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-1 50 "Warning, the name changecoords has be en redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 43 "Warning, the name arrow has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "f: =(x,y)->1+2*x*sin(x+y)*exp(-(x^2+y^2)/3); " }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&\"\" \"F.**\"\"#F.9$F.-%$sinG6#,&F1F.9%F.F.-%$expG6#,&*&#F.\"\"$F.*$)F1F0F. F.!\"\"*&#F.F=F.*$)F6F0F.F.F@F.F.F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Ber\344kning av gradientvektorn och sedan normering:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "v:=grad(f(x,y),vector([x,y]) ):vn:=normalize(v): " }{TEXT -1 28 " (skriv alltid ut det...!)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Nu kan vi rita gradientf\344ltet i " }{TEXT 264 2 "xy" }{TEXT -1 12 "-planet med " }{TEXT 278 9 "fieldpl ot" }{TEXT -1 138 " (g\366r det f\366rst f\366r att kolla att allt fun kar). Vi ritar p\345 en g\345ng gradientvektorn tillsammans med niv \345kurvorna vilkas antal v\344ljes med \"" }{TEXT 274 13 "contours = \+ n " }{TEXT -1 14 "\", default \344r " }{TEXT 275 6 " n= 10" }{TEXT -1 12 "). Gl\366m ej " }{TEXT 265 16 "axis=constrained" }{TEXT -1 63 " \+ (g\366r vi med musen f\366rst\345s) f\366r att se att de \344r ortogon ala. 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t.ex. " }{TEXT 268 1 "z" }{TEXT 304 1 " " }{TEXT 303 1 " =" }{TEXT 302 1 " " }{TEXT 301 5 "-0.3 " }{TEXT -1 118 "s\345 att defi nitionsm\344ngden med niv\345kurvorna ligger synligt under ytan, vidar e kan vi fylla ut definitionsomr\345det med " }{TEXT 270 11 "filled= true" }{TEXT -1 6 "; med " }{TEXT 279 12 "labels= ... 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F\366r att inte tangentplanet skall dominera kan du d\345 bes t\344mma hur mycket som skall ritas med " }{TEXT 276 4 "view" }{TEXT -1 2 ": " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "plot3d(\{f(x,y),tangpl an(x,y)\},x=-3..3,y=-sqrt(9-x^2)..sqrt(9-x^2),view=[-3..3,-2..3,0.5..2 .6],grid=[39,39]):" }{TEXT -1 31 " kolla! Vrid tills du \344r n\366jd! 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20 "6#>%.tangvektskruvG7%,&*&\"\"##\"\"\"F(-%$cosG6#%\"tGF*F* *(F(F)F.F*-%$sinGF-F*!\"\",&*&F(F)F0F*F**(F(F)F.F*F+F*F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,tangvektlinG7%\"\"\"F&\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "arbskruv:=int(dotprod(F(u(t)[1],u(t )[2],u(t)[3]),tangvektskruv),t=0..33*Pi/4): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ RaG " 0 "" {MPLTEXT 1 0 75 "arblin:=int(do tprod(F(l(t)[1],l(t)[2],l(t)[3]),tangvektlin),t=0..33*Pi/4): " }{TEXT -1 16 "\n(skriv ut det!)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+-&)H%H\"!\"&" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Anm\344rkning f\366r senare: \344r " }{TEXT 262 1 "F" }{TEXT -1 15 " konservativt?" }}{PARA 0 "" 0 "" {TEXT -1 56 "Till sist ber\344knar vi l\344ngden av v\345r koniska skr uvlinje:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "tv:=tangvektskr uv:ds:=sqrt(sum(tv[k]^2,k=1..3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#dsG*$,(\"\"%\"\"\"*$),&*&\"\"##F(F--%$cosG6#%\"tGF(F(*(F-F.F2F(-%$si nGF1F(!\"\"F-F(F(*$),&*&F-F.F4F(F(*(F-F.F2F(F/F(F(F-F(F(F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "langd:=evalf(int(ds,t=0..33*Pi/4)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&langdG$\"+wpqK[!\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 27 " EXEMPEL 3: Taylorutveckling" }{TEXT 297 1 " " }{TEXT -1 0 "" }{TEXT 298 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 300 1 " " }{TEXT -1 23 "Vi be traktar funktionen" }{TEXT 299 0 "" }}{PARA 0 "> " 0 "" {XPPEDIT 19 1 "g := x->16/sqrt(4+sin(2*x)^3*cos(x)-sin(x)):" "6#>%\"gGf*6#%\"xG7\"6$ %)operatorG%&arrowG6\"*&\"#;\"\"\"-%%sqrtG6#,(\"\"%F/*&-%$sinG6#*&\"\" #F/F'F/\"\"$-%$cosG6#F'F/F/-F76#F'!\"\"FAF,F,F," }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 41 "plot(g(x),x=-2..8,title=`kurvan y=g(x)`);" }} {PARA 13 "" 1 "" {GLPLOT2D 732 72 72 {PLOTDATA 2 "6&-%'CURVESG6$7aw7$$ !\"#\"\"!$\"3[L+3t10R:F-$\"3CS'H\\)QybrF-7$$!3!)**\\7` 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}{TEXT -1 8 " till " } {TEXT 287 3 "g " }{TEXT -1 9 "f\345s med " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "tp:=n->convert(taylor(g(x),x=0,n+1),polynom):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Titta t.ex. p\345 " }{XPPEDIT 18 0 "P[3];" "6#&%\"PG6#\"\"$" }{TEXT -1 7 " och " }{XPPEDIT 18 0 "P[6] ;" "6#&%\"PG6#\"\"'" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "tp(3);tp(6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*\" \")\"\"\"%\"xGF%*&#\"\"$\"#;F%*$)F&\"\"#F%F%F%*&#\"%@J\"$%QF%*$)F&F)F% F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0\"\")\"\"\"%\"xGF%*&#\"\" $\"#;F%*$)F&\"\"#F%F%F%*&#\"%@J\"$%QF%*$)F&F)F%F%!\"\"*&#\"&4D\"\"%'4% F%*$)F&\"\"%F%F%F4*&#\"(T]O*\"'?:\\F%*$)F&\"\"&F%F%F%*&#\")n+s^\"(S9i# F%*$)F&\"\"'F%F%F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Nu ritar vi " }{TEXT 288 1 "g" }{TEXT -1 35 " och n\345gra Maclaurinpolynom til l " }{TEXT 295 1 "g" }{TEXT -1 72 ", antingen som en m\344ngd funktio ner (som d\345 f\345r olika f\344rgar), eller med " }{TEXT 289 7 "disp lay" }{TEXT -1 74 ", d\345 kan man v\344lja f\344rg och tjocklek f\366 r varje kurva separat. Rita f\366rst " }{TEXT 290 1 "g" }{TEXT -1 32 " f\366r att f\345 bra intervall f\366r " }{TEXT 291 1 "x" }{TEXT -1 9 " resp. 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Du kan sedan rita det animerat, se nedan! 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-1 235 "Klicka p\345 figuren, d\345 kan du spela upp det (v \344lj l\344gre hastighet, eller g\345 fram\345t stegvis...), det \344 r v\344ldigt instruktivt, g\366r det!! Du kan \344ven printa ut alla \+ \"frames\". Tyv\344rr ritas d\345 b\344gge kurvor helt, \344ven om man skriver in t.ex. 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