{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }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 }1 0 0 0 8 4 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 1 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 6 "Quelle" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 414 "Dat einame: polynom.mws\nDateigr\366\337e: 44 KB\nName: Kerstin M\374ller \nSchule: Isolde-Kurz-Gymnasium\nKlasse: 13\nDatum: 05.04.99\nKategori e: Analysis / Differentialrechnung / Bestimmung von Funktionen\nThema: Polynom durch n Punkte\nStichw\366rter: Bestimmung von Funktionen\nKu rzbeschreibung: Bestimmung eines Polynoms, das durch Punkte und andere Angaben (Extrema, Wendestellen, Steigung) definiert und berechnet wir d, Spline-Fit" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Update auf Maple 8" }}{PARA 0 "" 0 "" {TEXT -1 24 "14.05.2004 Manuel M\374ller" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 24 "Polynom durc h n+1 Punkte" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "allgemein" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 141 "Wenn man n+1 Punkte hat, so erh\344lt man ein Polynom n- ten Grades. Man braucht also immer eine Angabe mehr als der Grad der F unktion sein soll." }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Polynom durch n+1 Punkte" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 44 "P1:=[1,4]; P2:=[3,2]; P3:=[2,6]; P4:=[4,3];\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Den Grad der Funktion angeben:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "grad:=3;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Die Funktion als Summe schreiben:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f:=unapply(add(a[i]*x^i,i=0..grad), x);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f(x);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "Die Funktion des x-Wertes ist der y-Wert \+ des Punktes. Also kann man vier Gleichungen aufstellen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "gl1:=f(P1[1])=P1[2];\ngl2:=f(P2[1]) =P2[2];\ngl3:=f(P3[1])=P3[2];\ngl4:=f(P4[1])=P4[2];\n\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Vier Unbekannte, vier Gleichungen -> Glei chungen nach a0, a1 usw. aufl\366sen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "solve(\{gl1,gl2,gl3,gl4\},\{a[0],a[1],a[2],a[3]\});\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Die Funktion hei\337t also:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f(x);\n" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "inter p([P1[1],P2[1],P3[1],P4[1]],[P1[2],P2[2],P3[2],P4[2]],x);\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "zeichnen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "f unktion:=plot(f(x),x=0..6):\npunkte:=plot(\{P1,P2,P3,P4\},x=0..6,style =point,color=blue,symbol=diamond):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "display(funktion,punkte);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 56 "Polynom definiert durch vier A ngaben (mit Steigung usw.)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "P1:=[2,4]; P2:=[5,7];\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Den Grad der Funktion angeben: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "grad:=3;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Die Funktion als Summe schreiben:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f:=unapply(add(a[i]*x^i,i=0. .grad),x);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f(x);\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "Die Funktion des x-Wertes ist der \+ y-Wert des Punktes. Also kann man vier Gleichungen aufstellen:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "gl1:=f(P1[1])=P1[2];\ngl2:=f (P2[1])=P2[2];\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "F\374r die St eigung braucht man die erste Ableitung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "fs:=diff(f(x),x);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "fs:=unapply(fs,x);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Also hei\337t die dritte Gleichung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "gl3:=fs(P1[1])=3;\n" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 53 "\304nderung der Steigung ist die zweite Ableitung. Also :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "fss:=diff(fs(x),x);\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "fss:=unapply(fss,x);\n " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Also hei\337t die vierte Glei chung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "gl4:=fss(P2[1])=1 ;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Vier Unbekannte, vier Glei chungen -> Gleichungen nach a0, a1 usw. aufl\366sen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "solve(\{gl1,gl2,gl3,gl4\},\{a[0],a[1],a[2 ],a[3]\});\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Die Funktion hei\337t also:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f(x);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "zeichnen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "funktion:=plot(f(x),x=0..6):\npunkte:=plot(\{P1,P2\},x=0..6,s tyle=point,color=blue,symbol=diamond):\ndisplay(funktion,punkte);\n" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Spline- Fit" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "P1:=[2,1]; P2:=[3,4]; P3:=[5,3];\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Den Grad der beiden Polynome angeben:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "grad:=3;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Die Funktion als Summe schreiben:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "f1:=unapply(add(a[i]*x^i,i=0..grad) ,x);\nf2:=unapply(add(b[i]*x^i,i=0..grad),x);\n" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 14 "f1(x);\nf2(x);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "Die Funktion des x-Wertes ist der y-Wert des Punktes. Al so kann man schon mal drei Gleichungen aufstellen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "gl1:=f1(P1[1])=P1[2];\ngl2:=f1(P2[1])=P2[2] ;\ngl3:=f2(P3[1])=P3[2];\n\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 198 " F\374r P2 gilt ja f\374r beide Polynome, da\337 die Funktion des x-Wer tes den y-Wert ergibt. Da der \334bergang m\366glichst glatt sein soll , m\374ssen die beiden Polynome im Punkt zwei denselben Funktionswert \+ haben:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "gl4:=f2(P2[1])=P2 [2];\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "F\374r die Steigung bra ucht man die erste Ableitung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "f1s:=diff(f1(x),x);\nf2s:=diff(f2(x),x);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "f1s:=unapply(f1s,x);\nf2s:=unapply(f2s,x);\n " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Da die beiden Steigungen in P 2 gleich gro\337 sein sollen, hei\337t die n\344chste Gleichung:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "gl5:=f1s(P2[1])=f2s(P2[1]); \n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Die Kr\374mmung ist die zwe ite Ableitung. Also:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f1s s:=diff(f1s(x),x);\nf2ss:=diff(f2s(x),x);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "f1ss:=unapply(f1ss,x);\nf2ss:=unapply(f2ss,x);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Also hei\337t die n\344chste Gl eichung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "gl6:=f1ss(P2[1] )=f2ss(P2[1]);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "Die letzten \+ zwei Gleichungen erhalten wir, indem wir einfach Werte f\374r die Able itungen der Funktionen an den Endpunkten angeben:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 19 "gl7:=f1s(P1[1])=1;\n" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "gl8:=f2s(P3[1])=2;\n" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 61 "Diese acht Gleichungen l\366sen wir nach den acht Varia blen auf:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "solve(\{gl1,gl 2,gl3,gl4,gl5,gl6,gl7,gl8\},\{a[0],a[1],a[2],a[3],b[0],b[1],b[2],b[3] \});\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Und diese Werte belegen wir jetzt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Und lassen die beiden Funktione n ausgeben:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f1(x);\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f2(x);\n" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 71 "Jetzt lassen wir die zwei Funktionen und die gegeb enen Punkte zeichnen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "f kt1:=plot(f1(x),x=1..6,0..10,color=green):\nfkt2:=plot(f2(x),x=1..6,0. .10,color=red):\npunkte:=plot(\{P1,P2,P3\},color=blue,style=point,symb ol=box):\ndisplay(fkt1,fkt2,punkte);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "Parabel durch drei Punkte" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 6 "normal" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Gegeben sind drei Punkte, durch die eine Parabel verlaufen soll." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "P1:=[-2,7]; P2:=[1,-1]; P3:=[5,8];\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "grad:=2;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f:=unapply(add(a[i]*x^i,i=0..grad),x);\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f(x);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "gl1:=f(P1[1])=P1[2];\ngl2:=f(P2[1])=P2[2];\ng l3:=f(P3[1])=P3[2];\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " solve(\{gl1,gl2,gl3\},\{a[0],a[1],a[2]\});\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f(x);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "fkt:=plot(f (x),x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "punkte:=plot(\{P 1,P2,P3\},style=point,symbol=box,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "display(fkt,punkte);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "\334berp r\374fung, ob die Funktion stimmt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "f:=x->interp([-2,1,5],[7,-1,8],x);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f(x);\n" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 40 "Parabelschar (ein Punkt wird vers choben)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "Gegeben sind drei Punk te, durch die eine Parabel verlaufen soll. Einer dieser Punkte soll ve rschoben werden, so da\337 man eine Parabelschar erh\344lt." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 36 "P1:=[-2,10]; P2:=[1,t]; P3:=[5,18];\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "grad:=2;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f:=unapply(add(a[i]*x^i,i=0..grad),x);\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f(x);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "gl1:=f(P1[1])=P1[2];\ngl2:=f(P2[1]) =P2[2];\ngl3:=f(P3[1])=P3[2];\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "solve(\{gl1,gl2,gl3\},\{a[0],a[1],a[2]\});\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f(x);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "fkt:=plot(\{seq(f(x),t=-2..2)\},x=-6..8,y=-4..20):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "punkte13:=plot(\{P1,P3\},style=poin t,symbol=box,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "punkt2:=plot(\{seq(P2,t=-2..2)\},style=point,symbol=box,color=blue ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "display(fkt,punkte13, punkt2);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "\334berpr\374fung, ob die berechnete Funk tion stimmt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "interp([-2, 1,5],[10,t,18],x);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 5 "\334bung " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 160 "Bestimme die ganzrational Funktion, deren Schaubild d ie Punkte W1(-1/8) und W2(3/-344) als Wendepunkte besitzt und an der S telle 0 eine waagrechte Tangente hat." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Den Grad der Funktion angeben:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "grad:=4;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 " Die Funktion als Summe schreiben:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f:=unapply(add(a[i]*x^i,i=0..grad),x);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f(x);\n" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 10 "fs:=D(f);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "fss:=D(fs);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Punkte:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "w1:=[-1,8]; w2:=[3,-344]; \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "gl1:=f(-1)=8;\ngl2:=f (3)=-344;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "F\374r die Wendepunkte brauche ich die zw eite Ableitung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "gl3:=fss (-1)=0;\ngl4:=fss(3)=0;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "F\374r die waagrechte Tange nte brauche ich die erste Ableitung, da die Steigung in diesem Punkt g leich null ist:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "gl5:=fs( 0)=0;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "solve(\{gl1,gl2,gl3,gl4,gl5\});\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Die Funktion hei\337t also:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f(x);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "zeichnen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "fu nktion:=plot(f(x),x=-2..6):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "punkte:=plot(\{w1,w2\},x=-2..3,style=point,color=blue,symbol=box): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "display(funktion,punkte );\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }