(SNLIYH00 (ZZLZZTLU[Z *OHW[LY! -µÕ>Ài ,Ì ÕVÌà >«ÌiÀÊx\Ê-µÕ>ÀiÊ,ÌÊÕVÌà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«ÌiÀÊx\Ê-µÕ>ÀiÊ,ÌÊÕVÌà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¶W\RX³VHHIRUHYHU"´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¿UVWKDYHWREHDWDFHUWDLQKHLJKWDQGWKHQ¿QG WKHGLVWDQFHWRDQREMHFW7KHKHLJKWLVWKHLQGHSHQGHQW YDULDEOHVLQFHWKHGLVWDQFH\RXVHHLVGHSHQGHQWRQKRZ KLJK\RXDUH 7KHFDOFXODWRUJUDSKLVVKRZQEHORZ >«ÌiÀÊx\Ê-µÕ>ÀiÊ,ÌÊÕVÌà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¿QG DQHTXDWLRQRIWKHIRUP Y A X LVPDGH6LQFHWKH GHSHQGHQWYDULDEOHLVGLVWDQFHGDQGWKHLQGHSHQGHQW YDULDEOHLVKHLJKWKWKHHTXDWLRQZLOOEHRIWKHIRUP D A H 7KHSRLQWLVXVHGWRGHWHUPLQHD SRVVLEOHYDOXHIRUD D A H A A z A $GGLWLRQDO$OJHEUD,,7(.6 $)RXQGDWLRQVIRU IXQFWLRQV7KHVWXGHQWXVHV SURSHUWLHVDQGDWWULEXWHV RIIXQFWLRQVDQGDSSOLHV IXQFWLRQVWRSUREOHP VLWXDWLRQV 7KHVWXGHQWLVH[SHFWHGWR $LGHQWLI\WKH PDWKHPDWLFDOGRPDLQV DQGUDQJHVRIIXQFWLRQV DQGGHWHUPLQH UHDVRQDEOHGRPDLQ DQGUDQJHYDOXHVIRU FRQWLQXRXVDQGGLVFUHWH VLWXDWLRQV %FROOHFWDQGRUJDQL]H GDWDPDNHDQGLQWHUSUHW VFDWWHUSORWV¿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«ÌiÀÊx\Ê-µÕ>ÀiÊ,ÌÊÕVÌà ) #AN 3EE &OREVER 4EACHER .OTES 5V[LZ &KHFNLQJE\ORRNLQJDWDJUDSKRI D H ZLWKWKH VFDWWHUSORWVKRZVDUHDVRQDEOH¿W RI[ I [ [ DQGWKH UHFLSURFDORI[I[ [ $4XDGUDWLFDQG VTXDUHURRWIXQFWLRQV7KH VWXGHQWIRUPXODWHVHTXDWLRQV DQGLQHTXDOLWLHVEDVHGRQ VTXDUHURRWIXQFWLRQVXVHV DYDULHW\RIPHWKRGVWR VROYHWKHPDQGDQDO\]HV WKHVROXWLRQVLQWHUPVRIWKH VLWXDWLRQ 7KHVWXGHQWLVH[SHFWHGWR (GHWHUPLQHVROXWLRQVRI VTXDUHURRWLQHTXDOLWLHV XVLQJJUDSKVDQGWDEOHV D D KM b H H r M <RXZRXOGKDYHWREHDWOHDVWPLQWKHDLU ([WHQVLRQ4XHVWLRQV % 'RHVWKHGRPDLQDQGUDQJHWKDWLVDSSURSULDWHIRUD URFNHWODXQFKKDYHWKHVDPHYDOXHVDVWKHGRPDLQDQG UDQJHRIWKHIXQFWLRQ\RXXVHGWRPRGHOWKHEDOORRQULGH" 1R:LWKDURFNHW\RXUKHLJKWFRXOGJHWH[FHSWLRQDOO\KLJK EXW\RXUYLHZZRXOGEHRIKDOIRIWKHHDUWK¶VVXUIDFHDQG JRLQJKLJKHUZRXOGQRWDOORZ\RXWRVHHDQ\PRUHRIWKH HDUWK¶VVXUIDFH % ([SUHVV\RXUPRGHODVDIXQFWLRQRIGLVWDQFHQRWKHLJKW % 8VLQJWKHGLDJUDPGHYHORSDPRGHOIRUWKLVHYHQWLQ WHUPVRIK KHLJKWLQPHWHUVG GLVWDQFH\RXFDQVHHLQ NLORPHWHUVDQGU UDGLXVRIWKHHDUWKLQNLORPHWHUV/HW U NP >«ÌiÀÊx\Ê-µÕ>ÀiÊ,ÌÊÕVÌà ) #AN 3EE &OREVER 4EACHER .OTES ¥ ´ § ¶ D H H % 8VHWKHPRGHOWKDWZDVMXVWGHYHORSHGWR¿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«ÌiÀÊx\Ê-µÕ>ÀiÊ,ÌÊÕVÌà ) #AN 3EE &OREVER 4EACHER .OTES GRHVWKHGLIIHUHQFHLQHVWLPDWLRQXVLQJWKHQHZRQHFRPHIURP" 7KHRULJLQDOPRGHO D H D H VKRXOGJLYHHVWLPDWHVDOLWWOH H ODUJHUEHFDXVHWKHDPRXQWEHLQJDGGHGE\WKH WHUPLQWKHQHZHTXDWLRQLV QHJOLJLEOHDQGWKHKLVOHVVWKDQK >«ÌiÀÊx\Ê-µÕ>ÀiÊ,ÌÊÕVÌà ) #AN 3EE &OREVER 3TUDENT 7ORK >«ÌiÀÊx\Ê-µÕ>ÀiÊ,ÌÊÕVÌà ) #AN 3EE &OREVER 3TUDENT 7ORK >«ÌiÀÊx\Ê-µÕ>ÀiÊ,ÌÊÕVÌà ,:DV*RLQJ+RZ)DVW" $FFLGHQWLQYHVWLJDWRUVXVHWKHUHODWLRQVKLS S D WRGHWHUPLQHWKH DSSUR[LPDWHVSHHGRIDFDUVPSKIURPDVNLGPDUNRIOHQJWKGIHHWWKDWLW OHDYHVGXULQJDQHPHUJHQF\VWRS7KLVIRUPXODDVVXPHVDGU\URDGVXUIDFH DQGDYHUDJHWLUHZHDU $SROLFHRI¿FHULQYHVWLJDWLQJDQDFFLGHQW¿QGVDVNLGPDUNIHHWORQJ $SSUR[LPDWHO\KRZIDVWZDVWKHFDUJRLQJZKHQWKHGULYHUDSSOLHGWKH EUDNHV" ,IDFDULVWUDYHOLQJDWPSKDQGWKHGULYHUDSSOLHVWKHEUDNHVLQDQ HPHUJHQF\VLWXDWLRQKRZPXFKGLVWDQFHGRHV\RXUPRGHOVD\LVUHTXLUHG IRUWKHFDUWRFRPHWRDFRPSOHWHVWRS" :KDWLVDUHDOLVWLFGRPDLQDQGUDQJHIRUWKLVVLWXDWLRQ" 'RHVGRXEOLQJWKHOHQJWKRIWKHVNLGGRXEOHWKHVSHHGWKHGULYHUZDV JRLQJ"-XVWLI\\RXUUHVSRQVHXVLQJWDEOHVV\PEROVDQGJUDSKV >«ÌiÀÊx\Ê-µÕ>ÀiÊ,ÌÊÕVÌà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ÀXHQFHUHDOLVWLFGRPDLQDQGUDQJHYDOXHV IRUWKHVLWXDWLRQ" :KDWLVLQYROYHGLQVWRSSLQJDFDU",VLWPRUHWKDQZKDWD VNLGPDUNVKRZV" &DQDFDUVWRSZLWKRXWOHDYLQJDVNLGPDUN" %HVLGHVVSHHGZKDWHOVHZRXOGFRQWULEXWHWRWKHOHQJWK RIDVNLGPDUN" 'LGWKHIXQFWLRQWKDWZDVJLYHQUHODWLQJVNLGPDUNWR VSHHGWDNHDQ\RWKHUFRQGLWLRQVLQWRDFFRXQW" :KDWPLJKW\RXFKDQJHDERXWWKHHTXDWLRQWRWDNHVRPH RIWKHVHRWKHUHOHPHQWVLQWRDFFRXQW" 6DPSOH6ROXWLRQV 7KHJLYHQYDOXHRIIHHWLVWKHOHQJWKRIDVNLGPDUN G6XEVWLWXWHIRUGLQWKHIRUPXOD S D S S z MPH 7KHJLYHQYDOXHRIPSKLVWKHDSSUR[LPDWHVSHHGRI WKHFDUV6XEVWLWXWHIRUVLQWKHIRUPXODDQGVROYHIRUG S D D D D z FEET 7KHIXQFWLRQKDVDGRPDLQDQGDUDQJHRIDOOSRVLWLYH UHDOQXPEHUV7KHVLWXDWLRQGLFWDWHVWKDWWKHUHLVVRPH UHDOLVWLFPD[LPXPOHQJWKDQGVSHHG >«ÌiÀÊx\Ê-µÕ>ÀiÊ,ÌÊÕVÌà ) 7AS 'OING (OW &AST 4EACHER .OTES 'RXEOLQJWKHOHQJWKRIWKHVNLGPDUNGRHVQRWGRXEOHWKH VSHHG7KHWDEOHEHORZVKRZVWKDWZKHQWKHVNLGYDOXHLV GRXEOHGWKHVSHHGYDOXHLVQRWGRXEOHG)RUH[DPSOH EXWLVQRWWZLFH VNLG VSHHG IW IW IW IW IW PSK PSK PSK PSK PSK 4HE GRAPH ALSO SHOWS THE FUNCTION IS NOT LINEAR .OTICE THE LABELED POINTS 7KHVWXGHQWLVH[SHFWHGWR $LGHQWLI\WKH PDWKHPDWLFDOGRPDLQV DQGUDQJHVRIIXQFWLRQV DQGGHWHUPLQHUHDVRQDEOH GRPDLQDQGUDQJHYDOXHV IRUFRQWLQXRXVDQGGLVFUHWH VLWXDWLRQV $4XDGUDWLFDQGVTXDUH URRWIXQFWLRQV7KHVWXGHQW IRUPXODWHVHTXDWLRQVDQG LQHTXDOLWLHVEDVHGRQVTXDUH URRWIXQFWLRQVXVHVDYDULHW\ RIPHWKRGVWRVROYHWKHP DQGDQDO\]HVWKHVROXWLRQVLQ WHUPVRIWKHVLWXDWLRQ 7RVKRZWKLVV\PEROLFDOO\OHWV¶UHSUHVHQWWKHVSHHGZKHQWKH VNLGPDUNOHQJWKLVGRXEOHG V G V G V G 7KHVSHHGLVFKDQJHGE\DIDFWRURIDSSUR[LPDWHO\,W LVQRWGRXEOHG %XTENSION 1UESTIONS % &OR WHAT TYPE OF FUNCTION WOULD DOUBLING THE LENGTH OF THE XVALUE DOUBLE THE YVALUE ,INEAR FUNCTIONS OF THE FORM Y MX WOULD BE SUCH THAT IF X IS DOUBLED THEN Y WILL BE DOUBLED Y MX >«ÌiÀÊx\Ê-µÕ>ÀiÊ,ÌÊÕVÌà %UHODWHUHSUHVHQWDWLRQV RIVTXDUHURRWIXQFWLRQV VXFKDVDOJHEUDLFWDEXODU JUDSKLFDODQGYHUEDO GHVFULSWLRQV &RQQHFWLRQ WR7$.6 2EMHFWLYH7KHVWXGHQW ZLOOGHPRQVWUDWHDQ XQGHUVWDQGLQJRIWKH SURSHUWLHVDQGDWWULEXWHVRI IXQFWLRQV 2EMHFWLYH7KHVWXGHQW ZLOOGHPRQVWUDWHDQ XQGHUVWDQGLQJRITXDGUDWLF DQGRWKHUQRQOLQHDUIXQFWLRQV 2EMHFWLYH7KHVWXGHQW ZLOOGHPRQVWUDWHDQ XQGHUVWDQGLQJRIWKH PDWKHPDWLFDOSURFHVVHVDQG WRROVXVHGLQSUREOHPVROYLQJ ) 7AS 'OING (OW &AST 4EACHER .OTES % %XPRESS THE LENGTH OF A SKID MARK AS A FUNCTION OF SPEED S D S D D S .OTE THAT THE SQUARE ROOT FUNCTION AND THIS QUADRATIC FUNCTION ARE INVERSES OF EACH OTHER FOR POSITIVE VALUES OF S % )NVESTIGATORS lND THAT A CAR THAT CAUSED AN ACCIDENT LEFT A SKID MARK FEET LONG $AMAGE TO THE CAR REVEALS THAT IT WAS MOVING AT A RATE OF MPH WHEN IT HIT THE OTHER CAR (OW FAST WAS THE CAR GOING WHEN IT STARTED TO SKID )F THE SKIDDING CAR HAD NOT BEEN STOPPED BY HITTING ANOTHER CAR IT WOULD HAVE NEEDED ANOTHER z FEET TO SKID BEFORE STOPPING 4HE TOTAL SKID WOULD HAVE BEEN FEET ! SKID MARK THIS LONG WOULD IMPLY THE CAR WAS GOING z MPH WHEN IT STARTED TO SKID % 7HAT BESIDES THE ACTUAL BRAKING DISTANCE DO YOU THINK AFFECTS THE TOTAL DISTANCE THAT IT TAKES TO STOP A CAR IN AN EMERGENCY 7KHGLVWDQFHWKDWWKHFDUWUDYHOVZKLOHWKHGULYHULVUHDFWLQJWRWKHHPHUJHQF\VLWXDWLRQ QHHGVWREHDGGHGWRWKHEUDNLQJGLVWDQFHWRGHWHUPLQHWKHWRWDOVWRSSLQJGLVWDQFH 7KHUHLVDEXLOGLQJRQDFRUQHURIWKHKLJKZD\WKDWEORFNVDGULYHU¶VYLHZIRUIHHW,I WKHVSHHGOLPLWRQWKLVVWUHWFKRIKLJKZD\LVPSKGRHVDGULYHUKDYHHQRXJKWLPHWR VWRSLIWKHUHLVDFDUEURNHQGRZQLQWKHKLJKZD\IHHWDURXQGWKHFRUQHU" 'HWHUPLQHWKHGLVWDQFHWRVWRSDWPSKKRXUXVLQJWKHUXOH S D D D D z FEET $VVXPLQJWKHFDUFRXOGEHJLQWRVWRSWKHLQVWDQWWKHGULYHUVDZWKHEURNHQGRZQFDULW ZRXOGWDNHIHHWWRVWRSIURPDVSHHGRIPSK+RZHYHURWKHUIDFWRUVWKDWDUHQRW WDNHQLQWRDFFRXQWKHUHVXFKDVDGULYHU¶VUHDFWLRQWLPHZRXOGLQFUHDVHWKLVVWRSSLQJ GLVWDQFH7KHUHIRUHWKHUHLVSUREDEO\QRWHQRXJKURRP >«ÌiÀÊx\Ê-µÕ>ÀiÊ,ÌÊÕVÌà 7LF7RF 7KHUHLVDW\SHRIZDOOFORFNWKDWNHHSVWLPHE\XVLQJZHLJKWVJHDUVDQGD SHQGXOXP7KHSHQGXOXPVZLQJVEDFNDQGIRUWKWRWXUQDVHULHVRIZKHHOV$V WKHZKHHOVWXUQWKHKDQGVDGYDQFH7KHOHQJWKRIWKHSHQGXOXPGHWHUPLQHV KRZIDVWLWVZLQJV7KHIDVWHUWKHSHQGXOXPVZLQJVWKHIDVWHUWKHFORFNJRHV 6XSSRVH\RXUFORFNLVUXQQLQJWRRVORZO\$V\RXDWWHPSWWR¿[\RXUFORFN\RX WU\GLIIHUHQWOHQJWKSHQGXOXPV<RXFUHDWHWKHIROORZLQJWDEOHE\UHFRUGLQJZKDW \RXREVHUYH7KHSHULRGRIDSHQGXOXPLVWKHOHQJWKRIWLPHGXULQJZKLFKLW VZLQJVIURPRQHVLGHWRWKHRWKHUDQGEDFNDJDLQWRWKHVWDUWLQJSRVLWLRQ /HQJWKRISHQGXOXP FP FP FP FP FP FP 7LPHRIRQH FRPSOHWHVZLQJ VHF VHF VHF VHF VHF VHF &UHDWHDVFDWWHUSORWRIWKLVGDWDZLWKWKHOHQJWKRIWKHSHQGXOXPRQWKH [D[LVDQGWKHSHULRGLQVHFRQGVRQWKH\D[LV 'HVFULEHYHUEDOO\WKHIXQFWLRQDOUHODWLRQVKLSEHWZHHQWKHOHQJWKRID SHQGXOXPDQGLWVSHULRG ([SHULPHQWZLWK¿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«ÌiÀÊx\Ê-µÕ>ÀiÊ,ÌÊÕVÌà 4IC 4OC 4EACHER .OTES 5V[LZ 0DWHULDOV *UDSKLQJFDOFXODWRU $OJHEUD,,7(.6)RFXV $4XDGUDWLFDQGVTXDUH URRWIXQFWLRQV7KHVWXGHQW IRUPXODWHVHTXDWLRQVDQG LQHTXDOLWLHVEDVHGRQVTXDUH URRWIXQFWLRQVXVHVDYDULHW\ RIPHWKRGVWRVROYHWKHPDQG DQDO\]HVWKHVROXWLRQVLQWHUPV RIWKHVLWXDWLRQ 7KHVWXGHQWLVH[SHFWHGWR 6FDIIROGLQJ4XHVWLRQV % :KHUHZRXOGGDWDFROOHFWLRQDQGDQDO\VLV¿WLQWKLV SUREOHP" % :RXOG\RXH[SHFWWKHSHULRGWREHDIXQFWLRQRIWKH SHQGXOXPOHQJWK" % 6XSSRVHWKHUHODWLRQVKLSZHUHQRWIXQFWLRQDO:KDWZRXOG WKDWPHDQDERXWWKHFORFNNHHSLQJWKHFRUUHFWWLPH" % 7KHIXQFWLRQWKDWPRGHOVWKLVVLWXDWLRQKDVDQLQ¿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¿WWKHGDWDSHUIHFWO\:KDWDUH VRPHUHDVRQVZK\WKLVZRXOGKDSSHQLILWLVWKHFRUUHFW PRGHO" % ,VWKHDQVZHU\RXIRXQGH[DFW" % ,I\RXGRQRWKDYHDQH[DFWDQVZHUZLOOWKHFORFNNHHS WKHH[DFWWLPH" 6DPSOH6ROXWLRQV 7KHVFDWWHUSORWLVFUHDWHGZLWKWKHKRUL]RQWDOD[LV UHSUHVHQWLQJOHQJWKLQFHQWLPHWHUVDQGWKHYHUWLFDOD[LV UHSUHVHQWLQJWLPHLQVHFRQGV 7KHWLPHLWWDNHVWRFRPSOHWHRQHVZLQJRIWKHSHQGXOXP LVGHSHQGHQWRQLWVOHQJWK$VWKHOHQJWKLQFUHDVHVWKH WLPHLQFUHDVHV >«ÌiÀÊx\Ê-µÕ>ÀiÊ,ÌÊÕVÌà 4IC 4OC 4EACHER .OTES %HFDXVHWKHGDWDSORWLVDQLQFUHDVLQJJUDSKWKDWLV FXUYHGGRZQWKHSDUHQWIXQFWLRQ Y X ZDVWULHG ,WGLGQRW¿WWKHIXQFWLRQEXWYDULRXVPXOWLSOHVZHUH WULHG Y X Y X Y X MODELS THE DATA Y CLOSELY X )URPWRWKHKHLJKWRIWKHFORFNIURPWKHÀRRU 7RXVHWKLVPRGHOZLWKWKHGDWDSORWWHGLQFHQWLPHWHUV WKH[YDOXHPXVWEHFRQYHUWHGWRPHWHUV X CM v Y P METER CM X 7KHVWXGHQWLVH[SHFWHGWR %FROOHFWDQGRUJDQL]H GDWDPDNHDQGLQWHUSUHW VFDWWHUSORWV¿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«ÌiÀÊx\Ê-µÕ>ÀiÊ,ÌÊÕVÌà 4IC 4OC 4EACHER .OTES X )URPWKHWDEOHIRUWKHPRGHOIXQFWLRQ Y P WKHYDOXHDWVHFRQGVLV FP 2QHFDQDOVRVROYHXVLQJWKHIRUPXOD P X X P X ¥ ´ § P ¶ ¥ ´ X z CM § P ¶ &RQVLGHUWKHRULJLQDOIXQFWLRQJLYHQZLWK[LQPHWHUV P P v >«ÌiÀÊx\Ê-µÕ>ÀiÊ,ÌÊÕVÌà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À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«ÌiÀÊx\Ê-µÕ>ÀiÊ,ÌÊÕVÌà 4IC 4OC 4EACHER .OTES % (DFKWLPHWKHSHQGXOXPPDNHVDFRPSOHWHVZLQJLWDGYDQFHVDJHDURQHWRRWK7KH JHDUKDVWHHWK:KHQWKHJHDUPDNHVRQHFRPSOHWHUHYROXWLRQWKHPLQXWHKDQG DGYDQFHVPLQXWH:KDWLVWKHOHQJWKRIWKHSHQGXOXPQHFHVVDU\IRUWKHFORFNWRNHHS DFFXUDWHWLPH" 7KHSHQGXOXPQHHGVWRPDNHFRPSOHWHWULSVDPLQXWH7KHSHULRGPXVWEH VHFRQGV ¥ X ´ P § ¶ v P X X ¥ ´ v § P ¶ ¥ ´ X v § P ¶ X z M OR CM % ([SUHVVWKHIXQFWLRQLQWKHIRUP Y A X P X P X z X >«ÌiÀÊx\Ê-µÕ>ÀiÊ,ÌÊÕVÌà (SNLIYH00 (ZZLZZTLU[Z *OHW[LY! Ý«iÌ>Ê>` }>ÀÌ V ÕVÌà >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà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«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà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«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà $ESERT "IGHORN 4EACHER .OTES 5V[LZ 5HIHUHQFHVXVHGLQWKLV VHFWLRQ +HQU\(0F&XWFKHQ 'HVHUW%LJKRUQ6KHHS KWWSELRORJ\XVJVJRYVW QRIUDPHUKWP 7H[DV3DUNVDQG:LOGOLIH 'HVHUW%LJKRUQVRQWKH 5LVHZZZELJJDPHKXQWQHW VHFWLRQV7H[DV'HVHUWB %LJKRUQVBRQBWKHB5LVHB KWPO 0DWHULDOV 6FDIIROGLQJ4XHVWLRQV :KDWDUHWKHLQGHSHQGHQWDQGGHSHQGHQWYDULDEOHVLQ WKLVVLWXDWLRQ" +RZFDQ\RXFRXQWWKH\HDUVWRPDNHWKHQXPEHU FRPSXWDWLRQHDVLHU" :KDWLVWKHLQLWLDOFRQGLWLRQLQHDFKPRGHO" :KDWW\SHRIHTXDWLRQPXVW\RXVROYHWR¿QGWKH EDVHV" +RZFDQ\RXXVHORJDULWKPVWRKHOS\RX¿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«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà $ESERT "IGHORN 4EACHER .OTES THE POPULATION IN 3 W z 4HE DATA POINTS ARE NOW AND 3UBSTITUTING IN THE GENERAL MODEL GIVES A .OW SUBSTITUTE IN 0 B T AND INTERCHANGE SIDES TO GET E 'GHWHUPLQHVROXWLRQV RIH[SRQHQWLDODQG ORJDULWKPLFHTXDWLRQV XVLQJJUDSKVWDEOHVDQG DOJHEUDLFPHWKRGV )DQDO\]HDVLWXDWLRQ PRGHOHGE\DQ H[SRQHQWLDOIXQFWLRQ IRUPXODWHDQHTXDWLRQRU LQHTXDOLW\DQGVROYHWKH SUREOHP E E E z 4HE MODEL FOR THE GROWING POPULATION IS 0 T 6LQFHWKHVHDUHH[SRQHQWLDOPRGHOVWKHPDWKHPDWLFDO GRPDLQIRUERWKPRGHOVLVWKHVHWRIDOOUHDOQXPEHUVDQG WKHUDQJHLVWKHVHWRIDOOSRVLWLYHUHDOQXPEHUV 7KHGRPDLQIRUWKHGHFUHDVLQJSRSXODWLRQPRGHOLVWKH VHWRILQWHJHUVIURPWRLQFOXVLYHFRUUHVSRQGLQJ WRWKH\HDUVIURPWR7KHUDQJHLVWKHVHW ^`ZKHUHLQDWDEOHRIYDOXHV ZHDUHURXQGLQJGRZQWRWKHQHDUHVWVKHHSLQWKHDQQXDO FRXQW 7KHGRPDLQIRUWKHLQFUHDVLQJSRSXODWLRQPRGHOLVWKH QRQQHJDWLYHLQWHJHUVZLWKW FRUUHVSRQGLQJWR 6LQFHWKHUHLQWURGXFWLRQSURMHFWFRQWLQXHVWKHUHDSSHDUV WREHQRXSSHUERXQGRQWKHGRPDLQ7KHUDQJHLVWKHVHW ^` 7KHGRPDLQDQGWKHUHIRUHWKHUDQJHZLOOEHUHVWULFWHG E\SUDFWLFDOFRQFHUQVVXFKDVVSDFHDQGDYDLODEOHIRRG DQGZDWHUIRUWKHVKHHS ,QWKHGHFUHDVLQJSRSXODWLRQPRGHOE WKH >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà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± )N THE GROWING POPULATION MODEL B THE ANNUAL PERCENTAGE INCREASE IS ABOUT BECAUSE 7RGHWHUPLQHKRZPDQ\\HDUVLWWDNHVWKHGHFUHDVLQJ SRSXODWLRQWRGURSWRDWPRVWVKHHSZHXVH WKHFDOFXODWRU¶VJUDSKRUWDEOHIXQFWLRQV/HW 9 X DQG 9 *UDSKWKHIXQFWLRQVDQG ¿QGWKHSRLQWRILQWHUVHFWLRQ 7KHJUDSKVKRZVWKDWLWWDNHVQHDUO\\HDUVIRUWKH SRSXODWLRQWRGHFUHDVHWRVKHHS7KHWDEOHVKRZV WKDWLQ\HDUVWKHSRSXODWLRQKDVGURSSHGWR VKHHSDQGLQ\HDUVLWKDVGURSSHGWR 7RVHHWKLVDOJHEUDLFDOO\ZHPXVWVROYHWKHHTXDWLRQ 7KHVWXGHQWLVH[SHFWHGWR W (GHWHUPLQHVROXWLRQVRI VTXDUHURRWLQHTXDOLWLHV XVLQJJUDSKVDQGWDEOHV W OQ W OQ W z 7RZDUGVWKHHQGRIVLQFH WKH VKHHSSRSXODWLRQZLOOEHDERXW)URPWKDWSRLQWRQLW ZLOOEHOHVVVLQFHWKHSRSXODWLRQLVGHFUHDVLQJ :HXVHWKHVDPHSURFHGXUHWRGHWHUPLQHZKHQWKH LQFUHDVLQJSRSXODWLRQZLOODJDLQUHDFKVKHHS/HW 9 X AND 9 >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà $ESERT "IGHORN 4EACHER .OTES &RQQHFWLRQ WR7$.6 ,WZLOOWDNH\HDUVDIWHUIRUWKHVKHHSSRSXODWLRQWR UHDFK 7RVKRZWKLVDOJHEUDLFDOO\ZHVROYHWKHHTXDWLRQ T T LN T LN T z 'XULQJVLQFH WKHSRSXODWLRQ VKRXOGEHFRPHDJDLQDQGWKHQFRQWLQXHWRLQFUHDVH 7KHPRGHOIRUWKHJURZLQJSRSXODWLRQVKRZVWKDWLQ WKHUHVKRXOGEHDERXWELJKRUQVKHHS7KHGDWDVDLG WKHUHZHUHRQO\7KHPRGHODVVXPHVWKDWDQQXDO JURZWKRFFXUVDWDFRQVWDQWUDWH7KLVDVVXPSWLRQLV SUREDEO\QRWUHDOLVWLFIRUWKLVVLWXDWLRQEHFDXVHDQXPEHU RIIDFWRUVFDQDIIHFWWKHVL]HRIWKHVKHHSSRSXODWLRQ² ZHDWKHUGLVHDVHSUHGDWRUVIRRGDQGZDWHUVXSSO\HWF >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà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¿WGDWDZHOO$QQXDOGDWDGHVFULEHVWKH SRSXODWLRQDVDSHUFHQWDJHRIWKHSUHYLRXV\HDU¶VSRSXODWLRQ7KLVLVVLPLODUWR LQWHUHVWHDUQHGLQWHUHVWSDLGZKLFKLVPRGHOHGE\ ! 0 p R T ZKHUHULVWKHSHUFHQWLQWHUHVWDQGWLVWKHQXPEHURI\HDUV ,QRXUIXQFWLRQV B p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«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà $ESERT "IGHORN 4EACHER .OTES $VULQFUHDVHVIURPWRWKHQXPEHURI\HDUVLWWDNHVWKHVKHHSSRSXODWLRQWR GRXEOHGHFUHDVHVIURP\HDUVWR\HDUV :HFDQDOVRXVHWKHFDOFXODWRU¶VWDEOHIXQFWLRQWRGHWHUPLQHGRXEOLQJWLPHV LN 9 /HW LN 8 ZKHUH; SHUFHQWDJHLQFUHDVHLQSRSXODWLRQVWDUWLQJZLWK; 7KHUHVXOWVDUHVKRZQEHORZ ,WZLOOWDNHEHWZHHQDQG\HDUVIRUWKHVKHHSSRSXODWLRQWRGRXEOHZLWK SRSXODWLRQLQFUHDVHUDWHVRIWR >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà &RPSDULQJDQ([SRQHQWLDO)XQFWLRQDQG,WV,QYHUVH 7ZRIULHQGV(PLO\DQG/RUUDLQHZHUHZRUNLQJRQDQDOJHEUDWDVNDVVLJQHGWR WKHLUWHDP7KH\ZHUHDUJXLQJDERXWWKHPHDQLQJRIWKHLQYHUVHRIDIXQFWLRQ 7KH\FRQVXOWHGDGLFWLRQDU\DQGIRXQGWKHIROORZLQJGH¿QLWLRQVIRULQYHUVH UHYHUVHGLQSRVLWLRQGLUHFWLRQRUWHQGHQF\ RSSRVLWHWRLQQDWXUHRUHIIHFW LQYHUWHGRUWXUQHGXSVLGHGRZQ +HUHLVWKHLUDVVLJQPHQW $QDO\]HDQGFRPSDUHWKHIXQFWLRQVEHORZ 9 LOGA X 9 A X 9 ¥ ´ §A¶ X 9 LOG X A LNX LNA LNX ¥ ´ LN §A¶ 7RKHOSLQWKHFRPSDULVRQVWKH\KDYHEHHQDVNHGWRFRPSOHWHWKUHHVHWVRI DFWLYLW\VKHHWVLQZKLFKDLVJLYHQWKUHHGLIIHUHQWYDOXHVDQGH $IWHUWKH\FRPSOHWHGWKHDFWLYLW\VKHHWV(PLO\FODLPHGWKDWWKHSDLUVRIIXQFWLRQV WKDWDUHLQYHUVHVRIHDFKRWKHUDUH 9 AND 9 9 AND 9 /RUUDLQHFODLPHGWKDWWKHSDLUVRIIXQFWLRQVWKDWDUHLQYHUVHVRIHDFKRWKHUDUH 9 AND 9 9 AND 9 >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà &RPSOHWHWKHLUDVVLJQPHQWWRGHWHUPLQHZKLFKVWXGHQWLVFRUUHFW &RPSDULQJDQ([SRQHQWLDO)XQFWLRQDQG,WV,QYHUVH6HW$ &UNCTION 'RAPH OR TABLE $OMAIN 2ANGE )NTERCEPTS 9 X 9 ¥ ´ § ¶ X 9 LOGX 9 LOG X >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà &RPSDULQJDQ([SRQHQWLDO)XQFWLRQDQG,WV,QYHUVH6HW$ &UNCTION )NCREASES OR DECREASES #ONCAVE UP OR DOWN PARENT FUNCTION 9 X ¥ ´ 9 § ¶ 4RANSFORMATION ON 9 X 9 LOGX 9 LOG X :KLFKSDLUVRIIXQFWLRQVDUHLQYHUVHVRIHDFKRWKHU" >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà &RPSDULQJDQ([SRQHQWLDO)XQFWLRQDQG,WV,QYHUVH6HW% &UNCTION 'RAPH OR TABLE $OMAIN 2ANGE )NTERCEPTS 9 X ¥ ´ 9 § ¶ X 9 ,OGX 9 LOG X >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà &RPSDULQJDQ([SRQHQWLDO)XQFWLRQDQG,WV,QYHUVH6HW% &UNCTION )NCREASES OR DECREASES #ONCAVE UP OR DOWN 9 X 9 4RANSFORMATION ON 9 PARENT FUNCTION ¥ ´ § ¶ X 9 ,OGX 9 LOG X :KLFKSDLUVRIIXQFWLRQVDUHLQYHUVHVRIHDFKRWKHU" >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà &RPSDULQJDQ([SRQHQWLDO)XQFWLRQDQG,WV,QYHUVH6HW& &UNCTION 'RAPH OR TABLE $OMAIN 2ANGE )NTERCEPTS 9 EX ¥ ´ 9 §E¶ X 9 LOG X E >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà &RPSDULQJDQ([SRQHQWLDO)XQFWLRQDQG,WV,QYHUVH6HW& &UNCTION )NCREASES OR DECREASES #ONCAVE UP OR DOWN 9 EX 4RANSFORMATION ON 9 PARENT FUNCTION ¥ ´ 9 §E¶ X 9 LOG X E :KLFKSDLUVRIIXQFWLRQVDUHLQYHUVHVRIHDFKRWKHU" >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà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±" :KDWLVWKHLQYHUVHRI\ [±" 'RHVHYHU\IXQFWLRQKDYHDQLQYHUVHWKDWLVDOVRD IXQFWLRQ" :KDWPXVWEHWUXHDERXWDIXQFWLRQIRULWWRKDYHDQ LQYHUVHWKDWLVDOVRDIXQFWLRQ" +RZFDQ\RXXVHWKHJUDSKRIDIXQFWLRQWR¿QGWKH JUDSKRILWVLQYHUVH" :KDWLVWUXHDERXWWKHGRPDLQDQGUDQJHRIWKH LQYHUVHRIDIXQFWLRQ" ,IWKHSRLQWLVRQWKHJUDSKRIDIXQFWLRQZKDW FRUUHVSRQGLQJSRLQWLVRQWKHJUDSKRILWVLQYHUVH" :LWKDQH[SRQHQWLDOIXQFWLRQZHLQSXWDQH[SRQHQW DQGRXWSXWDSRZHU:KDWLVWUXHDERXWDORJDULWKPLF IXQFWLRQ" 7KHVWXGHQWLVH[SHFWHGWR $GHYHORSWKHGH¿QLWLRQ RIORJDULWKPVE\H[SORULQJ DQGGHVFULELQJWKH UHODWLRQVKLSEHWZHHQ H[SRQHQWLDOIXQFWLRQVDQG WKHLULQYHUVHV >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà #OMPARING AN %XPO 4EACHER .OTES $GGLWLRQDO$OJHEUD,,7(.6 $$OJHEUDDQG JHRPHWU\7KHVWXGHQW FRQQHFWVDOJHEUDLFDQG JHRPHWULFUHSUHVHQWDWLRQVRI IXQFWLRQV 7KHVWXGHQWLVH[SHFWHGWR $LGHQWLI\DQGVNHWFK JUDSKVRISDUHQW IXQFWLRQVLQFOXGLQJOLQHDU I[ [TXDGUDWLFI[ [H[SRQHQWLDOI[ D[DQGORJDULWKPLF I[ ORJD[IXQFWLRQV DEVROXWHYDOXHRI[ I [ [ VTXDUHURRW RI[ I [ [ DQG UHFLSURFDORI[I[ [ 7KLVSDJHLQWHQWLRQDOO\OHIWEODQN &RQQHFWLRQ WR7$.6 2EMHFWLYH7KHVWXGHQW ZLOOGHVFULEHIXQFWLRQDO UHODWLRQVKLSVLQDYDULHW\RI ZD\V 2EMHFWLYH7KHVWXGHQW ZLOOGHPRQVWUDWHDQ XQGHUVWDQGLQJRIWKH SURSHUWLHVDQGDWWULEXWHVRI IXQFWLRQV 2EMHFWLYH7KHVWXGHQW ZLOOGHPRQVWUDWHDQ XQGHUVWDQGLQJRITXDGUDWLF DQGRWKHUQRQOLQHDU IXQFWLRQV >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà #OMPARING AN %XPO 4EACHER .OTES 6DPSOH6ROXWLRQV &RPSDULQJDQ([SRQHQWLDO)XQFWLRQDQG,WV,QYHUVH6HW$ &UNCTION 'RAPH OR TABLE $OMAIN 2ANGE )NTERCEPTS X CAN BE ANY REAL NUMBER Y CAN BE ANY POSITIVE REAL NUMBER XINTERCEPT NONE YINTERCEPT X CAN BE ANY REAL NUMBER Y CAN BE ANY POSITIVE REAL NUMBER XINTERCEPT NONE YINTERCEPT 9 LOGX X CAN BE ANY POSITIVE REAL NUMBER Y CAN BE ANY REAL NUMBER XINTERCEPT YINTERCEPT NONE 9 LOG X X CAN BE ANY POSTIVE REAL NUMBER Y CAN BE ANY REAL NUMBER XINTERCEPT YINTERCEPT NONE 9 X ¥ ´ 9 § ¶ X >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà #OMPARING AN %XPO 4EACHER .OTES &RPSDULQJDQ([SRQHQWLDO)XQFWLRQDQG,WV,QYHUVH6HW$ &UNCTION 9 )NCREASES X ¥ ´ 9 § ¶ )NCREASES OR DECREASES #ONCAVE UP OR DOWN 4RANSFORMATION ON 9 #ONCAVE UP PARENT FUNCTION X $ECREASES #ONCAVE UP 2EmECT 9 8 OVER YAXIS )NCREASES #ONCAVE DOWN 2EmECT 9 8 OVER Y X $ECREASES #ONCAVE UP 2EmECT 9 8 OVER YnAXIS 9 LOGX 9 LOG X 4HEN REmECT OVER Y X 7KHLQYHUVHIXQFWLRQSDLUVDUH<DQG<DQGDOVR<DQG<)RUHDFKSDLUWKHJUDSKRIRQH IXQFWLRQLVWKHUHÀHFWLRQRIWKHJUDSKRIWKHRWKHURYHUWKHOLQH\ [ >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà #OMPARING AN %XPO 4EACHER .OTES &RPSDULQJDQ([SRQHQWLDO)XQFWLRQDQG,WV,QYHUVH6HW% &UNCTION 'RAPH OR TABLE $OMAIN 2ANGE )NTERCEPTS X CAN BE ANY REAL NUMBER Y CAN BE ANY POSITIVE REAL NUMBER XINTERCEPT NONE YINTERCEPT X CAN BE ANY REAL NUMBER Y CAN BE ANY POSITIVE REAL NUMBER XINTERCEPT NONE YINTERCEPT 9 ,OGX X CAN BE ANY POSITIVE REAL NUMBER Y CAN BE ANY REAL NUMBER XINTERCEPT YINTERCEPT NONE 9 LOG X X CAN BE ANY POSTIVE REAL NUMBER Y CAN BE ANY POSITIVE REAL NUMBER XINTERCEPT YINTERCEPT NONE 9 X ¥ ´ 9 § ¶ X >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà #OMPARING AN %XPO 4EACHER .OTES &RPSDULQJDQ([SRQHQWLDO)XQFWLRQDQG,WV,QYHUVH6HW% &UNCTION 9 9 )NCREASES OR DECREASES )NCREASES X ¥ ´ § ¶ X #ONCAVE UP OR DOWN #ONCAVE UP 2ANGE VALUES INCREASE FASTER AND FASTER 4RANSFORMATION ON9 PARENT FUNCTION $ECREASES #ONCAVE UP 2ANGE VALUES INCREASE FASTER AND FASTER 2EmECT 9 OVER YnAXIS I±[ I[ )NCREASES #ONCAVE DOWN 2ANGE VALUES INCREASE MORE AND MORE SLOWLY 2EmECT 9 OVER Y X #ONCAVE UP 2ANGE VALUES INCREASE MORE AND MORE SLOWLY 2EmECT OVER \AXIS 4HEN REmECT OVER YX )F AB IS ON GRAPH OF 9 THEN BA IS ON GRAPH OF 9 9 ,OGX $ECREASES 9 LOG X )F AB IS ON GRAPH OF 9 THEN BA IS ON GRAPH OF 9 7KHLQYHUVHIXQFWLRQSDLUVDUH<DQG<DQGDOVR<DQG<)RUHDFKSDLUWKHLQSXWGRPDLQ YDOXHVDQGRXWSXWUDQJHYDOXHVRIRQHIXQFWLRQEHFRPHWKHRXWSXWYDOXHVDQGLQSXWYDOXHV RIWKHRWKHUUHVSHFWLYHO\ >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà #OMPARING AN %XPO 4EACHER .OTES &RPSDULQJDQ([SRQHQWLDO)XQFWLRQDQG,WV,QYHUVH6HW& &UNCTION X 9 LOG X E $OMAIN 2ANGE )NTERCEPTS X IS ANY Y IS ANY XINTERCEPT REAL NUMBER POSITIVE NONE REAL NUMBER YINTERCEPT 9 EX ¥ ´ 9 §E¶ 'RAPH OR TABLE X IS ANY Y IS ANY XINTERCEPT REAL NUMBER POSITIVE NONE REAL NUMBER YINTERCEPT X IS ANY Y IS ANY XINTERCEPT POSITIVE REAL NUMBER REAL NUMBER YINTERCEPT NONE X IS ANY Y IS ANY XINTERCEPT POSITIVE REAL NUMBER REAL NUMBER YINTERCEPT NONE >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà #OMPARING AN %XPO 4EACHER .OTES &RPSDULQJDQ([SRQHQWLDO)XQFWLRQDQG,WV,QYHUVH6HW& &UNCTION 9 E )NCREASES X ¥ ´ 9 §E¶ )NCREASES OR DECREASES #ONCAVE UP OR DOWN 4RANSFORMATION ON 9 #ONCAVE UP PARENT FUNCTION X $ECREASES #ONCAVE UP 2EmECT OVER 9 YAXIS )NCREASES #ONCAVE DOWN 2EmECT 9 OVER Y X $ECREASES #ONCAVE UP 2EmECT 9 OVER YAXIS 4HEN REmECT OVER Y X 9 LOG X E 7KHLQYHUVHIXQFWLRQSDLUVDUH<DQG<DQGDOVR<DQG<)RUHDFKSDLUWKHJUDSKRIRQH IXQFWLRQLVWKHUHÀHFWLRQRIWKHJUDSKRIWKHRWKHURYHUWKHOLQH\ [ )RUHDFKVHWWKHUHVXOWVDUHWKHVDPH7KHLQYHUVHIXQFWLRQSDLUVDUH<DQG<DQGDOVR< DQG</RUUDLQHDSSHDUVWREHFRUUHFW >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà #OMPARING AN %XPO 4EACHER .OTES ([WHQVLRQ4XHVWLRQV X ¥ ´ X :K\GR\RXVXSSRVHVRPHRQHPLJKWWKLQNWKDW F X DQG G X DUH § ¶ LQYHUVHIXQFWLRQV" /RWVRISHRSOHWKLQNRIDQG DVLQYHUVHVEHFDXVHWKH\DUHPXOWLSOLFDWLYH LQYHUVHVRU\RXLQYHUWWRJHW :KDWLVDFRPPRQVHQVHZD\RIGHVFULELQJWKHLQYHUVHRIDIXQFWLRQ" ,IDIXQFWLRQGHVFULEHVDVHTXHQFHRIRSHUDWLRQVRQDQLQSXWYDOXHWRJHWLWVRXWSXW YDOXHWKHLQYHUVHRIWKHIXQFWLRQXQGRHVWKRVHRSHUDWLRQVLQUHYHUVHRUGHU 'RHVHYHU\H[SRQHQWLDOIXQFWLRQKDYHDQLQYHUVHWKDWLVDIXQFWLRQ":K\RUZK\QRW" <HV$Q\H[SRQHQWLDOIXQFWLRQLVDRQHWRRQHIXQFWLRQ(DFKRXWSXWYDOXHFRPHV IURPH[DFWO\RQHLQSXWYDOXH,IDIXQFWLRQLVRQHWRRQHLWKDVDQLQYHUVHWKDWLVD IXQFWLRQ ,I F X A X WKHQI[ ORJD[:KDWFDQ\RXWHOOPHDERXWWKHGRPDLQDQGUDQJHRI F " 7KHGRPDLQRIIEHFRPHVWKHUDQJHRI F DQGWKHUDQJHRIIEHFRPHVWKHGRPDLQRI F )RUH[DPSOHZKDWKDSSHQVLIDELVRQWKHJUDSKRII":KDWKDSSHQVWR LQWHUFHSWV" ,IDELVRQWKHJUDSKRIIEDLVRQWKHJUDSKRII<LQWHUFHSWVRQWKHJUDSKRII EHFRPH[LQWHUFHSWVRQWKHJUDSKRIIDQGYLFHYHUVD :KDWGRHVDOOWKLVPHDQJHRPHWULFDOO\"+RZFDQ\RXXVHWUDQVIRUPDWLRQVWR¿QGWKH JUDSKRIWKHORJDULWKPLFIXQFWLRQWKDWLVWKHLQYHUVHRIDQH[SRQHQWLDOIXQFWLRQ" >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà #OMPARING AN %XPO 4EACHER .OTES *HRPHWULFDOO\\RXDUHVZLWFKLQJLQSXWDQGRXWSXWYDOXHVSRLQWE\SRLQW7RGRWKLV ZLWKWUDQVIRUPDWLRQV\RXUHÀHFWWKHJUDSKRIWKHH[SRQHQWLDOIXQFWLRQRYHUWKHOLQH \ [WRJHWWKHJUDSKRILWVLQYHUVHDORJDULWKPLFIXQFWLRQ 9 X 9 LOG X 9 ¥ ´ § ¶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«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà 3AVING -ONEY -AKING -ONEY 6XSSRVH\RXUHFHLYHIRUJUDGXDWLRQDJLIWRIIURP\RXUIDYRULWHUHODWLYH <RXDUHUHTXLUHGWRLQYHVWDWOHDVWRIWKHJLIWLQDQRZLWKGUDZDOVDYLQJV SURJUDPIRUDWOHDVWWZR\HDUV<RXKDYHGHVLJQHGWZRSODQVWRFRQVLGHU 3ODQ$)LUVW6DYLQJV%DQN)6%SD\VLQWHUHVWFRPSRXQGHGDQQXDOO\RQ VDYLQJVDFFRXQWV(PSOR\HH¶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«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà LQWHUHVWUDWHZRXOG(&8QHHGWRSD\IRU\RXWRKDYHWKHDPRXQW\RX FRXOGKDYHZLWK)6%LQ\HDUV" <RXVSHFXODWHDERXWPDNLQJQRDGGLWLRQDOGHSRVLWVDQGQR ZLWKGUDZDOVIURPWKHVDYLQJVDFFRXQWIRU\HDUV:KDWFRQWLQXRXV FRPSRXQGLQWHUHVWUDWHZRXOG(&8KDYHWRSD\VRWKDW\RXULQLWLDO GRXEOHVLQ\HDUV"+RZZRXOGWKDWFRPSDUHZLWKWKH DPRXQWLQ\RXUDFFRXQWDW)6%DIWHU\HDUV" >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà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¥ R´ ! 0 § N¶ NT ZKHUHQLVWKHQXPEHURIFRPSRXQGLQJSHULRGVSHU\HDU >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà 3AVING -ONEY 4EACHER .OTES ,QWKLVSUREOHPU Q DQGW :HQHHGWR GHWHUPLQH3VRWKDWZHEUHDNHYHQZLWK)6%7KHUHIRUH ZHQHHGWRVROYH ¥ ´ § ¶ 2XULQLWLDOGHSRVLWDW(&8ZRXOGQHHGWREHPRUH WKDQRXUGHSRVLWDW)6%IRUXVWRKDYHWKHVDPHDPRXQW LQ\HDUV 7KLVWLPHZHNQRZWKDWDW(&83 Q DQG W :HQHHGWRGHWHUPLQHWKHLQWHUHVWUDWHUVRWKDW ZHKDYHWKHVDPHDPRXQWDVZHZRXOGKDYHZLWK)6% 7KHUHIRUHZHQHHGWRVROYH R ´ ¥ § ¶ &RQQHFWLRQ WR7$.6 2EMHFWLYH7KHVWXGHQW ZLOOGHVFULEHIXQFWLRQDO UHODWLRQVKLSVLQDYDULHW\RI ZD\V 2EMHFWLYH7KHVWXGHQW ZLOOGHPRQVWUDWHDQ XQGHUVWDQGLQJRIWKH SURSHUWLHVDQGDWWULEXWHVRI IXQFWLRQV 2EMHFWLYH7KHVWXGHQW ZLOOGHPRQVWUDWHDQ XQGHUVWDQGLQJRITXDGUDWLF DQGRWKHUQRQOLQHDUIXQFWLRQV R ´ ¥ § ¶ H[SUHVVLRQVDQGWR WUDQVIRUPDQGVROYH HTXDWLRQV R R R 2EMHFWLYH7KHVWXGHQW ZLOOGHPRQVWUDWHDQ XQGHUVWDQGLQJRIWKH PDWKHPDWLFDOSURFHVVHVDQG WRROVXVHGLQSUREOHPVROYLQJ 7KHDQQXDOLQWHUHVWUDWHDW(&8ZRXOGQHHGWREH 1RZZHNQRZDW(&83 U DQG Q :HQHHGWRGHWHUPLQHWVRWKDWZHKDYH LQWKHVDYLQJVDFFRXQW7KHUHIRUHZHQHHGWRVROYH >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà 3AVING -ONEY 4EACHER .OTES ¥ ´ § ¶ T T T LN LN LN LN T z T ,WZRXOGWDNH \HDUVIRU\RXWRKDYHDVPXFKPRQH\DW(&8DV\RXZRXOGKDYHDW )6%DWWKHHQGRI\HDUV 7KHIXQFWLRQH[SUHVVLQJWKHDPRXQW$GROODUVLQDQDFFRXQWWKDWHDUQVLQWHUHVW FRQWLQXRXVO\LQWHUPVRIW\HDUVLV 3ODQ% ! 0E R T ZKHUHU WKHFRQWLQXRXVFRPSRXQGLQWHUHVWUDWH ,IU DQGW ZHQHHGWRVROYH 0 E E 0 0 E 0 2XUSULQFLSDODW(&8DJDLQZRXOGQHHGWREHWREUHDNHYHQZLWK)6%DIWHU \HDUVRIVDYLQJ 7KLVWLPHZHQHHGWRVROYH E R E R R LN R (&8¶VLQWHUHVWUDWHZRXOGQHHGWREH >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà 3AVING -ONEY 4EACHER .OTES )RU(&8ZHQHHGWRVROYHWKHHTXDWLRQ E R E R R LN LN R R %#5 WOULD HAVE TO PAY CONTINUOUS COMPOUND INTEREST !FTER YEARS AT &3" WE WOULD HAVE IN THE ACCOUNT ([WHQVLRQ4XHVWLRQV ,Q3ODQ$SUREOHPWKHIXQFWLRQUXOHVIRU)6%DQG(&8UHVSHFWLYHO\DUH X 9 X AND 9 7KHYDULDEOH[KDVEHHQXVHGWRUHSUHVHQW WLPHRQWKHJUDSKLQJFDOFXODWRU:KLFKJHQHUDWHVPRUHPRQH\"+RZGR\RXGHFLGH" 7KLVFDQEHVKRZQZLWKWDEOHV+HUHDUHWKHWDEOHVIRUWKHVHIXQFWLRQV )RU[OHVVWKDQWKH\YDOXHVDUHPRUHIRU<7KDWLVGXULQJWKH¿UVW\HDUVWKHUH LVPRUHPRQH\LQWKHDFFRXQWDW(&87KHDPRXQWVDUHDOPRVWHTXDODW[HTXDOV 7KH\EUHDNHYHQDW\HDUV)RU[YDOXHVJUHDWHUWKDQ<LVJUHDWHU$IWHUWKDW )6%LVWKHEHWWHUSODFHWRKDYH\RXUPRQH\ 7KHJUDSKDOVRVKRZVWKLVVROXWLRQ >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà 3AVING -ONEY 4EACHER .OTES !NOTHER WAY TO SEE THIS IS TO SOLVE THE EQUATION TO lND WHEN THE TWO FUNCTIONS HAVE THE SAME YVALUE X X ; = ; LOG X LOG X = LOG LOGX LOG LOG X LOG X LOG LOG X LOG X LOG X LOG LOG LOG X ;LOG LOG= LOG LOG X LOG LOG ;LOG LOG= )N ABOUT YEARS THE FUNCTIONS HAVE THE SAME AMOUNT % 7HAT KINDS OF EQUATIONS DID YOU HAVE TO SOLVE IN 0LAN ! 0ROBLEM HAD A LINEAR EQUATION ¥ ´ 0 FOR 0 § ¶ 0ROBLEM HAD AN EXPONENTIAL EQUATION R ´ ¥ § ¶ FOR R 0ROBLEM HAD EXPONENTIAL AND REQUIRED LOGARITHMS ¥ ´ § ¶ T FOR T % 3UPPOSE YOU ARE GIVEN THE VALUE OF RS IN THE CONTINUOUS COMPOUND INTEREST SITUATION 9 ! RS X (OW CAN YOU DETERMINE POSSIBLE VALUES OF RC IN THE COMPOUND INTEREST SITUATION THAT MAKE CONTINUOUS COMPOUND INTEREST THE BETTER DEAL >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà 3AVING -ONEY 4EACHER .OTES 7E WANT THE CONTINUOUSLY COMPOUNDED INTEREST AMOUNT EARNED TO BE GREATER THAN THE COMPOUND INTEREST AMOUNT !E RC X ! RS X E RC X RS X C E R RS RC LN RS #HOOSE RC SO THAT RC LN RS >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà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«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà ! 'RADUATION 0RESENT 4EACHER .OTES 5V[LZ 0DWHULDOV 6FDIIROGLQJ4XHVWLRQV ,ILQWHUHVWLVFRPSXWHGQWLPHVSHU\HDUZKDWZRXOG EHWKHIXQFWLRQUXOHH[SUHVVLQJWKHDPRXQWRIPRQH\ $GROODUVLQDVDYLQJVDFFRXQWDVDIXQFWLRQRIWKH QXPEHURI\HDUVWWKHPRQH\LVLQWKHDFFRXQW" *UDSKLQJFDOFXODWRU 1RWH3UHUHTXLVLWH H[SHULHQFHVIRUWKLVSUREOHP DUHWKDWVWXGHQWVZLOOKDYH EHHQH[SRVHGWRWKHLQWHUHVW IRUPXODV :KDWIXQFWLRQUXOHGR\RXXVHLILQWHUHVWLVFRPSXWHG FRQWLQXRXVO\" &RPSRXQGLQWHUHVW ¥ U ´ QW $ 3 ¦¦ µ § Q ¶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¥ R´ ! 0 § N¶ NT ZKHUH3LV\RXULQLWLDOGHSRVLWULVWKHDQQXDOLQWHUHVW UDWHDQGQLVWKHQXPEHURIFRPSRXQGLQJSHULRGVSHU\HDU 6LQFH3 Q W DQG$ ZHKDYH R ´ ¥ § ¶ R ´ ¥ § ¶ r r ¥ ´ ¦ µ R ¥ ´ § ¶ r § ¶ R r R r >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà ! 'RADUATION 0RESENT 4EACHER .OTES 4HEREFORE WE WOULD NEED TO lND A SAVINGS PROGRAM THAT PAID AT LEAST INTEREST COMPOUNDED MONTHLY )N THIS CASE WE KNOW 0 R N AND ! 7E MUST SOLVE FOR T AS FOLLOWS ¥ ´ § ¶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± >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà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¥ ´ z § ¶ 7KHDPRXQWHDUQHGLV± ,IWZLFHDVPXFKLVLQYHVWHGWKHDPRXQWHDUQHG LV ¥ ´ z § ¶ 7KHDPRXQWHDUQHGLV± >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà ! 'RADUATION 0RESENT 4EACHER .OTES 6KRZWKDWWKLVLVWUXHIRUDQ\DPRXQWLQYHVWHGDWDUDWHU FRPSRXQGHGQWLPHVSHU\HDUIRUW\HDUV 7KHRULJLQDODPRXQWLQYHVWHGLVUHSUHVHQWHGE\37KH DPRXQWHDUQHGLV ¥ R ´ NT ! 0 § N¶ 7KHLQWHUHVWHDUQHGLVWKHDPRXQWHDUQHGPLQXVWKH DPRXQWLQYHVWHG ¥ R´ 0 § N¶ NT 0 ,IWKHDPRXQWLVPXOWLSOLHGE\LWLV37KHDPRXQW HDUQHGLV ¥ R ´ NT 0 § N¶ 7KHLQWHUHVWHDUQHGLVWKHDPRXQWHDUQHGPLQXVWKH DPRXQWLQYHVWHG ¥ R´ 0 § N¶ NT « ¥ R´ 0 ¬0 ­ § N¶ NT º 0» ¼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«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà ! 'RADUATION 0RESENT 3TUDENT 7ORK >«ÌiÀÊÈ\ÊÝ«iÌ>Ê>`Ê}>ÀÌ VÊÕVÌà (SNLIYH00 (ZZLZZTLU[Z *OHW[LY! ,>Ì> ÕVÌà >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà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¿QHGE\WKHLQHTXDOLWLHVLQSUREOHP 'HVFULEHWKHORFDWLRQRIWKHSRLQWVRQ\RXUJUDSKZKHUHWKHGLPHQVLRQVRI WKHSDLQWLQJUHVXOWLQHDFKRIWKHIROORZLQJ D7KHSHULPHWHUDQGDUHDDUHDFFHSWDEOH E7KHSHULPHWHULVWRRVKRUWRUWRRORQJEXWWKHDUHDLVDFFHSWDEOH F7KHSHULPHWHULVDFFHSWDEOHEXWWKHDUHDLVWRRVPDOORUWRRODUJH G1HLWKHUWKHSHULPHWHUQRUWKHDUHDLVDFFHSWDEOH ([SODLQKRZ\RXDUULYHGDW\RXUUHVSRQVHV >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà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b X Y b " :KDWNLQGRIV\VWHPVPXVW\RXVROYHWRJHWWKH YHUWLFHVRIWKHERXQGDU\" :KDWDOJHEUDLFPHWKRGZLOO\RXXVHWRVROYHHDFK V\VWHP":K\" 3RLQWWRHDFKUHJLRQLQWKHSODQHGHWHUPLQHGE\ WKHERXQGDU\IXQFWLRQVDQGGHVFULEHZKHWKHUWKH SHULPHWHUDQGDUHDPHHWWKHUHTXLUHPHQWVLQWKDW UHJLRQ 6DPSOH6ROXWLRQV /HWO WKHOHQJWKLQLQFKHVRIDSDLQWLQJDQGZ WKHZLGWK LQLQFKHVRIWKHSDLQWLQJ 6LQFHWKHSHULPHWHURIWKHSDLQWLQJPXVWEHEHWZHHQ LQFKHVDQGLQFKHVZHNQRZWKDW ZKLFKJLYHV b L W b b L W b W b L b W >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà 0AINTINGS ON A 7ALL 4EACHER .OTES 6LQFHWKHDUHDRIWKHSDLQWLQJPXVWEHEHWZHHQ VTXDUHLQFKHVDQGVTXDUHLQFKHVZHKDYH 7KHVWXGHQWLVH[SHFWHGWR b LW b ZKLFKJLYHV bL b W W 2QWKHJUDSKLQJFDOFXODWRUWKHOHQJWKZLOOEHUHSUHVHQWHG E\<DQGWKHZLGWKZLOOEHUHSUHVHQWHGE\[ )XQFWLRQ5XOH&DOFXODWRU5XOH L W 9 X L W 9 X W L W L X 9 X 9 $LGHQWLI\WKH PDWKHPDWLFDOGRPDLQV DQGUDQJHVRIIXQFWLRQV DQGGHWHUPLQHUHDVRQDEOH GRPDLQDQGUDQJHYDOXHV IRUFRQWLQXRXVDQGGLVFUHWH VLWXDWLRQV %FROOHFWDQGRUJDQL]H GDWDPDNHDQGLQWHUSUHW VFDWWHUSORWV¿WWKHJUDSK RIDIXQFWLRQWRWKHGDWD LQWHUSUHWWKHUHVXOWVDQG SURFHHGWRPRGHOSUHGLFW DQGPDNHGHFLVLRQVDQG FULWLFDOMXGJPHQWV &RQQHFWLRQWR7$.6 +HUHLVWKHFDOFXODWRUJUDSKRIWKHV\VWHPZLWKZLQGRZ VHWWLQJV b X b b Y b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«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà $ % 2EMHFWLYH7KHVWXGHQW ZLOOGHPRQVWUDWHDQ XQGHUVWDQGLQJRIWZR DQGWKUHHGLPHQVLRQDO UHSUHVHQWDWLRQVRIJHRPHWULF UHODWLRQVKLSVDQGVKDSHV 2EMHFWLYH7KHVWXGHQW ZLOOGHPRQVWUDWHDQ XQGHUVWDQGLQJRIWKH PDWKHPDWLFDOSURFHVVHVDQG WRROVXVHGLQSUREOHPVROYLQJ 0AINTINGS ON A 7ALL 4EACHER .OTES 7KHLQWHUVHFWLRQRI 7KHLQWHUVHFWLRQRI 7KHLQWHUVHFWLRQRI 7KHLQWHUVHFWLRQRI 7R¿QGWKHYHUWLFHVDOJHEUDLFDOO\ZHVROYHWKHIROORZLQJV\VWHPVE\VXEVWLWXWLRQ 9 9 9 9 AND 9 & AND 9 ' AND 9 ( AND 9 ) 7KLVJLYHVWKH[FRRUGLQDWHIRUYHUWLFHV%DQG& 6XEVWLWXWHIRU[LQ<WRJHWWKH\FRRUGLQDWHV 7KLVJLYHVWKH[FRRUGLQDWHIRUYHUWLFHV'DQG$ 6XEVWLWXWHIRU[LQ<WRJHWWKH\FRRUGLQDWHV >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà 0AINTINGS ON A 7ALL 4EACHER .OTES 7KLVJLYHVWKH[FRRUGLQDWHIRUYHUWLFHV(DQG) 6XEVWLWXWHIRU[LQ<WRJHWWKH\FRRUGLQDWHV 5HPHPEHUWKDWWKHFRRUGLQDWHV[\RIWKHSRLQWVLQWKHSODQHUHSUHVHQWZLGWKOHQJWK RIWKHSDLQWLQJ7KHUHIRUHZHFDQFRQVLGHUSRLQWVRQO\LQWKH¿UVWTXDGUDQW D 7KHUHJLRQZKHUHWKHSHULPHWHUDQGDUHDDUHDFFHSWDEOHLVWKHFORVHGUHJLRQ EHWZHHQWKHOLQHVDQGWKHFXUYHV E)RUWKHSHULPHWHUWREHWRRVKRUWDQGWKHDUHDDFFHSWDEOHZHQHHG¿UVWTXDGUDQW SRLQWVEHORZWKHOLQH\ ±[EXWEHWZHHQRURQWKHFXUYHV >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà 0AINTINGS ON A 7ALL 4EACHER .OTES )RUWKHWKHSHULPHWHUWREHWRRORQJDQGWKHDUHDDFFHSWDEOHZHQHHG¿UVWTXDGUDQW SRLQWVDERYHWKHOLQH\ ±[EXWEHWZHHQRURQWKHFXUYHV F )RUWKHSHULPHWHUWREHDFFHSWDEOHDQGWKHDUHDWRRVPDOOZHQHHG¿UVWTXDGUDQWSRLQWV EHWZHHQRURQWKHOLQHVEXWEHORZWKHFXUYHY X >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà 0AINTINGS ON A 7ALL 4EACHER .OTES )RUWKHSHULPHWHUWREHDFFHSWDEOHDQGWKHDUHDWRRODUJHZHQHHG¿UVWTXDGUDQWSRLQWV EHWZHHQRURQWKHOLQHVEXWDERYHWKHFXUYH Y X G ,IWKHSHULPHWHUDQGDUHDDUHERWKXQDFFHSWDEOHZHQHHG¿UVWTXDGUDQWSRLQWVERWK RXWVLGHWKHOLQHVDQGWKHFXUYHV >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà 0AINTINGS ON A 7ALL 4EACHER .OTES ([WHQVLRQ4XHVWLRQV +RZLVWKHUHJLRQUHSUHVHQWLQJDFFHSWDEOHGLPHQVLRQVIRUDSDLQWLQJGLIIHUHQWIURP UHJLRQV\RXHQFRXQWHUHGLQOLQHDUSURJUDPPLQJSUREOHPV" 7KHERXQGDULHVDUHQRWDOOGH¿QHGE\OLQHDUIXQFWLRQV7KUHHERXQGDULHVDUHVHJPHQWV GH¿QHGE\WKHWZROLQHDUSHULPHWHUIXQFWLRQV7KUHHERXQGDULHVDUHGH¿QHGE\WKHWZR UHFLSURFDODUHDIXQFWLRQV :KLFKYHUWH[RUYHUWLFHVPLQLPL]HSHULPHWHUDQGDUHDVXEMHFWWRWKHUHVWULFWLRQV"+RZ GR\RXNQRZ" 7RPLQLPL]HERWKSHULPHWHUDQGDUHD\RXPXVWEHDWWKHLQWHUVHFWLRQRI 9 X AND 9 X EHFDXVHWKHVHJLYHWKHORZHUOLPLWVRQSHULPHWHUDQG DUHD7KLVLVYHUWH[%RU&7KHSDLQWLQJFDQKDYHGLPHQVLRQVLQFKHVE\ LQFKHV7RPD[LPL]HERWKSHULPHWHUDQGDUHD\RXPXVWEHDWWKHLQWHUVHFWLRQRI 9 X AND 9 X EHFDXVHWKHVHJLYHWKHXSSHUOLPLWVRQSHULPHWHUDQGDUHD 7KLVLVYHUWH[(RU)7KHSDLQWLQJFDQKDYHGLPHQVLRQVLQFKHVE\LQFKHV 'RHVWKHJUDSKKDYHDQ\V\PPHWU\" <HV,WLVV\PPHWULFZLWKUHVSHFWWRWKHOLQH\ [ 6XSSRVHWKHDUHDUHVWULFWLRQZHUHUHSODFHGZLWKWKHUHVWULFWLRQWKDWWKHGLDJRQDORID SDLQWLQJPXVWEHEHWZHHQLQFKHVDQGLQFKHVLQOHQJWK+RZZLOOWKLVFKDQJH\RXU UHVSRQVHVWRWKHSUHYLRXVWZRTXHVWLRQV" 7KHUHVWULFWLRQVRQWKHSHULPHWHUVWD\WKHVDPHEXWZHPXVWUHSODFHDUHDUHVWULFWLRQV ZLWKGLDJRQDOUHVWULFWLRQV >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà 0AINTINGS ON A 7ALL 4EACHER .OTES /HWG WKHOHQJWKRIWKHSDLQWLQJ¶VGLDJRQDOLQLQFKHV7KHQVLQFHD L W b D b b D b b L W b W b L b W W b L b W 7KHDUHDERXQGDU\HTXDWLRQVDUHUHSODFHGZLWK L W AND L W 7KHJUDSKRIWKHUHJLRQUHSUHVHQWLQJDFFHSWDEOHGLPHQVLRQVLVWKH¿UVWTXDGUDQWUHJLRQ EHWZHHQWKHOLQHVDQGEHWZHHQWKHVHPLFLUFOHV,WGRHVQRWLQFOXGHSRLQWVRQWKHD[HV VLQFHWKDWZRXOGJLYH\RX]HURZLGWKRU]HUROHQJWK 7R¿QGZKHUHWKHSHULPHWHUOLQHVDQGGLDJRQDOFLUFOHVLQWHUVHFWZHVROYHWKHV\VWHPV FRQVLVWLQJRI<DQG<DQGRI<DQG<)RUH[DPSOH 9 X 9 X ,ET 9 9 X X X X X X X $SSO\WKHTXDGUDWLFIRUPXODWRJHW[ LQFKHVRU[ LQFKHV >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà 0AINTINGS ON A 7ALL 4EACHER .OTES 6LPLODUO\ZHVROYHIRUWKHLQWHUVHFWLRQRI 9 AND 9 WRJHW[ LQFKHVRU[ LQFKHV 7KHYHUWH[SRLQWVDUHDQG >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà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«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà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¿QGLQJWKHGHFLPDO HTXLYDOHQWRIWKHUDWLRRIVDOWWRZKROHVROXWLRQ 7KHIROORZLQJWDEOHVKRZVWKHUHVXOWVRIWKHGLOXWLRQ SURFHVV 7KHVWXGHQWLVH[SHFWHGWR %FROOHFWDQGRUJDQL]H GDWDPDNHDQGLQWHUSUHW VFDWWHUSORWV¿WWKHJUDSKRID IXQFWLRQWRWKHGDWDLQWHUSUHW WKHUHVXOWVDQGSURFHHG WRPRGHOSUHGLFWDQG PDNHGHFLVLRQVDQGFULWLFDO MXGJPHQWV $$OJHEUDDQGJHRPHWU\ 7KHVWXGHQWFRQQHFWVDOJHEUDLF DQGJHRPHWULFUHSUHVHQWDWLRQV RIIXQFWLRQV >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà 3ALINE 3OLUTION 4EACHER .OTES 7KHDPRXQWRIWKHZKROHVROXWLRQLQFUHDVHVZKLOHWKH FRQFHQWUDWLRQRIVDOWGHFUHDVHV+RZHYHUWKHDPRXQWRI VDOWUHPDLQVFRQVWDQWWKURXJKRXWWKHSURFHVV7KH IXQFWLRQFDQEHPRGHOHGE\WKHIROORZLQJ #ONCENTRATION OF SALT /HW[ WKHDPRXQWRIZDWHUDGGHG/HW\ WKH FRQFHQWUDWLRQRIVDOW Y AMOUNT OF SALT AMOUNT OF WHOLE SOLUTION X 7KHZKROHVROXWLRQUHSUHVHQWVWKHRULJLQDOOLWHUSOXVWKH KDOIOLWHUVWKDWZHUHDGGHG (QWHULQJWKHYDOXHVIRUWKHDPRXQWRIZDWHUDGGHG[LQWR /DQGWKHVDOWFRQFHQWUDWLRQ\LQWR/RIWKHJUDSKLQJ FDOFXODWRUSURGXFHVWKHIROORZLQJWDEOHDQGVWDWSORW (QWHULQJWKHIXQFWLRQLQWRWKHJUDSKLQJFDOFXODWRUSURGXFHV DWUDQVIRUPDWLRQRIWKHJUDSKRIWKHIXQFWLRQ Y X 7KH SRUWLRQRIWKHJUDSKWKDWUHSUHVHQWVWKLVSUREOHPVLWXDWLRQLV WKHSRUWLRQRIWKHJUDSKLQWKH¿UVWTXDGUDQW 6SHFL¿FDOO\ Y IXQFWLRQ Y X LVDWUDQVIRUPDWLRQRIWKHSDUHQW X 7KHIXQFWLRQKDVEHHQWUDQVODWHGWRWKHOHIWRQHXQLWDQG FRPSUHVVHGYHUWLFDOO\ >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà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b ZKDWZRXOG ,IWKHIXQFWLRQ\RXZURWHWRPRGHODVLPLODUVLWXDWLRQKDGEHHQ Y X \RXNQRZDERXWWKHRULJLQDOVROXWLRQ" >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà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«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà 0IZZA 7ARS 0ART ,QDUHFHQWDGYHUWLVHPHQWWKHSL]]DUHVWDXUDQW/LWWOH1HUR¶VPDGHHDFKRIWKH IROORZLQJFODLPVFRPSDULQJWKHVL]HRILWVSL]]DVWRWKHVL]HRIFRPSDUDEO\ SULFHGSL]]DVDW'RQDWHOOR¶V/LWWOH1HUR¶VPDLQFRPSHWLWRU /LWWOH1HUR¶VJLDQWSL]]DLVELJJHUWKDQ'RQDWHOOR¶VODUJHSL]]D /LWWOH1HUR¶VH[WUDODUJHSL]]DLVELJJHUWKDQ'RQDWHOOR¶VPHGLXP SL]]D /LWWOH1HUR¶VODUJHSL]]DLVELJJHUWKDQ'RQDWHOOR¶VVPDOOSL]]D %\FODLPLQJIRUH[DPSOHWKDWLWVODUJHSL]]DLV³ELJJHU´WKDQ'RQDWHOOR¶V VPDOOSL]]D/LWWOH1HUR¶VLVDVVHUWLQJWKDWWKHDUHDRILWVODUJHSL]]DLV ELJJHUWKDQWKHDUHDRI'RQDWHOOR¶VVPDOOSL]]D 7KHGLDPHWHURI/LWWOH1HUR¶VODUJHSL]]DLVLQFKHVDQGWKHGLDPHWHURI 'RQDWHOOR¶VVPDOOSL]]DLVLQFKHV'HWHUPLQHZKHWKHUWKHFODLP³/LWWOH 1HUR¶VODUJHSL]]DLVELJJHUWKDQ'RQDWHOOR¶VVPDOOSL]]D´LVYDOLG D7KHGLDPHWHURI'RQDWHOOR¶VPHGLXPSL]]DLVLQFKHVDQG/LWWOH1HUR¶V H[WUDODUJHSL]]DLVLQIDFWELJJHUWKDQ'RQDWHOOR¶VPHGLXPSL]]D )LQGWRWKHQHDUHVWLQFKWKHGLDPHWHURI/LWWOH1HUR¶VH[WUDODUJHSL]]D +RZPXFKORQJHUWKDQWKHGLDPHWHURI'RQDWHOOR¶VPHGLXPSL]]DLV WKHGLDPHWHURI/LWWOH1HUR¶VH[WUDODUJHSL]]D"+RZPXFKORQJHULVWKH UDGLXV" E7KHGLDPHWHURI/LWWOH1HUR¶VJLDQWSL]]DLVLQFKHVDQG/LWWOH1HUR¶V JLDQWSL]]DLVLQIDFWELJJHUWKDQ'RQDWHOOR¶VODUJHSL]]D)LQG WRWKHQHDUHVWLQFKWKHGLDPHWHURI'RQDWHOOR¶VODUJHSL]]D:KDWFDQEH >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà VDLGDERXWWKHFRUUHVSRQGLQJGLDPHWHUVIRUDOOWKUHHSDLUVRIFRPSDUDEO\ SULFHGSL]]DVDW/LWWOH1HUR¶VDQG'RQDWHOOR¶V":KDWFDQEHVDLGDERXW WKHFRUUHVSRQGLQJUDGLL" $VVXPHWKDWIRUDQ\SL]]DDW'RQDWHOOR¶VWKHUDGLXVRIDFRPSDUDEO\SULFHG SL]]DDW/LWWOH1HUR¶VLVLQFKHVORQJHU D,IUUHSUHVHQWVWKHUDGLXVRIDSL]]DDW'RQDWHOOR¶V¿QGDQDOJHEUDLF H[SUHVVLRQWKDWJLYHVWKHH[DFWYDOXHLQWHUPVRI/LIQHHGHGIRUHDFK RIWKHIROORZLQJ LWKHDUHDRIDFRPSDUDEO\SULFHGSL]]DDW/LWWOH1HUR¶V LLWKHGLIIHUHQFHEHWZHHQWKHDUHDRIWKH'RQDWHOOR¶VSL]]DDQGWKHDUHD RIDFRPSDUDEO\SULFHG/LWWOH1HUR¶VSL]]D LLLDVDSHUFHQWDJHRIWKH'RQDWHOOR¶VSL]]DKRZPXFKELJJHUD FRPSDUDEO\SULFHG/LWWOH1HUR¶VSL]]DZRXOGEH E)LQGDUXOHIRUWKHIXQFWLRQ3UWKDWJLYHVWKHSHUFHQWDJHGHVFULEHGLQLLL DERYHDVDIXQFWLRQRIU F*LYHWKHGRPDLQDQGUDQJHRIWKLVIXQFWLRQIRUWKHSL]]DSUREOHPFRQWH[W G8VHWHFKQRORJ\WRSURGXFHDJUDSKRIWKLVIXQFWLRQ'HVFULEHZKDW WKHJUDSKUHSUHVHQWV7KHQYHULI\WKDWIRUHDFKRIWKHWKUHHSDLUVRI FRPSDUDEO\SULFHGSL]]DVJLYHQDERYHWKHJUDSKLQGLFDWHVWKHFRUUHFW SHUFHQWDJH H'HVFULEHDOODV\PSWRWLFEHKDYLRUIRUWKLVIXQFWLRQ7KHQH[SODLQWKH PHDQLQJRIWKLVEHKDYLRULQWKHJLYHQFRQWH[W >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà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¿UPWKDWIRU H[DPSOHZKHQWKHUDGLXVDW'RQDWHOOR¶VLVLQFKHVD FRPSDUDEO\SULFHGSL]]DDW/LWWOH1HUR¶VZLWKUDGLXV LQFKHVLVELJJHU" $VZHPRYHDORQJWKHJUDSKWRZDUGWKHYHUWLFDOD[LV ZKDWLVKDSSHQLQJWRU"7R3U" $VZHPRYHDORQJWKHJUDSKDZD\IURPWKHYHUWLFDO D[LVZKDWLVKDSSHQLQJWRU"7R3U" )URPWKHJUDSKZKDWFDQZHWHOODERXWGLIIHUHQFHVLQ SL]]DVRIWKHWZRUHVWDXUDQWV" 6DPSOH6ROXWLRQV )RU/LWWOH1HUR¶VODUJHSL]]D ,IG LQFKHVWKHQU LQFKHVDQG$ / / VTXDUHLQFKHV )RU'RQDWHOOR¶VVPDOOSL]]D ,IG LQFKHVWKHQU LQFKHVDQG$ / / VTXDUHLQFKHV >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà 0IZZA 7ARS 0T 4EACHER .OTES 3HUFHQWDJHLQFUHDVH P P P P P 7KLVFRQ¿UPVWKDWWKHFODLP³/LWWOH1HUR¶VODUJHSL]]DLV ELJJHUWKDQ'RQDWHOOR¶VVPDOOSL]]D´LVYDOLG D )RU'RQDWHOOR¶VPHGLXPSL]]D ,IG LQFKHVWKHQU LQFKHVDQG$ / / VTXDUHLQFKHV ,IULVWKHUDGLXVRI/LWWOH1HUR¶VH[WUDODUJHSL]]DWKHQ PR P P PR P P PR P P PR P P R R R p R z INCHES 6RWRWKHQHDUHVWLQFKWKHGLDPHWHURI/LWWOH1HUR¶VH[WUD ODUJHSL]]DLVLQFKHVZKLFKLVLQFKHVORQJHUWKDQWKH GLDPHWHURI'RQDWHOOR¶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«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà 0IZZA 7ARS 0T 4EACHER .OTES E)RU/LWWOH1HUR¶VJLDQWSL]]D ,IG LQFKHVWKHQU LQFKHVDQG$ / /VTXDUHLQFKHV ,IULVWKHUDGLXVRI'RQDWHOOR¶VODUJHSL]]DWKHQ P PR PR P PR PR P PR PR P PR PR PR P R R z INCHES R p 6RWRWKHQHDUHVWLQFKWKHGLDPHWHURI'RQDWHOOR¶VODUJHSL]]DLVLQFKHV)RUDOOWKUHH SDLUVRIFRPSDUDEO\SULFHGSL]]DVWKHGLDPHWHULVLQFKHVORQJHUDQGWKHUDGLXVLV LQFKHVORQJHUDW/LWWOH1HUR¶V D L PR PR R PR PR P LL PR PR PR PR P PR PR P PR LLL PR PR PR P R OR R PR PR R R E F 'RPDLQ R b 2 ZKHUH5LVDFRQVWDQWUHSUHVHQWLQJWKHODUJHVWSRVVLEOHUDGLXV RIDSL]]DWKDWFRXOGEHPDGHDW'RQDWHOOR¶VSHUKDSVDVFRQVWUDLQHGE\WKHVL]HRI WKHSL]]DRYHQ 0 R R R OR 0 R R R 5DQJH 0R b 0MAX ZKHUH3PD[LVDFRQVWDQWUHSUHVHQWLQJWKHODUJHVW SHUFHQWDJHGLIIHUHQFHLQFRPSDUDEO\SULFHGSL]]DVWKDWFRXOGEHPDGHDW'RQDWHOOR¶V ZLWKUDGLXVUDQG/LWWOH1HUR¶VZLWKUDGLXVU >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà 0IZZA 7ARS 0T 4EACHER .OTES G 7KHJUDSKDERYHVKRZVWKHSHUFHQWDJHGLIIHUHQFH<FRPSDULQJWKHVL]HRIDSL]]D DW/LWWOH1HUR¶VZLWKWKHVL]HRIDFRPSDUDEO\SULFHGSL]]DDW'RQDWHOOR¶VZKHUH;LVWKH UDGLXVRIWKH'RQDWHOOR¶VSL]]DLQLQFKHV$OWKRXJKPDQ\FKRLFHVIRUWKHZLQGRZZRXOG EHDSSURSULDWHWKHZLQGRZXVHGIRUWKHJUDSKDERYHLVJLYHQDWULJKWRIWKHJUDSK 7KHJUDSKDWOHIWVKRZVWKDWWKHSRLQWLVFORVHWR WKHJUDSKRIWKLVIXQFWLRQWKXVFRQ¿UPLQJWKDWZKHQWKH UDGLXVRID'RQDWHOOR¶VSL]]DLVLQFKHVRUWKHGLDPHWHULV LQFKHVDFRPSDUDEO\SULFHGSL]]DDW/LWWOH1HUR¶VZLWK UDGLXVLQFKHVGLDPHWHULQFKHVLVELJJHUWRWKH QHDUHVWZKROHSHUFHQW 7KHJUDSKDWOHIWVKRZVWKDWWKHSRLQWLVFORVHWR WKHJUDSKRIWKLVIXQFWLRQWKXVFRQ¿UPLQJWKDWZKHQWKH UDGLXVRID'RQDWHOOR¶VSL]]DLVLQFKHVRUWKHGLDPHWHULV LQFKHVDFRPSDUDEO\SULFHGSL]]DDW/LWWOH1HUR¶VZLWK UDGLXVLQFKHVGLDPHWHULQFKHVLVELJJHU1RWLFH WKDW/LWWOH1HUR¶VGLGQRWURXQGXSRWKHUZLVHWKHFODLP ³ELJJHU´ZRXOGKDYHEHHQIDOVH 7KHJUDSKDWOHIWVKRZVWKDWWKHSRLQWEHORQJVWR WKHJUDSKRIWKLVIXQFWLRQWKXVFRQ¿UPLQJWKDWZKHQWKH UDGLXVRID'RQDWHOOR¶VSL]]DLVLQFKHVRUWKHGLDPHWHULV LQFKHVDFRPSDUDEO\SULFHGSL]]DDW/LWWOH1HUR¶VZLWK UDGLXVLQFKHVGLDPHWHULQFKHVLVELJJHU H9HUWLFDODV\PSWRWHU $VWKHUDGLXVRIWKH'RQDWHOOR¶VSL]]DQHDUVWKHGLIIHUHQFHDVDSHUFHQWDJHEHWZHHQ WKH'RQDWHOOR¶VSL]]DDQGDFRPSDUDEO\SULFHG/LWWOH1HUR¶VSL]]DZLWKDUDGLXVLQFKHV ORQJHUEHFRPHVODUJH)RUH[DPSOHZKHQWKHUDGLXVRID'RQDWHOOR¶VSL]]DLVLQFKHV >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà 0IZZA 7ARS 0T 4EACHER .OTES DFRPSDUDEO\SULFHGSL]]DDW/LWWOH1HUR¶VZLWKUDGLXVRILQFKHVLVELJJHU +RUL]RQWDODV\PSWRWH3U $VWKHUDGLXVRIWKH'RQDWHOOR¶VSL]]DEHFRPHVODUJHUDQGODUJHUWKHGLIIHUHQFHDVD SHUFHQWDJHEHWZHHQWKH'RQDWHOOR¶VSL]]DDQGDFRPSDUDEO\SULFHG/LWWOH1HUR¶VSL]]D ZLWKDUDGLXVLQFKHVORQJHUQHDUV)RUH[DPSOHZKHQWKHUDGLXVRID'RQDWHOOR¶V SL]]DLVLQFKHVDFRPSDUDEO\SULFHGSL]]DDW/LWWOH1HUR¶VZLWKUDGLXVRI LQFKHVLVRQO\DERXWELJJHU ([WHQVLRQ4XHVWLRQV 'HWHUPLQHZKHWKHUWKHDUHDRIDVTXDUHZLWKSHULPHWHULQFKHVLVHTXLYDOHQWWRWKH DUHDRIDFLUFOHZLWKFLUFXPIHUHQFHLHSHULPHWHULQFKHV,IWKHWZRDUHDVDUHQRW HTXLYDOHQWGHWHUPLQHWKHSHUFHQWDJHFKDQJHZLWKUHVSHFWWRWKHDUHDRIWKHVTXDUH )RUDVTXDUHRISHULPHWHULQFKHVWKHVLGHOHQJWKVLV LQFKHV7KHUHIRUH WKHDUHDRIWKHVTXDUHLV VTXDUHLQFKHV)RUDFLUFOHRIFLUFXPIHUHQFHLQFKHV LQFKHV7KHUHIRUHWKHDUHDRIWKHFLUFOHLV P z P P P P VTXDUHLQFKHV7KHDUHDRIWKHFLUFOHLVDERXW z SHUFHQWELJJHUWKDQ WKHDUHDRIWKHVTXDUH WKHUDGLXVULV $VWKHFRPPRQSHULPHWHUFKDQJHVZLOOWKHDUHDRIWKHFLUFOHFRQWLQXHWREHODUJHUWKDQ WKHDUHDRIWKHVTXDUH",IVRZLOOWKHDUHDRIWKHFLUFOHDOZD\VEHODUJHUWKDQWKH DUHDRIWKHVTXDUH" ,QRUGHUWRDQVZHUWKH¿UVWTXHVWLRQZHFRXOGMXVWFKRRVHDQRWKHUFRPPRQSHULPHWHU QRWHTXDOWRLQFKHVDQGUHSHDWWKHSURFHVVDERYH7KLVZRXOGVKRZWKDWLQDWOHDVW RQHRWKHUFDVHWKHUHVXOWVWXUQRXWWKHVDPH²WKDWLVWKHDUHDRIWKHFLUFOHLV ODUJHUWKDQWKHDUHDRIWKHVTXDUH+RZHYHULQRUGHUWRVKRZWKDWWKLVLVWUXHLQJHQHUDO ZHPXVWVKRZWKDWWKHH[SUHVVLRQ AREA OF CIRCLE AREA OF SQUARE AREA OF SQUARE LVDSSUR[LPDWHO\IRUDOOYDOXHVRIWKHSHULPHWHU36XFKDJHQHUDODUJXPHQW IROORZV >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà 0IZZA 7ARS 0T 4EACHER .OTES ,I3UHSUHVHQWVWKHFRPPRQSHULPHWHURIDVTXDUHDQGDFLUFOHWKHQWKHDUHDRIWKH 0 0 0 0 VTXDUHLVJLYHQE\ ¥ ´ 7KHDUHDRIWKHFLUFOHLVJLYHQE\ P¥ ´ § ¶ § P ¶ P 0 0 7KHUHIRUHWKHDUHDRIWKHFLUFOHLVDOZD\VODUJHU+RZPXFK P ODUJHUDVDSHUFHQWDJHLVJLYHQE\WKHIROORZLQJH[SUHVVLRQ 6LQFH!/ ´ ´ ¥ ¥ 0 0 0 § ¶ § P P ¶ P ¥ ´ ¥ ´ 0 0 § ¶ § ¶ 7KHUDWLRGRHVQRWGHSHQGRQWKHYDOXHRIWKHSHULPHWHU3,WPXVWEHDFRQVWDQW &RQWLQXLQJWRVLPSOLI\ ´ ´ ¥ ¥ P § P ¶ § P ¶ P z P ¥ ´ ¥ ´ P § ¶ § ¶ 7KLVSURYHVWKDWWKHDUHDRIDFLUFOHLVDOZD\VDSSUR[LPDWHO\ODUJHUWKDQWKHDUHD RIDVTXDUHZLWKWKHVDPHSHULPHWHU >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà 0IZZA 7ARS 0T 3TUDENT 7ORK >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà 0IZZA 7ARS 0T 3TUDENT 7ORK >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà <RX¶UH7RDVW'XGH $WWKH<RX¶UH7RDVW'XGH7RDVWHU&RPSDQ\WKHZHHNO\FRVWLQGROODUVRI SURGXFLQJ[WRDVWHUVLVJLYHQE\&[ [ &RPSXWHDQGLQWHUSUHW D& E # # X 8VHWHFKQRORJ\WRSURGXFHDJUDSKRIWKHIXQFWLRQ Y X X LQ DQDSSURSULDWHYLHZLQJZLQGRZ'HVFULEHZKDWWKHJUDSKUHSUHVHQWV7KHQ GHVFULEHDOODV\PSWRWLFEHKDYLRUIRUWKLVIXQFWLRQDQGH[SODLQWKHPHDQLQJ RIWKLVEHKDYLRULQWKHJLYHQFRQWH[W 8VHWKHIROORZLQJWZRPHWKRGVWR¿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«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà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«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà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«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà 9OURE 4OAST $UDE 4EACHER .OTES D 7KHYLHZLQJZLQGRZRQWKHOHIWVKRZVWKDW X WKHLQWHUVHFWLRQRI Y AND Y X LVWKHSRLQW6RZKHQWRDVWHUV DUHSURGXFHGDZHHNWKHDYHUDJHFRVWSHU WRDVWHULV X X X X E X X TOASTERS Y X X X X X X VKRZVWKDWWKH X DYHUDJHFRVWZLOODOZD\VEHSOXVWKHTXDQWLW\ ,QRWKHUZRUGVWKHDYHUDJH X 1RWLFHWKDWWKHQHZUHSUHVHQWDWLRQRIWKLVIXQFWLRQ FRVWLVWKHVORSHRIWKHFRVWIXQFWLRQRUWKHYDULDEOHFRVWSHUWRDVWHUSOXVDQ DPRXQWGHWHUPLQHGE\GLVSHUVLQJWKHWKH\LQWHUFHSWRIWKHFRVWIXQFWLRQ RUWKH¿[HGFRVWVWKDWEHIRUHWKH³¿UVW´WRDVWHULVPDGHPXVWEHH[SHQGHGHTXDOO\ DPRQJWKHWRWDOQXPEHURIWRDVWHUVSURGXFHG[6LQFH JHWVVPDOOHUDQG X VPDOOHUDV[JHWVODUJHUDQGODUJHU X JHWVFORVHUDQGFORVHUWRDV[JHWV ODUJHUDQGODUJHU7KLVLVKRUL]RQWDODV\PSWRWLFEHKDYLRUZLWKKRUL]RQWDODV\PSWRWH \ >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà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«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà 9OURE 4OAST $UDE 4EACHER .OTES Y X X X X X X X X X X 1RWLFHWKDWWKHQHZUHSUHVHQWDWLRQRIWKLVDYHUDJHFRVWIXQFWLRQVKRZVWKDWWKH DYHUDJHFRVWZLOOHYHQWXDOO\EHKDYHPXFKOLNHWKHOLQHDUIXQFWLRQ\ [ZKHQ [EHFRPHVYHU\ODUJHVLQFHWKHH[SUHVVLRQ X EHFRPHVOHVVDQGOHVVVLJQL¿FDQW DV[JURZVODUJHUDQGODUJHU,QIDFWWKHOLQH\ [LVDVODQWRUREOLTXH DV\PSWRWHIRUWKHJUDSKRIWKLVDYHUDJHFRVWIXQFWLRQ >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà :KDW¶V0\(TXDWLRQ" )RUHDFKSUREOHPEHORZ\RXDUHJLYHQWKHJUDSKRIDUDWLRQDOIXQFWLRQDORQJ ZLWKVRPHDGGLWLRQDOLQIRUPDWLRQDERXWWKDWIXQFWLRQ&DUHIXOO\DQDO\]HHDFK JUDSKDQGWKHDGGLWLRQDOLQIRUPDWLRQWKHQVXSSO\DQDSSURSULDWHIXQFWLRQUXOH IRUHDFKUDWLRQDOIXQFWLRQZKRVHJUDSKLVVKRZQ'HVFULEHWKHGRPDLQDQG UDQJHIRUHDFKIXQFWLRQ $GGLWLRQDOLQIRUPDWLRQ\LVLQYHUVHO\SURSRUWLRQDOWRWKHVTXDUHRI[ $GGLWLRQDOLQIRUPDWLRQ7KHSDUHQWIXQFWLRQIRUWKLVJUDSKLV Y >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà X $GGLWLRQDOLQIRUPDWLRQ7KHSDUHQWIXQFWLRQIRUWKLVJUDSKLV Y X $GGLWLRQDOLQIRUPDWLRQ,IQRWIRUWKH³KROH´LQWKHJUDSKDWWKLVJUDSK FRXOGEHJHQHUDWHGZLWKMXVWDKRUL]RQWDOVKLIWXVLQJ Y DVWKHSDUHQW X IXQFWLRQ >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà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¿QGWKHXQNQRZQ SDUDPHWHUVWKDWDIIHFWWKHJUDSKRIWKHSDUHQW IXQFWLRQ" :KDWGRHVWKHJUDSKRI Y ORRNOLNH":KDW X WUDQVIRUPDWLRQVRI Y ZRXOGUHVXOWLQWKHJLYHQ X JUDSK" :KDWDOJHEUDLFPDQLSXODWLRQRIDQH[SUHVVLRQWKDW GH¿QHVWKHIXQFWLRQUHVXOWVLQDYHUWLFDOVKLIW"$ KRUL]RQWDOVKLIW"$UHÀHFWLRQDFURVVWKH[D[LV"$ UHÀHFWLRQDFURVVWKH\D[LV" +RZFRXOGZHXVHWKHJLYHQSRLQWWRKHOSXV¿QGDQ HTXDWLRQ" +RZFRXOGZHXVHWKHJLYHQSRLQWWRFRQ¿UPWKDWRXU HTXDWLRQLVFRUUHFW" DFURVVWKH[D[LV" X :KDWLVWKHUHÀHFWLRQRI Y X DFURVVWKH\D[LV" :KDWLVWKHUHÀHFWLRQRI Y &RPSDUHHDFKUHÀHFWLRQDQGGLVFXVVWKHUHDVRQVIRU WKHLUGLIIHUHQFHVDQGVLPLODULWLHV :KDWZRXOGWKHHTXDWLRQEHLIZH³¿OOHGLQWKHKROH"´ :KDWGRHVWKH³KROH´KDYHWRGRZLWKWKHGRPDLQ" :LWKWKHHTXDWLRQ" >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà 7HATS -Y %QUATION 4EACHER .OTES 6DPSOH6ROXWLRQV 6LQFH\LVLQYHUVHO\SURSRUWLRQDOWRWKHVTXDUHRI[WKHQ K Y IRUVRPHQRQ]HURFRQVWDQWN X K 6XEVWLWXWLQJWKHJLYHQSRLQWZHKDYH OR K 6RWKHUDWLRQDOIXQFWLRQZKRVHJUDSKLVVKRZQLVJLYHQ E\WKHHTXDWLRQ Y X 7KLVUXOHFDQEHYHUL¿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«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà 7KHVWXGHQWLVH[SHFWHGWR %H[WHQGSDUHQWIXQFWLRQV ZLWKSDUDPHWHUVVXFKDVD LQI[ D[DQGGHVFULEH WKHHIIHFWVRISDUDPHWHU FKDQJHVRQWKHJUDSKRI SDUHQWIXQFWLRQV &RQQHFWLRQWR7$.6 2EMHFWLYH7KHVWXGHQW ZLOOGHVFULEHIXQFWLRQDO UHODWLRQVKLSVLQDYDULHW\RI ZD\V 2EMHFWLYH7KHVWXGHQWZLOO GHPRQVWUDWHDQXQGHUVWDQGLQJ RIWKHSURSHUWLHVDQGDWWULEXWHV RIIXQFWLRQV 2EMHFWLYH7KHVWXGHQWZLOO GHPRQVWUDWHDQXQGHUVWDQGLQJ RITXDGUDWLFDQGRWKHU QRQOLQHDUIXQFWLRQV 7HATS -Y %QUATION 4EACHER .OTES 7KHUXOHFDQEHYHUL¿HGE\XVLQJDJUDSKLQJFDOFXODWRU 7KHGRPDLQLVWKHVHWRIDOOUHDOQXPEHUVH[FHSW 7KHUDQJHLVWKHVHWRIDOOUHDOQXPEHUVH[FHSW K 6WDUWLQJZLWKWKHSDUHQWIXQFWLRQ Y X DUHÀHFWLRQRYHUWKH[D[LVRUDUHÀHFWLRQRIWKH \D[LVIROORZHGE\DKRUL]RQWDOVKLIWWRWKHOHIWRIXQLWVDQGDYHUWLFDOVKLIWXSRIXQLW ZLOOUHVXOWLQWKHJLYHQJUDSK7KHUHVXOWLQJIXQFWLRQUXOHZLOOEH Y K X 7RGHWHUPLQHWKHYDOXHRINXVLQJWKHSRLQWVXEVWLWXWHIRU[DQGIRU\ K K X X Y OR Y X X X 6XEVWLWXWLRQRIWKHSRLQWLQWRWKLVUXOHLVIXUWKHUFRQ¿UPDWLRQ$OVRWKHSRLQW RQWKHSDUHQWIXQFWLRQZLOOVKLIWWRDIWHUWKHWKUHHWUDQVIRUPDWLRQVGHVFULEHG DERYH 7KHJUDSKRQWKHFDOFXODWRUDOVRPDWFKHVWKHRULJLQDOJUDSK 7KHGRPDLQRIWKHIXQFWLRQLVWKHVHWRIDOOUHDOQXPEHUVH[FHSW7KHUDQJHLVWKHVHW >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà 7HATS -Y %QUATION 4EACHER .OTES RIDOOUHDOQXPEHUVH[FHSW ,JQRULQJWKH³KROH´DQGVWDUWLQJZLWKWKHSDUHQWIXQFWLRQ Y X DKRUL]RQWDOVKLIWWRWKH OHIWRIXQLWVZRXOGUHVXOWLQWKHJLYHQJUDSK7KHUHVXOWLQJHTXDWLRQZRXOGEH Y +RZHYHUWKHJUDSKRI Y X GRHVQRWKDYHD³KROH´DW,QRUGHUWR³UHPRYH´ X WKLVSRLQWIURPWKHJUDSKZHQHHGWR³UHPRYH´[ IURPWKHGRPDLQRIWKHIXQFWLRQ 7KLVVXJJHVWVDQHHGWRKDYHWKHIDFWRU[±LQWKHGHQRPLQDWRU,QRUGHUIRUWKHUHVWRI WKHJUDSKWRUHPDLQXQFKDQJHGZHFDQLQWURGXFHWKHIDFWRU[±LQWKHQXPHUDWRUDV ZHOO$QRWKHUZD\WRWKLQNDERXWWKLVFKDQJHLQWKHHTXDWLRQLVWRWKLQNDERXWPXOWLSO\LQJ E\WKHH[SUHVVLRQ X ZKLFKLVHTXLYDOHQWWRIRUDOO[7KHUHVXOWLQJ X X HTXDWLRQZLOOEH Y X X X OR X X X X X X 7KHGRPDLQRIWKLVIXQFWLRQLVWKHVHWRIDOOUHDOQXPEHUVH[FHSWDQG7KHUDQJHLV WKHVHWRIDOOUHDOQXPEHUVH[FHSWDQGLVWKHYDOXHRI Y 7KHFDOFXODWRUJUDSKDQGWDEOHYHUL¿HVWKHDFFXUDF\RIWKLVUXOHGRPDLQDQGUDQJH ZKHQ[ X >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà 7HATS -Y %QUATION 4EACHER .OTES ([WHQVLRQ4XHVWLRQV &DUHIXOO\DQDO\]HWKHIROORZLQJJUDSKDQGWKHDGGLWLRQDOLQIRUPDWLRQJLYHQEHORZ WKHJUDSKWKHQVXSSO\DQDSSURSULDWHUXOHIRUWKHUDWLRQDOIXQFWLRQZKRVHJUDSKLV VKRZQ'HVFULEHWKHGRPDLQDQGUDQJHRIWKHIXQFWLRQ $GGLWLRQDOLQIRUPDWLRQ7KHUHDUHWZRYHUWLFDODV\PSWRWHVWKHLUHTXDWLRQVDUH[ DQG[ 7KHUHLVDKRUL]RQWDODV\PSWRWHLWVHTXDWLRQLV\ :LWKWKHWZRYHUWLFDODV\PSWRWHV[ DQG[ WKHGHQRPLQDWRURIWKLVUDWLRQDO IXQFWLRQPXVWKDYHWKHIROORZLQJIDFWRUV[±DQG[:LWKWKHKRUL]RQWDO DV\PSWRWH\ WKHGHJUHHRIWKHQXPHUDWRULVOHVVWKDQWKHGHJUHHRIWKH GHQRPLQDWRU,IZHDVVXPHWKDWWKHGHJUHHRIWKHGHQRPLQDWRULVWKHQWKHGHJUHH RIWKHQXPHUDWRULVHLWKHURU7KHVLQJOH[LQWHUFHSWLPSOLHVWKDWWKH GHJUHHLVDQGWKDWWKHQXPHUDWRULVRIWKHIRUPN[IRUVRPHFRQVWDQWN7KHUHIRUH WKHUDWLRQDOIXQFWLRQZKRVHJUDSKLVVKRZQKDVDQHTXDWLRQRIWKHIRUP Y KX KX OR Y X X X $OOZHQHHGWRGRQRZLVGHWHUPLQHDYDOXHIRUNVRWKDWWKHJUDSKRIWKLVIXQFWLRQ SDVVHVWKURXJKWKHSRLQW K K K >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà 7HATS -Y %QUATION 4EACHER .OTES 7KHUDWLRQDOIXQFWLRQZKRVHJUDSKLVVKRZQLVJLYHQE\WKHHTXDWLRQ Y X X 7KHGRPDLQLVDOOUHDOQXPEHUVH[FHSWDQG7KHUDQJHLVWKHVHWRIDOOUHDO QXPEHUV &DUHIXOO\DQDO\]HWKHIROORZLQJJUDSKDQGWKHDGGLWLRQDOLQIRUPDWLRQJLYHQEHORZWKH JUDSKWKHQVXSSO\DQDSSURSULDWHHTXDWLRQIRUWKHUDWLRQDOIXQFWLRQZKRVHJUDSKLV VKRZQ'HVFULEHWKHGRPDLQDQGUDQJH $GGLWLRQDOLQIRUPDWLRQ7KHGHQRPLQDWRULVTXDGUDWLFDQGKDVWKHIROORZLQJURRWV LDQGL7KHUHLVDKRUL]RQWDODV\PSWRWHLWVHTXDWLRQLV\ 6LQFHLDQGLDUHURRWVRIWKHGHQRPLQDWRU[±LDQG[LDUHIDFWRUVRIWKH GHQRPLQDWRU6LQFHWKHGHQRPLQDWRULVTXDGUDWLFZH¶OODVVXPHWKDWWKHGHQRPLQDWRU LVVLPSO\WKHSURGXFWRIWKHVHWZRIDFWRUV X I X I X I X >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà 7HATS -Y %QUATION 4EACHER .OTES :LWKWKHKRUL]RQWDODV\PSWRWH\ WKHGHJUHHRIWKHQXPHUDWRULVOHVVWKDQWKH GHJUHHRIWKHGHQRPLQDWRU²HLWKHURU:LWKQR[LQWHUFHSWWKHGHJUHHRIWKH QXPHUDWRUFDQ¶WEH6RWKHQXPHUDWRULVDFRQVWDQWDQGWKHUDWLRQDOIXQFWLRQKDV WKHIROORZLQJIRUP K Y IRUVRPHFRQVWDQWN X 8VLQJWKH\LQWHUFHSWZHFDQVROYHWKHIROORZLQJHTXDWLRQWR¿QGN K K 6RWKHUDWLRQDOIXQFWLRQZKRVHJUDSKLVVKRZQLVJLYHQE\WKHHTXDWLRQ Y X 7KHJUDSKFDQEHYHUL¿HGXVLQJDJUDSKLQJFDOFXODWRU 4HE DOMAIN IS THE SET OF ALL REAL NUMBERS 4HE RANGE IS THE SET OF ALL REAL NUMBERS b Y b >«ÌiÀÊÇ\Ê,>Ì>ÊÕVÌà (SNLIYH00 (ZZLZZTLU[Z *OHW[LY! Và >«ÌiÀÊn\Ê Và &RQWHPSODWLQJ&RPHWV $VDFRPHWPRYHVWKURXJKVSDFHLWVSDWKPD\EHWKDWRIDFRQLFVHFWLRQ $OOFRPHWVWUDYHODORQJSDWKVWKDWKDYHWKHLUVXQDWRQHIRFXVRIWKHFRQLF VHFWLRQ%DUEDUD¶VVFLHQFHIDLUSURMHFWIRFXVHVRQWKHSDWKVRIWZRSDUWLFXODU FRPHWV$OSKD]RLGDQG%HWDVWDULQWKH0HJDFHQWXULDQVRODUV\VWHPIRXQGLQ WKH2PHJDJDOD[\+RZHYHUVKHLVKDYLQJGLI¿FXOW\GLVWLQJXLVKLQJWKHVKDSHV RIWKHSDWKVRIWKHFRPHWV%DUEDUDKDVDVNHG\RXWRKHOSKHUPDWFKWKH FRUUHFWFRQLFVHFWLRQZLWKHDFKHTXDWLRQ7KHHTXDWLRQVRIWKHSDWKVRIWKHWZR FRPHWVDUHJLYHQEHORZLQELOOLRQVRIPLOHV $OSKD]RLG [\±[±\± %HWDVWDU \±[[±\± &RPSOHWHWKHVTXDUHWRGHWHUPLQHZKHWKHUWKHSDWKRIWKHHTXDWLRQRI WKHJLYHQFRPHWLVFLUFXODUSDUDEROLFHOOLSWLFDORUK\SHUEROLF*UDSKHDFK FRPHW¶VSDWKRQDVHSDUDWHJULG,IWKHSDWKLVFLUFXODUJLYHLWVFHQWHU DQGUDGLXV,ILWLVSDUDEROLFJLYHLWVYHUWH[DQGIRFXV,ILWLVHOOLSWLFDORU K\SHUEROLFJLYHLWVFHQWHUDQGIRFL 6LQFHWKHFRPHWVWUDYHODORQJSDWKVWKDWKDYHWKHLUVXQDVDIRFXVZKDW DUHWKHFRRUGLQDWHVRIWKHLUVXQ" &RXOGWKHVHWZRFRPHWVFROOLGH",IVRZKHUH"([SODLQ\RXUUHDVRQLQJ >«ÌiÀÊn\Ê Và #ONTEMPLATING #OMETS 4EACHER .OTES 5V[LZ 0DWHULDOV 6FDIIROGLQJ4XHVWLRQV +RZFDQ\RXGHWHUPLQHZKLFKFRQLFVHFWLRQWKH HTXDWLRQUHSUHVHQWVEHIRUHLWLVWUDQVIRUPHGLQWR VWDQGDUGIRUP" *UDSKLQJFDOFXODWRU $OJHEUD,,7(.6)RFXV $$OJHEUDDQG JHRPHWU\7KHVWXGHQW NQRZVWKHUHODWLRQVKLS EHWZHHQWKHJHRPHWULFDQG DOJHEUDLFGHVFULSWLRQVRI FRQLFVHFWLRQV +RZGR\RX¿QGWKHPLVVLQJWHUPVZKHQ\RX FRPSOHWHWKHVTXDUH" +RZGRHVWKHVWDQGDUGIRUPRIWKHHTXDWLRQKHOS\RX ¿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¿QGWKHGLVWDQFHIURPWKHFHQWHUWRWKHIRFLXVH F D±EZKHUHDLVWKHOHQJWKRIWKHVHPLPDMRUD[LV ELVWKHOHQJWKRIWKHVHPLPLQRUD[LVDQGFLVWKH GLVWDQFHIURPWKHFHQWHUWRWKHIRFL F ± F F D D§ E E§ >«ÌiÀÊn\Ê Và #ONTEMPLATING #OMETS 4EACHER .OTES ,IWKHFHQWHURIWKHHOOLSVHLVWKHSRLQWKNWKHFRRUGLQDWHV RIWKHYHUWLFHVRQWKHVHPLPDMRUD[LVDUHKNDDQG KN±DRUDQG 7KHFRRUGLQDWHVRIWKHYHUWLFHVRQWKHVHPLPLQRUD[LVDUH KENDQGK±ENRUDQG 7KHFRRUGLQDWHVRIWKHIRFLDUHKNFDQGKN±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¿QGWKHGLVWDQFHIURPWKHFHQWHUWRWKHIRFLXVH F DE D D >«ÌiÀÊn\Ê Và E E F F F #ONTEMPLATING #OMETS 4EACHER .OTES 7KHFRRUGLQDWHVRIWKHIRFLDUHKNFDQGKN±FRUDQG 7KHFRRUGLQDWHVRIWKHYHUWLFHVRQWKHWUDQVYHUVHD[LVDUHKNDDQGKN±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±\± [\±[±\± \±[[±\± [±[\±\± [[\±\± [±[\±\± [[\±\± >«ÌiÀÊn\Ê Và #ONTEMPLATING #OMETS 4EACHER .OTES 'LYLGHE\ \±\± 8VHWKHTXDGUDWLFIRUPXODWRVROYHZKHUHD E DQGF 7KHYDOXHVIRU\URXQGHGWRWKHQHDUHVWWHQWKDUH\ RU\ 7R¿QGWKHYDOXHRI[VXEVWLWXWHHDFKYDOXHRI\LQWRRQHRIWKHRULJLQDOHTXDWLRQV)LUVW \ DQGWKHHTXDWLRQIRU$OSKD]RLGLVXVHG 8VHWKHTXDGUDWLFIRUPXODWRVROYHIRU[ZKHUHD E DQGF [±[\±\± [±[±± [±[ [±[± X p p X OR X X URXQGHGWRWKHQHDUHVWWHQWK ,I%HWDVWDUWUDYHOVWKHQRUWKHUQEUDQFKRIWKHK\SHUEROLFSDWKWKHSRVVLEOHFROOLVLRQ SRLQWVDUHRU 1H[WXVLQJ\ VROYHIRU[7R¿QGWKHYDOXHRI[VXEVWLWXWHHDFKYDOXHRI\LQWR RQHRIWKHRULJLQDOHTXDWLRQV7KHHTXDWLRQIRU$OSKD]RLGLVDJDLQXVHG >«ÌiÀÊn\Ê Và [±[\±\± [±[±± [±[±± [±[± #ONTEMPLATING #OMETS 4EACHER .OTES 8VHWKHTXDGUDWLFIRUPXODWRVROYHIRU[ZKHUHD E DQGF X p p X OR X X URXQGHGWRWKHQHDUHVWWHQWK ,I%HWDVWDUWUDYHOVWKHQRUWKHUQEUDQFKRIWKHK\SHUEROLFSDWKWKHSRVVLEOHFROOLVLRQ SRLQWVDUHRU 7KHSDWKVRIWKHWZRFRPHWVFURVVHDFKRWKHUVRWKHUHLVDSRVVLELOLW\RIDFROOLVLRQ >«ÌiÀÊn\Ê Và #ONTEMPLATING #OMETS 4EACHER .OTES ([WHQVLRQ4XHVWLRQV %DUEDUD¶VUHVHDUFKRQWKHFRPHW$OSKD]RLGOHGWRDQLQWHUHVWLQJGLVFRYHU\6HYHUDO KXQGUHG\HDUVDJRWKHFRPHW¶VSDWKZDVUHSUHVHQWHGE\WKHHTXDWLRQ [[\\ 'HVFULEHWKHVKDSHDQGNH\HOHPHQWVRIWKHFRPHW¶VSDWK 7KHFRPHW¶VSDWKLVFLUFXODUEHFDXVHWKHFRHI¿FLHQWVRQWKH[DQG\VTXDUHGWHUPVDUH HTXDO&RPSOHWHWKHVTXDUHWRGHWHUPLQHWKHUDGLXVDQGFHQWHU X X Y Y X Y 7KHFHQWHULVDQGWKHUDGLXVLV 7KHIRUPXOD # PA B P A B A B FDQEHXVHGWRDSSUR[LPDWH WKHFLUFXPIHUHQFH&RIDQHOOLSVHZLWKPDMRUD[LVDDQGPLQRUD[LVE'HWHUPLQHWKH FLUFXPIHUHQFHRIWKH$OSKD]RLGRUELW 7KHYDOXHVIRUDDQGEGHWHUPLQHGLQSUREOHPDQGDUH A AND B 6XEVWLWXWHWKHVHYDOXHVLQWRWKHIRUPXOD # PA B P A B A B # P P 7KHDUHDRIWKHHOOLSVHLVDSSUR[LPDWHO\PLOOLRQPLOHV >«ÌiÀÊn\Ê Và >«ÌiÀÊn\Ê Và /RVWLQ6SDFH 7KHFRQLFVHFWLRQVSOD\DIXQGDPHQWDOUROHLQVSDFHVFLHQFH-RKDQQHV.HSOHU ZDVWKH¿UVW(XURSHDQWRWKLQNWKDWSODQHWVPRYHGLQHOOLSWLFDOUDWKHUWKDQ FLUFXODURUELWVDURXQGWKHVXQ+LVGHWDLOHGVWXG\RISODQHWDU\PRWLRQOHGWRWKH /DZVRI3ODQHWDU\0RWLRQZKLFKDUHNQRZQWRGD\DV.HSOHU¶V/DZV7KHODZV DSSO\WRDQ\FHOHVWLDOREMHFWWKDWRUELWVDQRWKHUREMHFWXQGHUWKHLQÀXHQFHRI JUDYLW\7ZRRI.HSOHU¶V/DZVDUHVKRZQEHORZ .HSOHU¶V)LUVW/DZ 7KHRUELWRIHDFKSODQHWLVDQHOOLSWLFDOSDWKZLWK WKH6XQDWRQHIRFXVRIWKHHOOLSVH .HSOHU¶V7KLUG/DZ)RUDSODQHWWKHFXEHRIWKHRUELW¶VVHPLPDMRUD[LV PHDVXUHGLQDVWURQRPLFDOXQLWV±$8LVHTXDOWR WKHVTXDUHRIWKHSODQHW¶VRUELWDOSHULRGPHDVXUHG LQ(DUWK\HDUV 7KHSRLQWDWZKLFKDSODQHWLVDWLWVFORVHVWWRWKHVXQLVFDOOHGSHULKHOLRQ $WSHULKHOLRQDQHZO\GLVFRYHUHGSODQHW$OSKDLVPLOOLRQPLOHVIURP WKHVXQ$SKHOLRQLVWKHSRLQWWKDWDSODQHWLVIDUWKHVWIURPWKHVXQ$OSKD LVPLOOLRQPLOHVIURPWKHVXQDWDSKHOLRQ D 8VHWKHJLYHQLQIRUPDWLRQWRVNHWFKDQGODEHODPRGHORI$OSKD¶VRUELW $VVXPHWKHIRFLOLHRQWKH[D[LVDQGWKHRULJLQLVWKHFHQWHURIWKHHOOLSVH E 'HWHUPLQHWKHOHQJWKVRIWKHPDMRUDQGVHPLPDMRUD[HVRIWKHHOOLSWLFDORUELW F 'HWHUPLQHWKHVXQ¶VGLVWDQFHIURPWKHFHQWHURIWKHHOOLSVH G ([SODLQKRZWRGHWHUPLQHWKHOHQJWKRIWKHVHPLPLQRUD[LVRI$OSKD¶VRUELW 5RXQGWRWKHQHDUHVWWHQWK H 8VH.HSOHU¶V)LUVW/DZWRKHOSGHWHUPLQHWKHHTXDWLRQRI$OSKD¶VRUELW 5RXQGWRWKHQHDUHVWWHQWK,QFOXGHWKHFRRUGLQDWHVRIWKHYHUWLFHVRIWKH HOOLSVHDQGRIWKHVXQ-XVWLI\\RXUDQVZHUV 8VH.HSOHU¶V7KLUG/DZWRGHWHUPLQHWKHOHQJWKRI$OSKD¶VRUELWDOF\FOHLQ (DUWK\HDUV5RXQG\RXUDQVZHUWRWKHQHDUHVWKXQGUHGWK1RWH,QRUGHU WRXVH.HSOHU¶V7KLUG/DZLWZLOOEHQHFHVVDU\WRH[SUHVVWKHOHQJWKRIWKH VHPLPDMRUD[LVUHODWLYHWRWKDWRIWKHHDUWKDVIROORZV (DUWKD[LV PLOOLRQPLOHV DVWURQRPLFDOXQLW$8 >«ÌiÀÊn\Ê Và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¿QGWKHGLVWDQFHIURPWKHRULJLQ WRWKHVXQ" 7KHVWXGHQWLVH[SHFWHGWR %VNHWFKJUDSKVRI FRQLFVHFWLRQVWRUHODWH VLPSOHSDUDPHWHU FKDQJHVLQWKHHTXDWLRQ WRFRUUHVSRQGLQJ FKDQJHVLQWKHJUDSK &LGHQWLI\V\PPHWULHV IURPJUDSKVRIFRQLF VHFWLRQV 'LGHQWLI\WKHFRQLF VHFWLRQIURPDJLYHQ HTXDWLRQ :KDWLVWKHUHODWLRQVKLSDPRQJWKHVHPLPDMRUD[LV FHQWHUVHPLPDMRUD[LVDQGWKHIRFL" 6DPSOH6ROXWLRQV D 8VLQJWKHJLYHQLQIRUPDWLRQLQPLOOLRQVRIPLOHVWKH IROORZLQJLVDVNHWFKRI$OSKD¶VRUELW$VVXPHWKH IRFLOLHRQWKH[D[LVDQGWKHRULJLQLVWKHFHQWHURIWKH HOOLSVH E 7RGHWHUPLQHWKHOHQJWKRIWKHPDMRUD[LVDGGOHQJWKV $6DQG63 PLOOLRQPLOHV7KHOHQJWK RIWKHVHPLPDMRUD[LVLVKDOIRI$3 $GGLWLRQDO$OJHEUD,,7(.6 1RQH PLOOLRQPLOHVZKLFKHTXDOVWKHOHQJWKVRI$2DQG23 >«ÌiÀÊn\Ê Và ,OST IN 3PACE 4EACHER .OTES F 7KHGLVWDQFHIURPWKHFHQWHURIWKHHOOLSVHLVIRXQG XVLQJVHJPHQWDGGLWLRQ2663 23,WLVNQRZQ WKDW26 WKHUHIRUH26 PLOOLRQ PLOHV G7KHOHQJWKRIWKHVHPLPLQRUD[LV2&FDQEHIRXQG XVLQJWKHIROORZLQJUHODWLRQVKLS7KHVWDQGDUGIRUPRI DQHOOLSVHZLWKFHQWHUDWWKHRULJLQDQGIRFLRQWKH[ D[LVLV X Y A B ZKHUHDLVWKHOHQJWKRIWKHPDMRUD[LVDQGELVWKH OHQJWKRIWKHPLQRUD[LV ,IWKHGLVWDQFHIURPWKHFHQWHUWRWKHIRFLLVFWKHQWKH UHODWLRQVKLSEHWZHHQDEDQGFLVE D±F ,QWKLVSUREOHPD 32 E 2& XQNQRZQ F 26 7KHUHIRUH &RQQHFWLRQ WR7$.6 2EMHFWLYH7KHVWXGHQW ZLOOGHPRQVWUDWHDQ XQGHUVWDQGLQJRIWZR DQGWKUHHGLPHQVLRQDO UHSUHVHQWDWLRQVRIJHRPHWULF UHODWLRQVKLSVDQGVKDSHV 2EMHFWLYH7KHVWXGHQW ZLOOGHPRQVWUDWHDQ XQGHUVWDQGLQJRIWKH PDWKHPDWLFDOSURFHVVHV DQGWRROVXVHGLQSUREOHP VROYLQJ 32 2&26 2& PLOOLRQPLOHV 2&URXQGHGWRWKHQHDUHVWWHQWK H .HSOHU¶V)LUVW/DZVWDWHVWKHSDWKRIWKHRUELWLVDQ HOOLSVH7KHUHIRUHWKHHTXDWLRQRI$OSKD¶VRUELWLV X Y 7KHFRRUGLQDWHVRIWKHYHUWLFHVRIWKHHOOLSVHDUHDW >«ÌiÀÊn\Ê Và ,OST IN 3PACE 4EACHER .OTES 7KHFRRUGLQDWHVRIWKHVXQIRFLDUHDOUHDG\NQRZQIURPSDUWFEHFDXVHWKHYDOXHRIF LVDQG 7KHVXQDSSHDUVDW .HSOHU¶V7KLUG/DZVWDWHVWKDWWKHFXEHRIDSODQHW¶VVHPLPDMRUD[LVLVHTXDOWRWKH VTXDUHRIWKHSODQHW¶VRUELWDOSHULRGZKHUHWKHVHPLPDMRUD[LVDQGWKHRUELWDOSHULRG DUHPHDVXUHGUHODWLYHWRWKRVHRI(DUWK2UELWDOSHULRGLVWKHWLPHWRFRPSOHWHRQHRUELW RIWKHVXQ(DUWK¶VRUELWDOSHULRGLVRQH\HDU ,QRUGHUWRGHWHUPLQHRQHRI$OSKD¶VRUELWDOF\FOHVLQ(DUWK\HDUVDQGXVH.HSOHU¶V /DZLWZLOOEHQHFHVVDU\WRH[SUHVVWKHOHQJWKRIWKHVHPLPDMRUD[LVUHODWLYHWRWKDWRI WKHHDUWKDVIROORZV(DUWK¶VVHPLPDMRUD[LV PLOOLRQPLOHV DVWURQRPLFDOXQLW $8 MIL MI MIL MI !5 X !5 X X !5 .HSOHU¶V7KLUG/DZVWDWHV $OSKD¶VVHPLPDMRUD[LVLQ$8V $OSKD¶VRUELWDOSHULRG [ ¥ ´ X § ¶ ¥ ´ X § ¶ 7KLVDQVZHUUHSUHVHQWVDSRUWLRQRID\HDU7RGHWHUPLQHWKHQXPEHURIGD\VWKH DQVZHUPD\EHPXOWLSOLHGE\WKHQXPEHURIGD\VLQD\HDU 7KHRUELWSHULRGLVDERXW\HDUVRUDSSUR[LPDWHO\GD\V >«ÌiÀÊn\Ê Và ,OST IN 3PACE 4EACHER .OTES ([WHQVLRQ4XHVWLRQV :KDWLVWKHHTXDWLRQRI$OSKD¶VRUELWLIWKHD[HVDUHSRVLWLRQHGVRWKDWORFDWLRQRI WKHVXQLVXVHGDVWKHRULJLQLQVWHDGRISODFLQJWKHFHQWHUDWWKHRULJLQ" X Y ,IWKHFHQWHURIWKHRULJLQLVPRYHG WRWKHIRFXVWKHQHZHTXDWLRQEHFRPHV 7KHHTXDWLRQIRXQGLQHZDV X Y 7KHHFFHQWULFLW\HRIDQHOOLSVHLVDPHDVXUHPHQWRILWV³ÀDWQHVV´7KHIRUPXOD H FDLVXVHGWR¿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¶VRUELW D 7KHOHQJWKRIWKHPDMRUD[LVRIWKHHOOLSVHLVIRXQGE\DGGLQJWKHFORVHVW GLVWDQFHDQGWKHIDUWKHVWGLVWDQFHRIWKHRUELW*LYHQWKDWWKHFORVHVWGLVWDQFH SHULKHOLRQLVELOOLRQPLOHVDQGWKHIDUWKHVWGLVWDQFHDSKHOLRQLV ELOOLRQPLOHVWKHOHQJWKRIWKHPDMRUD[LVLVRUELOOLRQPLOHV 7KHVHPLPDMRUD[LVLVKDOIRIWKDWYDOXHRUELOOLRQPLOHV DF ELOOLRQPLOHVWKHUHIRUHF ELOOLRQPLOHV 7KHHFFHQWULFLW\RI=HWD¶VRUELWLVGHWHUPLQHGE\ E C A 5RXQGHGWRWKHQHDUHVWKXQGUHGWKWKHHFFHQWULFLW\LVDSSUR[LPDWHO\ 7KHHFFHQWULFLW\RIDQHOOLSVHFDQEHDQ\YDOXHEHWZHHQDQGEHFDXVH LWLVDUDWLRRIWZRSRVLWLYHYDOXHVFDQGDDQGFD$QHOOLSVHZLWKDQ >«ÌiÀÊn\Ê Và ,OST IN 3PACE 4EACHER .OTES C HFFHQWULFLW\FORVHWRLVYHU\QDUURZEHFDXVHLI LVFORVHWRFLVFORVHWRDDQG A B A C PXVWEHYHU\VPDOO$QHOOLSVHZLWKHFFHQWULFLW\FORVHWRLVDOPRVWDFLUFOH EHFDXVHLI C LVDOPRVWHTXDOWR]HURDLVPXFKODUJHUWKDQFDQG B A C PXVWEH A DOPRVWHTXDOWRD=HWD¶VHFFHQWULFLW\LVZKLFKLVFORVHUWR]HURWKDQWRRQHWKHUHIRUHLW LVIDLUO\QDUURZ E %6 2& 2'±26 2& ± 2& 6' %' 7KHHTXDWLRQRI=HWD¶VRUELWLVDSSUR[LPDWHO\ 2' X Y >«ÌiÀÊn\Ê Và .DORWRQLF.DSHU /HVWHU/DUJHLVSODQQLQJDQRWKHUGLDEROLFDOSORWWRIHOOWKHVXSHUKHUR+HURLF +RUDFH:LWKWKHKHOSRIWKHEULOOLDQWVFLHQWLVW'U0DGG/HVWHUKDVGLVFRYHUHG DZD\WRWUDQVIRUP.DORWRQLF¶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ÀHFWV DQ\OLJKWGLUHFWHGWRZDUGRQHIRFXVRIWKHK\SHUERODWKURXJKWKHRWKHUIRFXV 'U0DGGKDVGHYLVHGKLVSODQRQDFRRUGLQDWHJULG+HKDVSODFHGWKH FHQWHURIWKHK\SHUERODDWWKHRULJLQDQGWKHYHUWH[RIWKHPLUURU¶VEUDQFK DW7KHIRFXVRIWKHPLUURU¶VEUDQFKLVDWZKLFKLVWKHGDQJHU ]RQHIRU/DQDDQG+HURLF+RUDFH7KHHTXDWLRQVRIWKHDV\PSWRWHVDUH Y X AND Y X 'U0DGGDQG/HVWHU¶VKLGLQJSODFHLVDWWKH FRRUGLQDWHV 8VLQJ'U0DGG¶VGHVFULSWLRQVNHWFKDJUDSKRIWKHK\SHUEROLFPLUURUHG VXUIDFHLWVLPDJLQDU\EUDQFKLHWKHRWKHUEUDQFKRIWKHPDWKHPDWLFDO K\SHUERODWKHORFDWLRQRIWKHKLGLQJSODFHDQGWKHGDQJHU]RQH 'HWHUPLQHWKHORFDWLRQRIWKHIRFLRIERWKEUDQFKHVRIWKHK\SHUEROD :ULWHDUXOHWKDWPRGHOVWKHPLUURUHGVXUIDFH 6XSSRVH'U0DGGDQG/HVWHU/DUJHSODQWRGLUHFWWKHEHDPRI.DORWRQLF IURPWKHLUORFDWLRQDWWRWKHIRFXVRIWKHLPDJLQDU\EUDQFK:KHUH ZLOOWKHEHDPRI.DORWRQLFLQWHUVHFWWKHK\SHUEROLFPLUURU" >«ÌiÀÊn\Ê Và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¶VEUDQFKDWDQGWKHIRFXVDW X AND Y X Y $V\PSWRWHVDUH 7KHIRFXVRIWKHLPDJLQDU\EUDQFKLVDW $GGLWLRQDO$OJHEUD,,7(.6 $4XDGUDWLFDQG VTXDUHURRWIXQFWLRQV7KH VWXGHQWIRUPXODWHVHTXDWLRQV DQGLQHTXDOLWLHVEDVHGRQ TXDGUDWLFIXQFWLRQVXVHVD YDULHW\RIPHWKRGVWRVROYH WKHPDQGDQDO\]HVWKH VROXWLRQVLQWHUPVRIWKH VLWXDWLRQ 7KHVWXGHQWLVH[SHFWHGWR %DQDO\]HDQGLQWHUSUHW WKHVROXWLRQVRITXDGUDWLF HTXDWLRQVXVLQJ GLVFULPLQDQWVDQGVROYH TXDGUDWLFHTXDWLRQVXVLQJ WKHTXDGUDWLFIRUPXOD >«ÌiÀÊn\Ê Và +ALOTONIC +APER 4EACHER .OTES 7KHVWDQGDUGIRUPRIDQHTXDWLRQRIDK\SHUERODZLWK FHQWHUDQGYHUWLFHVRQWKH[D[LVLV &RQQHFWLRQ WR7$.6 X Y A B 7KHHTXDWLRQVRIWKHDV\PSWRWHVDUH Y B B X AND Y X A A 7KHUHIRUHE DQGD 7KHHTXDWLRQRIWKHK\SHUERODLV X Y X Y 7KH.DORWRQLFEHDPZLOORULJLQDWHIURPWKHKLGLQJSODFHDW SRLQWDQGJRWRWKHIRFXVRIWKHLPDJLQDU\EUDQFK DW 2EMHFWLYH7KHVWXGHQW ZLOOGHPRQVWUDWHDQ XQGHUVWDQGLQJRIWKH SURSHUWLHVDQGDWWULEXWHVRI IXQFWLRQV 2EMHFWLYH7KHVWXGHQW ZLOOGHPRQVWUDWHDQ XQGHUVWDQGLQJRITXDGUDWLF DQGRWKHUQRQOLQHDU IXQFWLRQV 2EMHFWLYH7KHVWXGHQW ZLOOGHPRQVWUDWHDQ XQGHUVWDQGLQJRIWKH PDWKHPDWLFDOSURFHVVHV DQGWRROVXVHGLQSUREOHP VROYLQJ 7KHVORSHRIWKHOLQHMRLQLQJWKHVHSRLQWVLV >«ÌiÀÊn\Ê Và +ALOTONIC +APER 4EACHER .OTES 7KHHTXDWLRQRIWKLVEHDPLVDOLQHDVIROORZV 7R¿QGWKHSRLQWZKHUHWKHEHDPRI.DORWRQLFZLOOLQWHUVHFWWKHPLUURUWKHV\VWHPRI HTXDWLRQVK\SHUERODDQGWKHEHDPPXVWEHVROYHG7KHHTXDWLRQRIWKHEHDPLV VROYHGIRU\WKHUHIRUH[FDQEHIRXQGE\VXEVWLWXWLRQLQWRWKHK\SHUERODHTXDWLRQ X Y X X X X X X X X X X X X 8VHWKHTXDGUDWLFHTXDWLRQWRVROYHZLWKD E F B p B AC X A X p p X z OR X z X 7KHSRVLWLYHVROXWLRQ[ LVWKHRQO\UHDVRQDEOHVROXWLRQIRUWKLVVLWXDWLRQEHFDXVH RQO\KDOIRIWKHK\SHUERODUHSUHVHQWVWKHPLUURUHGVXUIDFH >«ÌiÀÊn\Ê Và +ALOTONIC +APER 4EACHER .OTES 7KLVYDOXHLVWKHQVXEVWLWXWHGLQWRWKHHTXDWLRQIRUWKH.DORWRQLFEHDP\ [ 7KHVROXWLRQVIRU[DQGWKHQ\DUHVKRZQRQWKHJUDSKLQJFDOFXODWRUVFUHHQEHORZ 7KHYDOXHRI\LVDSSUR[LPDWHO\ 7KHEHDPRI.DORWRQLFZLOOLQWHUVHFWWKHK\SHUEROLFPLUURUHGVXUIDFHDWDERXW >«ÌiÀÊn\Ê Và +ALOTONIC +APER 4EACHER .OTES ([WHQVLRQ4XHVWLRQ 6XSSRVH'U0DGGKDVHUUHGLQKLVFDOFXODWLRQVDQGSODFHGWKHPLUURUDWWKHZURQJ SRVLWLRQ+HGHWHUPLQHVKHPXVWUHORFDWHLWXQLWWRWKHZHVWRILWVFXUUHQWORFDWLRQ 7KHORFDWLRQRIKLVKLGLQJSODFHDQGKLVRULJLQDOWDUJHWWKHIRFXVRIWKHRULJLQDO LPDJLQDU\EUDQFKRIWKHK\SHUERODZLOOUHPDLQWKHVDPH'HWHUPLQHWKHFRRUGLQDWHV RIWKHSRLQWZKHUHWKHEHDPZLOOLQWHUVHFWWKHK\SHUEROLFPLUURU ,IWKHPLUURULVUHORFDWHGXQLWZHVWRILWVRULJLQDOSRVLWLRQWKHK\SHUERODLVVKLIWHG XQLWOHIWDORQJWKH[D[LV7KHFHQWHURIWKHQHZK\SHUEROLFPLUURULV7KH YHUWH[LVORFDWHGDWDQGWKHIRFXVDW 7KHHTXDWLRQRIWKHK\SHUEROLFPLUURUEHFRPHV X Y 7KHHTXDWLRQRIWKH.DORWRQLFEHDPUHPDLQVXQFKDQJHG\ [ 7RORFDWHWKHQHZSRLQWRILQWHUVHFWLRQ¿UVWPXOWLSO\WKHHTXDWLRQE\DQGWKHQ VXEVWLWXWH\IURPWKHEHDPLQWRWKHK\SHUEROLFHTXDWLRQ [±[ [[±[[ [[±[±[± [[± [[± [[± [ 2QO\[ PDNHVVHQVHIRUWKLVVLWXDWLRQ 7RGHWHUPLQHWKHYDOXHRI\VXEVWLWXWHIRU[LQWKHUXOHIRUWKHEHDP 7KHQHZSRLQWRILQWHUVHFWLRQLV \ >«ÌiÀÊn\Ê VÃ