Is Calculus Obsolete?

10 downloads 245 Views 113KB Size Report
domain. I want to consider two examples /, g and just to let you, the reader share the students inability (or unwillingn
,V &DOFXOXV 2EVROHWH" (G 'XELQVN\ 3XUGXH 8QLYHUVLW\ (GXFDWLRQ 'HYHORSPHQW &HQWHU 8QLYHUVLW\ RI 1HZ +DPSVKLUH 6R ZK\ GR ZH QHHG FDOFXOXV LI ZH KDYH DOO WKLV WHFKQRORJ\" , PHDQ LI \RX ZDQW PH WR nQG DOO WKH UHODWLYH PD[LPXP YDOXHV RI D IXQFWLRQ , OO MXVW GUDZ WKH JUDSK ]RRP LQ RQ WKH KXPSV DQG JHW DV FORVH DQ DSSUR[LPDWLRQ DV \RX ZDQW 'R \RX HYHU KHDU WKLV IURP \RXU VWXGHQWV" :KDW GR \RX DQVZHU" b @ 387 3/276  +(5( 6,'( %< 6,'( $ SUHWW\ UHDVRQDEOH nUVW JXHVV LV WKDW WKHVH WZR IXQFWLRQV DUH WKH VDPH EXW ZLWK GLmHUHQW VFDOHV _ SUREDEO\ WKH VDPH H[SUHVVLRQ ZLWK YHU\ GLmHUHQW FRHpFLHQWV ,W LV SUHWW\ FOHDU WKDW WKH [D[LV LV D KRUL]RQWDO DV\PSWRWH DQG WKDW D OLWWOH PRUH QHHGV WR EH GRQH WR PDNH VXUH DERXW ZKDW KDSSHQV QHDU WKH \D[LV /HW V ]RRP LQ DQG ORRN DW WKH WZR JUDSKV LQ WKH GRPDLQ >b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b   [  [ , WKLQN WKDW PRVW RI XV ZRXOG EH SUHWW\ KDSS\ LQ DQ HOHPHQWDU\ FRXUVH LI D VWXGHQW PDGH VXFK DQ LQYHVWLJDWLRQ DQG JDYH WKLV DQVZHU :KDW DERXW J" 7R VDYH VSDFH SHUKDSV \RX ZLOO WDNH P\ ZRUG IRU LW WKDW IRU VRPH WLPH DV WKH GRPDLQ LQWHUYDO JHWV VPDOOHU \RX JHW WKH VDPH JUDSK ,I D VWXGHQW PDGH VXFK DQ DQDO\VLV DQG FRQFOXGHG WKDW WKH JUDSK RI J ZDV SUHWW\ PXFK WKH VDPH DV WKDW RI I  EXW ZLWK GLmHUHQW VFDOLQJ RU SHUKDSV D KLJKHU H[SRQHQW \RX ZRXOG KDYH WR EH VDWLVnHG 6DPH DQDO\VLV VDPH DQVZHU %XW D UHDOO\ SHUVLVWHQW VWXGHQW PLJKW JR DV IDU DV WKH LQWHUYDO >b @ DQG JHW WKLV 387 3/27  +(5(  7KH UHDGHU ZLOO QRWLFH WKDW WKLV SDUWLFXODU WRRO SORWV RQO\ D GLVFUHWH VHW RI SRLQWV DQG GRHV QRW WU\ WR FRQQHFW WKHP ZLWK D

FRQWLQXRXV OLQH 7KH VDPSOLQJ RI SRLQWV LV ?LQHUDFWLYH LQ WKH VHQVH WKDW LQ UHJLRQV ZKHUH WKH IXQFWLRQ FKDQJHV PRUH UDSLGO\ PRUH SRLQWV DUH VDPSOHG , WKLQN WKHUH DUH JRRG SHGDJRJLFDO UHDVRQV IRU WKLV NLQG RI JUDSKLQJ SURWRFRO EXW WKDW LV DQRWKHU VWRU\ DQG \RX GRQ W KDYH WR DJUHH 7KH SRLQWV , ZLVK WR PDNH LQ WKLV DUWLFOH DUH MXVW DV YDOLG PD\EH HYHQ PRUH VR LI \RXU WRRO FRQQHFWV WKH SRLQWV VRPHKRZ



1RW WHUULEO\ LQIRUPDWLYH 7KH SUREOHP LV WKDW WKH JUDSK KDV D SRLQW DW DURXQG   FDQ \RX VHH LW" EXW PRVW RI WKH SRLQWV DUH PXFK FORVHU WR WKH [D[LV VR \RX GRQ W VHH YHU\ PXFK ,W GRHV VD\ WKDW J LV SUREDEO\ QRW YHU\ PXFK OLNH I  ,W DOVR VXJJHVWHV WKDW WKHUH LV VRPHWKLQJ JRLQJ RQ DURXQG [  ORRN DW KRZ WKH SRLQWV FOXVWHU VR PD\EH WKH VWXGHQW PLJKW WU\ WKH GRPDLQ >b @ 387 3/27  +(5( 1RZ ZH DUH JHWWLQJ VRPHZKHUH $IWHU ORRNLQJ DURXQG WKH VWXGHQW PLJKW FRQFOXGH WKDW WKH JUDSK ORRNV VRPHWKLQJ OLNH

)LUVW KDQG GUDZQ SLFWXUH 7KH GHVFULSWLRQ PLJKW WKHQ EH WKDW DJDLQ WKH [D[LV LV D KRUL]RQWDO DV\PSWRWH 7KH \D[LV LV DOVR D YHUWLFDO D\PSWRWH EXW ZLWK I LW ZDV RQ WKH QHJDWLYH VLGH DQG ZLWK J LW LV RQ WKH SRVLWLYH VLGH 0RUHRYHU WKHUH DUH WZR PRUH YHUWLFDO DV\PSWRWHV DW [ hD ZKHUH D LV VRPH IDLUO\ VPDOO SRVLWLYH QXPEHU $JDLQ WKHUH DUH QR WXUQLQJ SRLQWV EXW WKH FRQFDYLW\ GRHV FKDQJH IURP XS WR GRZQ DW WKH [ bD DV\PSWRWH DQG IURP GRZQ WR XS DW [ D 7KLV LV FHUWDLQO\ D PXFK ULFKHU UHVXOW RI WKRXJKWIXO XVH RI WKH JUDSKLFV WRRO ,W KDV WZR VHULRXV GUDZEDFNV  $V IDU DV , NQRZ WKHUH LV QR JUDSKLQJ WRRO WKDW ZLOO FUHDWH D JUDSK VXFK DV WKH DERYH ZKLFK ZDV GUDZQ E\ KDQG 7KH UHDVRQ LV WKDW LQ RUGHU WR VKRZ WKH YDULRXV IHDWXUHV YHU\ GLmHUHQW VFDOHV PXVW EH XVHG $ VFDOH WKDW LV DSSURSULDWH WR VKRZ RQH IHDWXUH VXFK DV ODUJH SRVLWLYH YDOXHV FORVH WR [  ZLOO FDXVH D VTXHH]LQJ LQ RWKHU SODFHV VXFK DV QHDU [  WKDW PD\ FRQFHDO LPSRUWDQW IHDWXUHV $OVR RQH FDQQRW VKRZ RQ WKH VDPH JUDSK WKH EHKDYLRU QHDU [  d b DQG QHDU [

 d b 

 7KH UHVXOW LV ZURQJ 1RZ ZH FDQ UHYHDO WKH GHnQLWLRQ RI J ,W LV JLYHQ E\

‚

J [

  b  [ [



$ OLWWOH FDOFXOXV ZLOO WHOO \RX YHU\ TXLFNO\ WKDW WKHUH LV D FULWLFDO SRLQW DW [  EXW DJDLQ \RX KDYH WR EH FDUHIXO LQ PDNLQJ D JUDSK ,I \RX XVH DQ LQWHUYDO OLNH >b @ WKHQ DJDLQ WKH ODUJH YDOXHV QHDU  ZLOO VTXHH]H RXW DOO WKH DFWLRQ QHDU  %XW LI \RX UHDOL]H 

IURP WKH H[SUHVVLRQ WKDW WKLV JUDSK ZLOO EH V\PPHWULF DERXW WKH \D[LV WKHQ \RX FDQ ORRN DW RQO\ RQH VLGH DQG JR D OLWWOH DZD\ IURP  E F@ ZKHUH E LV D VPDOO SRVLWLYH QXPEHU  @ ZKLFK JLYHV WKH IROORZLQJ JUDSK 387 3/27  +(5( $JDLQ WKH YDVWO\ GLmHUHQW VFDOHV SUHYHQW JUDSKLFV FXUYH DQG LW LV QHFHVVDU\ WR DQDO\]H SLHFHV DQG WKHQ XVLQJ FRQFHSWV IURP FDOFXOXV SRVVLEO\ ZLWK WKH KHOS FRPH XS ZLWK DQ\WKLQJ OLNH WKH IROORZLQJ VROXWLRQ WR

VRIWZDUH WR SUHVHQW D VLQJOH SLFWXUH RI WKH SXW WKHP WRJHWKHU ,W VHHPV WKDW RQO\ E\ RI JUDSKLQJ VRIWZDUH LV D VWXGHQW OLNHO\ WR WKH RULJLQDO SUREOHP

6HFRQG KDQG GUDZQ SLFWXUH $V D nQDO FRPPHQW WKH UHDGHU PLJKW WKLQN WKLV LV D SUREOHP PDGH XS WR LOOXVWUDWH P\ SRLQW 1RW VR , PDGH XS WKH FRHpFLHQWV  EXW WKH EDVLF H[SUHVVLRQ IRU J GHVFULEHV D FKHPLFDO UHDFWLRQ EHWZHHQ WZR PROHFXOHV LQ D FHUWDLQ VXEVWDQFH , WRRN LW IURP D &KHPLVWU\ ERRN , ZRQGHU LI WKDW PDWWHUV

 ZLWK VRPH KHOS IURP $QG\ *OHDVRQ