{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 10 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE " " -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{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 -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 }1 0 0 0 6 6 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 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE " " -1 -1 "" 1 24 0 0 0 0 0 0 1 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 34 "L\366sung der Abituraufgabe A2 1996 " }}{PARA 257 "" 0 "" {TEXT -1 39 "L\344ufer StD Kepler-Gymnasi um Freiburg ;" }{TEXT 257 25 "e-mail: v.laeufer.@gmx.de" }}{PARA 0 "" 0 "" {TEXT -1 20 " " }{TEXT 269 55 " Zu jedem t au s R ist eine Funktion f gegeben durch: " }{TEXT -1 5 " " }{TEXT 256 20 "f(x) = 2 - 2/(x^2+1)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "restart; with(plots): with(student):with(plottools):" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 13 "Teilaufgabe a" }}{PARA 0 "" 0 "" {TEXT -1 104 "Untersuchen Sie K auf Symmetrie, Asymptoten,gemeinsame Punkte mit der x-Achse, Extrem- und Wendepunkte." }}{PARA 0 "" 0 "" {TEXT -1 53 "Zeichnen Sie die Kurve K samt Asyptote f\374r [-3 ; 3 ]" }} {PARA 0 "" 0 "" {TEXT -1 108 "In welchem Bereich liegen die x-Werte de rjenigen Punkte von K, deren Abstand von der Asymptote < 0,05 ist ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 12 "Berechnungen" }}{PARA 0 "" 0 "" {TEXT -1 51 " \+ Definition der Funktion:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f:=x->2-2/(x^2+1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"xG6\"6$%)operatorG%&arrowGF(,&\" \"#\"\"\"*$,&*$9$F-F.F.F.!\"\"!\"#F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 " Symmetrie: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve(f(-u)=f(u),u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"uG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "d as bedeutet: ???" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 " \+ Asymptoten: a) waagrecht/schief b) sen krecht (Polstellen) ............. ? " }}{PARA 0 "" 0 "" {TEXT 258 111 " beide Antworten k\366nnen hier ohne Berechnungen eingetippt werden ! also bitte ! ?" }}{PARA 0 "" 0 "" {TEXT -1 18 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "As:=x->2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AsG\"\"#" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 35 " Nullstellen :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "x0:= [solve(f(x)=0)]: N:=[op(1 ,x0),0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG7$\"\"!F&" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 103 " Extremwe rte: Bedingung f'(x)=0 und f''(x) <>0 also zun\344chst die Ableitunge n: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f1_str:=diff(f(x),x);f1:=una pply(f1_str,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'f1_strG,$*&,&*$% \"xG\"\"#\"\"\"F+F+!\"#F)F+\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #f1G:6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&,&*$9$\"\"#\"\"\"F2F2!\"#F0F 2\"\"%F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f2_str:=diff( f1(x),x);f2:=unapply(f2_str,x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'f2_strG,&*&,&*$%\"xG\"\"#\"\"\" F+F+!\"$F)F*!#;*$F'!\"#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2G :6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&,&*$9$\"\"#\"\"\"F2F2!\"$F0F1!#; *$F.!\"#\"\"%F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 " \+ f ' = 0 " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "x_extr:=so lve(f1(x)=0,x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "hinreichend:=f2( x_extr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'x_extrG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,hinreichendG\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 " Aus dem Ergebnis der Gr \366\337e 'hinreichend' folgert man: .............................?" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Tp:=[x_extr,f(x_extr)];" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#TpG7$\"\"!F&" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 99 " Wendepunkte: f'' = 0 u nd f''' <> 0 oder Vorzeichenuntersuchung bei f''" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 28 "x_wende:=[solve(f2(x)=0,x)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "links:=subs(x=op(1,x_wende)-1,f2(x)):rechts:=subs(x=o p(1,x_wende)+1,f2(x)):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(x_wendeG7 $,$*$\"\"$#\"\"\"\"\"##F*F(,$F'#!\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "evalf(links);evalf(rechts);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+cF!Q8\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+_ &GH4'!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 " \+ .. daraus folgert man die Existenz der Wendestelle(n), weil symmetris ch !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "WP1:=[op(1,x_wende) ,f(op(1,x_wende))];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$WP1G7$,$*$\" \"$#\"\"\"\"\"##F*F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "W P2:=[op(2,x_wende),f(op(2,x_wende))];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$WP2G7$,$*$\"\"$#\"\"\"\"\"##!\"\"F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Abstand:=solve(As(x)-f(x)<0.05);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(AbstandG6$-%*RealRangeG6$,$%)infinityG!\"\"-%%Ope nG6#$!+)*z*\\C'!\"*-F'6$-F-6#$\"+)*z*\\C'F1F*" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 9 "Zeichnung" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "As:=x->2:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot(\{f(x),As\},x=-3 ..3,y=-1..3);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7S7$$!\"$\" \"!$\"\"#F*7$$!1+++vq@pG!#:F+7$$!1++D^NUbFF0F+7$$!1++]K3XFEF0F+7$$!1++ ]F)H')\\#F0F+7$$!1++D'3@/P#F0F+7$$!1++Dr^b^AF0F+7$$!1++D,kZG@F0F+7$$!1 ++Dh\")=,?F0F+7$$!1++DO\"3V(=F0F+7$$!1+++NkzViUC\"F0F+7$$!1++DhkaI6F0 F+7$$!1+++]XF`**!#;F+7$$!1++++Az2))FhnF+7$$!1++]7RKvuFhnF+7$$!1-+++P'e H'FhnF+7$$!1****\\7*3=+&FhnF+7$$!1)***\\PFcpPFhnF+7$$!1)****\\7VQ[#Fhn F+7$$!1)***\\i6:.8FhnF+7$$!1b+++v`hH!#=F+7$$\"1++](QIKH\"FhnF+7$$\"1** **\\7:xWCFhnF+7$$\"1,++vuY)o$FhnF+7$$\"1)******4FL(\\FhnF+7$$\"1)**** \\d6.B'FhnF+7$$\"1++](o3lW(FhnF+7$$\"1*****\\A))oz)FhnF+7$$\"1+++Ik-,5 F0F+7$$\"1+++D-eI6F0F+7$$\"1++v=_(zC\"F0F+7$$\"1+++b*=jP\"F0F+7$$\"1++ v3/3(\\\"F0F+7$$\"1++vB4JB;F0F+7$$\"1+++DVsYw7#F0F+7$$\"1++v)Q?QD#F0F+7$$\"1+++5jypBF 0F+7$$\"1++]Ujp-DF0F+7$$\"1+++gEd@EF0F+7$$\"1++v3'>$[FF0F+7$$\"1++D6Ej pGF0F+7$$\"\"$F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"F*F*-F$6$7in7$F($\"1++++ +++=F07$F.$\"1mCwf$\"1:Ry-O[q;F07$FA$\"1__UwKO Q;F07$FD$\"1*ykK'*z.g\"F07$FG$\"1O1u])Qob\"F07$FJ$\"1;(\\2p^]]\"F07$FM $\"1Y89YN__9F07$FP$\"1bA#zP\"H%Q\"F07$FS$\"1/tu5sY/8F07$FV$\"1s*\\.oG^ @\"F07$FY$\"13#zw,*3A6F07$Ffn$\"1hL\"yQlJ&**Fhn7$Fjn$\"1,ctoIHP()Fhn7$ F]o$\"1g93UhlprFhn7$F`o$\"1/J=$*3CxcFhn7$Fco$\"1\"o%p\"\\:B+%Fhn7$Ffo$ \"1k)[\"GxL)[#Fhn7$Fio$\"1n&\\N/%>i6Fhn7$$!1)**\\P9(\\$*=Fhn$\"1sDd&Rp C#p!#<7$F\\p$\"1^Efq6pRLF^z7$$!1()*\\i:sw%)*F^z$\"1&4b([Y!4#>F^z7$$!1$ ***\\(oKQm'F^z$\"1Rh))Q&o?%))Fap7$$!1'*\\7`H\">2&F^z$\"1ezzl$f;8&Fap7$ $!1***\\(=K**zMF^z$\"1Un`&*39>CFap7$$!1-]P%[t!))=F^z$\"1k*o+G-r7(!#>7$ F_p$\"1,G'y[DTv\"!#?7$$\"1&*\\7`MSd8F^z$\"1'e>/S4Wo$F[\\l7$$\"1&**\\il g4,$F^z$\"115_)\\M:\"=Fap7$$\"1&*\\Pfy^kYF^z$\"1Os&R&z4UVFap7$$\"1&*** \\i]2=jF^z$\"15%[\"e@(=&zFap7$$\"1&**\\(o%*=D'*F^z$\"1*)>U'4xe$=F^z7$F cp$\"1HFdN%o)*G$F^z7$$\"1******\\4+p=Fhn$\"1sZ)=DA0v'F^z7$Ffp$\"1_,prT 'z7\"Fhn7$Fip$\"1HJsu)3^R#Fhn7$F\\q$\"1j(HX8xe'RFhn7$F_q$\"1ufy2,_#f&F hn7$Fbq$\"1,-!*=u;MrFhn7$Feq$\"1>Xq/\\5D()Fhn7$Fhq$\"1/5A.f-,5F07$F[r$ \"1sb&pV=@7\"F07$F^r$\"17UPC'oz@\"F07$Far$\"1,&4Uhv*38F07$Fdr$\"1(fo'f S&HQ\"F07$Fgr$\"1UDv8Y\")\\9F07$Fjr$\"1%el>L+j]\"F07$F]s$\"1=!yb5Mub\" F07$F`s$\"1ppG]k6+;F07$Fcs$\"1jSZ6X7Q;F07$Ffs$\"12o7Vn.r;F07$Fis$\"1?P 6$Q(p(p\"F07$F\\t$\"1;Q'GB]Ys\"F07$F_t$\"1[E+ " 0 "" {MPLTEXT 1 0 0 "" }}}} }{SECT 0 {PARA 3 "" 0 "" {TEXT -1 13 "Teilaufgabe b" }}{PARA 0 "" 0 " " {TEXT -1 216 "P(u/v) mit u > 0 ist Kurvenpunkt von K und bildet zu sammen mit R( -u / 2) und den Koordinatenachsen ein Rechteck. Bestimme n Sie u so, da\337 der Inhalt des Rechtecks extremal wird. Bestimmen S ie die Art des Extremums." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Zuerst zeichnen wir eine Lageskizze:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "k:=plot(\{f(x),As(x),l\},x=- 4..4,y=0..2.5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "with(pl ottools):l:=line([2,f(2)],[2,2],color=blue):g:=line([2,f(2)],[-2,f(2)] ,color=black):h:=line([-2,f(-2)],[-2,2],color=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plots[display](g,l,h,k);" }}{PARA 13 "" 1 "" {INLPLOT "6)-%'CURVESG6$7$7$$\"\"#\"\"!$\"+++++;!\"*7$$!\"#F*F+-%'COLO URG6&%$RGBGF*F*F*-F$6$7$F'7$F(F(-F26&F4F*F*$\"*++++\"!\")-F$6$7$F.7$F/ F(F9-F$6$7S7$$!\"%F*F(7$$!1nmmmFiDQ!#:F(7$$!1LLLo!)*Qn$FKF(7$$!1nmmwxE .NFKF(7$$!1mmmOk]JLFKF(7$$!1MLL[9cgJFKF(7$$!1nmmhN2-IFKF(7$$!1+++N&oz$ GFKF(7$$!1nmm\")3DoEFKF(7$$!1+++:v2*\\#FKF(7$$!1LLL8>1DBFKF(7$$!1nmmw) )yr@FKF(7$$!1+++S(R#**>FKF(7$$!1++++@)f#=FKF(7$$!1+++gi,f;FKF(7$$!1nmm \"G&R2:FKF(7$$!1LLLtK5F8FKF(7$$!1MLL$HsV<\"FKF(7$$!1-++]&)4n**!#;F(7$$ !1PLLL\\[%R)FipF(7$$!1)*****\\&y!pmFipF(7$$!1******\\O3E]FipF(7$$!1KLL L3z6LFipF(7$$!1MLL$)[`PqM8FKF(7$$\"1++++.W2:FKF(7$$\"1LLLep'Rm\"FKF(7$$\"1+++S>4N=F KF(7$$\"1mmm6s5'*>FKF(7$$\"1+++lXTk@FKF(7$$\"1mmmmd'*GBFKF(7$$\"1+++Dc B,DFKF(7$$\"1MLLt>:nEFKF(7$$\"1LLL.a#o$GFKF(7$$\"1nmm^Q40IFKF(7$$\"1++ +!3:(fJFKF(7$$\"1nmmc%GpL$FKF(7$$\"1LLL8-V&\\$FKF(7$$\"1+++XhUkOFKF(7$ $\"1+++:o'pM=FK7$FV$\"1!HA-G-!==FK7$FY$\"1[)))Rg[-!=FK7$Ffn$\"1-7caZ5za)*pY\"FK7$F^p$\"1#f]w)zz)Q\"FK7$Fap$\"1))[Za!ycF\"FK7$Fdp$\"117RQP Of6FK7$Fgp$\"1Fip7$$!1LLLeGmCDFip$\"1<_Zc%*R)>\" Fip7$Fgq$\"1!e-b_16'e!#<7$$!1nm;aH-88Fip$\"1-#)Q10i*Q$F^]l7$$!11++]-6& )))F^]l$\"1aE\"3jOlc\"F^]l7$$!1tm;/1binF^]l$\"1<7+s+y/\"*F\\r7$$!1SLLe 4**RYF^]l$\"1#=mKt_mH%F\\r7$$!12+]78VL$3Fr)4=F^]l$\"1G5/\"yA\"\\l!#>7$$\"16LL3Uh9SF^]l$ \"1&3%Hs&Q#=KF\\r7$$\"1/L$e9d$>iF^]l$\"1t$*p)[siq(F\\r7$$\"1(HLL3+TU)F ^]l$\"1]DuC!3$49F^]l7$$\"1GL$efeLG\"Fip$\"1<'=%\\[kSKF^]l7$F^r$\"1K+iR HxudF^]l7$$\"1hmmm7+#\\#Fip$\"1^QS!)QRp6Fip7$Far$\"1TS`hU+@>Fip7$$\"1i mm\"*f#))3%Fip$\"1_`G1bvkGFip7$Fdr$\"1zTFGO;&*QFip7$$\"1jmm;(HXx&Fip$ \"1x;IWTL,]Fip7$Fgr$\"1q$3L2k$3hFip7$$\"1%*****\\C4puFip$\"1D9V+l)>;(F ip7$Fjr$\"1CA2bN@m\")Fip7$$\"1$****\\BL]`H<()=N=FK7$Fju$\"1Y>J\\:p[=FK7$F]v$\"1Ek,J3Qh=FK7$ F`v$\"1K>$)G(>@(=FK7$FcvF^w-F26&F4F*FgvF*-%+AXESLABELSG6$%\"xG%\"yG-%% VIEWG6$;FFFcv;F*$\"#DFiv" 2 484 484 484 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Hoehe:=2-subs(x=u,f(x));Breite:=2*u ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&HoeheG,$*$,&*$%\"uG\"\"#\"\"\"F+F+!\"\"F*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%'BreiteG,$%\"uG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "A_rechteck:=u->2*u*(2/(u^2+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+A_rechteckG:6#%\"uG6\"6$%)operatorG%&arrowGF(,$*&9$ \"\"\",&*$F.\"\"#F/F/F/!\"\"\"\"%F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "A1:=diff((4*u)/(u^2+1),u);A2:= diff(A1,u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G,&*$,&*$%\"uG\"\"#\"\"\"F+F+!\"\"\"\"% *&F)F*F'!\"#!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G,&*&,&*$%\"u G\"\"#\"\"\"F+F+!\"#F)F+!#C*&F)\"\"$F'!\"$\"#K" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "u_extr:=[solve(A1=0,u)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'u_extrG7$\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "extr1_bed:=subs(u=op(1,u_extr),A2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*extr1_bedG!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "extr2_bed:=subs(u=op(2,u_extr),A2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*extr2_bedG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 " Randwerte: A(0) und lim A(u) f\374r u \+ gegen unendlich" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "A_links:=A_recht eck(0);A_rechts:=limit((4*u)/(u^2+1),u=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(A_linksG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%) A_rechtsG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 " \+ Also ist A_rechteck(x_extr) ein Abs. Maximum !" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "A_max:=A_rechteck(op(1,u_extr));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&A_maxG\"\"#" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 13 "Teilaufgabe c" }}{PARA 0 "" 0 "" {TEXT -1 60 " \+ Es sei g(x) = 2 - 2/x^2 mit x <> 0 " }{TEXT 259 34 "Zeig e: g(x) < f(x) f\374r alle x <> 0" }}{PARA 0 "" 0 "" {TEXT -1 189 " \+ Untersuchen Sie damit, wie gro\337 der Inhalt der Fl \344che mindestens ist, die von K , der x-Achse und den \+ Geraden: x = 3 und x = 5 begrenzt wird." }}{PARA 0 "" 0 " " {TEXT -1 21 " " }{TEXT 260 52 "Anleitung:Definie re zun\344chst die neue Funktion g(x):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "g:=x->2-2/x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"gG:6#%\"xG6\"6$%)operatorG%&arrowGF(,&\"\"#\"\"\"*$9$!\"#F1F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Bereich:=solve(f(x)-g(x)>0,x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(BereichG6$-%*RealRangeG6$,$%) infinityG!\"\"-%%OpenG6#\"\"!-F'6$F,F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 " Beachte: der Befehl " }{TEXT 262 49 " Int(f(x),x=a..b) gibt die Integralschreibweise," }}{PARA 0 " " 0 "" {TEXT -1 53 " der Befehl: " }{TEXT 263 54 "int(f(x),x=a..b) gibt den Wert des Integrals zur \374ck !!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "a:=3: b:=5:" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "A_mind:=Int(g(x),x=a..b);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'A_mindG-%$IntG6$,&\"\"#\"\"\"*$%\"x G!\"#F-/F,;\"\"$\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "A_mind:=int(g(x),x=3..5); \+ A_mind:= evalf(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'A_mindG#\"# c\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'A_mindG$\"+LLLLP!\"*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 20 " " }{TEXT 261 67 "Mit diesem Programm k\366nne wir aber das Integral auch exakt l \366sen !!" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "A_exakt:=Int(f(x),x=a ..b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(A_exaktG-%$IntG6$,&\"\"#\" \"\"*$,&*$%\"xGF)F*F*F*!\"\"!\"#/F.;\"\"$\"\"&" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 26 "A_exakt:=int(f(x),x=a..b);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%(A_exaktG,(\"\"%\"\"\"-%'arctanG6#\"\"&!\"#-F)6#\" \"$\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A_exakt:=evalf( \");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(A_exaktG$\"+5+H^P!\"*" }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 13 "Teilaufgabe d" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 110 "Die Tangente an K im Punkt B(b/f(b)) (b>0) soll d urch den Ursprung gehen. Bestimmen Sie die Koordinaten von B." }} {PARA 0 "" 0 "" {TEXT -1 86 "Geben Sie die Tangentengleichung an und z eichnen Sie die Tangente zus. mit der Kurve K" }}{PARA 0 "" 0 "" {TEXT -1 109 "Weisen Sie durch Rechnung nach,da\337 au\337er B alle Pu nkte von K mit pos. Abszisse unterhalb der Tangente liegen." }}{PARA 0 "" 0 "" {TEXT -1 95 "Begr\374nden Sie, da\337 von allen Kurvenpunkte n im 1. Feld der Punkt B am n\344chsten bei A(0/2) liegt." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 46 "Tangentengleichung , Ber\374hrpunkt und Z eichnung" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "tg:=f1(b0)*(x-b0 )+f(b);tang:=unapply(tg,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#tgG, &*(,&*$%#b0G\"\"#\"\"\"F+F+!\"#F)F+,&%\"xGF+F)!\"\"F+\"\"%#\"#D\"#8F+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tangG:6#%\"xG6\"6$%)operatorG%& arrowGF(,&*(,&*$%#b0G\"\"#\"\"\"F2F2!\"#F0F2,&9$F2F0!\"\"F2\"\"%#\"#D \"#8F2F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "(0=subs(x=0,t g));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"!,&*&,&*$%#b0G\"\"#\"\"\" F+F+!\"#F)F*!\"%#\"#D\"#8F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Loes:=[solve(\",b0)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%LoesG7 &,$*$,&\"\"\"F)*&%\"IGF)\"#R#F)\"\"#\"\"%F-#F)\"\"&,$F'#!\"\"F1,$*$,&F )F)F*!\"%F-F0,$F6F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 " \+ nur die dritte L\366sung entspricht den Suchbedingungen ! Deshalb lautet die Tangentengl.:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 " F\374r die Tangentengl. verwendet man \+ die bekannte Formel: t(x)= f '(b)(x-b)+f(b)" }}{PARA 0 "" 0 "" {TEXT -1 25 " " }{TEXT 265 20 "zun\344chst als Term \+ .." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "t(x):=f1(1)*(x-1)+f(1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"tG6#%\"xGF'" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 25 " " }{TEXT 264 33 "und jetz t als Funktionsgleichung:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Tx:=un apply(t(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#TxG:6#%\"xG6\"6$% )operatorG%&arrowGF(9$F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 " \+ " }{TEXT 266 23 "Die Koordinaten lauten:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "B:=[1,f(1)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG7$\"\"\"F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 " Zeichnung:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot(\{f(x),Tx(x)\},x=-3..5,y=-2..6);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7S7$$!\"$\"\"!F(7$$!1nmmmFiDG!#:F,7$$!1LLLo!)* Qn#F.F07$$!1nmmwxE.DF.F37$$!1mmmOk]JBF.F67$$!1MLL[9cg@F.F97$$!1nmmhN2- ?F.F<7$$!1+++N&oz$=F.F?7$$!1nmm\")3Do;F.FB7$$!1+++:v2*\\\"F.FE7$$!1LLL 8>1D8F.FH7$$!1nmmw))yr6F.FK7$$!1-+++uR#***!#;FN7$$!1,+++5#)f#)FPFR7$$! 1++++E;!f'FPFU7$$!1nmm;G&R2&FPFX7$$!1MLLLF.rKFPFen7$$!1OLLLHsVF.F`q7$$ \"1******H%=H<#F.Fcq7$$\"1mmm1>qMBF.Ffq7$$\"1++++.W2DF.Fiq7$$\"1LLLep' Rm#F.F\\r7$$\"1+++S>4NGF.F_r7$$\"1mmm6s5'*HF.Fbr7$$\"1+++lXTkJF.Fer7$$ \"1mmmmd'*GLF.Fhr7$$\"1+++DcB,NF.F[s7$$\"1MLLt>:nOF.F^s7$$\"1LLL.a#o$Q F.Fas7$$\"1nmm^Q40SF.Fds7$$\"1+++!3:(fTF.Fgs7$$\"1nmmc%GpL%F.Fjs7$$\"1 LLL8-V&\\%F.F]t7$$\"1+++XhUkYF.F`t7$$\"1+++:o:Djg2&FP7$FX$\"1MQqlYz%4%FP7$$!1,++vF\\sT FP$\"1aR&\\sJc'HFP7$Fen$\"1v[0#Q'4L>FP7$$!1NLLLyP2DFP$\"1\"ff%QL,$=\"F P7$Fhn$\"1vJL7Up,f!#<7$$!1-+]Poc*H\"FP$\"1k7Od%\\;K$Ffz7$$!1!pmmT2Tb)F fz$\"17$F[o$\"15-7)Q()\\;# !#?7$$\"16L$3_\"y%H#Ffz$\"1&))zx6]E0\"F]o7$$\"1UmmT&[0E%Ffz$\"1l')4:m( Qi$F]o7$$\"1t**\\ibJEiFfz$\"1ji'obfMs(F]o7$$\"1/LL$e#3#>)Ffz$\"1!>[b#o DL8Ffz7$$\"1(***\\i;O77FP$\"1%p_2,gq*GFfz7$F_o$\"1Th\"y23e-&Ffz7$$\"1K LLe#=#oCFP$\"1472=_X[6FP7$Fbo$\"1>#[;t&R(*>FP7$$\"1,+++*=C:%FP$\"1lX8n :NTHFP7$Feo$\"1v*eNcIm'RFP7$$\"1MLLeF1JeFP$\"14J#4(3xu]FP7$Fho$\"1'o,& H%p8='FP7$$\"1nmmTrLvuFP$\"1$[,:Us'prFP7$F[p$\"1'Q$=u?A9\")FP7$$\"1*** **\\(*)[6\"*FP$\"1[2gZm=s!*FP7$F^p$\"1tp7s[Vg**FP7$Fap$\"1?Q\"=o]x:\"F .7$Fdp$\"1Nlyg\"))[F\"F.7$Fgp$\"1n&*G4B$*z8F.7$Fjp$\"1[z^,N#*o9F.7$F]q $\"1'*='\\e(QS:F.7$F`q$\"1#3:Ls3xf\"F.7$Fcq$\"1M;hpeW];F.7$Ffq$\"1'f#y ?D'**o\"F.7$Fiq$\"1#*G`/$[bs\"F.7$F\\r$\"1a07$R')Hv\"F.7$F_r$\"1#fUG*e qyjmH2*=F.7$Fjs$\"1\\n+Hb.**=F.7$F]t$\"111eL**p0>F.7 $F`t$\"1fjYcW67>F.7$Fct$\"1TBQ4$ow\">F.7$Fft$\"1Bp2Bp2B>F.-Fit6&F[uF*F \\uF*-%+AXESLABELSG6$%\"xG%\"yG-%%VIEWG6$;F(Fft;$!\"#F*$\"\"'F*" 2 474 474 474 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 145 -29960 0 0 0 0 0 0 }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 58 "Nachweis, da\337 f(x) unterhalb der Tange nte bleibt (f\374r x>0)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 " \+ Dazu mu\337 gezeigt werden,da\337 die Ungleichung : t(x )-f(x) > 0 g\374ltig ist f\374r alle x>0 " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "t(x)-f(x)>0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#2\" \"!,(%\"xG\"\"\"!\"#F'*$,&*$F&\"\"#F'F'F'!\"\"F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solve(\",x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%*RealRangeG6$-%%OpenG6#\"\"!-F'6#\"\"\"-F$6$F*%)infinityG" }}}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 29 "K\374rzeste Entfernung zu A(0/2) " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 301 " Nach dem \+ vorausgegangenen Nachweis ist der Abstand zur Tangente sicher immer k \374rzer, als der Abstand \+ zu jedem anderen Kurvenp unkt von K. Da aber AB senkrecht steht auf der Tangente, ist B " }} {PARA 0 "" 0 "" {TEXT -1 109 " der zu A n\344chstgel egene Kurvenpunkt . ( Bei dieser Argumentation entf\344llt eine Rechnu ng !)" }}{PARA 0 "" 0 "" {TEXT -1 119 " oder : man b estimmt den Abstand eines bel Kurvenpunktes zu A und l\344\337t diesen Abstand minimal werden." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Abstand :=sqrt((x-0)^2 +(f(x)-2)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Abs tandG*$,&*$%\"xG\"\"#\"\"\"*$,&F'F*F*F*!\"#\"\"%#F*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Abstand1:=diff(Abstand,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%)Abstand1G,$*&,&*$%\"xG\"\"#\"\"\"*$ ,&F(F+F+F+!\"#\"\"%#!\"\"F*,&F)F**&F-!\"$F)F+!#;F+#F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Loesungen:=[solve(\",x)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*LoesungenG7)\"\"!\"\"\"!\"\"*$,&!\"#F'*&% \"IGF'\"\"$#F'\"\"#F'F/,$F)F(*$,&F+F'F,F(F/,$F2F(" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 111 " Beachte: die einzige re elle und positive L\366sung ist 1, also die x-Koordinate von B !!" }} {PARA 0 "" 0 "" {TEXT -1 112 " damit ist der N achweis eigentlich schon fertig, aber wir wollen sicherheitshalber noc h " }}{PARA 0 "" 0 "" {TEXT -1 111 " \374ber d ie 2. Ableitung die Best\344tigung f\374r die Minimaleigenschaft unter suchen, zumal es" }}{PARA 0 "" 0 "" {TEXT -1 44 " \+ ja sehr sch\366n ist, " }{TEXT 267 41 "solche Ableitungen vornehme n zu lassen !!" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Abstand2:=diff(Ab stand1,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)Abstand2G,&*&,&*$%\"x G\"\"#\"\"\"*$,&F(F+F+F+!\"#\"\"%#!\"$F*,&F)F**&F-F1F)F+!#;F*#!\"\"F/* &F'#F6F*,(F*F+*&F-!\"%F)F*\"#'**$F-F1F4F+#F+F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 " ..." }{TEXT 268 62 " na, das w \344re doch etwas f\374r Ableitungsspezialisten gewesen !!" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "hinreichend:= subs(x=1,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,hinreichendG,$*$\"\"##\"\"\"F'#\"\"$F'" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 84 " da dieser Wer t positiv ist, folgt daraus die Minimaleigenschaft" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 5 "limit" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Abstand_ minimal:=subs(x=1,Abstand);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0Abst and_minimalG*$\"\"##\"\"\"F&" }}}}}}{MARK "4 5 3 1 0" 20 }{VIEWOPTS 1 1 0 1 1 1803 }