TIInterActive File) Funktionsundersgelse, asymptoter og lokale ekstrema Extrema:  Asymptoter: Monotoni:  Ovenfor er der tre MathBoxe med brugerdefinerede funktioner, som kan finde asymptoter, monotoniforhold og lokale ekstrema. Dobbeltklik p en af dem, hvis du vil se hvordan de er programmerede. Du kan frit kopiere dem over i egne TI InterActive dokumenter. Du kan ogs rette direkte i forskriften for f nedenfor. Se nedenfor om matematikken bag. Eksempel:     Matematikken bag En funktion f har skr asymptote med ligning y = ax + b hvis  I s fald har vi, at  hvoraf  Vi har videre, at  Asymptotens ligning kan alts bestemmes ud fra de to sidstnvnte grnseovergange. Omvendt ses let, at hvis de to sidstnvnte grnsevrdier er endelige tal, s har f skr asymptote med ligning y = ax + b. Denne fremgangsmde bruges til bestemmelse af skr asymptoter i den brugerdefinerede funktion, asymptoter. Ved samme lejlighed fanges eventuelle vandrette asymptoter, som kan opfattes som det specialtilflde af skr asymptote, hvor asymptotens hldning bliver nul. Metoden er rimeligt generel og kan fange skr asymptoter, som normalt ikke prsenteres p gymnasialt niveau, for eksempel  En polynomiumsbrk har lodrette asymptoter i nvnerens nulpunkter, sfremt disse ikke samtidig er nulpunkter i tlleren. Hvis en polynomiumsbrk har en fllesrod i nvner og tller, kan brken forkortes med (x - denne fllesrod). Dette gr TI InterActive automatisk, og nr brken er forkortet s meget som muligt p denne mde, er tilbagevrende nulpunkter i nvneren alle sammen lodrette asymptoter. Funktionen, asymptoter, finder ikke lodrette asymptoter i forbindelse med ln(x) og med tan(x) Lokale ekstrema I alle lrebger til gymnasialt B niveau kan lses, at det er en ndvendig betingelse for lokalt ekstremum i x = x0, at  Sdvanligvis lses denne ligning, hvorefter en monotoniundersgelse afslrer, hvilke af lsningerne der er lokale ekstrema. Det er ikke nemt at automatisere en monotoniundersgelse i TI InterActive, men vi kan i stedet for g frem som flger: I videregende matematikbger kan ses, at det er en tilstrkkelig betingelse for lokalt ekstremum i x = x0, at differentialkvotienten er lig med nul i x0, og at den anden afledede ikke er lig med nul i x0.  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3d028000 20 0/33 10001 0 0 0 0 13601 0 10000 0 0 0 0 1e000000d@?Define extrema(f,x)=Func ::Local e,n,r ::[x;y]=:r ::zeros(derivative(f,x),x)=:e ::For n,1,dim(e) :: If (derivative(f,x,2)|x=e[n]) /= 0 Then :: augment(r,[e[n];f|x=e[n]])=:r :: EndIf ::EndFor ::Return r ::EndFuncTimes New Roman "Done"Times New Roman = = x),x)=:e ::For n,1,dim(e) :: If (derivative(f,x,2)|x=e[n]) /= 0 Then :: augment(r,[e[n];f|x=e[n]])=:r :: EndIf ::EndFor ::Return r ::EndFuncTimes New Roman "Done"Times New Roman = wK x  -- 2 2< 2 I- --!:!:! .! L!Ij!f! - -- 2  2 Z- --!d!e?!rt!i M &   Z .Times New Roman؟ww wk fW- --Symbol ؟ww wk fW- --!=<! <4- --!"CR!DCx!oC!nC!eC<!"Cq''NANI!:- --#^#88^- --!x^!;!y- --##  - --!=!:Z!rq!:!:!z.!ej=Y @dd;7 CTxobjItemࡱ> 5  !"#$%&'()*+,-./0123489:;<=>?@ABCDEFGHIJK6Root EntryZn@3ContentsdOlePres0007*Z4a5b 64 64 1 1 1 0 ffff0007 b0002 796c4743 74536870 88656c79 ffffff 0 0 90000000 1 7010000 54000000 73656d69 77654e20 6d6f5220 6e61 0 0 0 0 ff00ff00 7800ffff 64000000 8000000 6e617453 64726164 0 1 80010006 ffffff88 0 0 0 190 1000000 7 626d7953 6c6f 0 0 0 0 0 0 ff0000 ffffff 78 64 64745309 626d7953 16c6f 10000 0 8001 0 0 0 2bc0000 0 70100 79530000 6c6f626d 0 0 0 0 0 0 0 ffff00ff 7800ff 640000 530d0000 79536474 6c6f626d 646c6f42 1 1 80010000 0 0 0 0 190 1000001 7 656d6954 654e2073 6f522077 6e616d 0 0 0 0 ff0000 ffffff 78 64 64745309 6c617449 16369 10000 0 8001 0 0 0 2bc0000 10000 70100 69540000 2073656d 2077654e 616d6f52 6e 0 0 0 0 ffff00ff 7800ff 640000 530d0000 74496474 63696c61 646c6f42 1 1 80010000 0 0 0 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71 0 1ffff 10000 0 10650000 2680 8f00 7100 0 1ffff00 1000000 0 22000000 0 0 ffff0000 140001 796c4743 69536870 656c706d 706d6f43 7265736f 0 0 48a 12c 0 1ffff 4743000a 6870796c 6d726554 1 8f ffffffff 0 1ffff 10000 0 0 0 0 60018 60000 60000 60000 0 2360000 cb0000 ffff0000 ffffffff ffffffff 1 1 0 0 0 1 0 3 454d4305 25453 3d028000 20 0(33 10001 0 0 0 0 13601 0 10000 0 0 0 0 0d@?Define asymptoter(f,x)=Func ::Local a,b,c,d,r,e,n ::{}=:r::limit(f/x,x,inf)=:a ::If -inf < a and a < inf Then :: limit(f-a*x,x,inf)=:b :: If -inf < b and b < inf Then :: {y=a*x+b}=:r :: EndIf ::EndIf ::limit(f/x,x,-inf)=:c ::If -inf < c and c < inf Then :: limit(f-c*x,x,-inf)=:d :: If -inf < d and d < inf Then :: If a /= c or b /= d Then :: augment(r,{y=c*x+d})=:r :: EndIf :: EndIf ::EndIf ::zeros(getDenom(f),x)=:e ::For n,1,dim(e),1 ::augment(r,{x=e[n]})=:r ::EndFor ::Return r ::EndFuncTimes New Roman "Done"Times New Roman = = ment(r,{y=c*x+d})=:r :: EndIf :: EndIf ::EndIf ::zeros(getDenom(f),x)=:e ::For n,1,dim(e),1 ::augment(r,{x=e[n]})=:r ::EndFor ::Return r ::EndFuncTimes New Roman "Done"Times New Roman = ~ @9 NANI') 4؟ww wi fQ- -- 2 C>uG 2 C>Gu- --!fu!,!x- -- 2 G 2 G- --!=!Fa!u!n!c!:!:!L.!')    @ g@ g.Times New Roman؟ww w f- --!D!eS!f!i!n!e! 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L!Ij!f! !-!i!n!fE! c!<! !d! !a=!nr!d! !d! =!<[! !i!n!f ! '!TE!h!e!n!:C!:C! C.! CL! Cj! C!IC!fC! C!aC! C=!/C[!=Cy! C!cC! C! C.!oCL!rC! C!bC! C!/C&!=CD! C!dC! C!TC!hCD!eCy!nC!:!:! .! L! j! ! ! !a!u!gL!m!e!n !t>- -- 2 a\ 2 a\a- --!r!,- -- 2  2 F 2 7a- --!y!=R!c!*!x!+<!d- -- 2  2 F 2 7a 2 9 2 9a- --!=9!:}!r!:a!:a! a.! aL! aj! a!Ea!na!da!Ia[!fa!:!:! .! L!Ej!n!d!I!fE!: !: !E .!n r!d !I !f !: !: !z .!e [!r !o !s - -- 2  $ VU 2  $U V - --!g V!e !t !D !e 1!n f!o !m - -- 2  /* aU 2  /*U a - --!f a- -- 2   U 2  U  - --!, !x - -- 2    @U 2   U @ - --!= @!: !e !: !: !F .!o r!r ! !n !, '!1 E!, !d !i !m - -- 2 JE | 2 JE |, - --!e |- -- 2   2  , - --!, !1 !:, !:, !a, .!u, c!g, !m, !e, !n, U!t, - -- 2 " , s 2 9 s  - --!r, !,, - -- 2 " (#, iU 2 - (#F i 2 9 (# i - --!x, i!=, !e, - --1 G1 ! ! G- --!n, G- --1 |1   | 2 " , U 2 - F  2 9   2 " , )s 2 9 s ) - --!=, )!:, m!r, !: !: !E .!n r!d !F !o '!r c!:J !:J !RJ .!eJ y!tJ !uJ !rJ !nJ '! J \!rJ z!: !: !E .!n r!d !F !u '!n \!c ''jY @dd;7 CTxobjItem'ࡱ> p  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnRoot EntryZn@3fdLzqContentsOlePres000a5e3 64 64 1 1 1 0 ffff0007 b0002 796c4743 74536870 88656c79 ffffff 0 0 90000000 1 7010000 54000000 73656d69 77654e20 6d6f5220 6e61 0 0 0 0 ff00ff00 7800ffff 64000000 8000000 6e617453 64726164 0 1 80010006 ffffff88 0 0 0 190 1000000 7 626d7953 6c6f 0 0 0 0 0 0 ff0000 ffffff 78 64 64745309 626d7953 16c6f 10000 0 8001 0 0 0 2bc0000 0 70100 79530000 6c6f626d 0 0 0 0 0 0 0 ffff00ff 7800ff 640000 530d0000 79536474 6c6f626d 646c6f42 1 1 80010000 0 0 0 0 190 1000001 7 656d6954 654e2073 6f522077 6e616d 0 0 0 0 ff0000 ffffff 78 64 64745309 6c617449 16369 10000 0 8001 0 0 0 2bc0000 10000 70100 69540000 2073656d 2077654e 616d6f52 6e 0 0 0 0 ffff00ff 7800ff 640000 530d0000 74496474 63696c61 646c6f42 1 1 80010000 0 0 0 0 190 1000000 7 4d204954 687461 0 0 0 0 0 0 ff0000 ffffff 78 64 64745305 14954 10000 0 8001 0 0 0 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ffff0000 140001 796c4743 69536870 656c706d 706d6f43 7265736f 0 0 456 10a4 0 1ffff 4743000a 6870796c 6d726554 1 8f ffffffff 0 1ffff 10000 0 0 0 0 60018 60000 60000 60000 0 2360000 cb0000 ffff0000 ffffffff ffffffff 1 1 0 0 0 1 0 3 454d4305 25453 3d028000 20 0'33 10001 0 0 0 0 3601 0 10000 0 0 0 0 0d@@Define monotoni(f,x)=Func ::Local r,t,n,a,d,i,s,x ::d:=derivative(f,x) ::r:=[[fra_x,til_x,y',y]] ::n:=zeros(getDenom(d),x) ::t:=zeros(getNum(d),x) ::a:=augment(n,t) ::sortA(a) ::If dim(a)=0 Then :: s:=[- inf , inf ] :: If (d|x=1)>0 Then :: s:=augment(s,[pos,voks]) :: Else :: s:=augment(s,[neg,aft]) :: EndIf :: r:=augment(r;s) ::ElseIf dim(a)=1 Then :: s:=[- inf ,a[1]] :: If (d|x=a[1]-1)>0 Then :: s:=augment(s,[pos,vok]) :: Else :: s:=augment(s,[neg,aft]) :: EndIf :: r:=augment(r;s) :: s:=[a[1], inf ] :: If (d|x=a[1]+1)>0 Then :: s:=augment(s,[pos,voks]) :: Else :: s:=augment(s,[neg,aft]) :: EndIf :: r:=augment(r;s) ::Else :: s:=[- inf ,a[1]] :: If (d|x=a[1]-1)>0 Then :: s:=augment(s,[pos,voks]) :: Else :: s:=augment(s,[neg,aft]) :: EndIf :: r:=augment(r;s) :: For i,1,dim(a)-1 :: s:=[a[i],a[i+1]] :: If (d|x=(a[i]+a[i+1])/2)>0 :: Then :: s:=augment(s,[pos,voks]) :: Else :: s:=augment(s,[neg,aft]) :: EndIf :: r:=augment(r;s) :: EndFor :: s:=[a[dim(a)], inf ] :: If (d|x=a[dim(a)]+1)>0 Then :: s:=augment(s,[pos,voks]) :: Else :: s:=augment(s,[neg,aft]) :: EndIf :: r:=augment(r;s) ::EndIf ::Return r ::EndFuncTimes New Roman "Done"Times New Roman = = ]+a[i+1])/2)>0 :: Then :: s:=augment(s,[pos,voks]) :: Else :: s:=augment(s,[neg,aft]) :: EndIf :: r:=augment(r;s) :: EndFor :: s:=[a[dim(a)], inf ] :: If (d|x=a[dim(a)]+1)>0 Then :: s:=augment(s,[pos,voks]) :: Else :: s:=augment(s,[neg,aft]) :: EndIf :: r:=augment(r;s) ::EndIf ::Return r ::EndFuncTimes New Roman "Done"Times New Roman = 0 1000000 1000000 0 61000000 35800d 8f0000 710000 0 0 1 1 0 75 b000000 14780 8f00 7100 0 ffff00 1000000 0 6000000 35800d00 8f000000 71000000 0 0 100 100 0 800d6700 53 8f 71 0 10000 10000 0 d6d0000 3580 8f00 7100 0 1000000 1000000 0 65000000 35800d 8f0000 710000 0 0 1 1 0 1e800d6e 8f000000 71000000 0 0 100 100 0 801f7400 37 8f ffffffff 0 10001 10000 0 280000 0 a9800b00 8f000000 71000000 0 ff00Z   M @@.Times New Roman؟ww w}f- --!"<!D<!o<i!n<!e<!"<''NANIjj&@dd5;676 CTxobjItem6ࡱ>  Root EntryZn@3@#4@ContentsLOlePres000 !11b7 64 64 1 1 1 0 ffff0007 b0002 796c4743 74536870 b1656c79 ffffff 0 0 90000000 1 7010000 54000000 73656d69 77654e20 6d6f5220 6e61 0 0 0 0 ff00ff00 7800ffff 64000000 8000000 6e617453 64726164 0 1 80010006 ffffff88 0 0 0 190 1000000 7 626d7953 6c6f 0 0 0 0 0 0 ff0000 ffffff 78 64 64745309 626d7953 16c6f 10000 0 8001 0 0 0 2bc0000 0 70100 79530000 6c6f626d 0 0 0 0 0 0 0 ffff00ff 7800ff 640000 530d0000 79536474 6c6f626d 646c6f42 1 1 80010000 ffffff88 0 0 0 190 1000001 7 656d6954 654e2073 6f522077 6e616d 0 0 0 0 ff0000 ffffff 78 64 64745309 6c617449 16369 10000 0 8001 0 0 0 2bc0000 10000 70100 69540000 2073656d 2077654e 616d6f52 6e 0 0 0 0 ffff00ff 7800ff 640000 530d0000 74496474 63696c61 646c6f42 1 1 80010000 0 0 0 0 190 1000000 7 4d204954 687461 0 0 0 0 0 0 ff0000 ffffff 78 64 64745305 14954 10000 0 8001 0 0 0 2bc0000 0 70100 49540000 74614d20 68 0 0 0 0 0 0 ffff00ff 7800ff 640000 53090000 49546474 646c6f42 1 1 30000 50004 70006 3930008 15e0000 de0000 0 0 0 1 0 1ffff 4743000a 6870796c 4854414d 393 15e de 0 ffff 10000 0 10000 1ffff 4743000a 6870796c 574f524d 393 15e de 0 ffff 10000 0 50000 1ffff 4743000e 6870796c 636e7546 656d614e 4b 8f 71 0 ffff 10000 0 10000 4b800b 8f0000 710000 0 ffff0000 0 1 0 ffff0001 a0001 796c4743 68436870 4b7261 8f0000 710000 0 30000 1 1 0 66 0 ff000000 140001ff 6c474300 53687079 6c706d69 6d6f4365 65736f70 72 0 124800 b4000 0 1ffff00 43000d00 70796c47 45464d68 4445434e ac 8f 71 0 ffff 10000 0 10000 ac800b 8f0000 710000 0 ffff0000 0 1 0 ffff0004 f0001 796c4743 61506870 4c6e6572 37746665 8f000000 ff000000 ffffff 1000000 100 100 0 80092800 3d 8f 71 0 ffff 10000 0 10000 3d800b 8f0000 710000 0 ffff0000 0 1 0 ffff0002 80001 796c4743 494d6870 3c 8f 71 0 ffff 10000 0 10000 3c800b 8f0000 710000 0 ffff0000 0 1 0 80100001 3c 8f 71 0 10003 10000 0 780000 0 0 1300 ff000000 a0001ff 6c474300 54687079 16d7265 8f000000 ff000000 ffffff ff000000 1ff 100 0 0 0 13000000 1ffff00 43001000 70796c47 72615068 69526e65 37746867 8f000000 ff000000 ffffff 1000000 100 100 0 801f2900 1 8f ffffffff 0 1ffff 10000 0 0 0 0 10013 180000 190022 1ffff 47430008 6870796c 974f4d 8f0000 710000 0 ffff0000 1 1 0 800b0001 97 8f 71 0 ffff 10000 0 40000 1e8010 8f0000 710000 0 ffff0000 0 1 0 17801020 8f000000 71000000 0 ff000000 ff 100 0 80103a00 44 8f 71 0 ffff 10000 0 103d0000 1e80 8f00 7100 0 ffff00 1000000 0 20000000 0 0 0 0 1ffff 4743000b 6870796c 4152464d 20443 15e00 de00 0 1ffff00 1000000 0 3000000 4801400 b1000002 93000000 0 ff000000 ff 100 0 b000100 80 0 0 0 ffff00 1000000 0 4000000 37801700 96000000 ff000000 ffffff 1000000 100 0 0 80092800 1cd b1 93 0 ffff 10000 0 10000 1cd800b b10000 930000 0 ffff0000 0 1 0 ffff0006 90001 796c4743 77506870 6b72 b100 9300 0 1ffff00 1000000 0 2000000 3d801400 8f000000 71000000 0 ff000000 ff 100 0 b000100 80 0 0 0 ffff00 1000000 0 4000000 801700 0 0 0 1000000 100 0 0 80092800 3d 8f 71 0 ffff 10000 0 10000 3d800b 8f0000 710000 0 ffff0000 0 1 0 801b0002 3c 8f 71 0 ffff 10000 0 10000 3c800b 8f0000 710000 0 ffff0000 0 1 0 80100001 3c 8f 71 0 10003 10000 0 780000 0 0 1300 1f000000 180 8f00 ffffff00 ff 1ffff00 1000000 0 0 0 0 21001300 80 0 0 0 1000100 0 0 29000000 801f 0 0 0 ffff0000 1 0 0 0 0 130000 0 3d0036 80140037 2e 69 53 0 ffff 10000 0 10000 800b 0 0 0 ffff0000 0 1 0 80170004 0 0 0 0 10001 0 0 9280000 2e80 6900 5300 0 ffff00 1000000 0 1000000 2e800b00 69000000 53000000 0 ff000000 ff 100 0 ff000200 80001ff 6c474300 4d687079 2d4e 6900 5300 0 ffff00 1000000 0 1000000 2d800b00 69000000 53000000 0 ff000000 ff 100 0 10000100 2d80 6900 5300 0 1ffff00 1000000 0 32000000 0 0 130000 0 1801f 690000 ffff0000 ffff ffff0000 1 1 0 0 0 130000 8021 0 0 0 10000 1 0 0 801f29 0 0 0 ff000000 1ff 0 0 0 0 13000000 0 49004100 4200 0 80802400 8f000000 71000000 0 ff000000 1ff 100 0 b000100 8080 8f00 7100 0 ffff00 1000000 0 3000000 1e801000 8f000000 71000000 0 ff000000 ff 100 0 80102000 44 8f 71 0 ffff 10000 0 102b0000 1e80 8f00 7100 0 ffff00 1000000 0 20000000 0 0 0 0 3c801b 8f0000 710000 0 ffff0000 0 1 0 800b0001 3c 8f 71 0 ffff 10000 0 10000 3c8010 8f0000 710000 0 30000 1 1 0 78 0 13000000 0 69802400 8f000000 71000000 0 ff000000 1ff 100 0 b000100 6980 8f00 7100 0 ffff00 1000000 0 3000000 1e801000 8f000000 71000000 0 ff000000 ff 100 0 80102000 2d 8f 71 0 ffff 10000 0 102d0000 1e80 8f00 7100 0 ffff00 1000000 0 20000000 0 0 0 0 3c8044 8f0000 710000 0 ffff0000 0 1 0 800b0001 3c 8f 71 0 ffff 10000 0 10000 3c8010 8f0000 710000 0 ffff0000 1 1 0 36 0 13000000 0 1801f00 b1000000 ff000000 ffffff ff000000 1ff 100 0 0 0 13000000 802100 0 0 0 1000000 100 0 0 801f2900 0 0 0 0 1ffff 0 0 0 0 0 30013 2f0000 30005c 1ffff 4743000b 6870796c 6e694c48 20465 1e00 f00 ffffff00 ffffff 1000000 0 14000000 11980 8f00 7100 0 ffff00 1000000 0 1000000 800b00 0 0 0 ff000000 ff 100 0 17000400 3780 9600 ffffff00 ff 1000100 0 0 28000000 e28009 8f0000 710000 0 ffff0000 0 1 0 800b0001 e2 8f 71 0 ffff 10000 0 40000 3c801b 8f0000 710000 0 ffff0000 0 1 0 800b0001 3c 8f 71 0 ffff 10000 0 10000 3c8010 8f0000 710000 0 30000 1 1 0 78 0 13000000 0 69802400 8f000000 71000000 0 ff000000 1ff 100 0 b000100 6980 8f00 7100 0 ffff00 1000000 0 3000000 1e801000 8f000000 71000000 0 ff000000 ff 100 0 80102000 2d 8f 71 0 ffff 10000 0 102d0000 1e80 8f00 7100 0 ffff00 1000000 0 20000000 0 0 0 0 3c8044 8f0000 710000 0 ffff0000 0 1 0 800b0001 3c 8f 71 0 ffff 10000 0 10000 3c8010 8f0000 710000 0 ffff0000 1 1 0 35 0 13000000 0 1801f00 8f000000 ff000000 ffffff ff000000 1ff 100 0 0 0 13000000 802100 0 0 0 1000000 100 0 0 801f2900 0 0 0 0 1ffff 0 0 0 0 0 30013 620000 630071 0 0 801f0000 1 15e ffffffff 0 1ffff 10000 0 0 0 0 220013 220000 220000 220000 0 3930000 15e0000 ffff0000 ffffffff ffffffff 1 1 0 0 0 1 0 3 454d4305 25453 8000 05f6 64 64 1 1 1 0 ffff0007 b0002 796c4743 74536870 88656c79 ffffff 0 0 90000000 1 7010000 54000000 73656d69 77654e20 6d6f5220 6e61 0 0 0 ff000000 ff000000 7800ffff 64000000 8000000 6e617453 64726164 0 1 80010006 ffffff88 0 0 0 190 1000000 7 626d7953 6c6f 0 0 0 0 0 0 ff ffffff 78 64 64745309 626d7953 16c6f 10000 0 8001 0 0 0 2bc0000 0 70100 79530000 6c6f626d 0 0 0 0 0 0 ff0000 ffff0000 7800ff 640000 530d0000 79536474 6c6f626d 646c6f42 1 1 80010000 ffffff88 0 0 0 190 1000001 7 656d6954 654e2073 6f522077 6e616d 0 0 0 0 ff ffffff 78 64 64745309 6c617449 16369 10000 0 8001 0 0 0 2bc0000 10000 70100 69540000 2073656d 2077654e 616d6f52 6e 0 0 0 ff0000 ffff0000 7800ff 640000 530d0000 74496474 63696c61 646c6f42 1 1 80010000 0 0 0 0 190 1000000 7 4d204954 687461 0 0 0 0 0 0 ff ffffff 78 64 64745305 14954 10000 0 8001 0 0 0 2bc0000 0 70100 49540000 74614d20 68 0 0 0 0 0 ff0000 ffff0000 7800ff 640000 53090000 49546474 646c6f42 1 1 30000 50004 70006 2980008 d20000 780000 0 0 0 1 0 1ffff 4743000a 6870796c 4854414d 1a8 96 78 0 ffff 10000 0 10000 1ffff 4743000a 6870796c 574f524d 1a8 96 78 0 ffff 10000 0 30000 1ffff 4743000b 6870796c 696c6544 626d 9600 7800 0 ffff00 1000000 0 1000000 62800b00 96000000 78000000 0 ff000000 ff 100 0 ff000200 a0001ff 6c474300 43687079 44726168 96000000 78000000 0 1000000 100 100 0 80103d00 1e 96 78 0 10001 10000 0 200000 0 0 1ffff00 43001400 70796c47 6d695368 43656c70 6f706d6f 726573 0 58000000 9c000011 9 ff000000 80001ff 6c474300 4d687079 14553 8f00 7100 0 ffff00 1000000 0 1000000 45800b00 8f000001 71000000 0 ff000000 ff 100 0 10000600 2680 8f00 7100 0 1ffff00 1000000 0 22000000 538010 8f0000 710000 0 ffff0000 1 1 0 3c801044 8f000000 71000000 0 ff000000 1ff 100 0 80106f00 35 8f 71 0 1ffff 10000 0 106e0000 3580 8f00 7100 0 1ffff00 1000000 0 65000000 268010 8f0000 710000 0 ffff0000 1 1 0 22 0 14000000 0 1ffff00 43000a00 70796c47 72655468 16d 8f00 ffffff00 ff 1ffff00 1000000 0 0 0 0 8001400 8000000 8000000 8000000 0 5b000000 7f000004 ff000001 ffffffff ffffffff 1ff 100 0 0 0 100 0 300 4d430500 2545345 2800000 1203d 30300000(33 10001 0 0 0 0 13601 0 10000 0 0 0 0 0d@@"f(x) := ((x)^(2) + x - 6)/(x - 5)Times New Roman "Done"Times New Roman = 2 0 14000000 0 1802900 69000000 ff000000 ffffff ff000000 1ff 100 0 0 0 14000000 802b00 0 0 0 1000000 100 0 0 80292900 0 0 0 0 1ffff 0 0 0 0 0 14 300000 310038 0 ffff0000 80001 796c4743 4f4d6870 80 8f 71 0 1ffff 10000 0 10000 80800b 8f0000 710000 0 ffff0000 0 1 0 80100003 1e 8f 71 0 ffff 10000 0 10200000 4480 8f00 7100 0 ffff00 1000000 0 2b000000 1e8010 8f0000 710000 0 ffff0000 0 1 0 20 0 0 0 3c802500 8f000000 71000000 0 ff000000 ff 100 0 b000100 3c80 8f00 7100 0 ffff00 1000000 0 1000000 3c801000 8f000000 71000000 0 3000000 100 100 0 7800 0 0 14 803a0000 69 8f 71 0 1ffff 10000 0 10000 69800b 8f0000 710000 0 ffff0000 0 1 0 80100003 1e 8f 71 0 ffff 10000 0 10200000 2d80 8f00 7100 0 ffff00 1000000 0 2d000000 1e8010 8f0000 710000 0 ffff0000 0 1 0 20 0 0 0 3c803300 8f000000 71000000 0 ff000000 ff 100 0 b000100 3c80 8f00 7100 0 ffff00 1000000 0 1000000 3c801000 8f000000 71000000 0 ff000000 1ff 100 0 3600 0 0 14 80290000 1 8f ffffffff 0 1ffff 10000 0 0 0 0 802b0014 0 0 0 0 10001 0 0 29290000 80 0 0 0 1ffff00 0 0 0 0 0 3001400 1b000000 1c004c00 1ffff00 43000b00 70796c47 694c4868 656e 1e0000 f0000 ffff0000 ffffffff 0 1 0 1198017 8f0000 710000 0 ffff0000 0 1 0 800b0001 0 0 0 0 ffff 10000 0 40000 37801a 960000 ffff0000 ffff 10000 1 0 0 e2800928 8f000000 71000000 0 ff000000 ff 100 0 b000100 e280 8f00 7100 0 ffff00 1000000 0 4000000 3c802500 8f000000 71000000 0 ff000000 ff 100 0 b000100 3c80 8f00 7100 0 ffff00 1000000 0 1000000 3c801000 8f000000 71000000 0 3000000 100 100 0 7800 0 0 14 803a0000 69 8f 71 0 1ffff 10000 0 10000 69800b 8f0000 710000 0 ffff0000 0 1 0 80100003 1e 8f 71 0 ffff 10000 0 10200000 2d80 8f00 7100 0 ffff00 1000000 0 2d000000 1e8010 8f0000 710000 0 ffff0000 0 1 0 20 0 0 0 3c803300 8f000000 71000000 0 ff000000 ff 100 0 b000100 3c80 8f00 7100 0 ffff00 1000000 0 1000000 3c801000 8f000000 71000000 0 ff000000 1ff 100 0 3400 0 0 14 80290000 1 8f ffffffff 0 1ffff 10000 0 0 0 0 802b0014 0 0 0 0 10001 0 0 29290000 80 0 0 0 1ffff00 0 0 0 0 0 3001400 52000000 53006100 0 0 29000000 180 8f00 ffffff00 ff 1ffff00 1000000 0 0 0 0 1d001400 1d000000 1d000000 1d000000 0 5b000000 7f000004 ff000001 ffffffff ffffffff 1ff 100 0 0 0 100 0 300 4d430500 2545345 2800000 1203d 0(33 10001 0 0 0 0 13601 0 10000 0 0 0 0 0d@@f := ((x)^(2) + x - 6)/(x - 4)Times New Roman ((x)^(2) + x - 6)/(x - 4)Times New Roman = 0 0 14000000 1d00 1d00 1d00 1d00 0 45b00 17f00 ffffff00 ffffffff 1ffffff 1000000 0 0 0 1000000 0 3000000 5000000 53454d43 254 203d0280 1 0(33 10001 0 0 0 0 13601 0 10000 0 0 0 0 0d@@f := ((x - 2)*(x + 3))/(x - 4)Times New Roman ((x - 2)*(x + 3))/(x - 4)Times New Roman = = T XNANI"(R:@dd ;(7( CTxobjItem(ࡱ>  Root EntryZn@3@#4@ContentsOlePres000+ a79 64 64 1 1 1 0 ffff0007 b0002 796c4743 74536870 88656c79 ffffff 0 0 90000000 1 7010000 54000000 73656d69 77654e20 6d6f5220 6e61 0 0 0 0 ff00ff00 7800ffff 64000000 8000000 6e617453 64726164 0 1 80010006 ffffff88 0 0 0 190 1000000 7 626d7953 6c6f 0 0 0 0 0 0 ff0000 ffffff 78 64 64745309 626d7953 16c6f 10000 0 8001 0 0 0 2bc0000 0 70100 79530000 6c6f626d 0 0 0 0 0 0 0 ffff00ff 7800ff 640000 530d0000 79536474 6c6f626d 646c6f42 1 1 80010000 ffffff88 0 0 0 190 1000001 7 656d6954 654e2073 6f522077 6e616d 0 0 0 0 ff0000 ffffff 78 64 64745309 6c617449 16369 10000 0 8001 0 0 0 2bc0000 10000 70100 69540000 2073656d 2077654e 616d6f52 6e 0 0 0 0 ffff00ff 7800ff 640000 530d0000 74496474 63696c61 646c6f42 1 1 80010000 0 0 0 0 190 1000000 7 4d204954 687461 0 0 0 0 0 0 ff0000 ffffff 78 64 64745305 14954 10000 0 8001 0 0 0 2bc0000 0 70100 49540000 74614d20 68 0 0 0 0 0 0 ffff00ff 7800ff 640000 53090000 49546474 646c6f42 1 1 30000 50004 70006 3d90008 8f0000 710000 0 0 0 1 0 1ffff 4743000a 6870796c 4854414d 3d9 8f 71 0 ffff 10000 0 10000 1ffff 4743000a 6870796c 574f524d 3d9 8f 71 0 ffff 10000 0 30000 1ffff 4743000e 6870796c 636e7546 656d614e 1f9 8f 71 0 ffff 10000 0 10000 1f9800b 8f0000 710000 0 ffff0000 0 1 0 ffff000a a0001 796c4743 68436870 357261 8f0000 710000 0 ffff0000 1 1 0 2d801061 8f000000 71000000 0 ff000000 1ff 100 0 80107300 35 8f 71 0 1ffff 10000 0 10790000 5380 8f00 7100 0 1ffff00 1000000 0 6d000000 3c8010 8f0000 710000 0 ffff0000 1 1 0 1e801070 8f000000 71000000 0 ff000000 1ff 100 0 80107400 3c 8f 71 0 1ffff 10000 0 106f0000 1e80 8f00 7100 0 1ffff00 1000000 0 74000000 358010 8f0000 710000 0 ffff0000 1 1 0 26801065 8f000000 71000000 0 ff000000 1ff 100 0 7200 0 0 1ffff 47430014 6870796c 706d6953 6f43656c 736f706d 7265 0 12480000 b400000 0 ffff0000 d0001 796c4743 464d6870 45434e45 1df44 8f00 7100 0 ffff00 1000000 0 1000000 df800b00 8f000001 71000000 0 ff000000 ff 100 0 ff000400 f0001ff 6c474300 50687079 6e657261 7466654c 37 8f ffffffff 0 10001 10000 0 9280000 17080 8f00 7100 0 ffff00 1000000 0 1000000 70800b00 8f000001 71000000 0 ff000000 ff 100 0 d000500 4b80 8f00 7100 0 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35800d74 8f000000 71000000 0 ff000000 1ff 100 0 800d6100 35 8f 71 0 1ffff 10000 0 ff780000 a0001ff 6c474300 54687079 16d7265 8f000000 ff000000 ffffff ff000000 1ff 100 0 0 0 ff000000 140001ff 6c474300 53687079 6c706d69 6d6f4365 65736f70 72 0 115800 99400 1300 1300 1300 1300 0 0 0 0 0 1ffff00 1000000 0 0 0 1000000 0 3000000 5000000 53454d43 254 203d0280 1 0/33 10001 0 0 0 0 13601 0 10000 0 0 0 0 30000000d@@.0 /= ((f(x))| x = ~"`msub_((x),(0))",(x),2)Times New Roman INPUT ERROR: SyntaxTimes New Roman = = ljeNANI y Times New RomanTimes New RomanTimes New RomanTimes New RomanTimes New RomandR@;(T7( CTxobjItem(ࡱ>  Root Entry{NW` 5pContentsKOlePres000  !,,?9@.@>@$@@|4!/l?-DT!@es-8R?-DT!@es-8R? Graph TitleArialXArialYArial?@@$@$$@$$@$$@$ArialCXYFunctionPlot y1(x):=f(x)CListBasedPlot??????x CGraphLabel Graph TitleArial XArial YArial,,6U M   U   ,,.__ ,, 2 ,,----**66BBNNZZffrr~~~~vvnnff^^VVNNFF>>66..&&'-Arial ؟ww w f,- **66BBNNZZffrr~~!-10!-81!-6I!-4a!-2y!2!4!6!8!10'-Arial ؟ww w f,- ~~vvnnff^^VVNNFF>>66..&&!-9!-6!-3!3!6!9n!12V!15>!18&'- !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ "%'*,./1234567788999::::::::::::::999988887776665 5 5 4 4 3---33- ---'NANI  "D 0u RqA)| = X&N"0Hd@;:d7: CTxobjItem&:ࡱ>  Root Entry{NW` h/x ContentsOlePres000&  !"#,,?$@$?$@$?@|4!/l?-DT!@es-8R?-DT!@es-8R? Graph TitleArialXArialYArial?@@$@$$@$$@$$@$ArialCXYFunctionPloty1(x):=sqrt(x^2-2x+2) y2(x):=1-x y3(x):=x-1CListBasedPlot??????x CGraphLabel Graph TitleArial XArial YArial,,6U M &*& M   ,,.__ ,, 2 ,,----**66BBNNZZffrr~~~~rrffZZNNBB66**'-Arial@ R؟ww wg f- **66BBNNZZffrr~~!-10!-81!-6I!-4a!-2y!2!4!6!8!10'-Arial ؟ww wg f- ~~rrffZZNNBB66**!-10!-8!-6!-4!-2!2v!4^!6F!8.!10'-++, -!."/#0$1%2&3'4(5)6*7+8,9-:.;/<0=1>2?3@4A5B6C7D8E9F:G;H<I=J>K?L@MANBOCPDQERFSGTHUIVJWKXLYMZN[O\P]Q^R_S`TaUbVcWdXeYfZg[h\i]j^k_l`manbocpdqerfsgthuivjwkxlymzn{o|o}p~qrstuvwxyz{||}~~}|{{zyxwvutsrqponmmlkjihgfedcba`_^]\[ZYXWVUTSRQPONMLKJIHGFEDCBA@?>=<;:9876543210/. - , + * )---))- --**+ ,!-".#/$0%1&2'3(4)5*6+7,8-9.:/;0<1=2>3?4@5A6B7C8D9E:F;G<H=I>J?K@LAMBNCODPEQFRGSHTIUJVKWLXMYNZO[P\Q]R^S_T`UaVbWcXdYeZf[g\h]i^j_k`lambncodpeqfrgshtiujvkwlxmynzo{p|q}r~stuvwxyz{|}~     ---- --** + , - . /0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~~}|{zyxwvutsrqponmlkjihgfedcba`_^]\[ZYXWVUTSRQPONMLKJIHGFEDCBA@?>=<;:9876543210/ . - , + *---**- ---'NANI00/=@{Gz?D