{VERSION 5 0 "IBM INTEL LINUX" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 } {PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 34 "Osittaisderivaattoja L/ osder.mws" }}}{EXCHG {PARA 19 "" 0 "" {TEXT -1 21 "L ke-to 13-14.2.02 \+ HA" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Pystyleikkaustasoja" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 38 "with(plots): setoptions3d(axes=BOXED):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "tasoleikkxz:=(f,y0,vali::ran ge)->spacecurve([x,y0,f(x,y0)],x=vali):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f:=(x,y)->x^2-y^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "tasoleikkxz(f,1,-2..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "tasoxz:=(y0,valix::range,valiz::range)->plot3d([x,y0, z],x=valix,z=valiz,grid=[5,5]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "display(tasoleikkxz(f,1,-2..2),tasoxz(1,-2..2,-1..3),style=wir eframe);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 134 "tangxz:=( f,x0,y0,xvali::range)->spacecurve([_x,y0,f(x0,y0)+D[1](f)(x0,y0)*(_x-x 0)],_x=op(1,xvali)..op(2,xvali),color=red,thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot3d(f(x,y),x=-2..2,y=0..2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "display(tangxz(f,1,1,-1..3) ,tasoleikkxz(f,1,-3..3),tasoxz(1,-3..3,-4..7),style=wireframe,labels=[ x,y,z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "display(tangxz (f,1,1,-1..3),tasoleikkxz(f,1,-3..3),tasoxz(1,-3..3,-4..7),plot3d(f(x, y),x=-2..3,y=-1..2),style=hidden,labels=[x,y,z]);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 30 "display(%,style=patchcontour);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Osittaisderivaattojen laskeminen" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Maple:ssa on kaksi derivointifunkt iota: 1) " }{TEXT 0 5 "diff " }{TEXT -1 6 "ja 2) " }{TEXT 0 1 "D" } {TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 108 "Edellinen operoi lause kkeeseen ja palauttaa lausekkeen. J\344lkimm\344isen argumenttina on f unktio ja se palauttaa" }}{PARA 0 "" 0 "" {TEXT -1 107 "derivaatatfunk tion. Edellinen sopii symbolisen lausekkeen k\344sittelyyn. J\344lkimm \344inen on k\344tev\344, kun halutaan" }}{PARA 0 "" 0 "" {TEXT -1 46 "laskea derivaatan arvoja annetuissa pisteiss\344." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Esim " }}{PARA 0 "" 0 "" {TEXT -1 20 "Aluksi lau seketyyli." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f:=sqrt(x+y^2) ; # K\344sitell\344\344n lausekkeena." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "fx:=diff(f,x); fy:=diff(f,y);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Lasketaan arvot pisteess\344 (0,2)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "simplify(subs(x=0,y=2,fx)); simplify(subs (x=0,y=2,fy));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "K\344sitell\344 \344n nyt funktiotyylill\344:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f:=(x,y)->sqrt(x+y^2); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f1:=D[1](f);f2:=D[2](f);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f1(0,2);f2(0,2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Esime rkiksi tangentin yht\344l\366 on hyvin k\344tev\344\344 muodostaa funk tiotyylill\344, kts. " }{TEXT 0 7 "tangxz " }{TEXT -1 1 "." }}}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "Korkeammat derivaatat" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "f:=x*exp(z)*arctan(y/x); # K\344sit ell\344\344n lausekkeena" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "fx:=diff(f,x): fx:=map(simplify,%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "fy:=simplify(diff(f,y));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "Voidaan jatkaa t\344st\344 derivointia. Maplen diff otta a argumentikseen jonon derivoimismuuttujia. Siten voidaan suoraan lask ea t\344h\344n tapaan:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f xy:=simplify(diff(f,x,y));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "fyx:=simplify(diff(f,y,x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Samat n\344kyv\344 olevan!" }}{PARA 0 "" 0 "" {TEXT -1 45 "Detivoi daan koko r\366ykki\366 (er\344s 7. derivaatta)" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 58 "simplify(diff(f,x,z,x,y,z,y,x)); # 3 x:\344 \344, 2 y:t\344, 2 z:aa." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "simplify(diff(f,x,x,x,y,y,z,z));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Sama tuli, kun j\344rjestettin uudestaan." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Tehd\344\344n viel\344 op eraattorityylill\344:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "F: =(x,y,z)->x*exp(z)*arctan(y/x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "D[1,1,1,2,2,3,3](F)(x,y,z): simplify(%);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "Ketjus\344\344nt\366" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f:=sqrt(u(x,y)^2 + v(x,y)^2);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "diff(f,x);diff(f,y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "Maple o saa soveltaa ketjus\344\344nt\366\344 ainakin yksinkertaisessa tapauks essa. Tietenkin se osaa soveltaa sit\344 yleisess\344kin tapauksessa. \+ " }}{PARA 0 "" 0 "" {TEXT -1 96 "Lopetan nyt toistaiseksi t\344h\344n. Jos jatkoa seuraa, niin menk\366\366n ty\366arkille osder2.mws. (13.2 .02)" }}}}}{MARK "2" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }