^u^agwtyilkgÁ - bumath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡...

30

Upload: others

Post on 04-Nov-2020

0 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

������������ ��������� ������ � ����� �"!$#&%�')(*,+.-0/1(2#43651798:(;8=<07>#�?@BAC;C>DFEHGJILK;GNMPOQGREQSUT;GWVXI�TGRYWSUT;GWE�IQDZSUV;YWOQGW[\EQG]KSUE�C>D\CA^_[`OQSaIcb\d9DFK�d>[\VT"KGR^UC1ILKGNGRYWDFV;DFeJbFdf b1gFSahiSUV;gZ[:jk[lbmEQD\enG�enD\VGRb\o�p�DqTDrILKSaEWd9ILKGJMPOQGWEHSUTGRVXI�K2[\E�sFtFtrTDF^U^U[\OQEuEHGWVXI�IQDZGRh\GWOvbIw[=xyC2[:bFGROWoPpuKSaEPjzSU^U^BYWGROHIw[`SUV^{bJKGR^UCyILK;G�GRYWDFV;DFeJb f GWYW[\AEQGuIL[|x}C2[:bFGROQE�jzSU^a^BCOHD f [ f ^abNEQ[:hFG[$^US{IQIL^aGND\~�ILKG}erDFVG�b�[\VT�EQC�GWV;T�ILK;GrOHGWEvI:or��^U^4ILKSaE�EQC�GWVTSaVg�jzSU^U^ f DiDFEvI�EL[\^aGWE��2gFAOHGWERdKGR^UC�eZ[`ViAB~ [\YRIQAOQSaVg}YWDFerC2[\VSaGWEz[\V;T1EHDr~0D\OHILK�opuKG"G���GRYRIqT;DBGREZVD\InEHIQDFC&IQKGWOHG�KDlj�GRhFGROWo�@iACC�DFEQG�[�C2[\OvILSUYRA^_[`ONIw[|x�C2[:bFGROqEL[:h\GWE�~�IQGWGWV�C>GROQYWGRVXI�DFO��FtrTDF^U^U[\OQEz[`VT�EHC>GRVTE�IQKGJOHGWen[\SUV;SUVgr�B�:t}IQDZgFG�I�KGRO�K;DFAEQG]C2[\SaV�IQGWT�opuKGRV�IQKG$C2[\SaV�IQGWONK[\E}[\V�G�xiILOQ[��B�lt"SUV;YWDFerG\o,@BACC�DFEHG$EQKG�EL[:h\GWEWd�EL[:b��=��C�GWOHYWGRV�IWd�D\O� s;���\t1TDF^U^U[\OQE�[\V;T�EQC�GWVT�IQKGZOHGWEHINDFV�[mA;EQGWT�YW[\OWo1puKGqC�GWOHEQDFV�jzKD�K2[\T,IQKGqAEHGWT�Y:[`OjzSU^a^)VDlj�K2[:hFGJG�xiILOQ[nY:[`EQK�IQDnEQ[lh\Gy[\VTmEQC�GWVT�d�[\V;T�EQDnDFV�oupuKG�ILD\IL[\^)SUerC2[\Y�I�D\~.IQKGNsFt\tTDF^a^_[\OHE�gFS{hFGWV�ILDr[yC2[\OvILSaYWA^U[\OkIw[=x�C2[:bFGRO�SaEzeNAYwK1gFOQGW[`ILGRO�ILK2[\VN�HAEvI�s\tFtyTDF^U^U[\OQERo� V�DFOHTGWO�ILD�GREHIQSUen[`ILGyILK;GnILD`Iw[\^�GR�>GWY�INDFV�ILKGnGWYRDFVDFeJb�D\~�G:[\YwK�s\tFtmTD\^U^_[`O�gFSa~�IWd�j�Gen[\�\G�ILK;G�~0DF^a^UDljzSUV;gy[\EQEHAerC;ILSaDFV�o� �[`YwK�C�GWOQEHDFVmjzSa^U^�EQ[:hFGr�=�}C�GWOHYWGRV�I�D\~�ILKG�enD\VGRbqIQKGRbOQGRYWGRSahFG�[\VT�EQC�GWVT�ILKG1OQGREHI:o¢¡�G1Y:[\V�~0DF^U^aD=j£KDlj¤[�C2[\OHIQSUYRA^_[\O}s\tFt�TDF^U^U[\OQENj�DFOH�iEnSaIQEjk[lbrILK;OQDFAg\KnILK;G�GWYRDFVDFe]bFo�¥�~)YWD\AOQEHG\d;D\VG�C�GWOHEQDFVZeZ[:brEQC�GWVTnILKGRSUO�jzSUVT;~ [`^U^2DFVnEQGRh\GWOQ[\^SaIQGWerEWd f A;I�j�G�[\EQEHAenG�IQK2[`IuGW[\YwK�C>GROQEHDFV�OQGWYRGWS{hBSaVg}[NC�DFOHIQSUDFVqD`~)ILKG�DFOHSUgFSaV2[\^�sFtFtNT;DF^U^U[\OQEjzSU^a^EQC�GWV;Tq¦X��C>GROQYRGWVXI:o�§�GRVYWG`diIQKGz�2OQEvIkC�GWOHEQDFVnEQC�GWVTE��B�lt�¨�tB��¦X��©Qs\tFt;o.¥�~>ILKSaE�enDFV;GRbFd�=�yC>GROQYRGWVXI�SaE�EQ[lh\GWT�[`VT"t;�ª¦X��©`�B�ltJ¨�«F¬F¬;���\tySUE�EQC�GWVXI:o�¥�~.IQKSUERd�t;��¦F��©=«X¬\¬;�­�`tN¨®¬F¦FsB�ª« � �SUEuEHC>GRVXI�[\V;T1EHDnD\V�o�§�GWV;YWG\d2IQKG�ILD\IL[\^�GR�>GWY�I�DFV1IQKG�GWYRDFVDFeJbqSUEuD\~4G:[\YwKmsFtFtyTD\^U^_[`OQE�SaE�B�lt�¯}«X¬F¬;���\t�¯n¬Fs\¦;�ª« � �9¯1�:�:�\¨°t;�ª¦X��©0sFtFt�¯rt;��¦F�)©v± t;�ª¦X��©0sFtFtX²l¯rt;��¦F�)©v± t;�ª¦X��©v±³t;�ª¦X��©0sFtFtX²H²l¯1�:�W�¨�sFtFt±³t;�ª¦X�u¯�tB��¦X�|´.¯�t;�ª¦X�`µ4¯¶�:�W��²·�¡,G�K2[:h\G]VDlj°G�xiILOQ[\YRIQGWT"[`VmSaV�IQGWOHGWEHIQSUVg}en[`ILK;GWen[`ILSaY:[\^�COQD f ^UGRem¸¹¡6K2[|IzSUE�ILK;G�EHAetB��¦X�z¯�t;��¦F�`´4¯�t;�ª¦X�`µ4¯¶�W�:�jzKGROQG$ILKGROQG�SUEyV;D�^aSUerSaI�ILD�ILK;G$ViA;e f GRO}D\~�EHAereZ[\V;TEWo,¡�G$��VDljºILK[`IyILKG$EHGW»�AGRVYWGt;�ª¦X�|¼NIQGWVT;E�IQDr½RGWOHDrh\GWOHb1»�ASaYw�i^{bFd f ABI�j�G][\OQG�[\T;TSUVgyIQDFgFGRIQKGWO�[\O f SaIQOL[\OHSU^{bnen[\VXbqIQGWOQerEWdEQD}ILK;GWOQG�SaEuVDng\A2[\OL[`V�IQGWG�ILK[`IuILKG�EHAeºjzSU^a^ f G��VSaIQG\o¡,Gn[\^aOQG:[`T;b"K2[:h\GrIQKG}eZ[`IQKGWen[`IQSUY:[`^.en[\YwKSUV;GWOHb�ILD�TG:[`^�jzS{ILK�ILK;SUE�COQD f ^UGRemoN¡6K2[|Ij�G]OHG:[\^a^abnenGW[\V f bZILKG�G�xBCOHGWEHEQSUD\VtB��¦X�z¯�t;��¦F� ´ ¯�t;�ª¦X� µ ¯¶�W�:�SUE�IQKG�EQGR»iA;GWVYRGJgFS{hFGWV f b ¾ ¿ ¨°t;�ª¦X�¾ ´ ¨°t;�ª¦X�u¯�t;��¦F� ´:À¾ µ ¨¶t;��¦F��¯�t;�ª¦X� ´ ¯�t;�ª¦X� µRÀ

Page 2: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

[\VTrEQD�DFV�o.puKSUE�SUEPYW[\^U^aGWTyILKG�Á /iÂ�Ã�/i#4ÄF/q79Å�+�(Æl8=< (- Á Ã�* Á\o.¡�Guj�[\VXI�ILD���VDlj¢ILKGz^aSUerSaID\~�ILKG�EHGW»�AGWV;YWGJD`~.C2[`OHILSU[\^�EHAenERdILK2[|IzSUEt;�ª¦X�z¯�t;�ª¦X�`´4¯�t;��¦F�`µ4¯¶�:�:�X¨ ^USae¼lÇzÈ ¾ ¼ �¡�GyY:[\V�~0DFOHeÉILK;GyEQAeÊD`~�IQKG}GW^aGWerGWVXILE�D\~�[\VXb�EHGW»�AGRVYWG`orpuK;GygFGWVGROL[\^PV;D\Iw[`IQSUDFV�SUE�ILK;G~0DF^a^UDljzSUVg;¸Ë /X̹#�<08l< 7�#�?uÍ SahFGRV�[yEHGW»�AGWV;YWGyÎ ¿ À Î ´ À Î µ À �:�:�ÏdILKG < #�Ì.#�<08l/ Á /�Æ|< / Á�[\EHEQDiYWS_[|ILGWT�jzSaIQK$ILKSaEEQGR»�AGWVYRGJSaEuTGWVD`ILGWT f b ÈмlÑ ¿ Î ¼ ¨�Î ¿ ¯�Î ´ ¯�Î µ ¯6�:�:�W�puKG�EQGR»iA;GWVYRG ¾ ¿ ¨ Î ¿ À¾ ´ ¨ Î ¿ ¯�Î ´¾ µ ¨ Î ¿ ¯�Î ´ ¯�Î µ ¯¶�:�:� Àooo À¾ Ò ¨ Òм:Ñ ¿ Î ¼ À �:�:� oooSUEuYW[\^U^aGWTqIQKG�Á /iÂ�Ã�/i#�ÄF/�79Å�+�(Æl8=<0(2- Á Ã�* Á�D\~4ILKG�EQGROQSaGWE�[\VTÈмlÑ ¿ Î ¼SUEuT;GR�2VGRTmIQD f G�ILKG�^aSUerSaI�D\~4IQKG�C2[\OvILS_[`^�EQAerEWdIQK2[`IzSaEÈм:Ñ ¿ Î ¼ ¨ ^USaeÒ ÇzÈ ¾ Ò ¨ ^USaeÒ ÇzÈ ÒмlÑ ¿ Î ¼ �¡�G�EQ[lb�IQKG�EHGWOHSUGWE ÄF7�#>Ó�/�Æ|Ô9/ Á�S{~;IQKG�^aSUerSaI�D\~ILK;G�C2[`OHILSU[\^FEHAenE�G�xBSaEHILE�±³[\E4[u�2VSaIQG�V�Ae f GROw²�o� ~.IQKG�^USaenS{IkD\~4C2[\OvILSU[\^�EHAenEzTDiGWEzVD`IzG�xBSUEvI:dILK;GWVmj�G�EL[:b$ILK;G�EHGWOHSUGWEzSaE 34<�Ó�/iÆ=Ô9/i#>8 o� V;�2VS{ILG�EQGROQSUGREk[`OQGz~0OQGW»�AGRVXIL^ab}YW[\^U^aGWT��HAEHI�EHGWOQSaGWERo�@iIw[\OvILSaVg�jzS{ILKn[\VXb}EQGR»iA;GWVYRG\d;j�G�YW[\V~0DFOHeÕ[�EQGWOHSUGRE f br[\T;TSUVg�AC}ILK;G�GR^UGRenGRV�IQE�D\~�IQKGzEQGW»�AGRVYWG�SUV}DFOHTGWOPILD�~0DFOHeÖILK;G�EHGW»�AGRVYWGDFOuC2[`OHILSU[\^�EHAerEWo�×.SUV;TSUVg]ILKG�h`[\^UA;G�D\~4IQKG�EQGROQSUGRE]± SØoÙG\o{d;IQKG�^USaenS{IkD\~�ILK;G�C[\OHIQS_[\^>EQAerEw²kSaEAEHA2[\^U^{b$hFGROHb"T;SaÚrYWA^{I:o � V�DFOHTGWO�ILDqTG�ILGROQerSUVG�ILKGJÛ ILK ILGROQe¤SUV�IQKGJEHGW»�AGRVYWG}D\~PC[\OHIQS_[\^EQA;enERdjkG�eNAEvI�[\TT�Û"ILGROQerEuD\~�ILKG�DFOHSUgFSaV2[\^�EQGR»iA;GWVYRG\o� ¹h\GWV�S{~)ILKG�DFOHSUgFSaV2[\^�EQGR»iA;GWVYRG]SaEgFS{hFGWV f bq[NEQSUerC^aG�~0DFOHeNA^U[]OQGR^_[`IQSUVg�ILKG�Û ILK IQDyÛ.diILKGROQG�SUE�VDygFA2[`OL[\VXILGRG�ILK2[|I�ILKGROQG�jzSa^U^f G�EHAYwK"[N~0DFOHeNA^U[J~0DFO�ILKG�C2[`OHILSU[\^�EHAerEWo���aÜ�Ý$/�7�*,/X8lÆ|<³ÄmÞ�/�Æ|< / Á ?

ß

Page 3: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

puKGROQGn[\OQG}[q~0GRjàSUV;�VSaIQGNEQGROQSUGRE�~0DFO�jzKSUYwK�ILK;GrC[\OHIQS_[\^4EQAerE�Y:[\V f GrYRDFerCA;ILGRT�G·x;[\YRIQ^abFopuKG�enD\EHIzSUerC�DFOHIL[\VXI�D\~�ILKGREQG][\OQG�IQKG�gFGWDFerGRIQOQSaY�EQGROQSUGREWoË /X̹#�<08l< 7�#�? � Ô9/i7�*,/X8lÆ|< Ä Á /�Æ|< / Á�SUEÈм:ÑáBâ ¼ ¨ â á ¯ â ¿ ¯ â ´ ¯¶�W�:�W�puKG�V�Ae f GWO â SUEuYW[\^U^aGWTqIQKG Æ|(B8=< 7 D\~�ILKG�EQGROQSaGWEWopuK2[`I�SaEWd|[ugFGWDFerGRIQOQSaYPEQGWOHSUGRE�SaE)DFVGP~0DFOHenGRT�~0OQDFe°IQKG�EQGR»iA;GWVYRG�� À â À â ´ À â µ À �:�:�W~0DFO�[uYRDFVEHIL[\VXIâ o�¡6KSa^UG�SaIzSaE�[r^US{IQIL^aG�EQ^aDFCCXbFd�[}EQGROQSUGRE�IQK2[`I�TDiGWE�VD\I�EHIw[`OHIzjzSaIQK â áz¨à�Fd f A;I�jzSaIQKmEHDFerGâ ¼N~0DFOuÛ�ã��]SaEz[\^UEHDyY:[\^a^UGRT1g\GWDFerGRIQOQSUY`o¡6K2[`I�en[\�\GRE�S{IuC�DFEQEHS f ^aG�ILD}YWDFerCA;IQG�ILKG�EHAeäD\~.[}g\GWDFerGRIQOQSUY�EQGWOHSUGRE�SaE�ILKG�~ [\Y�IuILK2[|Ij�G]YW[\V�d f b�[yYW^aGRhFGRO�IQOQSaYw��d;D f Iw[`SUV1[\V�G�x;[\Y�I�G·x;C;OQGWEHEQSaDFVm~0DFO�IQKG�Û IQK C[\OHIQS_[\^>EQAe1o�å�G�I¾ Ò ¨ Òм:ÑáBâ ¼ ¨Ö��¯ â ¯ â ´ ¯¶�:�:�:¯ â Ò �¡�G�VG·xBIkG�xBC^aDFOQG�ILKG�OQGW^U[`ILSaDFVEHKSUC f G�IcjkGRGWV ¾ Ò [\VT ¾ Ò.æ ¿ o¹puKGROQG�[`OQG�Icj�DJjk[:bBE�ILD]jzOHSaIQG¾ Ò.æ ¿ SUV$IQGWOQerEuD\~ ¾ Ò o�×.SaOQEvI:d;j�G�VD\ILG�IQK2[`I¾ Ò4æ ¿ ¨ç�k¯ â ¯ â ´ ¯¶�:�W�=¯ â Ò ¯ â Ò.æ ¿ ¨ ¾ Ò ¯ â Ò.æ ¿ ���^aEQDdBjkG�Y:[\V�jzOQS{ILG¾ Ò.æ ¿ ¨Ö��¯ â ¯ â ´¹¯6�:�:�l¯ â Ò ¯ â Ò.æ ¿ ¨Ö��¯ â ±v��¯ â ¯ â ´.¯6�:�:�=¯ â Ò ²P¨Ö��¯ â

¾ Ò �puKGREQG�IcjkDrj�[:biE�D`~4jzOHSaILSaVg ¾ Ò.æ ¿ gFS{hFG�AE�[\V�G�xBCOHGWEQEHSUDFV1ILK2[|I ¾ Ò EL[`IQSUEH�GWE¾ Ò ¯ â Ò4æ ¿ ¨���¯ â¾ Ò �@BDF^{hiSUVgNIQKSUE�~0DFO ¾ Ò d;j�G]D f Iw[\SaV ¾ Ò,è â

¾ Ò ¨Ö� è â Ò.æ ¿DFO ¾ Ò ¨ � è â Ò.æ ¿� è â ÀCOHD=hiSaTGWT â�é¨ä�\oNpuKSUE�SaE�[Z~0DFOQeJA^_[n~0D\O�ILKG}ê ILK C2[\OvILS_[`^.EQA;eÊD\~�ILKG}EHGWOQSaGWERoJ¡6SaIQK"ILKSaEG�xBCOHGWEHEQSUD\V�D\~zIQKG1ê ILK C[\OHIQS_[\^�EQAe1dPjkG$K2[:h\G�["KD\C>G$D\~�YRDFenC;A;ILSaVg"ILKGqh`[\^aAGqD\~�ILK;GEQGROQSaGWEWd ÈмlÑá â ¼ ¨ ^USaeÒ ÇzÈ ÒмlÑá â ¼¨ ^USaeÒ ÇzÈ ¾ Ò¨ ^USaeÒ ÇzÈ � è â Ò.æ ¿� è ⨠� è ^USUe Ò ÇzÈ â Ò4æ ¿� è â �¬

Page 4: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

ë Dlj�d â Ò.æ ¿ [\CCOHDX[\YwKGRE�½RGWOHDZ[\V;TmêÊIQGWVTEzILD}SaV;�2VS{IcbqSa~�ì â ìXí®�\o�@BDnS{~zì â ì�í°�Fdj�G�K2[:hFGÈмlÑá â ¼ ¨ �� è â �� ~uì â ìXî®��ILKGRVq^USae Ò ÇzÈ â Ò4æ ¿ TDiGWE�V;D\IkG·xBSUEHIu[\VTqEHDJILKG�YWDFOHOQGREQC�DFVTSaVgNgFGRDFenG�ILOHSUYzEQGROQSUGRETS{hFGWOHgFGWERo�×.SUV2[`^U^ab\dBS{~ â ¨Ö��ILKGRV Èм:Ñá â ¼ ¨ Èм:Ñá �[\VT�ILK;SUEuTS{hFGWOHgFGWERo � ~ â ¨ è ��ILKGRV1IQKG�C2[\OHIQS_[\^>EQAerEzD\~Èм:Ñá ± è �=² ¼[\^{ILGWOHV2[`IQG f G�IcjkGRGWVm½WGWOHDZ[`VT�DFVG\d;EQDyILKSaEuEQGROQSUGREzTSah\GWOQg\GWE�[\E�j�GW^U^³o.¡�G�YW[\V�EQAeren[\OQSa½WG�SaV[ïJð /�79Æ|/i*&? puKG]g\GWDFerGRIQOQSUY�EQGWOHSUGRE Èм:ÑáBâ ¼YWD\V�h\GWOHgFGWE�ILD �� è âSa~zì â ì�í®�\o � ~�ì â ì�ã���ILKGRV1IQKG�EQGROQSUGREzTSah\GWOQg\GWEWopuKGNgFGRDFerGRILOHSUY�EQGROQSUGRE�ILA;OQVE�D\A;I�IQD f G�hFGROHb"A;EQGR~0A;^4~0DFO�IcjkDqOHG:[\EHDFVERo�×.SUOHEHI:d�SaI�YWD\enGREAC"SaV�G·x[`enC^aGWE�[\VT"EHGWYWD\VT�d�S{I�SaE�DFV;GJD\~PIQKG]h\GWOHb1~0GRjçSaV;�2VS{ILG�EQGWOHSUGRE�~0DFOzjzKSUYwK"j�GNYW[\VYWD\enCABILG�ILKG�h`[\^UAGZ±0IQKG�^USUerSaI�D\~.IQKGJC2[`OHILSU[\^�EHAenEL²�ou¡�G]YW[\V�AEHGJDFA;O���VDljz^UGRTgFG]D\~¹ILK;GgFGRDFenG�ILOHSUYzEQGROQSUGREk[`E�[�EHIw[`OHILSaVg�C>DFSaVXI�IQD]D f Iw[\SaVnSaV;~0DFOQen[`IQSUDFV}[ f DFA;I�D\IQKGWO�EQGROQSUGREWd f A;I��2OQEvIj�G]YRDFenC;^UGRIQG�ILK;G�G·x;[\enC;^UG�[ f D=h\G\opuKGyILD\IL[\^¹G���GRYRI�D\~�gFSahiSaVg$sFtFtqT;DF^U^U[\OQE�IQD�[qIw[|x"C2[:bFGWO�[\E�S{I�j�DFOQ��E�SaIQE�j�[:b"IQKOQDFA;gFKILK;G�GRYWDFV;DFeJb$SaE

t;�ª¦X��©lsFt\t�¯�t;�ª¦X� ´ ©=sFtFtz¯�t;��¦F� µ ©=sFtFtu�:�:�F¨ ÈмlÑ ¿ t;�ª¦X� ¼ ©=s\tFt;�puKSaE�SUE�YR^UDFEHG\d f A;IuVD`IuG�x;[\Y�IL^abZ[NgFGRDFenG�ILOHSUY�EHGWOHSUGWERoP� f DlhFG�j�G��HAEHI�~ [`YRILD\OQGWT$D\A;I�ILK;G�sFtFt;opuKSaE�^aDBD\�iE4�HA;EHI�^USa�\G�DFOHTSUV[\OHbq[\OQS{ILKerGRIQSUY\d f A;Izj�G�eJAEHI f G�YW[\OQG�~0A^ØoPpuK;G�V;D\Iw[`IQSUDFVÈмlÑ ¿ t;�ª¦X� ¼ ©=s\tFt

«

Page 5: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

SUE�VD\I�[\V�D\OQTSaV2[\OHbm[\T;TSaIQSUDFV"C;OQD f ^aGWe1d�S{I�SUE�[q^USUerSaIzC;OQD f ^aGWe1o�§�GWV;YWG\d�[`V�bm~ [\YRIQDFOQSaVgZj�GTD}eNAEvI f Gk�HAEvILSa�GWT1AEQSaVgyILKG�OQA;^UGWEzD\~4^USaenS{ILERo ë D\ILG�ILK[`IÈмlÑ ¿ t;�ª¦X� ¼ ©=s\tFt ¨ ^USaeÒ ÇzÈ ñ Òм:Ñ ¿ tB��¦X� ¼ ©=sFtFt|ò¨ ^USaeÒ ÇzÈ ñ s\tFt ÒмlÑ ¿ t;�ª¦X� ¼ ò¨ sFtFt&^aSUeÒ ÇzÈ ñ ÒмlÑ ¿ t;�ª¦X� ¼ ò¨ sFtFt Èм:Ñ ¿ t;�ª¦X� ¼ �

¡�G�AEQGRT¶IQK2[`IqIQKG�^USaenS{InD`~J[�YRDFVEvIw[\VXI�±³s\tFt�SUV¢IQKSUEqYW[\EQG:²nILSaenGRE$[�EQGR»iA;GWVYRG�SaEZILK2[|IYWD\VEHIL[\VXI�IQSUerGWE�IQKG�^USUerSaI�D\~�ILKG�EHGW»�AGWV;YWG\o�¡,G�VDlj�K[lh\G�IQK2[`IÈмlÑ ¿ t;�ª¦X� ¼ ©=sFtFt�¨°sFtFt Èм:Ñ ¿ t;��¦F� ¼ �¡,G�VDlj�eJAEHIzYRDFenC;A;ILG ÈмlÑ ¿ t;�ª¦X� ¼ �

puKSaEuSUEz[\^aenDFEvIuSUVqIQKG�~0DFOQe£D\~.[NgFGWDFerGRIQOQSaY�EQGROQSUGREWo�puKG�DFV^{bnIQKSUVg}erSUEHEQSUV;gNSUE�IQKG�Û�¨¶tEQA;enen[\VT�on��gX[`SUV�d)j�GZY:[`VVD\Iu�HAEHIJ[\TT�ILKSaE�DFV�oZ¡�GneJAEHI�AEHGnILK;GnOHA^UGRE�~0DFO�^aSUerSaILE�D\~EQGR»�AGWVYRGWE�IQD}EL[:b ÈмlÑ ¿ t;��¦F� ¼ ¨ ^aSUeÒ Ç�È ÒмlÑ ¿ t;�ª¦X� ¼¨ ^aSUeÒ Ç�È ñ è ��¯6�k¯ ÒмlÑ ¿ t;�ª¦X� ¼ ò¨ è ��¯ ^USaeÒ ÇzÈ ñ ÒмlÑá t;�ª¦X� ¼ ò¨ è ��¯ ÈмlÑá t;�ª¦X� ¼ �

��I]^_[\EvI�j�GZK[lh\GnIQKGng\GWDFerGRIQOQSUYyEHGWOHSUGWE]G�x;[\Y�IL^ab�[`E]S{IJ[`CC>GW[\OQE�SUV�ILK;GrIQKGWDFOHGWe1oZ¡�GnYW[\VCA;IzIQKGWEHG]EvILGWC;E�IQDFgFGRIQKGWOz[`VTmA;EQG�ILKG�ILK;GWDFOHGWeóIQDrD f Iw[\SaVÈмlÑ ¿ t;�ª¦X� ¼ ©=s\tFt ¨ sFtFt ñ è �k¯ ÈмlÑá t;�ª¦X� ¼ ò¨ sFtFtyô è �k¯ �� è t;��¦F�9õ ¨°s\tFtyô è ��¯ �tB�a�=�2õ¨ sFtFt�©=s;�ªsFsu�:�:�F¨°¬\«Xt\t;�

Page 6: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

puK2[`I�SaEWd\IQKGuILD\IL[\^GR�>GWY�I�DFVyILKGzGWYRDFVDFe]bND\~�ILKG�sFt\t�gFSah\GWVyILD�DFVGzC�GWOHEQDFVrSUEP¬`«XtFt�TDF^a^_[\OHEWo���a��%�')(2*�+.-0/ Á 7�Åzö�(2- Ã�/ Á 79Å�Þ�/�Æ|< / Á ?÷ DFA¢EHKDFA^aT f G"[ f ^UGmIQD,COHGWTSaYRInILKGmVG�xiI$EvILGWC�o®§�[:hiSUV;g,TG��2VGWT�IQKG�VG�jøYWDFVYRGWC;IZD\~SUVB�2VSaIQGNEQGROQSaGWEWd)j�G}VGWGRT�IQD1T;GRhFGR^UDFC�en[\YwKSUV;GWOHbmILK2[|I�[\^U^aD=jzE�AE�ILD�AEHGrS{I:o]��E�AEQA[\^Ød�j�GVGRGWTJEHSUerC^UGPG·x;[\enC;^UGWE�jzKGROQG�jkG��iV;D=j,ILKG�h`[\^UAG�D\~BILKG�EQGROQSaGWE4[\VT�OQA^aGWE�~0DFO)eZ[`VSUCA;^_[`IQSUVgEQGROQSaGWEWoù V;~0D\OHILA;V2[`ILGR^ab\diIQKG�gFGRDFerGRILOHSUY�EQGROQSUGREkYRDFenC;^UGRIQGWE�DFAO�^USaEHI�D\~)EQSaenC;^UGzG�x;[\erC^UGRE�D`~)EQGROQSUGRE~0DFOPjzKSaYwKnILK;G�SUVB�2VSaIQGuEQAeɱ0^USUerSaI.D\~�C[\OHIQS_[\^2EHAenEL²�YW[\V f G�YWDFerCA;IQGWTrG�x;[\YRIQ^ab\oPpuKGWOHG�[\OHGD\IQKGWO�EHGWOHSUGWE�IQK2[`I�j�GyY:[`V�YWDFerCA;IQGNG�x;[\Y�IL^ab\d f A;I�ILKG�b�[\OHGJhFGROHb�EHC>GRYWS_[`^.DFO�hFGWOvb�TS{ÚnYRA^aI[\VT�ILK;G�YRDFerCA;Iw[|ILSUD\VEu[\OHG�T;OQSah\GWV f bqhFGROHb�^UA;Yw��b�YRDFSUV;YWSUT;GWVYRGWEWo�°gFGRVGWOQ[\^�OQA^aG�[\CC;^USUGRT f bneZ[|ILKGReZ[`IQSUYRS_[\VE¹IQK2[`IkSUE�AEQA2[`^U^ab}V;D\Iu[\^a^UDlj�GWTn~0D\O�EvILATGRVXILESUEZú � ~Pb\DFA"Y:[`VVD\I�EQDF^{hFGN[nCOQD f ^UGWe1d9YwK2[\Vg\GJILKG]COQD f ^UGRemoÙû&puKSaE�SaE�jzK2[|I�j�GNTDn~0D\O�ILK;GEHIQAT;bZD\~)EQGROQSaGWEWo�@BSaVYWG�jkG�Y:[\VV;D\IuKDFC�G�IQDJ�2VTqIQKG�G·x[`YRI�h`[\^aAG�D\~�enDFEvI�EHGWOQSaGWERd;jkG�EQGRIHIL^aG~0DFO�TGRIQGWOHenSaVSUVgrSa~.ILKG]EQGWOHSUGRE�YWD\V�h\GWOHgFGWE�DFO�TS{hFGWOHgFGWERo�puK2[|I�SUERd2jkGNEHGRIHIL^UG�~0DFOz�2VTSaVgZDFABISa~�ILKG�^USUerSaI�D`~�C2[\OHIQS_[\^2EHAerE�G�xBSUEvILE�D\O�VD`I:o � ~)bFD\A$^UDiDF�y[`I�ILKG�G·xBGWOQYRSUEHGWEuD\~�IQKSUE�DFO�[`^UerDFEHI[\VXbnD\ILKGROuü�[\^aYWA^aAE f DiDF�9dib\DFAqjzSU^a^2EQGWG�D\V^abr[]~0G�j¶»iA;GWEHIQSUDFV;Eu[\EQ��SaVg�úv¡6K[`I�SaE�ILK;G�h`[\^UAGD\~PIQKG]~0DF^a^UDljzSUV;gnSaV;�2VS{ILG�EQGWOHSUGREwý:û6[\VT"IQKGWEHGy[\^UerDFEvI�[`^ajk[lbiE�SaVXhFDF^{hFGN[ngFGWDFerGRIQOQSaY]EHGWOQSaGWERo÷ DFA�K2[:hFG�en[\VXbZG·xBGWOQYRSUEHGWE�[\EQ��SUV;g,úHþ�DJILKG�~0DF^a^UDljzSUV;gJEQGROQSaGWEuYWD\V�h\GWOHgFG�DFOuTSah\GWOHgFGlý:û� Ph\GWVSa~2b\DFAyY:[`VVD\I¹�VTr[\VyG�x;[\Y�I¹h`[\^UAG�~0DFO¹[�EHGWOQSaGWERdFSaI.SUE¹VSaYWGkIQD���VDlj&S{~2ILKGkG�x;[\YRIPh`[\^aAGkG·xBSUEHIQEDFO�VD\I:o�§�SaTTGWV�SUVmIQKGN[`OQgFAerGWVXI�D\~PjzKGRIQKGWO�[ZC2[\OvILSUYRA^_[`O�EHGWOHSUGWE�YWDFVXhFGROQgFGRE�DFO�T;SahFGROQgFGREDFVGu~0OHGW»�AGWVXIQ^abN�2V;TE�[ f SaI�D\~�G�xiIQOL[�SUV;~0DFOHeZ[|ILSUD\VJGWEvILSaeZ[`IQSUVg�ILKGzEQSa½WGuD\~�ILK;G�^aSUerSaI4D\~�EQGROQSUGRESa~4SaIzG·xBSUEHIQE�D\OzIQKG f GRK2[:hBSaDFOzD\~.IQKG�EQGW»�AGRVYWGJD\~¹C2[`OHILSU[\^�EHAenEzS{~¹SaIzT;DBGRE�VD`I:o�¡,G�eZ[`�\G�[C�DFSUVXIzD\~4^UDiDF��SUVg]~0DFO�ILKSaEuSUV;~0D\OQen[`ILSaDFV$[\Euj�GW^U^³o���Ïÿ��$Ã�- / Á Å 79ÆmÞ�/�Æ`<0/ Á ?puKG�OQA^aGWE�~0DFOuen[\VSaCA^U[`ILSaVgJEHGWOQSaGWEzEHIL[\OHI�SaVm[N~ [\erSU^aS_[\O�jk[lb\oïJð /�79Æ|/i*&?uÍ S{hFGWV�Icj�DnEHGW»�AGWV;YWGWE�Î ¿ À Î ´ À Î µ À �:�:�[\VT�� ¿ À � ´ À � µ À �:�:�BEQAYwK1ILK2[`IÈмlÑ ¿ Î ¼ ¨��¢[`VT Èм:Ñ ¿ � ¼ ¨�� ÀILK[`IzSUERd f D\IQK�EQGWOHSUGRE�YRDFVXhFGROQgFG][\VT�jkG���VDlj°IQKGWSaOu^USUerSaIQE�IQKGWV� IQKG�EQGWOHSUGRE ȼlÑ ¿ ± Î ¼ ¯�� ¼ ²�[`^UEQD}YRDFVXhFGWOHgFGWE�[\VTÈмlÑ ¿ ±³Î ¼ ¯�� ¼ ²P¨���¯ � À

[\V;T� S{~��$SUEz[`V�bqYWDFVEvIw[\VXIWd9IQKGWV ȼ:Ñ ¿ �;Î ¼ YWDFVXh\GWOQg\GWE�[`VTÈмlÑ ¿ �;Î ¼ ¨����u�s

Page 7: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

puK2[`IJSUEWd�ILK;GZSUVB�2VSaIQG}EQAe D\~�IcjkD"EHGW»�AGRVYWGREySUE�ILKGnSUV;�2V;SaILG}EHAe D\~kIQKGZEHAe D\~�ILK;GEQGR»�AGWVYRGWE][\VT"IQKGNSUVB�2VSaIQG]EHAe D`~�[ZYWDFV;EHIw[`V�I�IQSUerGWE�[qEQGR»iA;GWVYRGrSaE�IQKGNYWD\VEHIL[\VXI�ILSUerGWEILK;G�SaV;�2VS{ILG�EHAemoPpuK[`I�SUEWd;SUV;�2V;SaILG�EHAerE�[`OQG�^USaVG:[`OWop4D$h\GWOQS{~�bmIQKGWEHGyILKGRDFOQGRenE�j�GyeNAEvI�OQGRY:[\^a^4ILK[`I�[\V�SUVB�2VSaIQGJEHGWOQSaGWE�SaE�OQGW[\^U^{b1IQKGN^aSUerSaID\~�ILKG�C2[`OHILSU[\^�EHAerEWo�§�GRVYWG`d Èм:Ñ ¿ ±³Î ¼ ¯�� ¼ ²�¨ ^USaeÒ ÇzÈ ÒмlÑ ¿ ± Î ¼ ¯�� ¼ ²ë Dlj�ILKGzOQSagFKXIPK2[\V;TnEHSUTGuYRDFVXIw[\SaVE�[�EQAeÕD\~9�2V;SaILGR^abJeZ[\VXbJViA;e f GROQERdiEHD�j�GzAEQGzILK;GuAEQA2[`^OQA;^UGWEzD\~.[\OHSaILK;GWerGRILSaY}± YRDFereNA;Iw[|ILSah\G�^_[:j�²�IQDrOHG:[\OHOL[\V;gFG�ILKG�EQA;emd

¨ ^aSUeÒ Ç�È ÒмlÑ ¿ ± Î ¼ ¯�� ¼ ²�¨ ^USaeÒ ÇzÈ ñ Òм:Ñ ¿ Î ¼ ¯ ÒмlÑ ¿ � ¼ ò¡�G�YW[\V1VDlj�[\CC;^abZIQKG�EQAeäOHA^UG�~0D\Ou^USUerSaIQEWdiD f IL[\SUVSaVg

^aSUeÒ Ç�È ñ ÒмlÑ ¿ Î ¼ ¯ Òм:Ñ ¿ � ¼ ò�¨ ^aSUeÒ Ç�È ÒмlÑ ¿ Î ¼ ¯ ^USUeÒ ÇzÈ Òм:Ñ ¿ � ¼ ¨���¯ � À[\VT�ILK;SUEuYWD\enC^aGRIQGWE�ILK;G][`OQgFAerGWVXIWoë D\IQG�IQK2[`I�ILKG��\G�bnIQD�IQKSUE�[\OHgFAerGWVXI�j�[\E�ILD�OQGRenGRe f GRO�jzK[`I�[`VZSUVB�2VSaIQGzEQGWOHSUGRE�OHG:[\^a^abSUERd;ILK;G�^aSUerSaI�D\~�ILKG�C2[`OHILSU[\^�EHAerEWo���VDljzSUVgyILK;SUEWdBjkG�Y:[`V�[\C;C^abrILKG�EHAeºOQA;^UG�~0DFOu^aSUerSaILERopuKG�EL[\erG�SUT;G:[NjkD\OQ��E�~0D\OueNA^aIQSUC^aSUYW[`ILSaDFV f b$YRDFVEvIw[\VXILEJ± EHGWG] .x;GROQYRSUEQGREw²·oå)SU�\G�[\^a^FILKGRDFOQGRenERd`ILK;G�ILK;GWDFOHGWeç[ f D=h\G�YRDFerGWE4jzS{ILK�KXbBC�D\IQKGWEHGWEWoPpuK;GWEQG�[\OQG�YWDFV;TSaIQSUDFVEILK[`I�eNAEvI f GuerGRI f GR~0D\OQG�j�GuY:[\Vr[\CC;^abJIQKG�ILK;GWDFOHGWe1o.puKSaV�ND\~>[�ILK;GWDFOHGWeÕ[\E�[�YRDFVXILOQ[\YRIWdú � ~¹b\DFA"TDrILKSaE�j�DFOH��d � jzSa^U^)gFS{hFG�bFDFA�IQKSUE�[\erDFAVXI�D\~PerDFVG�bFoÙû�puKG]KXbBC�D\IQKGWEHGWE�D\~4ILKSaEILK;GWDFOHGWe�[\OHG¹ILK2[|I�IQKGPEQGROQSaGWE� ȼ:Ñ ¿ Î ¼ [`VT� ȼlÑ ¿ � ¼ f D\ILK�YWD\V�h\GWOHgFG\o � ~GWS{ILKGRO)D\~iILKGREQG�EQGROQSUGRETS{hFGWOHgFG]IQKGWV"IQKG]IQKGWD\OQGWe¤EQ[lbiE�[ f EQDF^aA;ILGR^ab�VD\IQKSUV;gZ[ f DFA;I�ILKG]EQAe ȼlÑ ¿ ± Î ¼ ¯�� ¼ ²�oJ± � ~bFD\A"TDqVD`I�TDZ[\^U^)ILK;G]j�DFOQ�9d � en[:b�DFO�en[:b�VD\I�gFSah\G�b\DFA�[\VXb1D\~¹ILK;GJerDFVGRb\o­² � I�SaE�VD`IILDiD1TSaÚrYWA;^aI�ILD1YWDFerG}AC�jzS{ILK�G�x;[\erC^UGRE�jzK;GWOQGnG:[\YwK,D\~ ȼlÑ ¿ Î ¼ [\VT ȼlÑ ¿ � ¼ TSah\GWOQg\Gf A;I ȼlÑ ¿ ± Î ¼ ¯�� ¼ ²�YRDFVXhFGWOHgFGWERo¡,G�[\^a^���VDlj°IQK2[`I ±³Î ¿ ¯�Î ´ ²W±�� ¿ ¯�� ´ ² 騰Π¿ � ¿ ¯�Î ´ � ´ ÀEQD}ILK;GWOQG�SUEuV;DnEHSUerC^UG�OHA^UG�~0DFO�IQKG�COQDiTAY�I�D\~.Icj�DnSaV;�2VS{ILG�EHAenERo�puK;SUEzSUEzOQGR^_[`IQGWT$ILD}ILK;G~ [\Y�I�ILK2[|I�ILKGROQG�SaE�VDyEHSUerC^UG�OQA^aG�~0DFO�ILKG�SaVXILGWg\OL[\^>D\~)ILKG�C;OQDiTAYRI�D\~)IcjkDy~0AV;YRILSaDFVERoPpuKGúHCOHDBT;AYRI�OQA;^UGWûN~0D\OuSUVXILGRgFOL[\^aEuSUE�IQKG�ILGWYwK;VSU»�AG�D\~4SUVXILGRgFOL[`IQSUDFV f b$C[\OHIQEWo�������$Ã�- / Á Å 79Æ��]7�#>Ó�/�Æ|Ô9/i#�ÄF/"79Å�Þ�/�Æ|< / Á ? Û 8 ð ï /�Æ|* ï / Á 8X?puKG�OHGWEvIPD\~9DFAO4OQA^aGWE.~0D\O4EHGWOQSaGWE¹SaVXhFDF^{hFG�YRDFVXhFGWOHgFGWV;YWGzD\~2ILK;G�EQGROQSUGREWo�¡6KSU^aG�ILK;GRb�~0OQGW»�AGRVXIL^ab[\^aEQDNgFS{hFG�EQDFerG�GWEvILSaeZ[`IQG�D\~)IQKG�h`[\^aAG�D\~)IQKG�EHGWOQSaGWE�±0Sa~)S{I�YWD\V�h\GWOHgFGWEL²�dILK;G�eZ[`SUVZg\DX[\^�SUE�IQDTG�ILGWOHenSaVG�Sa~.[ygFSah\GWV1EQGROQSUGREzYWDFVXhFGROQgFGRE�DFOuT;SahFGROQgFGREWopuKG��2OQEvIrIQGWEvIr~0D\OyYWDFVXhFGROQgFGRVYWG�SUENILK;G1EHSUerC^UGREHINIQD,[`CC^ab\d f A;In[\^UEHD�gFSah\GWEyILKG�^aG:[\EvISUVB~0DFOQen[`ILSaDFVq[ f DFABIuILKG�EQGROQSaGWEWo � IzSUEuYW[\^U^aGWT$ILK;G�úvÛ ILK IQGWOQe£ILGWEvILûo

Page 8: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

ïJð /�79Æ|/i*&?uÍ S{hFGWVm[yEQGR»iA;GWVYRGNÎ ¿ À Î ´ À Î µ À �:�W�{d;Sa~ÈмlÑ ¿ Î ¼YWD\V�h\GWOHgFGWE�ILKGRV ^USae¼lÇzÈ Î ¼ ¨�t;�¡,G�YW[\Vmg\SahFG�[\V�SaV;~0DFOQen[\^�[\OQgFA;enGRV�Iu~0D\O�ILKSaEz[\Eu~0DF^U^aD=jzER¸Èм:Ñ ¿ Î ¼ ¨ ^USaeÒ ÇzÈ ÒмlÑ ¿ Î ¼ À

ILK;G]EHGWOQSaGWE�SUEzILKG�^USaenS{IuD\~.IQKGJC[\OHIQS_[\^�EHAerEWo � ~¹ILK;G]C2[`OHILSU[\^)EQA;enE�[\OQG�ILGRVTSaVgnIQDq[}^USUerSaIWdEL[:b��udILK;GWV�ILKG�EQA;e ÒмlÑ ¿ Î ¼eNAEvI f G�h\GWOHb1YR^UDFEHG�ILD���~0DFOz[\^a^�^_[`OQgFG�ê�o � VmDFOHTGWOu~0DFO f D\IQKÒмlÑ ¿ Î ¼[\VT Ò.æ ¿Ð¼lÑ ¿ Î ¼ILD f G�YR^UDFEHG�IQD��,j�G�eJAEHI�K2[:hFG�Î Ò4æ ¿ h\GWOHbrEHeZ[\^a^Øo.@BD]~0D\O�Û�^U[\OQgFG`diÎ ¼ eNA;EHI f GzhFGROHbrEQen[\^a^ØopuKSaE�SaE�ILKG�SaV;~0DFOQen[\^9j�[:b�D\~4EL[:biSUV;g ^USae¼lÇzÈ Î ¼ ¨�t;�¡�GNY:[\V�eZ[\�`GJILK;SUE�h\GWOHSa�2YW[`ILSaDFV"COHGWYWSaEQG f bmAEQSaVgZIQKGNTG��2VSaIQSUDFVmD\~�^USaenS{I:ozpuKSaE�en[\�\GRE�[VSaYWG�G�xBGWOHYWSaEQG\o¥�VZILKGz~ [\YRG�D\~)S{I:dXILK;SUE�^UDiDF��E�erDFOHG�^aSU�\G�[�j�[:brILDJA;EQG�IQKG�~ [`YRI�ILK2[`Ik[]EHGWOHSUGWE�YWDFVXh\GWOQg\GWEILK[\VmS{IzTDiGWE�[Nj�[:b$ILD}ILGREHIz~0DFOzYWD\V�h\GWOHgFGWVYRG\o�puK;G��`GRb1SUE�IQD}AEQG�ILKGmúHYRDFVXILOQ[\C>D\EQSaIQSah\GWû}D\~ILK;SUE�EHIL[`ILGRenGRV�IWoqpuKGrILKGRDFOQGRe EL[:biE]ILK[`I�Sa~�ILKGrEQGROQSUGREJYRDFVXhFGWOHgFGWE�ILKGRV,ILK;Gr^aSUerSaI�D\~�ILK;GEQGR»�AGWVYRG�D\~)EHAenen[\VT;E�eNA;EHI f G�½WGROQDo�@BD;dBS{~�j�G�TSaEQYWDlh\GWOkIQK2[`I�ILKG�^USaenS{IPD\~�IQKG�EHGW»�AGRVYWGD\~4EQAeren[\VTEzSUEuV;D\Iz½WGROQD}ILKGRV1IQKG�EQGWOHSUGRE�eJAEHI�TS{hFGWOHgFG\dIQK2[`I�SUEïJð /�79Æ|/i*&?uÍ S{hFGWVm[yEQGR»iA;GWVYRGNÎ ¿ À Î ´ À Î µ À �:�W� À S{~^USae¼lÇzÈ Î ¼ é¨�t ÀILK;GWV ÈмlÑ ¿ Î ¼¦

Page 9: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

TS{hFGWOHgFGWERopuKSUE}ILGREHInSUE}VSaYWG f GRY:[\A;EQG�S{I}SUEy~0OQGR»�AGWVXIL^{b�G:[\Evb�ILD�[\CC^{bFo � I ÄF(2#�#�798 ILGR^U^uAErSa~�[EQGR»�AGWVYRGJYRDFVXhFGWOHgFGWERo � ~ ^USae¼lÇzÈ Î ¼ 騰tILK;GWV�jkG���VDlj®ILKG�EHGWOQSaGWE ÈмlÑ ¿ Î ¼TS{hFGWOHgFGWERo � ~ ^USae¼lÇzÈ Î ¼ ¨°tILK;GWVm[\^U^�j�G���VDlj�SUE�IQK2[`IuILK;G�EHGWOHSUGWE ÈмlÑ ¿ Î ¼erSUgFKXIuYRDFVXhFGWOHgFG�DFOzSaIuerSUgFKXIuT;SahFGROQgFG`o�puK2[`I�SUERd;jkG�TD}VD\I�gFG�I�[\VXb�VGRj°SUV;~0D\OQen[`ILSaDFV�o×2DFOuG�x;[\erC^UG`d;j�G]YW[\VmEQ[:b$SUerenGRTS_[|ILGW^{b}ILK2[`IÈм:Ñ ¿ ¬\Û ´ ¯&�`Ûr¯6�¦\Û ´ ¯�«TS{hFGWOHgFGWE f GWYW[\AEQG ^aSUe¼:Ç�È ¬\Û ´ ¯��`Ûr¯¶�¦`Û ´ ¯�« ¨ ¬¦[\VTZ¬ �\¦ é¨�tBoP§�Dlj�GRhFGROWd;IQKGzÛ IQK IQGWOQe�IQGWEvIkgFS{hFGRE�AE�VD]TGR�2V;SaILS{hFGzSUVB~0DFOQen[`ILSaDFV}[ f DFA;I�ILK;GEQGROQSaGWE ÈмlÑ ¿ �ÛDFO ÈмlÑ ¿ �Û ´EQSaVYWG f D\ILK�EHGW»�AGWV;YWGWEJ�!�`Û�[\VT��!�`Û ´ K[lh\G]^aSUerSaI�½WGROQDo����" ïJð /#�]7�*,+�(Æ|< Á 7�# ï / Á 8n(#43�$·8 Á � ð < -³3�Æ=/i#�?¥�AO�VG�xiI�YRDFVXhFGROQgFGRVYWG�IQGWEvIkSaE�IQKG�erDFEvI�SaenC�DFOvIw[\VXI:d�[`VTnIQKG f [\EQSaE�~0DFO�en[\VXb}D\~�IQKG�D`ILKGROILGREHIQEWo � IzSUEuY:[`^U^UGRT$ILKG]ükDFerC2[\OHSUEQD\V$p4GREHIïJð /�79Æ|/i* %&�]7�*,+�(Æ|< Á 7�# ï / Á 8('=? @BACC�DFEHG}j�Gr[\OQGyg\SahFGRV�IcjkD�EQGR»�AGWVYRGWEJÎ ¿ À Î ´ À Î µ À �:�:�[\VT�� ¿ À � ´ À � µ À �:�W�{o�@iACC�DFEQG�ILK2[|I�t�)6Î ¼ )*� ¼ ~0DFOz[\^a^�Û.oPpuKGRVmIQKG�~0DF^U^aD=jzSaVgNKD\^UT�¸� � ~ Èм:Ñ ¿ � ¼

Page 10: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

YRDFVXhFGWOHgFGWE�ILKGRV Èм:Ñ ¿ Î ¼YRDFVXhFGWOHgFGWERo� � ~ Èм:Ñ ¿ Î ¼TS{hFGROQgFGREzIQKGWV Èм:Ñ ¿ � ¼TS{hFGROQgFGREWopuKSUEzOHGWEQA;^aI�OQGR^USUGREzDFV1IQKG�~ [\YRIWd2TSUEHYWAEHEQGRT�[`IuILK;G]GRVTmD`~�ükK2[\CBILGWO ß dIQK2[`I�[ f DFA;VTGWT�dSUV;YWOQGW[\EQSaVg�EQGR»�AGWVYRG�[\^{j�[:biE�K2[\E1[�^USaenS{I:o�¡,G�[`OQG�gFS{hFGWV6ILK[`I$ILKG�Î ¼,+ E�[\V;T�� ¼,+ E1[\OHGVDFV.-cVGRgX[`ILS{hFG`d2EQDyILKG�YRDFOQOHGWEQC�DFVT;SUVgrEQGR»iA;GWVYRGWE�D\~4C2[\OvILSU[\^�EHAenEÒмlÑ ¿ Î ¼

[\VT ÒмlÑ ¿ � ¼[\OHG�SaVYWOHG:[\EHSUVgr±�jkG�gFD�~0OQD\eàDFVGuIQGWOQeÖILD�ILKGzVG·xiI f b}[\T;TSUVg�DFVrVDFV.-cVGRgX[`ILS{hFGuV�Ae f GWOHEw²·o@BAC;C>DFEHG ÈмlÑ ¿ � ¼ ¨ ^USaeÒ ÇzÈ ÒмlÑ ¿ � ¼G�xBSaEHILE�[\V;TmGR»iA[\^UERd2EL[:b��Öo�puKGWV ÒмlÑ ¿ Î ¼ ) ÒмlÑ ¿ � ¼ )/�~0DFO4[\^U^�ê�d\EHD�ILKG�EQGR»�AGWVYRGuD\~2C2[`OHILSU[\^XEQA;enE4D\~ILK;GkÎ ¼,+ E4SUE4[ f DFAVTGRT�dFSaVYWOHG:[\EHSUVg�EQGR»iA;GWVYRG\o§�GRVYWG`d ÈмlÑ ¿ Î ¼G�xBSaEHILERok��E�[ f D\ViA;EWd2jkG�gFG�I�IQK2[`I�[}^USaIHIL^aG f SaIzD`~¹SUVB~0DFOQen[`ILSaDFV$[ f DFA;IuIQKG�EQAeäD\~4ILKG�Î ¼,+ ERdV2[\erGW^{b ÈмlÑ ¿ Î ¼ ) ÈмlÑ ¿ � ¼ ¨����

�lt

Page 11: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

¥�V�IQKGyD\ILK;GWO�K2[\VT�d�jkGn[\^UOHG:[\TBb"��VDljàIQKG}EQGW»�AGRVYWGnD\~�C2[\OvILS_[`^.EQA;enE�D\~�ILKG}Î ¼�+ E�SaESUV;YWOQGW[\EQSaVgdEHDnS{~ ÈмlÑ ¿ Î ¼TDiGWEyVD`IyG�xBSUEvINILK;GWV�SaIJeNAEHI f G f GWYW[\AEHG�IQKGqEHGW»�AGWV;YWG1D\~�C2[\OHIQS_[\^�EQA;enEND`~�IQKG�Î ¼�+ ENSaEAV f DFAVTGRT�o�§�GWV;YWG\d ^USaeÒ ÇzÈ Òм:Ñ ¿ Î ¼ ¨�0°�1 A;Izj�G�[\^UEHDrK[lh\G ÒмlÑ ¿ Î ¼ ) ÒмlÑ ¿ � ¼EQD}j�G�eJAEHI�K2[:hFG ^USaeÒ ÇzÈ ÒмlÑ ¿ � ¼ ¨�0 ÀDFO ÈмlÑ ¿ � ¼TS{hFGWOHgFGWERopuKGnükDFerC2[\OQSaEQDFV�p�GWEvIJ[`^U^UDljzE�A;E�IQD�AEQG}SUVB~0DFOQen[`ILSaDFVmj�G}��VDljÕ[ f DFA;I�DFV;GyEQGWOHSUGRE�IQDgX[\SaV$SUV;~0D\OQen[`ILSaDFV$[ f DFA;I�[\V;D\ILKGROWo�×2DFOuG·x[`enC^aG\d;j�G][\^UOHG:[\T;bZ��VDlj®ILK2[|IÈмlÑá ô ���õ ¼YWD\V�h\GWOHgFGWE f GWY:[`AEQG$S{INSUEJ["gFGWD\enG�ILOQSaYnEHGWOHSUGWENjzS{ILK�OQ[`ILSaD�^aGWEQE]ILK2[\V�D\VG\o�§�GRVYWG`dPjkG$YW[\VYWD\VYW^aATG�SUerenGRTS_[|ILGW^{b}ILK2[`I ÈмlÑá ô ��z¯�Û�õ ¼[\^aEQD}YWDFVXh\GWOQg\GWE f GRY:[\AEHG ô ��u¯�Û õ ¼ )¤ô �� õ ¼~0DFOz[\^a^�Û.o¹¡�G�TDrVD\Iz��VDlj®ILK;G�h`[\^aAG�D\~ÈмlÑá ô ��z¯�Û õ ¼ Àf A;Izj�G�Y:[\V1EL[:b$~0DFOuEHAOQG�ILK[`IÈмlÑá ô ���¯�Û�õ ¼ ) ÈмlÑá ô ���õ ¼ ¨ �� è ± �!�F�F² ¨ �« �� IqSaEnV;D\Iq[�YWDFSaVYWSaTGWV;YWGmIQK2[`InILKGmEQGWOHSUGREnj�G�YWD\enC2[`OQGWT�ILD,SaV�IQKG�G·x[`enC^aG�[ f DlhFGmSUEn[gFGRDFenG�ILOHSUYZEHGWOHSUGWERo,puKG$g\GWDFerGRIQOQSUYZEQGWOHSUGRE}[\OQGZIQKG$EQGROQSaGWEN~0DFOJjzK;SUYwK�jkG$YW[\V�enD\EHIyGW[\EQSa^ab

�F�

Page 12: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

TG�ILGWOHenSaVG}YWDFVXh\GWOQg\GWVYRGq[\VT�TSah\GWOHgFGWVYRG\o � V�~ [\Y�I:d�AEQSaVg�ILKG}YWD\enC2[`OQSUEHDFV�ILGREHI�jzSaILK�ILK;GgFGRDFenG�ILOHSUY�EHGWOQSaGWE�SUEuEHDrEvIw[\V;T2[\OQT�ILK[`I�SaIugFG�ILEzSaIQEuDljzV�V2[\erG\oÍ S{hFGRV�[yEQGROQSaGWE ÈмlÑ ¿ Î ¼j�G�jk[\VXI�IQDJTG�ILGWOHenSaVGzSa~�SaI�YWD\V�h\GWOHgFGWE�D\O�TS{hFGROQgFGRE f brYWDFerC2[\OHSUVg�S{I�IQDJ[�gFGWD\enG�ILOQSaYuEQGROQSUGREÈмlÑ ¿ â ¼ �1 A;I�jzKSaYwK�gFGRDFenG�ILOHSUY�EQGROQSUGREwý�puK2[`I�SUEWd2jzK2[|I�YwKD\SUYWG�D\~ â jzSa^U^)[`^U^UDlj�AE�IQDnA;EQG]IQKGJükDFe2-C2[\OHSUEHDFV�puKGWDFOHGWeZý¡,GR^U^³d;SUVZDFOQTGRO�ILDyAEHG�ILKG�ükDFerC2[\OHSUEHDFVqpuKGRDFOQGRemd;j�G�VGWGRT$ILDy��VDlj°IQKG�OQGR^_[`IQSUDFVEHKSUCf GRIcj�GWGWV â ¼N[\V;T�Î ¼ ~0DFO�[`^U^9Û.o � V$DFOHTGWO�ILDyEL[:bZ[\VXb�ILKSaVgNILK2[|I�KDF^aTEk~0DFOk[\^U^9Û4d;jkG�V;GWGWTqIQD��VDlj�IQK2[`IzÎ ¼ [`VT â ¼]ILGRVT$ILDN½RGWOQDr[`Iu[\CCOHDlxBSaeZ[`IQGW^{brIQKG�EQ[\enG�OL[`IQG}± S{~�Î ¼ TSUTqV;D\I�ILGRVTILD}½RGWOQD}ILK;GWV�jkG�jkD\A^UT���VDlj�[\A;IQDFen[`ILSaY:[\^a^abyILK[`IÈмlÑ ¿ Î ¼TS{hFGWOHgFGWT�o­²}§�GRVYWG`d�S{~�jkG�Y:[\V�YRDFenC[\OQG�Î ¼ ILD â ¼N~0DFOz[\^a^�Û�ILK;GWV�ILKG�OHGW^_[|ILSUD\VEQKSaC f G�IcjkGRGWVÎ ¼ [\VT1Î ¼ æ ¿ EHKDFA^aT f G�[ f DFA;IuIQKG�EL[\erG�[\E�IQKG�OQGW^U[`ILSaDFVEHKSUC f GRIcj�GWGRV â ¼ [\V;T â ¼ æ ¿ d;ILK2[|ISUEzÎ ¼ æ ¿ �\Î ¼�3 â oPpuK;SUEzenD`ILSah`[`IQGWEkIQKG54�[|ILSUDyp�GWEvI:oïJð /�79Æ|/i* %6�q(;8=<07 ï / Á 8('l? @BAC;C>DFEHGy[ZEHGW»�AGWV;YWG}Î ¿ À Î ´ À Î µ À �:�W��EHAYwK"ILK[`I�Î ¼ ã®tn~0DFO�[\^U^Û�[\VT ^USae¼lÇzÈ Î ¼ æ ¿Î ¼ ¨ â �� ~ â í°��ILKGRV ÈмlÑ ¿ Î ¼YWD\V�h\GWOHgFGWERo � ~ â î°��ILKGRV ÈмlÑ ¿ Î ¼TS{hFGWOHgFGWERo � ~ â ¨Ö��IQKGWV ÈмlÑ ¿ Î ¼erSUgFKXIuYRDFVXhFGWOHgFG�DFOzTSah\GWOQg\G\opuKG�[\OQg\AenGRVXIuEQKDljzSaVgyILK2[`IkILKSaEkILKGRDFOQGReºSUE�ILOQA;G�SUEu[ f S{IuEQA f IL^aG\o¹¡�G�jkDFA;^UT$^USa�\G�IQDYWD\enC2[`OQG ÈмlÑ ¿ Î ¼

� ß

Page 13: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

ILD Èм:Ñ ¿ â ¼jzKGROQG â SUE�IQKG}^USaenS{I�D`~�Î ¼ æ ¿ �\Î ¼ d f A;I�j�G}DFV^{b"��VDljàIQK2[`I�Î ¼ æ ¿ �\Î ¼ SaE�YW^aDFEQGNIQD â SUV�ILK;G^USaenS{I:o�×2DFO�[\VXbZC2[`OHILSaYWA^U[\O�Ûmh`[\^aAG\d�jkG�TDNVD\Ik��VDlj�S{~�Î ¼ æ ¿ �`Î ¼ SUE�^_[\OHgFGWO�DFO�EQen[\^U^aGWOPILK[\Vâ d;jkG�DFV^{bq��VDlj®IQK2[`IzSaIzSaEuYW^aDFEQG�IQD â oP¡,G�TGW[\^�jzSaIQK$ILKG�Y:[`EQGWE�DFVG�[`I�[NIQSUerG\o@BACC�DFEHG â í �Fo puK;GWV�jkG���VDlj IQK2[`I�~0DFO�EQA;ÚrYWSaGWVXIL^{b�^_[`OQgFG�Û.dJj�G�eNAEHI�K[lh\GÎ ¼ æ ¿ �`Î ¼ í£�Fo}¡�GrYW[\V�eZ[\�`GyILKSaE�enDFOHG}COQGRYWSUEHG f b�^aGRIHILSUV;g87$¨¤� è â o}puKGRV�j�Gn��VDlj~0OQD\eºILKGNT;GR�2VS{ILSaDFVmD`~PILKG]^USUerSaI�D\~�[ZEHGW»�AGWV;YWGNILK[`I�IQKGWOHGNSUE�[\V�ê EQAYwK"IQK2[`I�Sa~¹Û�ã°êILK;GWV 9999 Î ¼ æ ¿Î ¼ è â

9999 í 7ß �puKSaE�SaenC;^USUGRE�IQK2[`Iu~0DFOuÛ�ã6êt�) Î ¼ æ ¿Î ¼ ) â ¯ 7ß í°�F�

å)GRI â ¿ ¨ â ¯�7(� ß o¹¡,G�VDlj°K2[:hFG�ILK2[`I Î Ò4æ ¿Î Ò í â ¿DFO Î Ò.æ ¿ í â ¿ Î Ò �@BSaenSa^_[\OH^abFd Î Ò.æ ´ í â ¿ Î Ò.æ ¿ í â ´ ¿ Î Ò[\VT Î Ò4æ µ í â ¿ Î Ò.æ ´ í â µ¿ Î Ò �ükDFVXILSaV�ASUVg}SaV�IQKSUE�~ [`EQKSaDFV�d;j�G]D f Iw[\SaVÎ Ò.æ ¼ í â ¼ ¿ Î Ò~0DFOz[\^a^�Û�ã��Foë Dlj°j�G][\OQG�OQGW[\T;bqIQD}AEQG�ILKG]ükDFerC2[\OHSUEHDFV$p4GREHIWo�¡�G��iV;D=j�ILK2[|I�IQKG�EQGROQSUGREÈм:Ñ Ò â ¼ ¿YWD\V�h\GWOHgFGWERo�§�GRVYWG`d2ILKG�EQGROQSaGWE ÈмlÑ Ò Î ¼[\^aEQD}YWDFVXh\GWOQg\GWE f b$ILK;GNükDFerC2[\OHSUEHDFV�p4GREHI:ok��TTSaVg}�2VSaIQGW^{bqen[\VXbqILGROQerE�IQDZ[}EQGROQSUGREzTDBGREVD\IzYwK[\VgFG�jzKGRIQKGWO�DFOuVD`IzSaIzYWD\V�h\GWOHgFGWEN± EQGRG� ¹xBGROQYWSaEQGr�=²�d2EHDrj�G�K2[:hFGÈмlÑ ¿ Î ¼�l¬

Page 14: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

YWD\V�h\GWOHgFGWERo� ~2ILKGk^USUerSaI�D`~2ILKG�OL[`IQSUDFE4Î ¼ æ ¿ �\Î ¼ SUE.g\OQG:[|ILGWO�ILK2[`VNDFVGkIQKGWVJGRhFGRVXILA2[\^a^ab�ILKGREQG�~0OQ[\YRIQSUDFVEeNAEvI f G�gFOHG:[`IQGWO$ILK[\V�D\VG\oÕpuKG�TS{hFGWOHgFGWV;YWG�D\~JIQKG�YWDFOHOQGREQC�DFVTSaVg�gFGWDFerGRIQOQSaY"EQGROQSUGRE~0DF^a^UDljzEk~0OQD\eóILKG]ükDFerC2[\OHSUEHDFV$puKGWD\OQGWe1o� ~kIQKGn^aSUerSaI�D\~kIQKG}OL[`IQSUDFE�Î ¼ æ ¿ �\Î ¼ SUE�DFVGyIQKGWV�jkGrY:[\V�VD\I�EQ[lb�~0D\O�EHAOQGrSa~kÎ ¼ æ ¿ �`Î ¼SUE]^UGREQEJIQK2[\V�DFO]gFOQGW[`ILGRO]IQK2[\V�DFVGr~0DFOJ^U[\OQg\GnÛ4o�§�GRVYWG`d�j�GqYW[\V�EL[:b�VD\ILK;SUVg�[ f DFABIJILK;GYWD\V�h\GWOHgFGWVYRGJD\~�ILK;G�YRDFOQOHGWEHC>DFV;TSUVgrEQGROQSaGWEWopuKG:4�[`ILSaD�p�GWEvI�eZ[:b�ILGR^U^�AE¹S{~9[�EQGROQSUGRE.YRDFVXhFGWOHgFGWE�DFO4TSah\GWOHgFGWERd f A;I¹S{I¹TDiGWE4VD\IPT;SUOQGRYRIQ^abgFS{hFGN[`V�b�SUV;~0D\OQen[`ILSaDFV�YWDFVYRGWOHVSUVgnILKGJh`[\^aAGJD`~PILKGNEHGWOHSUGWE�Sa~PS{I�TDiGWE�YRDFVXhFGROQgFG`o�¡�GJD\V^abVGRGWTmIQDnEHAYWYRGWEHEH~0A^a^ab1YWDFerC2[\OHG�ILKG�EQA;enen[\VTEzSaV1IQKG�Iw[\Sa^�D\~P[}EQGROQSaGWEWd2IQK2[`I�SUERdILKG�ILGROQerE~0DFO�Û�^_[\OHgFG\d�SaV�DFOHTGWO�IQDmEHKDljÕYWDFVXh\GWOQg\GWVYRG\omükK2[\Vg\SUVgm[��VSaIQGrV�Ae f GRO�D`~kILGROQerE�TDBGREVD\IzSaV�;2AGRVYWG�YRDFVXhFGWOHgFGWV;YWG�DFVG�jk[lbqDFO�ILKG�D`ILKGROWdKDlj�GRh\GWOWd9j�G�YW[\V�YwK2[\Vg\G�IQKG�h`[\^UA;G�D\~[yYWDFVXh\GWOQg\GWVXI�EHGWOQSaGWEuILDr[\VXb�ILK;SUVg}jkG�jk[\VXI f b$YwK[\VgFSaVgZ[yEHSUVgF^aG�ILGROQe1op4D$A;EQGyILKG<4�[`IQSUDnp4GREHI][\E�EvIw[`IQGWT�[ f D=h\G\d�j�G}eNAEvI f GN[ f ^UG�ILD$YRDFenC;A;ILGJIQKGy^USaenS{IzD\~ILK;G$OL[`IQSUD"D\~�EQA;YWYWGREQEHSahFG�ILGROQerEyDFV�ILK;G$gFSah\GWV�EQGROQSUGREWo�×2DFOJEQDFerGqEQGROQSaGWEyILKSaENSUEJOQGW^U[`ILS{hFGR^abG:[`EHbFo�×2DFOuG·x[`enC^aG\d;~0D\O�ILKG�EQGROQSaGWE ÈмlÑ ¿ Û ´ß ¼j�G]YRDFenC;A;ILG^USae¼lÇzÈ ± Ûn¯¶�=² ´ � ß ¼ æ ¿Û ´ � ß ¼ ¨ ^USae¼lÇzÈ ± Ûr¯¶�l² ´Û ´ ß ¼ß ¼ æ ¿¨ ^USae¼lÇzÈ Û ´ ±v��¯6�!�`Û�² ´Û ´ �ߨ �ß �

puKG�EQGROQSUGREzYWDFVXhFGROQgFGRE f bqIQKG54�[`IQSUDyp�GWEvI f GWYW[\AEQG}�!� ß í°�FopuKG1^USUerSaIQENSUVXh\DF^ah\GWT�jzSaILK�4�[`ILSaD�p�GWEvIZYW[\V�d�D\~�YRDFAOQEHG\d f G�enD\OQG�YWDFerC^aSUY:[|ILGWT�o�×2DFOG�x;[\erC^aG\d;IQDrA;EQG�ILKG54�[`ILSaDyp4GREHI�DFV Èм:Ñ ¿ ß ¼¬ ¼ ¯�Û µj�G]eNA;EHIzYWD\enCABILG�ILKG�^aSUerSaI^USUe¼:ÇzÈ ± ß ¼ æ ¿ �±³¬`¼ æ ¿ ¯°± Ûr¯6�=² µ ²ß ¼ �±³¬ ¼ ¯�Û µ ² �

p�DnYRDFenC;A;ILG�IQKSUEu^aSUerSaI�jkG��2OQEvI�EQSUerC^aSa~�bn[\^Ug\G f OQ[\SUYW[\^U^{bZ[`EueNAYwKm[\EzC>DFEHEQS f ^UG^aSUe¼:Ç�È ß ¼ æ ¿ �±³¬|¼ æ ¿ ± ��¯°± Ûr¯6�=² µ �\¬`¼ æ ¿ ²Q²ß ¼ �;±³¬ ¼ ±v��¯�Û µ �\¬ ¼ ²H² ¨ ^aSUe¼:Ç�È ß �± ¬2±v��¯�± Ûr¯6�=² µ �`¬`¼ æ ¿ ²H²�!�± ��¯�Û µ �\¬ ¼ ² �

�:«

Page 15: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

ë Dlj�d ^USae¼lÇzÈ Û µ¬ ¼ ¨�tf GWY:[`AEQGZIQKGqTGRVDFerSUV2[|ILDFO�ILGRVTEJIQD"SUVB�2VSaIcb�eJAYwK�enD\OQGr»iA;SUYw��^ab�ILK2[`V,ILKGnV�AenGROL[`IQDFOWo@BDd f bqIQKG�OQA^aGWEu~0DFOu^aSUerSaILERdXjkG�K2[:hFG^USae¼lÇzÈ ß ¼ æ ¿ �;±³¬`¼ æ ¿ ±v��¯°±0Ûr¯¶�=² µ �\¬`¼ æ ¿ ²Q²ß ¼ �±³¬ ¼ ± ��¯�Û µ �`¬ ¼ ²Q² ¨ ߬ �

ë Dlj�d ߬ í°� ÀEQD}ILK;G�EHGWOHSUGWE Èм:Ñ ¿ ß ¼¬ ¼ ¯�Û µYWD\V�h\GWOHgFGWERo¥�V�ILKGJD\ILKGRO�K2[`VT�d>IQKG<4�[`IQSUDZp�GWEvI�TDiGWE�VD\I�EHDF^ah\GN[\^a^�COQD f ^UGWerEWo�×2DFO�EQGWOHSUGRE�D\~PILK;G~0DFOHe ÈмlÑ ¿ �Û>=~0DFOz[yYWD\VEHIL[\VXI@?�î�tNj�G]YRDFerCA;ILG^aSUe¼lÇzÈ �!�±0Ûr¯¶�=² =�A�`Û>= ¨ ^USae¼lÇzÈ Û =±0Ûr¯¶�=²B=¨ ^USae¼lÇzÈ Û =Û = ±v��¯6�!�`Û)² =¨ ^USae¼lÇzÈ �± ��¯¶�!�|Û)² =¨ �F�puKSaEPSUE¹ILK;G�jkDFOHEHI�C�DFEQEHS f ^aGuDFA;ILYRDFerGk~0DFO¹ILK;Gu^USUerSaI4EQSaVYWG�IQKGC4�[|ILSUD�p4GREHI�gFSah\GWE�VD�SUVB~0DFOQen[D-ILSaDFVrDFVGzj�[:brDFO�[\VD\IQKGWO�[ f DFA;I�ILKG�EQGROQSaGWEWo¹¡�G�EHILSa^U^2VGRGWTqerDFOHGzIQGWYwKV;SU»�AGWE�ILD�TGRIQGWOQerSUV;GSa~ dB~0DFOuG�x;[\erC^UG`d ÈмlÑ ¿ �ÛYWD\V�h\GWOHgFGWERo���FEG$W*�+4Æ=79+P/�ÆH$R#>8l/�Ô9Æ=(2- Á ?puKG�EQAereZ[|ILSUD\V¶V;D\Iw[`IQSUDFV6SaE1A;EQA2[\^a^ab�SaVXILOQDiTAYRGWT°[ f D\A;I�K[\^a~�j�[:b6ILK;OQDFAg\K®[�IcbiCSaY:[\^ü�[\^aYWA^aAE}YW^_[`EQEWo � InYWD\enGREnSaV�K2[`VT;b�jzKGWV&TG:[\^ujzS{ILK�[\CCOHDlxBSaeZ[`IQSUDFV;ErD`~�[\OHG:[\ErAVTGROgFOQ[\CKERoä×2DFO$G�x;[\erC^UG`d�ILK;G,[\C;COQD:xBSUen[`ILG�[\OHG:[�AVTGRO1IQKG�gFOL[\C;K°D\~JI¹±LK>²�¨ �!�DK°~0D\O�M)NKO)��ltySUE PмlÑ ¿ �Û 3RQ ¿ á¿ �KTS K��

�=�

Page 16: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

puKSaENGW»�A2[`IQSUDFV�YRDFOQOHGWEHC>DFV;TEyILD�IQKG$CSUY�ILAOHG f GW^UDlj£[\VT�SaENY:[`^U^UGRT,ILK;GU4�SaGWen[\VV�EHAe D\OACC�GWOzEHAemo

×4SUgFAOHG�¬;o{�F¸�ükDFerC2[\OQSaEQDFV$D`~V P¼lÑ ¿ ± �!�`Û)²�[`VTXW ¿ á¿ ± �!�DK>² S K�opuKGNEvILGRC�EHSU½RG�±WúZY<K9û�²zSUE�DFVGJ~0DFO�ILKSaE�EHAemd�jzKSUYwK�SaE�AV�AEHA2[\^a^abm^_[\OHgFG\o�puKGNEQen[\^a^UGWOzILK;GEHIQGWC1EQSa½WG\dIQKG�enD\OQG�[\YRYWAOQ[`ILG�ILK;G][`CCOQD:xBSUen[`IQSUDFV$D\~�ILK;G][`OQG:[Boù EHA2[\^U^{b�j�GyILKSaV��D\~�ILK;GyEQAe [\E�[`V�[`CCOQD:xBSUen[`IQSUDFV�~0D\O�ILKG}[\OHG:[$AVT;GWO�ILKG}gFOQ[\CK�o� V�IQKSUEyEHGWY�ILSUD\V�j�G1IQKSUV�,D\~�IQKG�[`OQG:[�AVT;GWO}ILKG�gFOQ[\CK�[\EZ[`V&[\C;COQD:xBSUen[`ILSaDFV�~0DFOyILK;GEQA;emo]puKGyC>D\EQSaIQSUDFV�D\~�ILKG}GWV;T�C�DFSaV�IQE�D\~�ILK;GrSaVXILGWOvh`[\^.D\~�SaV�IQGWgFOQ[`ILSaDFV"gFS{hFGNIQKGy^USaenS{ILEzD\~ILK;G�EQAeren[`ILSaDFV�o.p�Dy[\C;COQD:xBSUen[`ILG�[\VqSaV;�2VS{ILG�EHAe1dBj�G�eNAEHIkK2[:hFG�[\V$SaV;�2VS{ILG�SUVXILGROHh`[\^�D\~SUVXIQGWgFOQ[`ILSaDFV�dBIQK2[`IzSaEWd ÈмlÑ ¿ �Û 3RQ È¿ �K S K��

[ < Ô�Ã�Æ|/"���UÜ�?5$W#)Ì.#4<�8l/1(Æ=/i(,ÄF79Æ|Æ=/ Á +P7�#�34< #�Ô�8l7�< *,+�Æ=7�+P/�Æn<³#�8l/�Ô�Æ=(2- W È¿ ±v�!�!K>² S K¡6KSU^aGqSaINSUENGW[\EHb�IQD"jzOQS{ILG$CKOQ[\EQGRE}^USU�`G�úvIQKG�[\OQGW["D\~zILKG�gFOQ[\CK�AV;TGWOyILK;G$SUV;�2V;SaILGSUVXIQGWOHh`[\^aû}[\VT�~0DFOQeJA^_[\E Q È¿ �KTS K ÀSaIZSUE$[`VD\ILK;GWOqILK;SUVg,IQD�EL[:b�COHGWYRSUEQGR^ab�jzK2[`I$IQKGRb�erG:[`V�o�¡,G�VGRGWT�ILD�K[lh\G�[�COQGRYWSUEHGTG��2VSaIQSUDFV$jzKSaYwKmOHG\;2GRYRIQE�DFA;OuSUVXILAS{ILS{hFG�VD\IQSUDFV�o�ls

Page 17: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

Ë /X̹#�<08l< 7�#�? puKG�SUVXILGRgFOL[`^�DlhFGWO�[\V1SUVB�2VSaIQG�SUVXIQGWOHh`[\^)D`~^]}¨�I¹±LK>²�SaEuTGR�2V;GWT f bQ È_ I¹±`K>² S K�¨Ê^USaea ÇzÈ Q a_ I¹±`K>² S K��

� V;~0DFOHeZ[`^U^ab\d|j�GuY:[`^U^�ILKSaE4ILK;GrúQ[\OQGW[�AVT;GWO¹ILK;G�gFOL[`CKyD\~>I¹±LK>²�DFVNILK;GuSUVXILGROHh`[\^BÎT)�KO)�0°oÙû@BAYwK�[\V�SUVXILGRgFOL[`^�SUEuYW[\^U^aGWT <³*�+4Æ=79+P/�Æ opuKG�ILGWOHe¤úHSaenCOHDFC�GWOQûkSUE�ILDzTGREQYROQS f G�[\V�SUVXILGRgFOL[`IQSUDFV�COQD f ^UGWe�jzKGWV�GWS{ILKGRO�ILKG�SUVXIQGWOHh`[\^D\~�SUVXILGRgFOL[`IQSUDFV�EvILOQG�ILYwKGREqILD�SUV;�VSaIcb�SaV�D\VGmD\O f D\ILK�TSaOQGWY�ILSaDFVErDFOyILKG�~0AVY�ILSUD\V f GWSUV;gSUVXIQGWgFOQ[`ILGRT�IQGWVT;E�IQDrSaV;�2VS{IcbqSaV�IQKG�SUVXILGROHh`[\^�D`~.SaV�IQGWgFOQ[`ILSaDFV�o��EzG�x;[\erC^aGWEWd2YRDFVEHSUTGROQ È¿ �KTS K ¨ ^aSUea ÇzÈ Q a¿ �KbS K¨ ^aSUea ÇzÈ ^aV)±`K>²=ì a ¿¨ ^aSUea ÇzÈ ^aV)±��R² è ^aV�±v�l²¨ ^aSUea ÇzÈ ^aV)±��R²·�

ë Dlj�diILKSaE�^aSUerSaIPTDiGWE�VD\I�G�xBSUEvI f GWYW[\AEQG�^UV)±���²�ILGRVTE�ILDJSaV;�2VS{Icbr[\Ec��IQGWVTE�ILD]SUV;�2V;SaIcbFoP@BDj�G�EL[:bZIQK2[`I�ILKG�SaenC;OQDFC�GWO�SUVXILGRgFOL[`^>TDiGWEkVD\I�G·xBSUEHIWd;DFO�TSah\GWOHgFGWERdDFO�IQGWVT;E�ILDNSaV;�2VS{IcbZ[`VTj�G�jzOHSaILG Q È¿ �K S K�¨�0°�¡�GrSaV;~0DFOHeZ[\^a^ab�SaV�IQGWOHCOQG�I�IQKSUE�IQD�enGW[\V�ILK2[`I�IQKGr[\OQGW[$AVTGRO�IQKG}gFOL[\C;K�D`~d]�¨¤�A�DK�~0D\O�M)NKO)/0 SUEuSUVB�2VSaIQG\o@BSUerSU^U[\OQ^{bFdQ È¿ �K ´ S K ¨ ^USaea Ç�È Q a¿ �K ´ S K¨ ^USaea Ç�È è �K ´ ì a ¿¨ ^USaea Ç�È è ô � � è �� õ¨ ^USaea Ç�È è �� ¯6�¨ �F�¡�G$SUVXILGROQCOHGRINIQKSUEN~0DFOHeNA^U[1ILD�erG:[`V�ILK2[`IyIQKG$[\OHG:[�AVT;GWONILK;G$gFOL[\C;K�D\~C]�¨ �!�DK ´ ~0D\O�M)NKO)/0 SUEuDFVG`oe (Æ|#4< #�Ô�? puKSUE�TGR�2V;SaILSaDFV�D\~�ILK;G}[\OQGW[qAVTGRO�[qgFOL[\C;K�Dlh\GWO�[\V�SaV;�2VS{ILGNSaV�IQGWOvh\[`^.EQGRGWerEC�GWOH~0GRYRIQ^abnOQG:[`EQDFV2[ f ^UG`oP§�Dlj�GRhFGROWd2SaV;�2VS{IcbnSaE�[JEvILOQ[\VgFG�C;^_[\YRG\o¹¡�G�eNAEvIz[\^{j�[:biE f G�Y:[`OQGR~0A;^jzKGRVygFDFSUV;g�IQKGWOHG�ILD�TDFA f ^aG�YwK;GWYw�JG�hFGROHb�ILKSaVg�j�G�ILK;SUV��D\~9[`E.D f hiSUDFA;E.SUV�ILKG��2VS{ILG�jkDFOH^UT�o

� �

Page 18: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

×2DFO�G�x;[\erC^UG`d)OQGRY:[\^a^.ILK2[|I�j�G}Y:[\V�YWD\enCABILGNILK;G}h\DF^UA;enGyD\~�OQG�hFDF^aA;ILSaDFV�D\~�ILKG}OQGRgFSUD\V~0DFOHenGRT f b�OQG�hFDF^{hiSUVgzILKG�gFOQ[\CK]D\~f]N¨�I¹±LK>²)[ f DFA;I�IQKG�Kf-c[|xBSUE�DlhFGRO�IQKG�SaVXILGWOvh`[\^�ÎT)�KO)/�f bqILK;G�~0D\OQeNA^U[ Q a_�g ±hI¹±LK>²H² ´ S K��puKSaE�~0DFOHeNA^U[zYRDFerGWE�~0OQDFe�[`TTSUV;g�AC�ILKG�hFD\^UAerG�D\~2TSaEQYWE�~0DFOHenGRT f b�OQG�hFDF^{hBSaVg�OQGWY�Iw[\V;gF^UGRE[\CC;OQD:xBSUen[`ILSaVg1IQKGqOHGWgFSaDFV,AV;TGWONIQKGZgFOQ[\CK�D\~:I¹±LK>²J[`EJEQK;D=jzV f GW^aD=j�o" �[`YwK�TSUEH��K2[`EhFD\^UAerG g ILSaenGREJILKG$OQ[\TSaAE2I¹±`K>²JEQ»�A2[\OHGWT�ILSUerGbY<K�o���EyILKGqjzSaT;ILK�D\~�ILK;G$OQGWY�Iw[\V;gF^UGREILGRVTEzILD}½WGROQD�±hY<K8i tX²·dILKG�EQA;eóILGRVTEzILDyILKG�SaV�IQGWgFOQ[\^Øo

×.SUg\AOQG�¬;oª¬;¸�j�DF^aAerG�D\~4OQGRh\DF^UABILSUD\V�~0D\OQerGWT f b$OQG�hFDF^{hBSaVg2]}¨ç�A�DK�[ f DFA;IkK"[|xBSUERo§�GRVYWG`d>ILK;G]hFD\^UAerG�D\~POQG�hFDF^aA;ILSaDFV1D\~¹ILK;GJ[\OHG:[rAVTGRO�ILK;G]gFOQ[\CKmD\~V]n¨��!�DK�DlhFGRO�ILK;GSUVB�2VSaIQG�SUVXILGROHh`[\^¹�M)�K�í�0 eJAEHI f G

Q È¿ g ô �KPõ ´ S K�¨ g Q È¿ �K ´ S K��¡�G�YWDFerCA;IQGWT}[ f DlhFGkIQK2[`I¹IQKG�SaenCOHDFC�GWO�SUVXILGRgFOL[\^iDFVNIQKG�OQSagFKXI.SaE¹DFVG`o¹§�GWVYRG\dFILK;G�hFDF^aAenGD\~4OQG�hFDF^aA;ILSaDFV�D\~4IQKG�[\OQGW[yAVTGRO�IQKG�gFOL[\C;K�]y¨ç�!�DKm~0DFO��M)�Kl)/0 SUEuGW»�A2[`^�ILD g o

×4SUgFAOHG�¬;oÙ«¸�ükDFerC2[\OHSUEQD\V1D`~4h\DF^UAerG�D\~4OQG�hFDF^aA;ILSaDFV1[\VT�YWOHDFEQEB-cEQGRYRIQSUDFVm[\OQGW[;o�l¦

Page 19: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

1 A;Izjk[\SaIWo¹¡,G�YWD\enCABILGWTm[ f DlhFG�ILK2[|IQ È¿ �K S K$¨�0

DFOzILKG][\OHG:[}AVTGRO�ILK;G]gFOQ[\CKmD\~�]n¨��!�DK�[ f DlhFG�ILKG�SUVXIQGWOHh`[\^P��)/K#)�0 SUE�SUVB�2VSaIQG\o�@BDjzK2[`I�SaE�g\DFSUVg}DFV2ý"¡�G][\OQG�EL[:biSUV;grIQK2[`I�Sa~�jkG�Iw[`�\G�ILKG�SUV;�VSaIQG�[\OQGW[rAV;TGWOzILKG�gFOQ[\CK1D\~]}¨��!�DKr~0DFOu��)�KX)/0 [`VT}OQGRh\DF^ah\GzSaI�[ f DFA;I¹IQKGkKf- [|xBSaEWdFj�G�g\GRI�[�OQGRgFSUDFVyIQK2[`I�K2[\E¹�VSaIQGhFD\^UAerG�GR»�A2[\^>IQD g o � VZD\ILKGRO�j�DFOQT;EWd;S{I�j�DFA^aTZIw[`�\G�[\VnSUV;�VSaIQG�[\erDFAVXI�D\~�C2[\SaVXI�IQDNYWDlhFGROILK;G�;9[`I�OQGRgFSUD\VmAV;TGWO�ILKG]gFOL[\C;KX]Z¨Õ�A�DK�~0DFO��J)�K#)m0°d f ABI�DFV^{b1[r�2V;SaILG][\erDFAVXI�D\~C2[\SaVXIuILD}�2^U^�IQKG�SUV;�VSaIQG�ILOQA;enC�GRIzEHK2[\C�GWT�OHGWgFSaDFV$~0DFOQerGWT f b�OQG�hFDF^{hiSUVgyILK;SUEz[\OHG:[r[\OQDFA;VTILK;GMKf-c[|xBSUERoë D\IQKSUV;g1SaE�jzOQDFV;gmKGROQG`o 1 D`ILK�Y:[\^aYWA^U[`ILSaDFVE�[ f DlhFGr[\OHG}C>GROH~0GWY�IL^{b�YWDFOHOQGRYRI:or ¹hFGRV,ILK;GGWEvILSaeZ[`IQGWE�DFV�ILK;GZ[\erDFAVXI�D\~�C2[\SaVXI]V;GWYWGREQEQ[\OHb�ILD1C2[\SaVXI�DFO��2^U^¹IQKG}OQGRgFSUDFV;E�SaVXhFDF^{hFGWT�[\OHGYWD\OQOQGRYRIq±³[`I�^aG:[\EvI:d�Sa~�bFDFA�[\YRYWGWCBI]IQKGnSaTG:[$D\~kC2[\SaV�IQSUVg�SaV;�2VS{ILGNjk[\^U^aE�DFO��2^U^aSUVgqSaV;�2VS{ILGR^abTGRGWC$YWD\V�IL[\SUV;GWOQEL²�o¹¡6KGRVq�2^U^aSUVg�ILK;G�SUV;�VSaIQG�IQA f G�ILKG�jzSUT;IQKqD\~�C2[\SaVXI f GWYRDFenGREkhFGROHbnILKSaVjzKGRV#K�SUE�^U[\OQgFG`d>jzKSa^UGJjzK;GWV�C2[\SUVXIQSUVgqIQKGyYWOHDFEQEB-cEQGRYRIQSUDFV�d)IQKGNILKSaYw��VGWEHE]D\~�IQKGy^_[:bFGRO�D\~C2[\SaVXIkSaE�YRDFVEHIL[\VXIk~0DFO�[\^a^fK�oPpuKG�erDFOL[`^�SaEWdXjzKGRV$TG:[\^aSUVg�jzSaIQKZSaV;�2VS{IcbyjkG�Y:[\VV;D\I�G·x;C�GWY�IDFAOuA;EQA2[\^�SaV�IQASaIQSUDFV;E�ILD f G�[\EzOQGW^aS_[ f ^UG�[\EuSaV$ILKG��2VS{ILG�j�DFOQ^aT�o����n)?5oN(2Ä!p�8l7�Þ�/�Æ`<0/ Á ?¡�GZY:[`V�AEQGZjzK[`IJj�Gq��VDlj£[ f D\A;INYRDFenC;A;ILSaVg�SaenCOHDFC�GWO�SUVXILGRgFOL[`^UE�ILD"gFG�INSaV;~0DFOQen[`IQSUDFV[ f DFA;IzYRDFVXhFGWOHgFGWV;YWGJD`~.SaV;�2VS{ILG�EQGROQSaGWEWo�×2DFOuG·x[`enC^aG\dYRDFVEHSUTGROzIQKG�EQGROQSUGREÈм:Ñ ¿ �Û �puKSaEkSUE�Y:[\^a^UGRTnILK;G ð (Æ|*,7�#4< Ä Á /�Æ|< / Á\o ë D\IQG�ILK2[|Ik~0DFO�GRh\GWOHbZÛ.diILKG�[`OQG:[]AVTGROkILKG�gFOL[`CK]}¨��!�DK f GRIcj�GWGRV"Û�[\VT1Ûr¯¶��SUEu^aGWEHEzIQK2[\V��!�`Û�±0EQGWG�×.SagFAOHG�²�o

[ <0Ô�Ã�Æ=/"�)����?<�]7>#�*,+�(Æ|< Á 7>#¢79Šȼ:Ñ ¿ ±v�A�`Û)²rq <�8 ð W È¿ ± �!�DK>² S K �§�GRVYWG`d2jkG�Y:[\V�YRDFenC[\OQG�ILK;G�C[\OHIQS_[\^>EQAeºD`~.ILK;G�K[\OQerDFVSaY�EQGROQSUGREuILDr[\V1SUVXILGRgFOL[\^ÒVs ¿Ð¼:Ñ ¿ �Û î Q Ò¿ �KTS K���l�

Page 20: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

¡�G]eNA;EHI�K2[:hFG�ILKG�ACC�GWO�^USaenS{I�D\~4ILKG�EQAe f G�ê è ��SUV1DFOQT;GWOz~0DFOuILKG�OQGRYRIw[`VgF^UGREuILDr^USUG[ f DlhFG�ILKG�gFOQ[\CKm[\EzSUV�×.SagFAOHG£o

[ < Ô�Ã�Æ|/"���t"�?NÞ�Ã4* ÒVs ¿¼:Ñ ¿ ±v�!�|Û)²kî W Ò¿ ±v�!�!K>² S K§�GRVYWG`d2jkG�K2[:hFG ÈмlÑ ¿ �Û ã Q È¿ �KTS K��puK2[`I�SaEWd>Sa~PILK;GySUV;�2V;SaILG]EQAe DFV"ILKGJ^UGR~�I�K2[\E�[Zh`[\^aAGJILK;GWV�SaI�eNAEHI f GNgFOHG:[`IQGWO�IQK2[\V�ILK;Gh`[\^UA;G�D\~4IQKG�SUerCOQD\C>GRO�SUVXILGRgFOL[`^�DFV�ILKG�OQSagFKXI:o¹��E�[ f Dlh\G�j�G��iV;D=j°ILK2[`I

Q È¿ �KTS K ¨ ^aSUea ÇzÈ Q a¿ �KbS K¨ ^aSUea ÇzÈ ^aV)±��R² è ^aV�±v�l²[\VT}ILK;SUEP^USaenS{I.T;DBGRE�V;D\I�G�xBSUEvI f GWYW[\AEHG�^aV)±h��².ILGRVTE�ILD�SUVB�2VSaIcbN[\E���ILGRVTE�ILD�SUV;�VSaIcb\o.puKSaESUerC^aSUGWE�ILK2[|I�IQKG�^USaenS{IkD\~�ILK;G�C[\OHIQS_[\^>EQAerEzD\~.IQKG�K2[\OQerDFVSaY�EQGROQSaGWEuILGRVTEzILD}SUV;�VSaIcb\o�@BDILK;G�K[\OQerDFVSaY�EQGROQSUGREzTSah\GWOQg\GWEWoë D\IQG�IQK2[`I.IQKGkK[\OQerDFVSaY�EHGWOQSaGWE4C2[\EQEHGWE¹IQKGkÛ ILK IQGWOQe�IQGWEHIWd f A;I.EHIQSU^U^�TDiGWE4VD\I.YWDFVXhFGROQgFG`opuKG�EQGR»�AGWVYRGZ�!�|Û�TDBGREuVD\IzgFDyIQDy½WGWOHDr~ [`EHIzGWV;DFAgFK$~0DFO�IQKG�EQGROQSUGREuILD}YWDFVXh\GWOQg\G\o � V�~ [`YRI:dILK;G][ f DlhFG�[\OHgFAerGWVXI�EQKDljzEzILK2[|IuILKG�C2[\OvILSU[\^�EQA;enEÒus ¿Ð¼lÑ ¿ �ÛILGRVT�ILD}SUV;�VSaIcbq[`I�^UG:[`EHI�[\Eu~ [\EvI�[\EuILKG�EQGR»�AGWVYRGJ^aV)±³ê�²�op4DrAEHG�IQKG�EL[\erG�SUTGW[yDFV$ILKG�EQGROQSaGWE Èм:Ñ ¿ �Û ´ Àj�G]VD`ILG�ILK[`I ÒVs ¿Ð¼:Ñ ¿ �Û ´ î Q Ò¿ �K ´ S K

ß t

Page 21: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

EQD ÈмlÑ ¿ �Û ´ ã Q È¿ �K ´ S K��§�Dlj�GRh\GWOWd9~0OQD\e£ILK;G][ f DlhFG�j�G���VDlj®IQK2[`I

Q È¿ �K ´ S K�¨Ö� ÀEQDn[\^a^>ILK;G][`OQgFAerGWVXI�[ f DlhFG�gFSah\GWEzSUE�IQK2[`IWdSa~4SaIuG·xBSUEHIQEWdÈмlÑ ¿ �ê ´ ã��F�¡�G�YW[\V1VD\IzEL[:b1[\VXb�ILK;SUVgybFG�I�[ f DlhFG�YRDFVXhFGWOHgFGWV;YWGJD`~.ILK;G�EHGWOHSUGWERo@BSUV;YWGPILKG�SUerCOHDFC>GRO�SUVXIQGWgFOQ[\^FD\~��!�DK ´ ~0OQDFeÖ��IQD�SaV;�2VS{Icb�SUE��VSaIQG\d|SaI�SUE)V;D\I�AVOQGW[\EQD\V2[ f ^aGILDZgFAGWEHE�ILK[`I�IQKGNSaV;�2VS{ILG�EQAe¤D\~z�!�|Û ´ SaE�[\^aEQDr�2VSaIQG\o�¡,G]Y:[\V"YRDFerC2[\OQG�ILKG]SUVXILGRgFOL[`^�IQDILK;G�EHAeºSaVm[`VD\ILK;GWOuj�[:b1[\EzCSaYRIQAOQGRT1SaV1×.SUgFA;OQG�¬;o � o

[ < Ô�Ã�Æ|/"���`E>?�5m728l/ Ò¼:Ñ ´ ±v�!�|Û ´ ²�í W Ò¿ ±v�!�!K ´ ² S K �×2OQDFe�IQKSUE��2gFA;OQGzjkG�V;D\ILGzILK2[|I��!�|Û ´ SaE�^aGWEHE�ILK[\VnIQKG�[\OHG:[�AVTGRO�ILK;G�gFOQ[\CKZD`~P�!�DK ´ ~0OQDFeÛ è ��ILDrÛ.o�§�GWVYRG\d2jkG�Y:[`VmCABI�[\VmA;CC>GRO f DFA;VTmD\V�IQKG�C2[\OvILS_[`^�EQAe f bÒмlÑ ´ �Û ´ í Q Ò¿ �K ´ S K��ë D`ILG�ILK[`I�j�G�K[lh\G�ILD}EHIL[\OHI�IQKG�EQAeó[`IzÛm¨ ß SUV$D\OQTGRO�~0DFOkILK;G�OQGWY�Iw[\V;gF^UGREuSUV�×.SagFAOQG®IQD^USaG f GW^UDlj¶ILKG�gFOQ[\CK�op.[`�iSaVgyILKG�^aSUerSaI�D`~ f D\ILK�EHSUTGRE�[`E�êÊIQGWVTEzILD}SaV;�2VS{IcbFd;j�G�D f Iw[`SUVÈмlÑ ´ �Û ´ ) Q È¿ �K ´ S K��§�GRVYWG`d2jkG���VDlj°ILK[`I ÈмlÑ ´ �Û ´

ß �

Page 22: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

YWD\V�h\GWOHgFGWE f GRY:[\AEHG�IQKG�SUerCOQD\C>GRO�SUVXILGRgFOL[`^�YWDFVXh\GWOQg\GWEWo� ¹hFGRV"enD\OQG\dBjkG�[\^aEQD}gFGRIuIQK2[`IÈмlÑ ´ �Û ´ )��EQSaVYWG�ILK2[|I�SaE�IQKG�h`[\^aAG�D\~¹ILK;G]SaenCOHDFC�GWOzSUVXILGRgFOL[`^Øo���TTSUV;gnILK;GJÛ�¨Õ�]IQGWOHe �A�B� ´ ¨��]IQDf D\ILK�EQSaTGWERd2jkG�gFG�I�IQK2[`I ÈмlÑ ¿ �Û ´ ) ß �§�GRVYWG`dBj�G�K2[:h\G�EQKDljzVZIQK2[`I�ILKG�EQGROQSUGREkYRDFVXhFGWOHgFGWE�[\VTqK2[`E�h`[\^aAG�^aGWEHE�IQK2[\VZD\O�GR»iA[\^�IQD ß opuKG�C;OQDiYWGWEHE�T;GWEQYROQS f GRTqSUVyILK;G�G·x[`enC^aGWE�[ f DlhFG�SaE�Y:[\^a^UGWTyIQKG $R#>8l/�Ô9Æ=(2- ï / Á 8 o.¡�G�YW[\VjzOQS{ILG�IQKSUE�[\Ez[y~0DFOQen[\^�ILKGRDFOQGReä[\Eu~0DF^U^aDljzEW¸ïJð /�79Æ|/i* %v$R#>8l/�Ô9Æ=(2- ï / Á 8w'=? @BAC;C>DFEHG�ILKG�EQGR»�AGWVYRGyÎ ¿ À Î ´ À Î µ À �W�:�EL[|ILSUEv�2GWEÎ ¼ ¨�I.± Û)²jzKGROQG�I¹±LK>²kSaEz[rVD\V,-cVGRgX[`IQSahFG`dTGRYWOQGW[\EQSaVgrYWDFVXILSaV�ADFAEz~0AVY�ILSUD\V�~0D\OcK"ã��Fo� � ~�ILKG�SaenCOHDFC�GWO�SaV�IQGWgFOQ[\^ Q È¿ I.±LK>² S KTS{hFGROQgFGREzIQKGWV1ILKG�EHGWOQSaGWE Èм:Ñ ¿ Î ¼[\^aEQD}TS{hFGWOHgFGWERoyxmDFOQGRDlhFGWORd2ILKG�C2[\OvILSU[\^�EHAenEÒVs ¿Ð¼:Ñ ¿ Î ¼IQGWVT�ILD}SUVB�2VSaIcbq[`I�^UGW[\EHI�[\E�~ [`EHI�[\E

Q Ò¿ I¹±`K>² S K[\E�êÊILGRVTEuILDrSUVB�2VSaIcb\o� � ~�ILKG�SaenCOHDFC�GWO�SaV�IQGWgFOQ[\^ Q È¿ I.±LK>² S KYRDFVXhFGWOHgFGWE�ILKGRV1IQKG�EQGROQSUGRE Èм:Ñ ¿ Î ¼YRDFVXhFGWOHgFGWERokxmD\OQGWDlh\GWOWd Èм:Ñ ¿ Î ¼ )6Î ¿ ¯ Q È¿ I¹±`K>² S K��ßFß

Page 23: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

� V1COL[\Y�ILSaYWG\d9S{IzSUEuAEHA2[\^U^{b$G:[`EQSUGROuILDrTOL[:j°IQKG�CSUY�ILAOHGWE�[ f D=h\G]YRDFerC2[\OQSaVgyILKG�[\OHG:[}AVTGROILK;G�g\OL[\CK�D\~�I¹±`K>²�IQDyILKG�EQA;eºILK2[\V�S{IzSUE�ILD}OHGWerGWe f GWO�IQKG�TGRIL[\SU^aEuD\~4IQKG�ILKGRDFOQGRemo �[\YwK,ILSaenGyj�GZEHAYWYRGWEQEv~0A^U^{b,EHKDlj YWDFVXh\GWOQg\GWVYRG$DFO�TSah\GWOQg\GWVYRGqD\~u[1EQGROQSUGRE]j�GZD f Iw[\SaV[\VD`ILKGROzIQDBD\^>ILDrAEHG�jzS{ILK$ILK;GJükD\enC2[`OQSUEHDFV�p4GREHIWo�puK;G]K2[`OQerDFVSUY�EQGWOHSUGRE�[`VT1IQKG�EQGROQSUGREÈмlÑ ¿ �Û ´[\OHG�EHC>GRYWSU[\^�Y:[`EQGWE�D\~�jzK2[`I�[\OHG�YW[\^U^aGWT<z�- EQGROQSUGREÈм:Ñ ¿ �Û�{ �ù EHSUVg�ILKG�YWDFerC2[\OHSUEHDFVrIQGWEvI:dBj�G�K[lh\G�SUererGWTSU[`ILGR^ab�ILK2[`I�ILKGuz�- EQGWOHSUGREkYWD\V�h\GWOHgFGWE�~0DFO|zmã ß[\VT1TS{hFGWOHgFGWEu~0DFO�z�)��FoPpuKG�Y:[`EQGWE���í}z1í ß [\OQG�YRD=h\GWOHGWT�SaV$ILKG� .xBGWOQYRSUEHGWEWo����~,!b� Á 7>- Ã)8l/G�]7�#>Ó�/�Æ=Ô�/i#�ÄF/9?��^a^�D\~ILK;G�ILK;GWDFOHGWerE¹[\VTJG�x;[\erC^UGRE.D\~2EHGWOHSUGWE4EQD�~ [`O4K2[:h\G�K2[\Ty[`^U^XVDFV.-cVGRgX[`ILS{hFG�EQAereZ[`VTEWo§�Dlj�GRh\GWOWd�en[\VXb�EQGW»�AGRVYWGREyK2[:hFG$C�DFEQS{ILS{hFGn[\VT,V;GWgX[`IQSah\GnILGROQerEWd¹EHD�j�GqVGWGRT,ILD f GZ[ f ^UGILD�T;G:[\^kjzS{ILK�SaV;�2VS{ILGqEHAenEyD`~�IQKGWEHG$EQGR»iA;GWVYRGWEWo � IySaENOQGW[\EQD\V2[ f ^aGqILD�G�xBC�GWY�IrIQK2[`IyS{INSaEúHGW[\EQSaGWOQûq~0D\O�[qEQGWOHSUGRE�K[lhiSaVg f D\IQK�C>D\EQSaIQSah\Gy[\VT�VGRgX[`ILS{hFG}EQA;enen[\VTE�ILD�YWDFVXh\GWOQg\GnEHSUVYRGILK;GWOQG�SaEuY:[\VYRGW^a^_[`IQSUDFV f GRIcj�GWGRV�ILKG�C�DFEQS{ILS{hFG�[\V;TmVGRgX[`IQSahFG�IQGWOQerEWoË /X̹#�<08l< 7�#�?�Í Sah\GWV�[rEQGW»�AGRVYWG}Î ¿ À Î ´ À Î µ À �:�:�4±0jzKGROQG�ILKGJÎ ¼,+ E�Y:[\V f G]C>D\EQSaIQSah\G]D\~¹VGWgX[!-ILS{hFG:²�dj�G�EQ[:b$ILKG�EQGROQSaGWE ÈмlÑ ¿ Î ¼SUE (�� Á 7�- Ã�8:/B-`��ÄF7�#>Ó�/iÆ=Ô9/i#>8 Sa~�ILKG�EHGWOQSaGWE Èм:Ñ ¿ ì Î ¼ ìYWD\V�h\GWOHgFGWERo�¡�G�EQ[:bqILKG�EHGWOQSaGWEzSUE ÄF7�#�34<�8=<07>#�(2- -L��ÄF7�#>Ó�/iÆ=Ô9/i#>8 Sa~ÈмlÑ ¿ Î ¼YWD\V�h\GWOHgFGWEvI f A;I Èм:Ñ ¿ ì Î ¼ ìTS{hFGWOHgFGWERo@BD�[�EQGROQSaGWE$SUEq[ f EQD\^UA;IQGW^ab�YWD\V�h\GWOHgFGWVXImSa~�SaIZSUE$EvILSU^a^�YRDFVXhFGROQgFGRV�Im[`~�IQGWOqj�G�eZ[`�\G�[\^a^�ILK;GEQA;enen[\VTEuC�DFEHSaILS{hFG f b$Iw[`�iSaVgr[ f EHDF^UA;IQG�h`[\^UAGREWo¥�AO�SaV�IQASaIQSUDFV�ILK[`I�S{I�SaE�ûFK[\OQTGROQûrILDqYRDFVXhFGWOHgFGySa~4jkGJeZ[\�`GN[\^a^)ILKG�ILGWOHenE�C>D\EQSaIQSah\G]SaEYWD\V;�2OQerGWT f bqILK;G�~0D\^U^UDljzSaVgJILK;GWDFOHGWe1o

ß ¬

Page 24: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

ïJð /�79Æ|/i*&? @BACC�DFEQG�ILK;G�EHGWOHSUGWE Èм:Ñ ¿ ì Î ¼ ìYWD\V�h\GWOHgFGWERd�IQKGWV�ILKG�EHGWOQSaGWE ÈмlÑ ¿ Î ¼YWD\V�h\GWOHgFGWERokpuK2[`I�SaEWd2[\V1[ f EQDF^aA;ILGR^abqYRDFVXhFGROQgFGRV�I�EQGROQSUGREzSUEuYWD\V�h\GWOHgFGWVXI:opuKG�h\GWOHSa�2YW[`ILSaDFVZD\~�ILKSaE�~ [`YRI�SUE�EQAOHCOQSaEQSUV;gF^abyEHA f IQ^UG\o � I�OQG�hFDF^{hFGRE�[\OHDFAVTZIQKGz~ [\YRI�ILK2[|IÒм:Ñ�� ì Î ¼ ì�ã 99999 ÒмlÑ�� Î ¼ 99999 �puKGnEQAerEJDFV�ILKGn^UG�~�I]eJAEHI f GngFGRIHILSaVg1h\GWOvb,EHeZ[\^a^�[\E�� gFGRIQEJ^U[\OQgFG f GWYW[\AEQGrILKGnEQAeD\~¹IQKGN[ f EQDF^aA;ILG�h`[\^UA;GWE�YWDFVXhFGROQgFGREWo�§�GWVYRG\d�ILKG]EQAerE�DFV1ILKG]OQSagFKXI�[\OHGJ[`^UEQDngFGRIHILSaVgrh\GWOvbEQen[\^a^Øo � IzIQAOQVEzDFABI�IQK2[`IWdCOQD\C>GROQ^abZEHIL[`ILGRT�dILKSaEzSUEuEQABÚnYRSUGRV�IzIQDnEHKDlj°ILK[`IzILKG�EQGROQSaGWEÈмlÑ ¿ Î ¼YWD\V�h\GWOHgFGWERouükDFVEHSUTGROuILKSaEz[\V1[\T;hFGROHIQSUEQGRenGRVXI�~0DFOz[`V1��T;h`[\VYRGNü�[\^aYWA^aAEuYWD\AOQEHG\où EHSUVg�ILK;SUEqILK;GWDFOHGWe j�G�YW[\V¶G�xiILGRVT�D\AO$ILGWEvILE1[ f DlhFG�ILD�en[\VXb�enD\OQG"EHGWOQSaGWERo£×2DFOG�x;[\erC^aG\dïJð /�79Æ|/i* %v�q(B8=< 7 ï / Á 8('l?uÍ SahFGRV�[yEHGW»�AGWV;YWGyÎ ¿ À Î ´ À Î µ À �:�:�;jzS{ILK

^USUe¼:ÇzÈ9999 Î ¼ æ ¿Î ¼

9999 ¨ â �� ~ â í°��ILKGRV ÈмlÑ ¿ Î ¼YWD\V�h\GWOHgFGWE�[ f EQD\^UA;IQGW^abq[\VT�Sa~ â î®��IQKGWV ÈмlÑ ¿ Î ¼TS{hFGWOHgFGWERo� ~�ILKGNOQ[`ILSaD$D\~�ILKG}[ f EQDF^aA;ILG]h\[`^UAGRE�D\~�EQAYRYWGREQEQS{hFGrILGWOHenE�SUE�^UGREQE�IQK2[\V�DFVGNIQKGWV�IQKGyEQAeD\~zILK;Gm[ f EQDF^aA;ILGnh\[`^UAGREyYWDFVXhFGROQgFGRE f b�ILKG�COHGRhiSUDFA;E24�[`IQSUD"p�GWEvIrpuKGWDFOHGWe1dPILK2[`I}SaEWdPILK;GEQGROQSaGWE�SUEz[ f EHDF^UABILGW^{bqYRDFVXhFGWOHgFGWVXIWo�puKSUEuh\GWOQEHSUDFVmD\~�ILKG�4�[|ILSUDyp�GWEvI�[`^U^UDljzEuAEuIQDnTGW[\^�jzSaIQKEQGR»�AGWVYRGWE�SaVXhFDF^{hBSaVg f D\ILK�C�DFEQS{ILSah\G�[\VT1VGWgF[`ILS{hFG�EQAeren[\VTERo×.SUV;TSUVg,[,EQGROQSaGWEZIQK2[`IqSaEZYRDFVTS{ILSUD\V2[\^U^{b�YWD\V�h\GWOHgFGWVXI�SUEnVD\IZEQAYwK¶[\V�GW[\EHb�COQD f ^UGRemo¡�GqeNAEvIy�2VT�["EQGROQSaGWEJIQK2[`IyYRDFVXhFGWOHgFGWE f GWY:[`AEQG$D\~zYW[\VYRGW^U^U[`ILSaDFV f GRIcj�GWGRV�C�DFEHSaILS{hFG$[`VT

ß «

Page 25: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

VGRgX[`ILS{hFG�IQGWOHenE$[`VT�TSah\GWOHgFGWEqjzKGRV�ILKG"IQGWOQerE$[\OHG"en[\TG�[\^a^�C�DFEHSaILS{hFG`o�p4D�YRDFVEHIQOQAY�IEQA;YwK"[}EQGWOHSUGREWd2SaIzSaEzOQG:[`EQDFV2[ f ^UG�IQDnEvIw[\OvI�jzS{ILK1[}EQGWOHSUGREzIQK2[`I¹�HAEvI f [\OQGR^ab$TS{hFGROQgFGREWd2^USa�\G�ILK;GK2[\OHenD\VSUY�EHGWOHSUGWERoükDFVEHSUTGROzIQKG�EQGROQSUGRE ÈмlÑ ¿ ± è �=² ¼ æ ¿Û ¨ç� è �ß ¯ �¬ è �« ¯¶�:�:�R�å)GRI ¾ Ò ¨ ÒмlÑ ¿ ± è �=²c¼ æ ¿ÛTGRVD\ILGuIQKGzC2[\OHIQS_[\^BEQAerEWo 1 GWYW[\AEQGzILK;G�EHSUgFV;EPD\~>IQKGuILGROQerEPSUVNILK;G�EHGWOHSUGWE�[\^{ILGROQV2[`IQGuSUVyEHSUgFV�dj�G]��VDlj°IQK2[`I ¾ ¿ î ¾ ´ À ¾ ´ í ¾ µ À ¾�� î ¾>� À �W�:�W�¡�GN[\^aEQDZ��VDlj�ILK[`I�IQKG]IQGWOQerE�[`OQGJT;GWYWOHG:[\EHSUVg;d>ILK[`I�SaEWd�ì Î ¼ æ ¿ ì9í ì Î ¼ ìB~0D\O�[\^a^�Û4o���TTSaVgÎ Ò.æ ¿ ILD ¾ Ò d;gFSah\GWE�[ f G�IcjkGRGWV ¾ Ò [\V;T ¾ ÒVs ¿ puK[`I�SUEWdj�G�K2[:hFG¾ ¿ î ¾ µ î ¾>� î°�:�W��[`VT ¾ ´ í ¾�� í ¾>� í°�:�W�:�§�GRVYWG`d�IQKG"EQGR»�AGWVYRG ¾ ´ í ¾�� í ¾>� í �W�:��~0D\OQerE$[\V�SUV;YWOQGW[\EQSaVg�EHGW»�AGWV;YWG�jzKSUYwK�SaEf DFAVTGRT f b ¾ ¿ d�KGRVYWGNS{I�eJAEHI�K2[:hFG}[r^USUerSaIk�zo�puKGNC[\OHIQS_[\^�EQA;enE ¾ ¿ î ¾ µ î ¾>� îç�:�:�~0DFOHeÊ[$TGWYROQGW[\EQSaVgmEHGW»�AGRVYWGrjzKSUYwK�SUE f DFAV;TGWT f GR^UDlj f b ¾ ´ puKSaE�EQGW»�AGRVYWGreNAEvI][\^UEHDK2[:hFG][}^USUerSaIWdi[\VT f GRY:[\AEHG1ì ¾ Ò.æ ¿ è ¾ Ò ì`¨ ì Î Ò.æ ¿ ìXIQGWVTEzILDr½WGROQDn[\E�êÊILGWV;TE�IQDnSaV;�2VS{IcbFdILK;G�^aSUerSaIkD\~4IQKG�GRhFGRV�C2[`OHILSU[\^�EHAenE�[\V;TmILK;G�^aSUerSaI�D\~.IQKG�DBT;TmC2[`OHILSU[\^�EHAenEzeNA;EHI f G�ILK;GEL[`enGr±0EQGWG]×.SagFAOQG:²�o2§�GWVYRG�IQKG�EQGROQSUGRE ÈмlÑ ¿ ± è �=²c¼ æ ¿ÛYWD\V�h\GWOHgFGWERo

¡,GzYW[\VnG�hFGRVqEL[:b}[�^US{IQIL^aGkerDFOQG`oP@BSaVYWGuIQKGzC2[\OHIQS_[\^BEQAerE�[\^aIQGWOQV[`ILGz[\OHDFAVTyIQKGWSaOP^USUerSaIWdj�G]gFG�IuILK2[|I ��î ÈмlÑ ¿ ± è �=² ¼ æ ¿Û î°� è �ß ¨ �ßß �

Page 26: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

[\VT �ß í ÈмlÑ ¿ ± è �=² ¼ æ ¿Û í�� è �ß ¯ �¬ ¨ �s �puKG�EHAe£D`~�IQKG�EHGWOQSaGWEuSaE�[\^{j�[:biE f GRIcj�GWGWV�ILK;G�C2[\OvILS_[`^>EHAenE ¾ Ò [\VT ¾ Ò4æ ¿ ~0DFOu[\VXbZê�dEHDj�G]YW[\V�GWEHIQSUen[`ILG�IQKG�h`[\^UAG�D`~4IQKG�EQGWOHSUGRE�[\EuYR^UDFEHGW^{b$[\Euj�G]^aSU�\G`o¡,G�[\^aOQGW[\T;bq��VDlj®IQK2[`IuIQKG]K[\OQerDFVSaY�EQGROQSUGREÈмlÑ ¿ �ÛTS{hFGWOHgFGWERd9KGWVYRG ÈмlÑ ¿ ± è �=²c¼ æ ¿ÛSUEz[`VmG·x[`enC^aG�D\~.[}EHGWOHSUGWEuIQK2[`I�YWDFVXh\GWOQg\GWE�YWDFVT;SaILSaDFV2[\^a^ab\opuKG][\OQgFA;enGRV�I�[ f DlhFG�YW[\V f G�ILAOHVGWT1SUVXIQDn[}erDFOQG�g\GWVGROL[\^�IQKGWDFOHGWe¤[\Eu~0DF^a^UDljzEW¸ïJð /�79Æ|/i* %H!$-08l/�Æ|#�(;8=< #�Ô�Þ�Ã�* ï / Á 8('=? @BACC�DFEHGnILK;GZEQGR»�AGWVYRG�Î ¿ À Î ´ À Î µ À �:�:�)EQ[`ILSaEH�2GREILK;G�~0D\^U^UDljzSaVgNYWD\VTSaIQSUDFV;E� ì Î ¼ æ ¿ ì�)�ì Î ¼ ìX~0DFOz[\^a^�Û.dIQK2[`IzSUERd;ILK;G�EHGW»�AGRVYWGND`~¹[ f EQDF^aA;ILG�h`[\^aAGWEzSUEuT;GWYWOHG:[\EHSUVg;d� IQKG�EQSagFVE�D`~.Î ¼ [\VT�Î ¼ æ ¿ [\OHG�TSa�>GWOHGWVXI�~0DFOk[\^U^�Û.dBILK2[`I�SUEWdiILKG�EQSUg\VE�D\~�ILK;G�ILGROQerE�D\~IQKG�EQGW»�AGRVYWGJ[\^aIQGWOHV2[`ILG f GRIcj�GWGRV"C^aAEz[\VT1enSaV�AEWdILK;GWV ÈмlÑ ¿ Î ¼YWD\V�h\GWOHgFGWERopuKG�[\OHgFAerGWVXIPhFGROQSa~�biSaVg�ILKSaE¹ILKGRDFOQGRe�SUE¹IQKGzEL[\erGz[\E¹ILK[`I�gFSah\GWV}~0DFO¹ILK;GuG�x;[\erC^UGz[ f DlhFG`o%�')/�Æ|ÄX< Á / Á ?×2DFO�G:[\YwKqD`~�ILKG�~0DF^U^aDljzSUVg�EQGROQSaGWE�diTG�ILGWOHenSaVG�Sa~)S{I�YRDFVXhFGROQgFGREuDFO�T;SahFGROQgFGREWo � ~)S{IkYRDFVXhFGWOHgFGWERdgFS{hFGm[\EneJAYwK¢SUVB~0DFOQen[`ILSaDFV�[\ErbFDFA�Y:[\V¢[ f D\A;I}ILKG�h`[\^UAG1D\~�ILKGmEQGWOHSUGREWo � ~�S{InT;SahFGROQgFGREILD�SaV;�2VS{IcbFdPEHIL[`ILG1[\EyeJAYwK&[`ENbFDFA�Y:[\V�[ f DFA;INKDljó~ [\EvIyILKG�EQGR»�AGWVYRGmD`~�C2[`OHILSU[\^kEHAerEYWD\V�h\GWOHgFGWE�ILD}SUV;�VSaIcb\o�Fo Èм:Ñ ¿ �|¼ s,�« ¼ß o ÈмlÑ s � �ß ¼

ß s

Page 27: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

¬;o ÈмlÑ ¿�� ß ¼¬ ¼ s ¿«o Èм:Ñ ¿ �ß Ûr¯6��Bo ÈмlÑ ¿ �Û ß ¼s;o ÈмlÑ ¿ Û â ¼ s ¿ ~0DFO â [}YRDFVEvIw[\VXI� o Èм:Ñ ¿ â ¼ æ ¿Ûn¯6�¦;o ÈмlÑ ¿ �|¼Û ¼�;o Èм:Ñ ¿ ^aV)± Û�²Û ´�lt;o ÈмlÑ ´ �Û�^UV)±0Û)²�F�Fo Èм:Ñ ´ �ÛP± ^aV�± Û�²Q² ´

� ß ozp�D}EQKDlj�YRDFVXhFGWOHgFGWV;YWG]IQKG54�[|ILSUDyp�GWEvI�D\V�IQKG�EQGROQSUGREÈм:Ñ ¿ Î ¼OHGW»�ASUOHGWENIQK2[`IJIQKGq^aSUerSaI�D\~uILK;GqEQGR»iA;GWVYRG�ì Î ¼ æ ¿ �\Î ¼ ì�G�xBSaEHILE}[`ENÛ&ILGRVTENIQD�SUV;�2V;SaIcb[\V;TnSaE�^aGWEHE�IQK2[\V1�Fo � E�ILKGROQG�[\VD\IQKGWO�[�jkGW[\�\GROkYWD\VTSaIQSUDFV}DFV}ILK;G�EHGW»�AGWV;YWG�D\~�OL[`IQSUDFEIQK2[`IzSUerC^aSUGWE�YWDFVXh\GWOQg\GWVYRGlý�puKSUV�q[ f D\A;IzILKG�EQA;eÈм:Ñ ¿ � ¼

ß �

Page 28: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

jzKGROQG � ¼ ¨�� ¿´�� Sa~4Û�SaEuGRh\GWV�d¿µ�� Sa~4Û�SaEuDBT;T�d ��l¬;oz��VD\ILK;GWOuj�[:b$IQDrIQGWEHI�S{~¹Î ¼�3 â ¼ SUE�ILDrYWD\enCABILG�Î ¿h� ¼¼ ILKG�Û IQK OHDBD`IuD\~PÎ ¼ o�ü�[\V1b\DFATGREQSagFV1[rYRDFVXhFGWOHgFGWV;YWG]IQGWEvI f [\EQGRT�DFV$ILK;SUEuSUT;G:[Xý�:«o�@BK;D=j�~0DFOz[\VXb$h`[\^UA;G�D\~¹ê�dILKG�EHGWOQSaGWE Èм:Ñ Ò �ÛTS{hFGROQgFGRE�o�=�Bo�@BK;D=j�ILK2[|I�S{~ Èм:Ñ ¿ Î ¼TS{hFGROQgFGREWdILK;GWV Èм:Ñ Ò Î ¼TS{hFGROQgFGREz~0D\Ou[\VXb�ê�oyx�[`�\G�[yEQSUerSU^U[\O�EHIw[|ILGWerGWVXIu~0DFOuYRDFVXhFGROQgFGRVYWG`o�± � oÙG\o{d2EQK;D=j�ILK2[|IúHD\V^abrILKG�IL[\SU^>D\~�ILKG�EHGWOHSUGWEuen[`IHILGWOHE�~0DFO�YRDFVXhFGWOHgFGWV;YWG][\VT�TSah\GWOHgFGWVYRG�»iA;GWEHIQSUDFV;EQûo�²�ls;o�þ�G�ILGWOHenSaVG�YRDFVXhFGWOHgFGWV;YWGNDFOuT;SahFGROQgFGRVYWG�~0DFO�IQKG�EQGWOHSUGREÈм:Ñ ¿ �Û,{~0DFO���í}zmí ß o%�')/�Æ|ÄX< Á / Á 8:7���7�#�3�/�Æ�Fo�@BA;CC>D\EQG Èм:Ñ ¿ Î ¼YRDFVXhFGWOHgFGWERouükDFV;EHILOHAYRI�[\V;D\ILKGROzEQGWOHSUGRE Èм:Ñ ¿ � ¼jzS{ILK�� ¼ î�Î ¼ ~0DFOz[\^a^�Û�jzKSUYwKm[\^UEHDyYWDFVXh\GWOQg\GWEWo�@BACC�DFEHGÈм:Ñ ¿ Î ¼

ß ¦

Page 29: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

TS{hFGROQgFGREWo�ükDFVEvILOQA;YRI�[\VD\IQKGWOzEQGROQSaGWE Èм:Ñ ¿ � ¼jzS{ILK�� ¼ î�Î ¼ ~0DFOz[\^a^�Û�jzKSUYwKm[\^UEHDyTSah\GWOHgFGWERoß o]± M.[\A^�§�[\^aenDFEL²�@iACC�DFEQG�DFV;G�T2[:bZb\DFA�K2[\CC�GWV$ILDJY:[`IQYwKm[Ng\SUgX[\VXIQSUY��`[`VgX[\OHDBDo ÷ DFA[`IHIw[\YwK�D\VGNGWV;T�D\~�[qOQA f;f GRO f [\VT"jzS{ILK"^aGWVg`ILK��NerSU^aG�IQDqILKGJ�`[`VgX[\OHDBD + E�^aGWg$[`VT�2OHen^{b}[`IHIw[\YwKZIQKG�D`ILKGROkGWV;TZILD�ILKG�gFOQDFA;VT�o�@BA;CC>D\EQG�[M;2G:[JK[\CC�GWVEkIQD f G�EHSaIHILSUV;gDFV�IQKG$OQA ff GRO f [\VT�[|INILKG$C�DFSaV�IJjzKGWOHG$SaINSaEy[`IHIw[\YwKGRT�ILD"ILK;G$gFOQDFA;VT�o � IyEQGRGWEIQKGJ�|[\VgF[\OQDiDq[\V;Tr�HAerCE�DFV;GJSUV;YwK�ILDljk[\OQT�S{I:d9^_[\V;TSUVgrDFV�IQKGNOHA ff GWO f [\VT�o�puKG�|[\VgF[\OQDiD�EQGRGWE$ILK;G�[\CCOHDX[\YwKSaVg�;2G:[�[\V;T��HAenC;E���enSa^UG�ILD�gFG�I1[ljk[:bFoº±Ø@BD�ILK;GOHA ff GWO f [\VT�VDlj�K2[`E�IQD\Iw[`^P^UGWV;g\ILK ß enSa^UGWE�[\VT�ILKG2;2G:[�SUE�Icj�D�SaVYwKGRE�~0OHDFe ILK;G�;xBGRTrGRVTyD\~2IQKGuOQA f;f GRO f [\VT�o­²]puK;Gc;2G:[}±0AVTG�ILGWOHOQGRT9²9�HAerCEP[\VD`ILKGRO¹SUVYwK]ILDlj�[`OQTIQKG��`[`VgX[\OHDBDd;[\gX[\SaV�^_[\VT;SUVgyDFV$IQKG�OQA ff GWO f [`VT�o�puKG��`[`VgX[\OHDBD��HAerCE�[\VD`ILKGROerSU^aGz[\VTqEHD]~0DFOvILK�oPþ�DiGWE�ILKG:;2GW[JY:[`IQYwKZILK;G��|[\VgX[`OQDiD�ý���CCOHDlxBSaeZ[`IQGW^{b}KDlj6eZ[\VXb�HAerCE�jzSa^U^�S{I�Iw[\�`Glý¬;ou¡�G��iV;D=j°ILK2[`I ÈмlÑáiâ ¼ ¨ �� è âEHDrjzKXb�SUEHV + I ÈмlÑá ± è �=² ¼ ¨ �ß ÀDFOuSaEuSaIwý«o � ~ Èм:Ñ ¿ Î ¼YRDFVXhFGWOHgFGWE�EQKDlj�ILK2[`Iz~0DFOz[`V�bU��î�tNIQKGWOHG�SaEz[\VX� EHAYwKmIQK2[`I

ì ÒмlÑ�� Î ¼ ì�í �~0DFOz[`V�b$ê îN�ÖoP¡6K2[`I�Y:[\V�bFD\AmEQ[lbqSa~ Èм:Ñ ¿ Î ¼TS{hFGROQgFGREwý

ß �

Page 30: ^U^aGWTyILKGÁ - BUmath.bu.edu/people/rockford/chapter3-127.pdf · gjc2[`ohilsu[\^ eha enel² ou¡ g]yw[\v a ehgjdfa;o v dljz^ugrt gfg]d\~¹ilk;g gfgrdfeng ilohsuyzeqgroqsugrek[`e

�Bo�þ�DiGWE�IQKG�YWDFereNA;IL[`ILS{hFG4^_[:j,K;DF^UT�~0DFO�SUV;�2V;SaILG4EQGROQSUGREwý1@BACC�DFEHG�IQKGPEQGR»�AGWVYRGkÎ ¿ À Î ´ À �:�:�SaE)[uOQG:[`OQOL[`VgFGWerGWVXI)D`~BIQKGPEQGROQSUGREP� À �!� ß À �!�\¬ À �:�:�­dlIQK2[`I�SUERd:~0DFO�GW[\YwK]Û4d|Î ¼ ¨à± è �=²B� æ ¿ �w�~0DFOuEHDFerG��Ê[\VT�~0DFOuGW[\YwKmÛ4d�± è �=² ¼ æ ¿ �`Ûm¨°Î � ~0DFOuG·x;[\YRIQ^ab$DFV;G5��oyxmA;EHIÈм:Ñ ¿ Î ¼YRDFVXhFGWOHgFGlý

¬Ft