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Tanko tai k\366ysi on kiinnitetty molemmista p\344ist\344\344n ki tkattomien nivelten avulla. L\344ht\366asetelmana olkoon tilanne, joss a hinaaja sijaitsee origossa ja hinattava et\344isyydell\344 " }{TEXT 260 1 "a" }{TEXT -1 62 " y-akselilla. Hinaajan l\344htee liikkeelle x- akselin suuntaan. \n" }}{PARA 256 "" 0 "" {BITMAP 257 171 171 1 "?TM?G PB:nH>?FZ;jO[:^:>:::[B:;B::vy][:vy:vyyyYyA;J::::::::CDKCDKLJ\\^B:::::: :::::::::::::JZB:C\\>C;^B:::::::::::::::::::JZB:C\\>C;^B:::::::::::::: :::::JZB:C\\>C;^B:::::::::::::::::::;J\\BKL:C<::::::::::::::::Z<_ >L>Z:;;J\\BK;J:::::::::::::::J:Z>cJ?cBCCKL>J:>B;B:C\\:_b>^Z>CK: :::::::::::::::;J?KLBKJZL:;:::::::::::::::::J;DLKD<:;Z:^b>? 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C<:::::::::::::::::JZB:C\\>C;^BjZJ\\::::::::::::::::::>B ::::::::::::::::::;\\:^BL^>J\\Z;[>;:::::::::::::::::JZB:C\\>C;^BRJBK:: ::::::::::::::::>BJ\\jZJ\\::: :::::::::::::::JZB:C\\>C;^B^::::::::::::::::::>B;::::: :::::::::::::;\\:^BL^>J\\RJBK::::::::::::::::::JZB:C\\>C;^BL;C<::::::: :::::::::::>BJ\\FB^B::::::: :::::::::::JZB:C\\>C;^BcZ>;::::::::::::::::::>BJ\\@>L>::::::::::::::::::JZB:C\\>C;^bNJ\\::::::::::: ::::::::>BC;FB^B:::::::::::::::: :::;\\:^BL^>JDBK:::::::::::::::::::>B;::::::::::::::::::JZB :C\\>C;@>L>:::::::::::::::::::;\\:^BCDKLZ;[>;::::::::::::::::::JZB:CL \\L^>B?^B:::::::::::::::::::>BC;L;C<:::::::::::::::::::;\\:^BCD KLjZL>C<:::::::::::::::::::;\\:^b:>:L:^B:::::::::::::::::::>BB:::::::::::::::::::>Bb>^BBK:::::::: :::::::::::JZB:CDJ;B;Z>;:::::::::::::::::::;L:::::::::::: ::::::::>B:B;CDKC<::::::::::::::::JZB:C\\>S:BK:Z>J\\::: ::::::::::::::>Bb>>b:^BLNZZ:^b>_b>?ZJ^Z>_Z>?b>?b:_b:K>b>[>J\\^:CL\\LN\\::::::::::;D:_Z>_ Z>JZZ>C;CL<_Z:CJ;Z:_b:B;CDKC<:::::::::;D:;B:CD KCL\\LKZLN?b:_b:?BCBKC:CDKCB;L:BK;DKCD;^Z:_b:CK;DJLJ\\L^>_Z:CKLJ^Z>_Z>?b:^Z:K:LJZ^b:BKB;LJZ^Z:?b:^Z>C:L>; B;;\\:>Z:>BCBJCDKK\\Z:^BLN_B:L:BKB;^BZ:C;^B:CBJCDKL>:::::::::JZ?b>?b:_b>_ b>[>^b>BKBKB;Z>[:C;^B:CBJCDKL>:::::::::JZBBBBBBBBBBBB<5: " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Koska hinauset\344isyys pysyy vakiona (= " }{TEXT 261 1 " a" }{TEXT -1 66 "), saadaan Pythagoraan lauseella johdettua tilanteen \+ liikeyht\344l\366:\n" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "diff(y,x) = - y/sqrt(a^2-y^2);" "6#/-%%diffG6$%\"yG%\"xG,$*&F'\"\"\"-%%sqrtG6#,&*$% \"aG\"\"#F+*$F'F2!\"\"F4F4" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 88 "Laskujen aluksi on syyt\344 h \344vitt\344\344 mahdollisista aiemmista laskuista j\344\344neet muutt ujat. \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Ol koon hinausk\366yden pituus" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a:= 1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "M\344\344ritell\344 \344n differentiaaliyht\344l\366." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "yht:= diff(y(x), x)=-y(x)/sq rt(a^2-y(x)^2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Luonnollinen alkuehto olisi " }{TEXT 262 1 "y" } {TEXT -1 6 "(0) = " }{TEXT 263 1 "a" }{TEXT -1 137 ", mutta t\344m\344 on yht\344l\366ss\344 nimitt\344j\344n nollakohta eik\344 numeerinen \+ ratkaiseminen onnistu. Pieni huijauksenomainen muutos alkuehdossa autt aa:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "alkuehto:= y(0.001)=a-0.01:" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Ratkaistaan differentia aliyht\344l\366." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "rtk:= dsolve(\{yht, alkuehto\}, y(x), numeric , output=listprocedure);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "kaari:= subs(rtk, [x, y(x)]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 40 "Piirret\344\344n kuvaaja saadulle rat kaisulle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 48 "plot([kaari[], 0.0001..3], scaling=constrained);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 " Saatua k\344yr\344\344 kutsutaan nimell\344 " }{TEXT 256 8 "traktrix" }{TEXT -1 84 ". Sit\344 voidaan k\344ytt\344\344 miss\344 tahansa j \344ykiss\344 veto-, ty\366nt\366- tai seuraustilanteissa." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 8 "Teht\344vi \344" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 419 " Jos vet\344j\344 etenee jatkuvasti samaan suuntaan, alussa eri suuntaa n kulkenut hinattava saavuttaa t\344sm\344lleen saman kulkusuunnan vas ta \344\344rett\366myydess\344. Tarkastele tilannetta, miss\344 vene l \344htee vet\344m\344\344n lauttaa 50 metrin joustamattoman hinausk \366yden avulla edell\344 kuvatulla tavalla. Kuinka pitk\344\344n hina usta pit\344\344 suorittaa, ennen kuin lautan poikkeama suoraan etenev \344n veneen kulkusuunnan m\344\344ritt\344m\344st\344 suorasta on all e 5 cm?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "Tutki, voidaanko probleeman differentiaaliyht\344l\366 ratkaista s ymbolisesti Maplen " }{TEXT 264 6 "dsolve" }{TEXT -1 110 "-komennolla. Voidaanko ratkaisu lausua alkeisfunktioiden avulla? Onko yht\344l\366 ratkaistavissa k\344sin laskemalla? " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 51 "Ratkaise edell\344 tarkasteltu proble ema valitsemalla " }{TEXT 257 1 "y" }{TEXT -1 88 " riippumattomaksi mu uttujaksi, jolloin hinattavan reitti\344 kuvaava tuntematon funktio on " }{TEXT 265 1 "x" }{TEXT -1 1 "(" }{TEXT 258 1 "y" }{TEXT -1 182 "). Muodosta tilannetta vastaava differentiaaliyht\344l\366 alkuehtoineen ja ratkaise se numeerisesti. Joudutaanko jossakin kohdassa vastaavaan ongelmaan kuin edell\344 alkuehtoa asetettaessa?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 24 "JP & SKK & MS 12 .07.2001" }{TEXT -1 0 "" }}}}{MARK "0 2 0" 24 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }