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Diese wer den aus einem einzigen St\374ck Pappe" }{TEXT 209 0 "" }}{PARA 218 "" 0 "" {TEXT 209 94 "durch Falten und Verkleben hergestellt. Nebenstehen de Figur zeigt das Netz einer solchen T\374te." }{TEXT 209 0 "" }} {PARA 218 "" 0 "" {TEXT 209 59 "Die R\344nder mit den Breiten a=0,9 c m dienen zum Verkleben." }{TEXT 209 0 "" }}{PARA 218 "" 0 "" {TEXT 209 96 "a) Stellen Sie den Materialverbrauch als Funktion von x und \+ y dar, wenn das Volumen der T\374te" }{TEXT 209 0 "" }}{PARA 224 "" 0 "" {TEXT 209 9 " 1000 " }{XPPEDIT 18 0 "cm^3" "6#*$%#cmG\"\"$" } {TEXT 209 10 " betr\344gt." }{TEXT 200 0 "" }}{PARA 224 "" 0 "" {TEXT 209 44 "b) Berechnen Sie den Materialverbrauch f\374r " } {XPPEDIT 18 0 "matrix([[`x (in cm)`, 2, 5, 10, 10, 11], [`y (in cm)`, \+ 2, 5, 10, 5, 6]]);" "6#-%'matrixG6#7$7(%*x~(in~cm)G\"\"#\"\"&\"#5F+\"# 67(%*y~(in~cm)GF)F*F+F*\"\"'" }{TEXT 209 2 " ." }{TEXT 200 0 "" }} {PARA 218 "" 0 "" {TEXT 209 90 "c) F\374r welche Abmessungen x und \+ y wird der Materialverbrauch minimal? Wie gro\337 ist er?" }{TEXT 209 0 "" }}{PARA 218 "" 0 "" {TEXT 209 5 " " }{TEXT 209 0 "" }} {PARA 228 "" 0 "" {GLPLOT2D 561 423 423 {PLOTDATA 2 "6I-%'CURVESG6#7$7 $\"\"!F(7$$\"$v$!\"\"F(-F$6#7$7$F($\"#:F,7$F*F1-F$6#7$7$F($\"+++++=!\" )7$F*F8-F$6#7$7$F($\"+++++AF:7$F*F@-F$6#7$7$F($\"++++]BF:7$F*FG-F$6#7$ F'FF-F$6#7$7$$\"\"&F(F(7$FQFG-F$6#7$7$$\"#BF(F(7$FXFG-F$6#7$7$$\"#JF(F (7$FinFG-F$6#7$7$$\"#OF(F(7$F`oFG-F$6#7$F)FI-F$6$7$7$F($\"#bF,7$FQFjo- %&COLORG6&%$RGBG$\"#5F,$F(F,Fcp-F$6$7$F\\p7$$\"#8F(Fjo-F^p6&F`pFcpFcpF ap-F$6$7$Fgp7$FXFjoF]p-F$6$7$F_q7$FinFjoFjp-F$6$7$Fcq7$F`oFjoF]p-F$6$7 $Fgq7$F*FjoFjp-F$6$7$7$FhpF(7$FhpF1F]p-F$6$7$F`rFgpFjp-F$6$7$Fgp7$FhpF 8F]p-F$6$7$Fgr7$FhpF@Fjp-F$6$7$F[s7$FhpFGF]p-%%TEXTG6$7$$\"#DF,$\"#kF, Q%~y/26\"-Fas6$7$$\"#!*F,FfsQ#~xFis-Fas6$7$$\"#=F(FfsQ#~yFis-Fas6$7$$ \"#FF(FfsF_t-Fas6$7$$\"$N$F,FfsFhs-Fas6$7$$\"2++++++]n$!#:FfsQ#~aFis-F as6$7$$\"21++++++O\"Feu$\"#v!\"#Ffu-Fas6$7$$\"2/++++++S\"Feu$\"#NF,Q%~ x/2Fis-Fas6$7$Fju$\"%v6F^vQ#~hFis-Fas6$7$Fbv$\"#?F(Ffv-Fas6$7$Fju$\"%v AF^vFfu-%*AXESSTYLEG6#%%NONEG-%*THICKNESSG6#\"\"#-%(SCALINGG6#%,CONSTR AINEDG-%*GRIDSTYLEG6#%,RECTANGULARG-%+PROJECTIONG6#Fap-%*LINESTYLEG6#F (" 1 2 2 0 2 2 2 6 1 1 1 1.0 45.0 45.0 1 0 "Curve 1" "Curve 2" "Curve \+ 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve \+ 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Cur ve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 2 9" "Curve 30" "Curve 31" "Curve 32" "Curve 33" }}{TEXT 230 0 "" }} {PARA 218 "" 0 "" {TEXT 209 0 "" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 0 22 "restart: with(plots): " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 0 66 "matrix(2,6,[[`x (in cm)`,2,5,10,1 0,11],[`y (in cm)`,2,5,10,5,6]]);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 0 27 "A := (2*x+2*y+a)*(x+h+2*a);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 0 11 "V := x*y*h;" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 0 23 "h = s olve(V = 1000, h);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 0 10 "subs(%,A);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 218 "" 0 "" {TEXT 209 0 "" }}{PARA 218 "" 0 "" {TEXT 209 91 "Analog zu Aufga be 1 weisen wir der Variablen a erst am Ende der symbolischen Rechnu ng den" }{TEXT 209 0 "" }}{PARA 218 "" 0 "" {TEXT 209 25 "expliziten W ert 0.9 zu." }{TEXT 209 0 "" }}{PARA 218 "" 0 "" {TEXT 209 0 "" }}} {EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 0 18 "a := 0.9; # 0.9 cm" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 0 21 "f := \+ unapply(%%,x,y);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 0 7 "f(2,2);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 218 "" 0 " " {TEXT 209 0 "" }}{PARA 229 "" 0 "" {TEXT 217 26 "Erzeugen der Wertet abelle:" }{TEXT 217 0 "" }}{PARA 218 "" 0 "" {TEXT 209 0 "" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 0 23 "X := [2,5,10,10,11,25];" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 0 21 "Y := \+ [2,5,10,5,6,25];" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 0 36 "XYZ := zip((x,y)->[x,y,f(x,y)],X,Y);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 0 11 "array(XYZ);" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 0 28 "with( linalg): transpose(%%);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 218 "" 0 " " {TEXT 209 0 "" }}{PARA 230 "" 0 "" {TEXT 214 18 "Graphische L\366sun g:" }{TEXT 214 0 "" }}{PARA 218 "" 0 "" {TEXT 209 0 "" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 0 48 "contourplot(f(x,y),x=4..12,y=5..2 0,contours=20);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 218 "" 0 "" {TEXT 209 0 "" }}{PARA 218 "" 0 "" {TEXT 209 101 "Nach Anklicken der Zeichnu ng oben kann man etwa folgende N\344herungswerte f\374r die Extremstel le ablesen:" }{TEXT 209 0 "" }}{PARA 218 "" 0 "" {TEXT 209 0 "" }}} {EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 0 11 "# 6.3, 11.5" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 0 13 "f(6.3, 11.5);" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 0 62 "plot3 d(f(x,y),x=4..12,y=5..20,shading=zhue,style=patchcontour," }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 33 "\n orientation=[18,38],axes=box);" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 218 "" 0 "" {TEXT 209 0 "" }}{PARA 231 "" 0 "" {TEXT 213 19 "Analytische L\366sung:" }{TEXT 213 0 "" }} {PARA 218 "" 0 "" {TEXT 209 0 "" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 0 21 "fx := diff(f(x,y),x);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 22 "\nfy := diff(f(x,y),y);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 0 33 "solve(\{fx=0,fy=0,x>0,y>0\},\{x,y\});" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 0 23 "A_mi n = subs(%,f(x,y));" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {PARA 232 "" 0 "" {TEXT 231 0 "" }}{PARA 233 "" 0 "" {TEXT -1 0 "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }