¢��ø#��q0?>#@DN@�2KCD8#0?FR0;:<4#.#OJ0?>#@�C�OK@�IEÆ1@�O�p�Fa.#6/pylUFa25@y6:EOK0K:�.#9�0�^GXOu:�.&03>#@%F1S4RÆ1@*9�FROK@1q[\@yIE@�0+�H�^p ¤,  pyl�OK4S�Q��W Fa.#6/�s���#�s�¡��^¬x@�.#9�@1q�0?>#@DFRIE0K@�2K.FR0;:|.#L�L#25478N_:EOJO;:|CkN#I<@*W�472ul��f i^A#@�CD:E6:�25@�9�0V��25476�8#9�0;O-�0x:<Oy>#@�I�NaWY8#I(0;4=:�.10?2j476�8#9�@DOK47C�@%410?>#@�2r9�47.#OK0?2K8#9�0;:<4#.#Oy4RWvL#25478N#O�^A8NN4aOK@�� �¡Ç ÿ�±[J>#@�25@VÇÈ:EOy.#472KC=F�Ic:�.���^��u>#@�.�[\@y9�F1.z[b4#2K]=[x:<03>�� F�Ou47256@�25@�6mNFR:�25O�e

���i ��� Ï�Íÿ7���#©��©��s�Å� ¨�Q�� ��7©�����©u�¡�i ����� ?�7©(� ¤¢   ���� ��©R§A#4�q[\@y9�F1.z6@3TU.#@D0?>#@VCD8#IE0K:�N#IE:<9�F�0K:E47.Q4#._47256@�25@�6mNFR:�25O¶F�O�e

�Y�   ¦��   �(ÿ7�Y� © ¦�� © �s�Z���   i\�Q�   ���Y� © ��¦��   � © ��¦[J>#@�25@�ib���¨���QZ��(���`Zo� ¤¢  ^�u>#@*:|.aÆ1@�25O;@%4RWJ����¦��R��:<OXL1:EÆ1@�.mSaBõ�Qi\�Q� ¤,  �K� ¤¢  ¦�� ¤,  ��^òx410K@y0?>FR0\[\@X9�F1._0?25@�FR0�i�FROJFD>#47CD4#2KN>#:EO?C W°2547C±0?>#@yL#25478N5��:�.10;4=0?>#@%F18#0K4#CD472KN>#:EO?CÉ4RWvÇ*^-°.õW°FR9�0�q�0?>#:EO=:EO=FRIEIb[\@�.#@�@�6®0;4&6@3TU.#@�O?8#9�>�FHL72j478NR�_e%[\@�F125@=La:<Æa@�.�0�[\4�L72j478N#O=Ç F1.#6c�F1.#6®F&>#47C�47CD472KN>#:EO?C WY254#C�� :�.10;4�Gy8#01��ÇJ��^��x>#@�25@�O38#I<0;:|.#L�L7254#8NQ:EOD9�FRIEIE@�6�0?>#@ º�»��DÀ�Ñ1À�¼3»I�¿ñU¼3Á#ÑUðNI�¿ 4�W�� [u:E0?>HÇ%^

0

Page 7: áUÙ ãDß ê1ß Ý Ù3Ú ÛÍëbÝ5æ êaÜ7Ú Û è?éXåuÝ …rboyer/courses/math533/week06.pdf · ØcÙ?Ú ÛÜ7Ý5Ù*Þsß à Ùȧ§§ §§§ §§§§ §§§ §§§ §RáUâRãDãDÙ

ØcÙ?Ú�Û�Ü7Ý5Ù*Þsß�à�Ù®§�§�§�§�§�§�§�§�§�§�§�§�§�§�§�§�§3áUâRãDãDÙ�Û�Ý3ßäÚzåuÝ�æaÜ�ç�è3é\áUÙ�ãVß�ê1ß�Ý5Ù?Ú�Û(ëbÝ5æ�ê1Ü7Ú�Û�è?éXåuÝ5æ1Ü�çQìXÚ�Û�ß�æ1íè�*�c�X�õ~��d���N���5���y�u>#@�6:�>#@�6�2KF�IuL725478N#O��=h�F125@�FRIEI\OK@�CD:E6:�25@�9�0�N254#6�8#9�0KOD[J>#@�25@&Ç ��4shQFa.#6� ��4s©%� µ`F¦��R·*[u:E0?>m0?>#@=C=F1N�i\�Q�R���QZ��u�Ê�5×y������Zcq,[J>#@�2j@SZ&��F_472%�a^���PQ@=C=F�B&FRIEOK4�[J2j:<0;@i\�5�����QZ��(�Z��l�×®���(ÿ;Zc^f,@�0�Z&Si@VFDL1@�.#@�2;FR0K472x4RWsÇ���4Íh�Fa.#6&[J25:E0K@%�D�Z��Zc¦�Fa�bFa.#6�ö\� ��F¦�����^cPQ@ÐTU.#6ce

�7öu�±�QZc¦�F1��ÿ7�QF¦������±�QZ=�!ib�5���(ÿF¦�F# ?�(� �QZp¦����ö��V�±�QF¦�����ÿ1�QZp¦�F1�(�±�QF��>i\�5���(ÿ;Zp¦��3�>Fa�(� �5×�Zp¦�F1��§-°. NF1250K:E9�8#I�F12�qu��F1.#6�ö_O?FR0K:EO5W�B�03>#@&25@�I�FR0K:E47.#O_W�472*0?>#@&6:�>#@�6�2KFRIXL7254#8N,e&� h ��xa¦?ö © ��xaqbFa.#6�#öb�Èö�� ¤,  ^���1�xA#:|.#9�@DGy8#0¨��4sh�\:EOX:<O;47CD472KN>#:E9D0K4��=��l,��q[\@%C=F�B�25@�FR6:EIEB&L1@�.#@�2KF�I<:Ew�@=03>#:<OX@�ËF1C=N#IE@1^���1�*f,@�0VÇ ��4s©=6�4Í©_F1.#6c�û�Ég,Î1^&òx410K@1e*G¶8#0¨��4Í©�6k4s©���' � g,Î1^��u>#@�25@�O?8#IE0K:�.#LQL#25478N�>FRO47256@�2r4RWÍ��.�^�r.7�pf,@�03��S@uF�T�@�IE6=0?>#@�.*[\@s9�F1.¶W�4#2KC¡03>#@bO;@�CD:E6:|2j@�9�0bN254#6�8#9�0(4RWh� [u:E0?>=����[u:E0?>�ib�Q�,�oZ=�q��ÿ�Z[J>#@�25@��m�/���%F1.#6HZH�/�De

���1 �¦�Z, ?�(ÿ#�Q�R©R¦�Z�©����Z�Q�a ��>i\�QZ¢ Ï���R©1¦�Z, �Z�©���§�Q 1�x�\47.#OK:E6@�2x0?>#@%OK@�C�:<6:�25@�9�0%N2j476�8#9�0y4RW3� h [x:<03>&M%f���l�¦��y�uSaBUe

�Q�   ¦K²   ��ÿ7�Q� © ¦;² © ���Z�Q�   ��²   �   ¦;²   ² © ��§òx410K@y[u:E0?>Hl����=Fa.#6/����4s©aq[\@y47S#0?FR:�.&Fa.#410?>#@�2\L72j478N�4RW(47256@�2x�`.�^�Q01�ÍPQ@X9�F1._9�47.#OK0?2;8#9�0rF*L72j478N���4�W�47256@�2\�:F*S1B�F*OK@�C�:<6:�25@�9�0yN254#6�8#9�0u4RW343%x[u:E0?>�4$-¶Si@�9�F18#OK@Gy8#0¨�Q�3%�� '� 4$-7e ���i ¨¦3ö� j�(ÿ#���7©R¦3ö�©��(�Z���i ��>i\��ö� Ï�;�7©R¦3ö� T�®ö�©���§òx410K@1e��Ê>F�O%L1@�.#@�2KF�0K4725O=Z�F1.#6kdzO?FR0K:EO5W�B7:�.#LRZ % ��d - ��x_F1.#6kZod�ZQ��d�^V�x>#:<OD@�ËF1C=N#IE@_:<ODFO?Ni@�9�:|F�I(9�FROK@y4RW(0?>#@VN2j479�@�6�825@D:�.���.7�\[u:E0?>��¡��43%1^Gy.#410?>#@�2D25@�FRIE:Ew�FR0K:E47.�4RWu0?>#:EODL7254#8NQ:EOD0?>#@=O?8S#L#25478Nn4RWJgg%=La@�.#@�2KFR0K@�6®S1Bn0?>#@=05[\4�9�B79�IE@�O�pm��5�1¦3�¦?�i¦�.�¦� 1��[x:<03>/��� ���¦?�i¦� ¦M.7��OK4=0?>F�0u0?>#@y9�47.�tK8#L7FR0K@*4RW$p�SaBA�õ:<OyF=Ni4¨[\@�2\4�W3pU^�Qa1�7�y.#@y9�F1._O3>#4¨[Å0?>FR0\0?>#@yL7254#8N_4RW�47256@�2r@�:<L#>10u4�Wv8.#:E0&�8FR0K@�2K.#:E47.#OX9�F1..#410XS@y@�ËN25@�OKOK@�6mFROFDOK@�C�:<6:�25@�9�0DN254#6�8#9�0�^�Q11�XÌ(ËF1C=N#IE@�O��r.7�XF1.#6��Q01�XF125@VNFa250K:E9�8#I�F12y:�.#OK0?Fa.#9�@�O%4RWuF �_»�¿j¾:Ib�hIKîEÀrI L725478NH[J>#:E9�>&:EO*L1:EÆ1@�.mFRO0?>#@XOK@�CD:E6:�25@�9�0yN25476�8#9�0\4RWe4Íh�[u:E0?>=:E0KOuF18#0;47CD472KN>#:EO?C L725478N�����l,��^(òx410K@X0?>FR0�0?>#@¶C�@�0?FR9�B#9�I<:E9L7254#8ND9�F1._S@XÆ7:E@�[\@�6&FRObFÐO?8S#L7254#8N=4RWc0?>#@xOKB�CkCD@�0?25:E9XL725478N_gch,^�PQ@u[u:EIEIcOK@�@x0?>#:EObL#25478NkI|F�0K@�2:�.�4782xOK0?8#6B_4RWsM%FRIE41:EOX0?>#@�4725B�:�.z9�47..#@�9�0K:E47.m[u:E0?>z0?>#@%254#410KOu4�W@Z h ×��y��F^

a

Page 8: áUÙ ãDß ê1ß Ý Ù3Ú ÛÍëbÝ5æ êaÜ7Ú Û è?éXåuÝ …rboyer/courses/math533/week06.pdf · ØcÙ?Ú ÛÜ7Ý5Ù*Þsß à Ùȧ§§ §§§ §§§§ §§§ §§§ §RáUâRãDãDÙ

ØcÙ?Ú�Û�Ü7Ý5Ù*Þsß�à�Ùȧ�§�§�§�§�§�§�§�§�§�§�§�§�§�§�§�§RáUâRãDãDÙ�Û�Ý3ßäÚ�åuÝ5æ1Ü�ç�è3é\áUÙ�ãDß�ê1ß�Ý�Ù3Ú�ÛÍëbÝ5æ�êaÜ7Ú�Û�è?éXåuÝ�æaÜ�çQìJÚ�Û�ßäæaíè�Q,1�(�x>#@�25@u:EObFX9�F1.#4#.#:<9�F�IUO;@�CD:E6:|2j@�9�0rN2j476�8#9�0uFROKO;479�:�FR0K@�6k0K4%F1.aBVL#25478N���^(f,@�0\lm­|F^�f,@�0bgX��l,�FR9�0u4#._0?>#@%N254#6�8#9�0JL72j478N5� h �w�~6�� 6mÿ�ÿ�ÿN6��û��li¯°0K:�CD@�O3�JSaB�Ni@�2KCD8#0K:�.#L�9�4#47256:�.FR0;@�O�q�03>FR0:EO�q�: W��Q�mgJ��l,��qiI<@�0��Hÿ7�QZ, �¦�Z�©R¦�§�§�§;¦�Z�h#�\� ��Z ób¡  b¢ ¦R�QZ�£ ¡ ©�¢ ¦�§�§�§?¦�Z £ ��l,����^u�u>#@V25@�O38#I<0;:|.#LHOK@�CD:E6:�25@�9�0N254#6�8#9�0sL72j478ND:EO(9�FRIEIE@�6�0?>#@�¤ ¼3»�¾7¿EGÐñU¼3Á#ÑUðNI�¿ 4RW¢gph%[x:<03>=�_^(-�0b>F�O(47256@�2(l�{¨� ��� h ^v�x>#@uL725478N*4RWºKÀÃï�Â�»�Ñ N@�2;CV8#03FR0K:E47.mC=FR0?2j:<9�@�OyF1.#6H0?>#@%OKBC=CD@�0325B�L725478Nz4RWs0?>#@*9�8S@=F125@y@�ËF1C=N#IE@�O%4�Wv[J2j@�FR0?>N254#6�8#9�0KO�^$k���m��`¥(e&-�0_:EO�:�.a0K@�25@�OK0;:|.#L®0;4Q0?25B�0K4QIE:EOK0_0?>#@&6:#Vc@�25@�.a0�@�ËF1C=N#IE@�O�[\@�9�F1.�.#4R[Ê8#O;@&0K4QLa:<Æa@@�ËF1C=N#IE@�Oy4RWs.#47.a¯�F1S@�IE:�F1.mL#25478N#OX4RWv4#256@�2u�`.�^�\[\4�C=F�t�4#2b2j@�C=FR:�.#:�.#L�0K47N#:E9�OuW�472\4#82uOK038#6B�4RW(L7254#8N#O¶F12j@1e�5����0?>#@uA#B#IE4¨[ 0?>#@�4725BD[J>#:E9�>=6@�O;9�25:�S@�O\0?>#@u@�Ë#:EOK0K@�.#9�@yF1.#6=OK0?2;8#9�0?825@u4RW,O?8S#L7254#8N#Ob[X>#41OK@47256@�2uFa25@VNi4¨[\@�25Ox4RWÍFkN25:�CD@1o���1�\9�I�FRO;OK:ÃT�9�FR0K:E47.H4RW�TU.#:<0;@�I<BmL1@�.#@�2;FR0K@�6�FaS@�IE:�F1.mL7254#8N#O�^

1