{VERSION 4 0 "IBM INTEL LINUX22" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 } } {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 27 "Sarjat, osa 2 funktiosarj at" }}{PARA 19 "" 0 "" {TEXT -1 40 "V2/2002 tammi-helmikuu HA L/sarja t2.mws" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Osassa 1 k\344sittelemm e vakiotermisi\344 sarjoja, nyt on funktiosarjojen vuoro." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 70 "Sarjakehitelmien johtamista algebralliste n ja der/int-lauseiden avulla" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "Sum(z^k,k=0..infinity): %=value(%); # Maple osaa geometrisen s arjan etu- ja takaperin." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "series(1/(1-z),z=0,10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "poly9:=convert(%,polynom);" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 32 "K errotaan ja derivoidaan sarjoja" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "f:=1/(1-z); sarja:=series(f,z=0,7); poly7:=convert(sarja,polyn om);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "f^2=expand(poly7^2 )+R; #Maple ei tee sievennyksi\344 sarjamuodossa, siksi joudumme oper oimaan polynomilla ja lis\344\344m\344\344n omin p\344in virhetermin R ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f^3=expand(poly7^3)+R; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 239 "Maple j\344rjest\344\344 mit en sattuu. Polynomin j\344rjest\344minen kasvavien potenssien mukaan k \344y hyvin kehitt\344m\344ll\344 se Taylorin polynomiksi, jonka astel uku on v\344hint\344\344n polynomin oma asteluku. (Polynomifunktio s \344ilyy t\344ss\344 operaatiossa identtisen\344.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "taylor(poly7^3,z=0,22); # expand ei ole t \344ss\344 tarpeen." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "Maple tek ee \"likaisen ty\366n\" k\344den k\344\344nteess\344, meid\344n tulee \+ tiet\344\344 rajat, joissa tulokset ovat voimassa." }}{PARA 0 "" 0 "" {TEXT -1 46 "Katsotaan, mit\344 saadaan derivoimalla 4 kertaa:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "d4f:=diff(f,z$4);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "sarja:=series(f,z=0,15);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "d4f=diff(sarja,z$4); # Map le suostuu derivoimaan sarjaa, ei siis tarvitse katkaista, kuten edell \344." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "Maple tekee, mit\344 k \344sket\344\344n. Me tied\344mme [PSD]-lauseen nojalla, ett\344 opera ation on voimassa suppenemiskiekossa | z | < 1 ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "map(lauseke->lauseke/24,%); # Elegantti tapa \+ jakaa puolittain 24:ll\344." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "N \344in saatiin sarjakehitelm\344n alku." }}{PARA 0 "" 0 "" {TEXT -1 23 "Ryhdyt\344\344n piirtelem\344\344n." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eps:=0.1: plot(f,z=-1..1-eps);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 228 "sarjakuva3:=plot(convert(series(f,z=0,3),polynom),z= -1..1-eps,color=blue):\nsarjakuva5:=plot(convert(series(f,z=0,5),polyn om),z=-1..1-eps,color=green):\nsarjakuva10:=plot(convert(series(f,z=0, 10),polynom),z=-1..1-eps,color=black):" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "display(plot(f,z=-1..1- eps),sarjakuva3,sarjakuva5,sarjakuva10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "display(plot(f,z=-1..1-eps),display(sarjakuva3,sarjak uva5,sarjakuva10,insequence=true));" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 15 "Taylorin sarjat" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 154 "A loitetaan exp-funktiosta. Maple on erinomainen derivointi- ja sievenny sapulainen. Voimme tehd\344 omin k\344sin Taylorin polynomia niin pitk \344lle kuin haluamme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "f:=exp(x); n:=10:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "derivaatat:=[seq(diff(f,x$k),k=1..n)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "kertoimet:=eval(subs(x=0,derivaatat ));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "kertoimet:=eval(sub s(x=0,f)),op(kertoimet); # Maple ei tunne 0:tta derivaattaa, siksi li itettiin 1 eteen." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "kertoi met:=seq(kertoimet[k]/(k-1)!,k=1..n+1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Pn:=sum(kertoimet[k]*x^(k-1),k=1..n+1);" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 30 "Vaihtoehtoisesti for-lauseella" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "restart:f:=exp(x); n:=10:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "d[0]:=f: a[0]:=subs(x=0,f): for k from 1 to n do d[k]:=simplify(diff(d[k-1],x));a[k]:=eval(subs(x =0,d[k]))/k!; od: k:='k':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Pn:=sum(a[k]*x^k,k=0..n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 189 "T\344m\344 on laskennallisesti sik\344li j\344rkev\344mpi, ett\344 de rivaatat lasketaan per\344j\344lkeen, eik\344 edellisell\344 tuhlailev alla tavalla. Toinen etu on, ett\344 nyt voidaan indekskointi aloitta a my\366s 0:sta." }}{PARA 0 "" 0 "" {TEXT -1 150 "(Maplen taulukko ja \+ lista hoidetaan hakasulkuindeksoinnilla. Taulukon indeksointi on aivan vapaa, voidaan k\344ytt\344\344 vaikka symbolisia indeksej\344 (kuten " }{TEXT 0 14 "plots[display]" }{TEXT -1 3 " )." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Verrataan valmii seen Tayloriin:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "taylor(f ,x=0,11);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "series(f,x=0,1 1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "Voimme nyt kehitt\344\344 \+ periaatteessa mink\344 vain funktion. Mukavammalta tuntuu ensin tehd \344 omin avuin:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 297 "restar t:f:=sin(x); n:=15:\nderivaatat:=[seq(diff(f,x$k),k=1..n)];\nkertoimet :=eval(subs(x=0,derivaatat));\nkertoimet:=eval(subs(x=0,f)),op(kertoim et); # Maple ei tunne 0:tta derivaattaa, siksi liitettiin 1 eteen.\nk ertoimet:=seq(kertoimet[k]/(k-1)!,k=1..n+1);\nPn:=sum(kertoimet[k]*x^( k-1),k=1..n+1);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot([ f,Pn],x=-2.5*Pi..2.5*Pi,color=[red,blue]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "taysinkuva:=n->plot(convert(taylor( sin(x),x=0,n+1),polynom),x=-Pi..Pi):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "display(plot(sin(x),x=-Pi..Pi,color=blue),seq(taysink uva(n),n=1..11));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "displ ay(plot(sin(x),x=-Pi..Pi,y=-1..1,color=blue),display(seq(taysinkuva(n) ,n=1..11),insequence=true));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 24 " Funktio/operaattorityyli" }}{PARA 0 "" 0 "" {TEXT -1 72 " K\344tev\344 mmin saadaan tuo omaper\344inen ratkaisutapa aikaan m\344\344rittelem \344ll\344 " }{TEXT 19 1 "f" }{TEXT -1 54 " funktioksi ja k\344ytt \344m\344ll\344 derivaatta-operaattoria D." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 9 "restart: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f:=cos:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "seq((D@@k)(f) (0),k=0..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "n:=4: sum( ((D@@k)(f)(z0)/k!)*(z-z0)^k,k=0..n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Rakennetaan t\344m\344 " }{TEXT 0 6 "taypol" }{TEXT -1 60 "-fun ktioksi, joka on aivan yht\344 helppok\344ytt\366inen kuin valmis" }} {PARA 0 "" 0 "" {TEXT 0 7 "taylor " }{TEXT -1 88 "(taylorissa on \"sar ja-tietorakenne\", me tyydymme polynomiin, sit\344 useimmin tarvitsemm e)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "taypol:=(f,z0,n)->sum(((D@@k)(f)(z0)/k!)*(z-z0)^k, k=0..n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "taypol(sin,0,10 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "taypol(x->sin(x)*exp( -x),0,4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "taylor(sin(x)* exp(-x),x=0,5);" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "7" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }