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-%+AXESLABELSG6&%\"xG%\"yGQ!F/-%%FONTG6#%(DEFAULTG" 1 2 0 1 10 0 2 1 1 3 2 1.000000 48.000000 49.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 102 "Vi ser att F(x,y) \344r ungef\344r noll d \345 (x,y) \344r n\344ra (1,1) (vilket \345terspeglar att F \344r kon tinuerlig). " }}{PARA 261 "" 0 "" {TEXT -1 84 "Dessutom finns en kurva (som n\344stan ser ut l\344ngs en linje) l\344ngs vilken F(x,y) --- \+ " }}{PARA 262 "" 0 "" {TEXT 260 10 "niv\345kurvan" }{TEXT -1 67 " F(x, y)=0. Om vi l\345ter (x,y) r\366ra sig vinkelr\344t mot denna kurva, " }}{PARA 263 "" 0 "" {TEXT -1 60 "s\345 ser vi dock approximationen \+ F(x,y)=0 snabbt blir d\345lig. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 264 "" 0 "" {TEXT -1 82 "Med hj\344lp av partiella derivator kan vi dock finna ett plan som ger mycket b\344ttre " }}{PARA 266 "" 0 " " {TEXT -1 28 "approximation n\344ra (1,1,0). " }{TEXT 258 1 " " } {TEXT -1 13 "Detta kallas " }{TEXT 259 29 "tangentplanet i denna punk t " }{TEXT -1 29 "och \344r grafen till funktionen" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "G:=(x,y)->F(1,1)+D[ 1](F)(1,1)*(x-1)+D[2](F)(1,1)*(y-1);# " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GGj+6$%\"xG%\"yG6\"6$%)operato rG%&arrowGF),(-%\"FG6$\"\"\"F1F1*&--&%\"DG6#F16#F/F0F1,&9$F1F1!\"\"F1F 1*&--&F66#\"\"#F8F0F1,&9%F1F1F;F1F1F)F)F)6$\"\"!FE" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 268 "" 0 "" {TEXT -1 14 "Vi ritar F \+ och" }{MPLTEXT 1 0 0 "" }{TEXT -1 71 " G i samma diagram. Som synes f \345r vi nu mycket b\344ttre \366verensst\344mmelse." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "plot3d([F(x,y),G(x,y)],x=0.5..1.5,y=0.5..1.5,ori entation=[52,75],axes" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "=boxed,col our=[red,blue]);" }}{PARA 13 "" 1 "" {GLPLOT3D 486 486 486 {PLOTDATA 3 "6'-%%GRIDG6&;$\"\"&!\"\"$\"#:F)F&X,%)anythingG6\"6\"[gl'!%\"!!#\\bm \":\":3FF80000000000003FF59B8E38E38E393FF31555555555573FF0680000000002 3FEB1C71C71C71CD3FE505555555555D3FDD0000000000143FCE071C71C71CA53CDF1C 71C71C71C5BFD01FFFFFFFFFDBBFE0B8E38E38E378BFEA05555555553ABFF1FFFFFFFF FFF0BFF759C71C71C70FBFFD15555555554AC0019BFFFFFFFFFBC004E38E38E38E35C0 0863FFFFFFFFFEC00C200000000000C0100D1C71C71C72C0122AAAAAAAAAADC0146A00 00000004C016CC71C71C71CCC01953555555555CC01C0000000000083FF9B92F684BDA 113FF745ED097B425E3FF4B17B425ED0983FF1F684BDA12F693FEE1F684BDA12F93FE7 EF684BDA12FC3FE1525ED097B42E3FD47B425ED097C83FB53425ED097BABBFC5F684BD 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%(DEFAULTG-%+PROJECTIONG6%$\"$X\"F8$\"#&)F8\"\"\"" 1 2 0 1 10 0 2 1 1 2 2 1.000000 75.000000 52.000000 1 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 269 "" 0 "" {TEXT -1 90 "Vi skall nu speciellt studera hur F och G varierar l\344ngs linjer av formen (x,y)=(1,1)+t*v " }}{PARA 270 " " 0 "" {TEXT -1 93 "d\344r v=(cos(theta),sin(theta)) \344r en enhetsv ektor s\345 att avst\345ndet fr\345n (x,y) till (1,1) blir" }}{PARA 271 "" 0 "" {TEXT -1 14 "beloppet av t." }}{PARA 272 "" 0 "" {TEXT -1 0 "" }}{PARA 273 "" 0 "" {TEXT -1 59 "Vi b\366rjar med att definiera d enna parametrisering i Maple :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "X:=t->1+t*cos(theta); Y:=t->1+t*sin (theta);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 274 "" 0 "" {TEXT -1 93 "Nu ritar vi fallet theta=0 vilket ger (x,y)=(1,1)+(t,0), dvs li njen \344r parallell med x-axeln:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "the ta:=0;plot([F(X(t),Y(t)),G(X(t),Y(t))],t=-0.5..0.5,thickness=3,color=b lack);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XGj+6#%\"tG6\"6$%)operatorG%&arrowGF(,&\"\"\"F -*&9$F--%$cosG6#%&thetaGF-F-F(F(F(6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YGj+6#%\"tG6\"6$%)operatorG%&arrowGF(,&\"\"\"F-*&9$ F--%$sinG6#%&thetaGF-F-F(F(F(6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%&thetaG\"\"!" }}{PARA 13 "" 1 "" {GLPLOT2D 486 486 486 {PLOTDATA 2 "6(-%'CURVESG6#7S7$$!3++++++++]!#=$!3+++++++D6!#<7$$!3NLLLe%G?y%F*$! 3]#Gog12m3\"F-7$$!3OmmT&esBf%F*$!3kHm[qmw_5F-7$$!3LLL$3s%3zVF*$!3pHu;i F<95F-7$$!33LL$e/$QkTF*$!3LO0a-G#pu*F*7$$!3!pm;/\"=q]RF*$!3XpSc!yBrM*F *7$$!3TLL3_>f_PF*$!3XC/CqX#)p*)F*7$$!3))***\\(o1YZNF*$!3&e\">*e%f\">d) F*7$$!3]LL3-OJNLF*$!3%4PzNH3@:)F*7$$!3C++v$*o%Q7$F*$!3N[Z#*z[pCxF*7$$! 3ammm\"RFj!HF*$!3!RNi+GC^F(F*7$$!3JLL$e4OZr#F*$!3&[i3!z-KqoF*7$$!3?+++ 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***\\x%f\")oF*7$F[w$\"3&o**\\P/-a[(F*7$F`w$\"3<)**\\(=Yb;\")F*7$Few$\" 3s)****\\i@Ot)F*7$Fjw$\"3h)**\\PfL'z$*F*7$F_x$\"3>+++!*>=+5F-7$Fdx$\"3 .++DE&4Q1\"F-7$Fix$\"3&***\\P%>5p7\"F-7$F^y$\"3\"******\\:$*[=\"F-7$Fc y$\"34++Dr\"[8D\"F-7$Fhy$\"3++++Ijy58F-7$F]z$\"32+]P/)fTP\"F-7$Fbz$\"3 1+]i0j\"[V\"F-7$Fgz$\"3++++++++:F--%'COLOURG6&%$RGBG\"\"!FedlFedl-%*TH ICKNESSG6#\"\"$-%+AXESLABELSG6$Q\"t6\"Q!F^el-%%VIEWG6$;$!\"&!\"\"$\"\" &Ffel%(DEFAULTG" 1 2 0 1 10 3 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 275 "" 0 "" {TEXT -1 92 "Som \+ synes blir restriktionen av G en r\344t linje som \344r tangent till g rafen f\366r restriktionen " }}{PARA 277 "" 0 "" {TEXT -1 65 "av F i p unkten som svarar mot (1,1,0). Linjens lutning \344r precis" }}{PARA 276 "" 0 "" {TEXT -1 60 "D[1](F)(1,1), dvs den partiella derivatan med avseende p\345 x." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 97 "H\344rn\344st v\344ljer vi theta=Pi/2 vilket ger linjen \+ (x,y)=(1,1)+(0,t) som \344r parallell med y-axeln : " }{TEXT -1 1 " \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "theta:=Pi/2;plot([F(X(t ),Y(t)),G(X(t),Y(t))],t=-0.5..0.5,thickness=3,color=black);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&thetaG,$*&\"\"#!\"\"%#PiG\"\"\"F*" }} {PARA 13 "" 1 "" {GLPLOT2D 486 486 486 {PLOTDATA 2 "6(-%'CURVESG6#7S7$ $!3++++++++]!#=$\"3+++++++vL!#<7$$!3NLLLe%G?y%F*$\"3tZ[?)>@)fKF-7$$!3O mmT&esBf%F*$\"3q)))f9,+$eJF-7$$!3LLL$3s%3zVF*$\"3%))G-lG=D/$F-7$$!33LL $e/$QkTF*$\"3og@wSo2CHF-7$$!3!pm;/\"=q]RF*$\"3&4ApT8PT!GF-7$$!3TLL3_>f _PF*$\"3AF@2rt%4p#F-7$$!3))***\\(o1YZNF*$\"3([dnPyu:d#F-7$$!3]LL3-OJNL F*$\"3R6Q2)[KcW#F-7$$!3C++v$*o%Q7$F*$\"3RCu(RY3uJ#F-7$$!3ammm\"RFj!HF* $\"3G1(=SGPD=#F-7$$!3JLL$e4OZr#F*$\"3d(e-P3'4h?F-7$$!3?+++v'\\!*\\#F*$ \"3Tl#\\`J^7#>F-7$$!32+++DwZ#G#F*$\"3b()[;:2Kx$\"3=v!)*G7DJ5(F*7$$!3K****\\iXg#G'Fjq$\"33?A*)Go&\\U&F*7$$!3WlmmT& Q(RTFjq$\"3(R4p8To]i$F*7$$!3:nm;/'=><#Fjq$\"3gKnUC2tE>F*7$$!3uDMLLe*e$ \\!#@$\"3PC,qE[%3W%!#?7$$\"3Zem;zRQb@Fjq$!3yE&p&e+-o>F*7$$\"3'[***\\(= >Y2%Fjq$!32E;Zm=!)oPF*7$$\"3Qhmm\"zXu9'Fjq$!3Fy.M6tTmdF*7$$\"3\\'***** *\\y))G)Fjq$!3?5zmQ%3$*)yF*7$$\"3'*)***\\i_QQ5F*$!3FIVMr-g-5F-7$$\"3@* **\\7y%3T7F*$!3UMK&foG^@\"F-7$$\"36****\\P![hY\"F*$!3&\\>+'*[jzX\"F-7$ $\"3jKLL$Qx$o;F*$!3&y$>JJ.[#o\"F-7$$\"3!)*****\\P+V)=F*$!3igg/Mn(*G>F- 7$$\"3?mm\"zpe*z?F*$!3*[GbqkJ&e@F-7$$\"3%)*****\\#\\'QH#F*$!3p9bm(4(R; CF-7$$\"3GKLe9S8&\\#F*$!3KCV*)eSwlEF-7$$\"3R***\\i?=bq#F*$!3sdzO]+70gF-7$$\"3+++++ +++]F*$!3+++++++vjF--F$6#7S7$F($\"3++++++++XF-7$F/$\"3%)****\\7c#QI%F- 7$F4$\"3H**\\(oKNJ8%F-7$F9$\"3;++v[i$\"3b***\\7uWzu$F-7$FC$\" 35+]PH;jbNF-7$FH$\"3,+](ovKtP$F-7$FM$\"37+](=g9F>$F-7$FR$\"3)***\\(=C# y,IF-7$FW$\"3;+]P/AY6GF-7$Ffn$\"3x****\\_Yp:EF-7$F[o$\"34++D'[iKW#F-7$ F`o$\"3=++]2Z9\\AF-7$Feo$\"3<++]i)HU0#F-7$Fjo$\"3'*****\\#H$Rm=F-7$F_p $\"39+](=p>ep\"F-7$Fdp$\"3?++]#=\"*H\\\"F-7$Fip$\"3T+++I)o6K\"F-7$F^q$ \"3G+](oe)H@6F-7$Fcq$\"3#[+++b&zV%*F*7$Fhq$\"31)**\\(oLr-vF*7$F^r$\"3% ****\\i5WVl&F*7$Fcr$\"3!*)***\\(okds$F*7$Fhr$\"3W++vVnsa>F*7$F]s$\"3<$ 3++D1BW%Fbs7$Fds$!3j#**\\7eX)R>F*7$Fis$!3P&**\\(os:nOF*7$F^t$!3D&***\\ 77qKbF*7$Fct$!3%o*****\\1**fuF*7$Fht$!3m!***\\itYX$*F*7$F]u$!3H**\\7.j (p6\"F-7$Fbu$!3?***\\PBL&>8F-7$Fgu$!3P*****\\kR:]\"F-7$F\\v$!3s****\\P .(ep\"F-7$Fav$!3f**\\7GG'>(=F-7$Ffv$!3j****\\K%yW1#F-7$F[w$!3&*)*\\781 iXAF-7$F`w$!3n**\\i&Qm\\V#F-7$Few$!3i****\\(['3?EF-7$Fjw$!3e**\\7y+*Q \"GF-7$F_x$!37+++qfa+IF-7$Fdx$!3k***\\(y&G9>$F-7$Fix$!3j**\\7$eI2Q$F-7 $F^y$!3a*****\\YzYb$F-7$Fcy$!3\")***\\P^WSv$F-7$Fhy$!3b*******)*eB$RF- 7$F]z$!3A+]78%zC7%F-7$Fbz$!3v**\\(o\"*[WI%F-7$Fgz$!3++++++++XF--%'COLO URG6&%$RGBG\"\"!FedlFedl-%*THICKNESSG6#\"\"$-%+AXESLABELSG6$Q\"t6\"Q!F ^el-%%VIEWG6$;$!\"&!\"\"$\"\"&Ffel%(DEFAULTG" 1 2 0 1 10 3 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 278 "" 0 "" {TEXT -1 94 "Grafen till restriktionen av G \344r allts \345 \345terigen tangent till grafen till restriktionen av F." }} {PARA 279 "" 0 "" {TEXT -1 68 "Riktningskoefficienten \344r den partie lla derivatan med avseende p\345 y." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 262 88 "Om vi v\344ljer andra vinklar f\345r vi ocks\345 tangentlinjer till kurvor (observera skalorna):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "theta:=Pi/4 ;plot([F(X(t),Y(t)),G(X(t),Y(t))],t=-0.5..0.5,thickness=3,color=black) ;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %&thetaG,$*&\"\"%!\"\"%#PiG\"\"\"F*" }}{PARA 13 "" 1 "" {GLPLOT2D 486 486 486 {PLOTDATA 2 "6(-%'CURVESG6#7S7$$!3++++++++]!#=$\"3gzi&)Qq4)H\" !#<7$$!3NLLLe%G?y%F*$\"3KGUj_:yo7F-7$$!3OmmT&esBf%F*$\"33L([+_`f_PF*$\"3cuh3#\\WN5\"F-7$$!3) )***\\(o1YZNF*$\"3EsbHdc\"[1\"F-7$$!3]LL3-OJNLF*$\"34(H,Et_D-\"F-7$$!3 C++v$*o%Q7$F*$\"3woy6:$)4\"y*F*7$$!3ammm\"RFj!HF*$\"33#4qfS#)*)H*F*7$$ !3JLL$e4OZr#F*$\"3Z\\7(yEvE&))F*7$$!3?+++v'\\!*\\#F*$\"3y%\\p44*=D$)F* 7$$!32+++DwZ#G#F*$\"3Khj1)R:!oxF*7$$!3.+++D.xt?F*$\"3(QIV#)ywT?(F*7$$! 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