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I " }{TEXT 286 5 "maple" }{TEXT -1 14 " g\345r det med " }{TEXT 287 18 "insequence = true " } {TEXT -1 14 "(l\344s \344ven om " }{TEXT 288 7 "animate" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 26 "Vi tar ett enkelt exempel:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ff:=x->sin(x)+cos(x):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "display(seq(plot([convert(t aylor(ff(x),x=0,3*i),polynom),ff(x)],x=-14..14,y=-2..2,axes=frame,tick marks=[2,2]),i=1..12),insequence=true):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 229 "Klicka p\345 figuren, d\345 kan du spela upp det (v\344l j l\344gre hastighet, eller g\345 fram\345t stegvis...), det \344r v \344ldigt instruktivt, g\366r det!! Du kan \344ven printa ut alla \"fr ames\". 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F*7$F`[x$!+xbDu?F*7$Fj\\[l$!+efDu?F*7$F_][l$!+EiDu?F*7$Fd][lFdigm7$Fi] [l$!+;kDu?F*7$F^^[l$!+\"RcU2#F*7$Fc^[l$!+PjDu?F*7$Fh^[l$!+biDu?F*7$F]_ [l$!+VhDu?F*7$Fb_[l$!+NeDu?F*7$Fe[x$!+8aDu?F*7$Fj_[l$!+QHDu?F*7$Fj[x$! +f\"\\U2#F*7$Fb`[l$!+4lCu?F*7$Fg`[l$!+#eVU2#F*7$F\\a[l$!+]/Cu?F*7$F_\\ x$!+)=PU2#F*7$Fda[l$!+BUBu?F*7$Fia[l$!+u7Bu?F*7$F^b[l$!+'RGU2#F*7$Fd\\ x$!+YcAu?F*7$Ffb[l$!+7HAu?F*7$F[c[l$!+V/Au?F*7$F`c[l$!+#H=U2#F*7$Fi\\x $!+1l@u?F*7$Fhc[l$!+v^@u?F*7$F]d[l$!+SU@u?F*7$Fbd[l$!+FR@u?F*7$Fgd[l$! +@P@u?F*7$F\\e[l$!+?O@u?F*7$F^]x$!+FO@u?F*7$Fde[l$!+k[@u?F*7$Fc]x$!+_z @u?F*Fjco-F]do6$7$FfdoF\\^wFado-F]do6$7$Faf[lF`^wFgdo-%*AXESSTYLEG6#%% NONEG-%*AXESTICKSG6$\"\"#Ffghm" 1 2 0 1 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 13 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 256 50 "EXEMPEL 4: funktionalmatris, funk tionaldeterminant" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "Vi forts\344 tter med samma funktion (labb1 ex1). Vi skall ber\344kna station\344ra punkter och deras typ:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " with(linalg):v:=grad(f(x,y),vector([x,y])):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(\{v[1],v[2]\},\{x,y\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"xG\"\"!/%\"yGF&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "OBS: Vi f\345r naturligtvis ej alla l\366sningar, l\366sningen \+ (" }{XPPEDIT 18 0 "0,Pi;" "6$\"\"!%#PiG" }{TEXT -1 17 ") t. ex. f \345s med" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "fsolve(\{v[1], v[2]\},\{x,y\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"yG$\"+aEfTJ! \"*/%\"xG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "F\366r att bes t\344mma av vilken typ den station\344ra punkten (0,0) \344r ber\344kn ar vi Hessematrisen i punkten (0,0) (se anvisningarna):" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 40 "H1:=subs(x=0,y=0,hessian(f(x,y),[x,y]));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H1G-%'matrixG6#7$7$,$*&-%$cosG6#\" \"!\"\"\"-%$expGF.F0\"\"#F+7$F+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "H:=simplify(H1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"HG-%'matrixG6#7$7$\"\"#\"\"\"7$F+\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Anm: Att ber\344kna Hesse-matrisen g\345r s\344kert snabb are \344n att ber\344kna alla derivator " }}{PARA 0 "" 0 "" {TEXT -1 5 "(med " }{TEXT 260 1 "D" }{TEXT -1 1 "[" }{TEXT 261 1 "i" }{TEXT -1 1 "," }{TEXT 262 1 "k" }{TEXT -1 2 "](" }{TEXT 257 1 "f" }{TEXT -1 2 " )(" }{TEXT 258 1 "x" }{TEXT -1 1 "," }{TEXT 259 1 "y" }{TEXT -1 4 ") , " }{TEXT 263 1 "i" }{TEXT -1 1 "," }{TEXT 264 1 "k" }{TEXT -1 66 " = \+ 1,2). Du kan f\366rst ber\344kna Hessematrisen i en godtycklig punkt: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Hess:=hessian(f(x,y),[x ,y]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Och sedan plocka fram " }{TEXT 268 1 "D" }{TEXT -1 1 "[" }{TEXT 269 1 "i" }{TEXT -1 1 "," } {TEXT 270 1 "k" }{TEXT -1 2 "](" }{TEXT 265 1 "f" }{TEXT -1 2 ")(" } {TEXT 266 1 "x" }{TEXT -1 1 "," }{TEXT 267 1 "y" }{TEXT -1 6 ") med " }{MPLTEXT 1 0 9 "Hess[i,k]" }{TEXT -1 27 " ! I punkten (0,0) f\345s d \345 " }{TEXT 271 2 " H" }{TEXT -1 7 " s.o.:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "H:=simplify(subs(x=0,y=0,eval(Hess))); " } {TEXT -1 17 "(pr\366va, gl\366m ej " }{TEXT 276 4 "eval" }{TEXT -1 1 ")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG-%'matrixG6#7$7$\"\"#\"\" \"7$F+\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "D\345 kan vi nu av g\366ra om " }{TEXT 272 1 "H" }{TEXT -1 56 " \344r positivt definit, n egativt definit eller indefinit: " }{TEXT 283 6 "maples" }{TEXT -1 9 " svar \344r " }}{PARA 0 "" 0 "" {TEXT -1 1 "'" }{TEXT 273 5 "false" } {TEXT -1 9 "' eller '" }{TEXT 274 4 "true" }{TEXT -1 12 "' eller, om \+ " }{TEXT 275 1 "H" }{TEXT -1 84 " inneh\345ller parametrar, villkor so m m\345ste vara uppfylda f\366r att ... skall vara sant:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "definite(H,positive_def);definite(H,negative _def);definite(H,positive_def);definite(H,negative_def);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%&falseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&fa lseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Efters om det inte st\344mmer, s\345 \344r " }{TEXT 277 1 "H" }{TEXT -1 132 " indefinit och (0,0) allts\345 sadelpunkt. Man kan f\345 det enklare \+ genom att ber\344kna egenv\344rdena eller genom att ber\344kna determi nanten:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eigenvalues(H);d et(H);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&\"\"\"F$*$-%%sqrtG6#\"\"# \"\"\"F$,&F$F$F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Eftersom " }{TEXT 278 1 "H" } {TEXT -1 53 " har ett positivt och ett negativt egenv\344rde s\345 \+ \344r " }{TEXT 280 2 " H" }{TEXT -1 19 " indefinit. Eller:" }}{PARA 0 "" 0 "" {TEXT -1 42 "Eftersom determinanten \344t negativ, s\345 \+ \344r " }{TEXT 279 2 "H " }{TEXT -1 31 " indefinit. Analogt f\345s at t (0," }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 16 ") \344r sadelpunkt ." }}{PARA 0 "" 0 "" {TEXT -1 24 "Vi pr\366var en punkt till:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "statpkt:=fsolve(\{v[1],v[2] \},\{x,y\},\{x=-3..-1\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(statpk tG<$/%\"yG$!+,#)G:;!\"*/%\"xG$!+iAruAF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "H2:=subs(statpkt[1],statpkt[2],eval(Hess)):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eigenvalues(H2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+gg())z)!#6$\"+=5,ye!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "Eftersom b\344gge egenv\344rdena \344r positiv a, s\345 \344r punkten en lokal minimipunkt! Eller du r\344knar med de terminanten (d\345 m\345ste du \344ven kolla " }{TEXT 282 1 "A" } {TEXT -1 4 " = " }{XPPEDIT 18 0 "diff(f(x,y),x,x);" "6#-%%diffG6%-%\" fG6$%\"xG%\"yGF)F)" }{TEXT -1 14 " i punkten!):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "det(H2);A:=evalf(subs((statpkt[1],statpkt[2]) ,diff(f(x,y),x,x)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+V!*)><&!#6 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG$\"+wll!\\$!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Determinanten \344r positiv, " }{TEXT 281 1 "A" }{TEXT -1 54 " \344r positivt, allts\345 \344r punkten en l okal minimipunkt!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot(g (x),x=-2..8,title=`kurvan y=g(x) i ex.3`);" }}{PARA 13 "" 1 "" {GLPLOT2D 388 102 102 {PLOTDATA 2 "6&-%'CURVESG6$7aw7$$!\"#\"\"!$\"1L+ 3t 10R:F-$\"1S'H\\)QybrF-7$$!1+]7`'Gd[\"F-$\"16([7#=IerF-7$$!1n\"zpo1CV\" F-$\"1#*\\ne!*GkrF-7$$!1LL$3s%3z8F-$\"1$y^M(3vvrF-7$$!1LLL$)Qtr7F-$\"1 Nlr_T%eA(F-7$$!1LL$e/$Qk6F-$\"1ixJSL5GtF-7$$!1+]7GCad5F-$\"1EA!Q1K'*[( F-7$$!1nm;/\"=q]*!#;$\"1vAOtD8'p(F-7$$!1++]7)ok^)F\\p$\"1LC#36e9*yF-7$ $!1LL$3_>f_(F\\p$\"1-'pvq8?/)F-7$$!1****\\7848qF\\p$\"15%3/*)Q$)3)F-7$ $!1nm;/JE+lF\\p$\"1B/FL(*=4\")F-7$$!1,+++!\\QC'F\\p$\"1>h'p\\k,6)F-7$$ !1NL$e*[V()fF\\p$\"1#yAGg(G0\")F-7$$!1omm\"z?5t&F\\p$\"1.:Z&))=^4)F-7$ $!1,+](o1YZ&F\\p$\"1>1[0lO!3)F-7$$!1mm;a8(QT%F\\p$\"18SG/Vk*)zF-7$$!1K L$3-OJN$F\\p$\"1VzYhs_**yF-7$$!1lmTNE!))3$F\\p$\"1%*Gr`Tr$)yF-7$$!1)** ***\\#pW#GF\\p$\"18pD&[v<(yF-7$$!1KLeke8gDF\\p$\"1(=\\$[P4kyF-7$$!1lm; zC!eH#F\\p$\"1@[\"ez84'yF-7$$!1***\\P4p9.#F\\p$\"12GAX;MiyF-7$$!1KLL3d 8n=j!>b)3)F-7$$\"1qm\"HK/dT\"F\\p$\"11pseyDA\")F-7$$\"1.+]iD o%*=F\\p$\"1P!=mQFB9)F-7$$\"1q;H#orT8#F\\p$\"1)[pzvyj9)F-7$$\"1OL3-3mt BF\\p$\"1#e(*)Q#Gi9)F-7$$\"1.](=#*\\Jh#F\\p$\"1dWz,$*)=9)F-7$$\"1pmmT! 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