sol_ch5_functions

sol_ch5_functions - .kdswhu 8 Ixqfwlrqv 814 41 Ixqfwlrqv...

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Fkdswhu 8 Ixqfwlrqv 814 41 +d, Wkh grpdlq ri i> G + i , @ i 5 > 4 > 3 > 4 > 5 j 9 @ [= Khqfh/ i lv qrw d ixqfwlrq rq [= +e, Wkh grpdlq ri G + i , @ i 6 > 5 > 4 > 3 > 4 > 5 j 9 @ [= Qrz + 5 > 6, 5 i dqg + 5 > 9, 5 i= Wkhuhiruh/ i + 5,@6 dqg vlpxowdqhrxvo| i + 5,@9 = Wkxv/ i lv qrw zhoo gh qhg1 Khqfh/ i lv qrw d ixqfwlrq1 +f, Wkh grpdlq ri G + i , @ i 6 > 5 > 4 > 3 > 4 > 5 j @ [= Pruhryhu/ iru dq| d 5 [ wkhuh duh qr wzr glvwlqfw hohphqwv e dqg f lq \ vxfk wkdw + d> e , 5 i dqg + d> f , 5 Khqfh/ i lv d ixqfwlrq1 +g, Wkh grpdlq ri G + i , @ i 6 > 4 > 3 > 5 j @ [= Pruhryhu/ iru dq| d 5 [ wkhuh duh qr wzr glvwlqfw hohphqwv e dqg f lq \ vxfk wkdw + d> e , 5 i dqg + d> f , 5 Khqfh/ i lv d ixqfwlrq1 +h, Wkh grpdlq ri G + i , @ [ @ wkh vhw ri doo lqwhjhuv1 Pruhryhu/ iru dq| d 5 [ wkhuh duh qr wzr glvwlqfw hohphqwv e dqg f lq \ vxfk wkdw + d> e , 5 i dqg + d> f , 5 Khqfh/ i lv d ixqfwlrq1 +i, Wkh grpdlq ri G + i , @ [ @ wkh vhw ri doo lqwhjhuv1 Pruhryhu/ iru dq| d 5 [ wkhuh duh qr wzr glvwlqfw hohphqwv e dqg f lq \ vxfk wkdw + d> e , 5 i dqg + d> f , 5 Khqfh/ i lv d ixqfwlrq1 +j, Wkh grpdlq ri G + i , @ [ @ wkh vhw ri doo udwlrqdo qxpehuv1 Qrz 5 7 @ 7 ; >i + 5 7 ,@9 dqg i + 5 7 , @ i + 7 ; ,@45 = Wkdw lv/ i + 5 7 dqg i + 5 7 = Wkxv/ i lv qrw zhoo gh qhg1 Khqfh/ i lv qrw d ixqfwlrq1 61 Khuh i + { ,@ { 5 . { .4 iru doo { 5G + i U 1Q rz / { 5 . { .4@+ { . 4 5 , 5 . 6 7 1 Iru doo { 5 U > zh kdyh + { . 4 5 , 5 3 1 Wkhuhiruh/ { 5 . { { . 4 5 , 5 . 6 7 6 7 1 Khqfh/ wkh udqjh ri i lv doo wkrvh uhdo qxpehuv wkdw duh juhdwhu ru htxdo wr 6 7 1 81 +Uhfdoo wkdw p +prg q , lv wkh uhpdlqghu zkhq p lv glylghg e| q , 1 Qrz 6 5+prg :,@9 > 6 6+prg :,@5 > 6 7+prg :,@8 > 6 8+prg :,@4 > 6 9+prg :,@7 / 6 :+prg :,@3 > dqg 6 ;+prg :,@6 = Khqfh/ i @ i +5 > 9, > +6 > 5, > +7 > 8, > +8 > 4, > +9 > 7, > +: > 3, > +; > 6, j = Qrz/ iurp wkh uhsuhvhqwdwlrq ri i dv rughuhg sdluv zh vhh wkdw i lv rqh0rqh1 Dovr/ wkh udqjh ri Lp+ i , @ i 3 > 4 > 5 > 6 > 7 > 8 > 9 j @ \= Wkhuhiruh/ i lv rqwr \ 1 :1 +Uhfdoo wkdw p +prg q , lv wkh uhpdlqghu zkhq p lv glylghg e| q= , Qrz +5 6,.6+prg 8,@7 > +5 7,.6+prg 8,@4 > +5 8,.6+prg 8,@6 > +5 9,.6+prg 8,@3 / +5 :,.6+prg 8,@5 > dqg +5 ;,.6+prg 8,@7 = Khqfh/ i @ i +6 > 7, > +7 > 4, > +8 > 6, > +9 > 3, > +: > 5, > +; > 7, j = Qrz 6 9 @; dqg i +6,@7@ i +;, = Wkxv/ i lv qrw rqh0rqh1 Dovr wkh udqjh ri Lp+ i , @ i 3 > 4 > 5 > 6 > 7 j @ Khqfh/ i lv rqwr <1 +d, Khuh/ 3 9 @ 4 dqg i +3,@3@ i + 4, = Wkxv/ i lv qrw rqh0rqh1 Qrz/ ohw q 5 ] 1 Vxssrvh wkdw q 3 1 Wkhq wkhuh h{lvwv q 5 ] vxfk wkdw i + q , @ q= Qh{w vxssrvh wkdw q? 3 1 Wkhq q 4 5 ] vxfk wkdw i + q 4, @ q 1 Khqfh/ i lv rqwr ] 1 +e, Khuh/ i +3 > 4,@4@ i +4 > 3, exw +3 > 4, 9 @ +4 > 3, = Khqfh/ i lv qrw rqh0rqh1 Ohw q 5 ] = Wkhq q @ q .3@ i + q> 3, = Khqfh/ i lv rqwr ] 1 96
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97 81 Ixqfwlrqv 441 +d, Wr vkrz wkdw i lv rqh0rqh/ ohw d> e 5 U dqg i + d ,@ i + e , = Qrz i + d i + e , lpsolhv wkdw d s 5@ e s 5 1 Wklv lpsolhv wkdw d @ e 1 Khqfh/ i lv rqh0rqh1 Ohw d 5 U 1 Wkhq wkhuh h{lvwv d I 5 5 U vxfk wkdw i + d I 5 d I 5 s d= Khqfh/ i lv rqwr U 1 Frqvhtxhqwo|/ i lv d rqh0wr0rqh fruuhvsrqghqfh1 +e, Wr vkrz wkdw i lv rqh0rqh/ ohw d> e 5 U dqg i + d , @ i + e , = Qrz i + d , @ i + e , lpsolhv wkdw 5 d 4 6 @ 5 e 4 6 = Wklv lpsolhv wkdw d @ e 1 Khqfh/ i lv rqh0rqh1 Ohw d 5 U 1 Wkhq wkhuh h{lvwv 6 d .4 5 5 U vxfk wkdw i + 6 d .4 5 5+ 6 d .4 5 , 4 6 @ d= Khqfh/ i lv rqwr U 1 Frqvhtxhqwo|/ i lv d rqh0wr0rqh fruuhvsrqghqfh1 +f, Wr vkrz wkdw i lv rqh0rqh/ ohw d> e 5 U dqg i + d i + e , = Qrz i + d i + e , lpsolhv wkdw d 6 4@ e 6 4 1 Wklv lpsolhv wkdw d @ e 1 Khqfh/ i lv rqh0rqh1 Djdlq ohw d 5 U 1 Wkhq wkhuh h{lvwv + d .4, " $ 5 U vxfk wkdw i ++ d " $ , @ ++ d " $ , 6 d= Khqfh/ i lv rqwr U 1 Frqvhtxhqwo|/ i lv d rqh0wr0rqh fruuhvsrqghqfh1 461 Khuh i + { , @ { 5 9 { iru doo { 5G + i , @ U 1Q rz / { 5 9 { @ + { 6, 5 <
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This note was uploaded on 11/11/2009 for the course CSE 2353 taught by Professor Bhanukapoor during the Spring '09 term at SMU.

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sol_ch5_functions - .kdswhu 8 Ixqfwlrqv 814 41 Ixqfwlrqv...

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