{VERSION 4 0 "HP RISC UNIX" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 "" 0 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE " " -1 -1 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 20 "Das Riemann-Integral" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart : with(plots):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 288 "twerte := [seq(i *0.5, i = -1..12)]:\nvwerte := [2.0, 2.5, 4.0, 6.1, 7.8, 9.1, 10.4, 11 .5, 12.6, 13.4, 13.7, 13.9, 13.95, 15.0]:\nv := unapply(interp(twerte, vwerte, t), t):\nAnfang := 0: Ende := 5:\nHoehe1 := 14: Hoehe2 := 13: Hoehe3 := 12:\nKurve := plot(v(t), t = Anfang..Ende, thickness=3):" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 3332 "utreppe := proc(n)\nglobal v, An fang, Ende;\nlocal i, delta, Flaeche, dunkelgruen;\ndelta := (Ende - A nfang)/n;\ndunkelgruen := COLOR(RGB, 0.2, 0.4, 0.2);\nFlaeche := sum(v (Anfang+i*delta)*delta, i=0..n-1);\nplots[display]([Kurve, seq(\nPOLYG ONS([[Anfang+i*delta,0],\n [Anfang+(i+1)*delta,0],\n \+ [Anfang+(i+1)*delta,v(i*delta)],\n [Anfang+i*delta,v(i*delta )]], dunkelgruen),\n i=0..n-1),\ntextplot([Anfang+0.1, Ho ehe1, sprintf(\"Fl\344cheninhalt = %8.5f\", Flaeche)],\nalign=\{RIGHT, BELOW\}, font=[HELVETICA, 24], color=dunkelgruen)]);\nend:\notreppe := proc(n)\nglobal v, Anfang, Ende;\nlocal i, delta, Flaeche, hellgruen; \ndelta := (Ende-Anfang)/n;\nhellgruen := COLOR(RGB, 0.7, 0.9, 0.7);\n Flaeche := sum(v(Anfang+i*delta)*delta, i=1..n);\nplots[display]([Kurv e,seq(POLYGONS(\n [[Anfang+i*delta,0],\n [Anfang+(i+1)*delt a,0],\n [Anfang+(i+1)*delta,v((i+1)*delta)],\n [Anfang+i*d elta,v((i+1)*delta)]], hellgruen),\n i=0..n-1),\ntextplot([A nfang+0.1, Hoehe1, sprintf(\"Fl\344cheninhalt = %8.5f\", Flaeche)],\na lign=\{RIGHT,BELOW\}, font=[HELVETICA, 24], color=hellgruen)]);\nend: \ntreppe := proc(n)\nglobal v, Anfang, Ende;\nlocal i, delta, FlaecheL , FlaecheR, dunkelgruen, hellgruen;\ndelta := (Ende-Anfang)/n;\ndunkel gruen := COLOR(RGB, 0.2, 0.4, 0.2);\nhellgruen := COLOR(RGB, 0.7, 0.9, 0.7);\nFlaecheL := sum(v(Anfang+i*delta)*delta, i=0..n-1);\nFlaecheR \+ := sum(v(Anfang+i*delta)*delta, i=1..n);\nplots[display]([Kurve, seq(P OLYGONS(\n [[Anfang+i*delta,0],\n [Anfang+(i+1)*delta,0],\n \+ [Anfang+(i+1)*delta,v(i*delta)],\n [Anfang+i*delta,v(i*delta)]], \+ dunkelgruen),\n i=0..n-1),\nseq(POLYGONS(\n [[Anfang+i*delt a,v(i*delta)],\n [Anfang+(i+1)*delta,v(i*delta)],\n [Anfang+(i +1)*delta,v((i+1)*delta)],\n [Anfang+i*delta,v((i+1)*delta)]], hel lgruen),\n i=0..n-1),\ntextplot([Anfang+0.1, Hoehe1, sprintf( \"Fl\344cheninhalt = %8.5f\", FlaecheL)],\nalign=\{RIGHT,BELOW\}, font =[HELVETICA, 24], color=dunkelgruen),\ntextplot([Anfang+0.1, Hoehe2, s printf(\"Fl\344cheninhalt = %8.5f\", FlaecheR)],\nalign=\{RIGHT,BELOW \}, font=[HELVETICA, 24], color=hellgruen)])\nend:\ntreppe2 := proc(n) \nglobal v, Anfang, Ende;\nlocal i, delta, FlaecheL, FlaecheR, dunkelg ruen, hellgruen;\ndelta := (Ende-Anfang)/n;\ndunkelgruen := COLOR(RGB, 0.2, 0.4, 0.2);\nhellgruen := COLOR(RGB, 0.7, 0.9, 0.7);\nFlaecheL := sum(min(v(Anfang+i*delta), v(Anfang+(i+1)*delta))*delta, i=0..n-1);\n FlaecheR := sum(max(v(Anfang+i*delta), v(Anfang+(i+1)*delta))*delta, i =0..n-1);\nplots[display]([Kurve, seq(POLYGONS(\n [[Anfang+i*delta, 0],\n [Anfang+(i+1)*delta,0],\n [Anfang+(i+1)*delta, (min(v(i* delta), v((i+1)*delta)))],\n [Anfang+i*delta, (min(v(i*delta), v(( i+1)*delta)))]], dunkelgruen),\n i=0..n-1),\nseq(POLYGONS(\n \+ [[Anfang+i*delta, min(v(Anfang+i*delta), v(Anfang+(i+1)*delta))],\n \+ [Anfang+(i+1)*delta,(min(v(Anfang+i*delta), v(Anfang+(i+1)*delta)) )],\n [Anfang+(i+1)*delta,(max(v(Anfang+i*delta), v(Anfang+(i+1)*d elta)))],\n [Anfang+i*delta,(max(v(Anfang+i*delta), v(Anfang+(i+1) *delta)))]], hellgruen),\n i=0..n-1),\ntextplot([Anfang+0.1, H oehe1, sprintf(\"Fl\344cheninhalt = %8.5f\", FlaecheL)],\nalign=\{RIGH T,BELOW\}, font=[HELVETICA, 24], color=dunkelgruen),\ntextplot([Anfang +0.1, Hoehe2, sprintf(\"Fl\344cheninhalt = %8.5f\", FlaecheR)],\nalign =\{RIGHT,BELOW\}, font=[HELVETICA, 24], color=hellgruen)])\nend:" }}} {EXCHG {PARA 257 "" 0 "" {TEXT -1 83 "Ein Auto startet, wobei seine Ge schwindigkeit durch die folgende Kurve gegeben sei:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "display([Kurve, plot(0, t=0..5)]);" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 137 "Welchen Weg hat es in diesen f\374nf S ekunden zur\374ckgelegt?\nWir betrachten die numerischen Werte f\374r \+ die Geschwindigkeit zu vollen Sekunden:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "matrix([['i'$'i'=0..5],[vwerte[2*'i'+2]$'i' =0..5]]);" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 14 "Erster Ansatz:" } {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Sum('v'('i'), ' i'=0..4) = sum(vwerte[2*'i'+2], 'i'=0..4);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "utreppe(5);" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 151 "Berechnet wurde die dunkelgr\374n eingezeichnete Fl\344che, d.h. \+ die Geschwindigkeit wurde konstant untersch\344tzt. Alternativ l\344 \337t sie sich auch \374bersch\344tzen:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "otreppe(5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Sum( 'v'('i'), 'i'=1..5) = sum(vwerte[2*'i'+2], 'i'=1..5);" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 55 "Mit kleineren Intervallen erh\344lt man sch\344rfere Grenzen:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "display([seq(display([treppe(5*i),\ntextplot([0.1, H oehe3, sprintf(\"%d Rechtecke\", 5*i)],\nalign=\{RIGHT,BELOW\}, font=[ HELVETICA, 24], color=tan)]), i=1..40)], insequence=true);" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 109 "Bei noch feineren Unterteilungen ist d ie Graphik \374berfordert; wir beschr\344nken uns daher auf Rechenerge bnisse:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 316 "Digi ts := 30:\nprintf(\" Rechtecke Fl\344che 1 Fl\344che 2\\n\");\nfor n in [5, 10, 20, 50, 100, 200, 500, 1000, 2000, 5000, 10000, 20000, 5 0000, 100000, 200000, 500000, 1000000] do\ndelta := 5/n; i := 'i';\npr intf(\"%10d %8.5f %8.5f\\n\",\nn, sum(v(i*delta)*delta, i=0..n-1), sum(v(i*delta)*delta, i=1..n))\nod:\nDigits := 10:" }}}{EXCHG {PARA 263 "" 0 "" {TEXT -1 36 "Tats\344chliche Fl\344che unter der Kurve:" } {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "int(v(t), t=0.. 5);" }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 60 "Entsprechender Ansatz f \374r die Fl\344che unter einer Sinuslinie:" }{MPLTEXT 1 0 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "v := sin:\nAnfang := 0: Ende := ev alf(Pi): i := 'i':\nHoehe1 := 1.3: Hoehe2 := 1.2: Hoehe3 := 1.1:\nKurv e := plot(v(t), t=Anfang..Ende, thickness=3):\ndisplay(Kurve);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "utreppe(6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "otreppe(6);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 32 "sum(v('i'*Pi/6)*Pi/6, 'i'=0..5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "display([seq(display([treppe(5*i),\ntextplot([0.1 , Hoehe3, sprintf(\"%d Rechtecke\", 5*i)],\nalign=\{RIGHT,BELOW\}, fon t=[HELVETICA, 24], color=tan)]), i=1..10)], insequence=true);" }}} {EXCHG {PARA 265 "" 0 "" {TEXT -1 40 "So geht es also nicht! Richtiger Ansatz:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "dis play([seq(display([treppe2(5*i),\ntextplot([0.1, Hoehe3, sprintf(\"%d \+ Rechtecke\", 5*i)],\nalign=\{RIGHT,BELOW\}, font=[HELVETICA, 24], colo r=tan)]), i=1..10)], insequence=true);" }}}{EXCHG {PARA 266 "" 0 "" {TEXT -1 27 "Allgemein sieht das so aus:" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1593 "f := x -> sin(x+0.5) + sin (6*x+3)/3:\nmin1 := fsolve(diff(sin(x+1/2) + sin(6*x+3)/3, x) = 0, x, \+ 0..0.5):\nmax1 := fsolve(diff(sin(x+1/2) + sin(6*x+3)/3, x) = 0, x, 0. 5..1):\nmin2 := fsolve(diff(sin(x+1/2) + sin(6*x+3)/3, x) = 0, x, 1..1 .5):\nmax2 := fsolve(diff(sin(x+1/2) + sin(6*x+3)/3, x) = 0, x, 1.5..2 ):\na1 := f(min1): a2 := f(0.25): a3 := f(0.5): a4 := f(1):\na5 := f(1 .25): a6 := f(min2): a7 := f(1.5): a8 := f(2):\nb1 := f(0): b2 := f(0. 5): b3 := f(0.75): b4 := f(max1):\nb5 := f(1): b6 := f(1.5): b7 := f(1 .75): b8 := f(max2):\nxi1 := 0.033: xi2 := 0.35: xi3 := 0.68: xi4 := 0 .95:\nxi5 := 1.06: xi6 := 1.46: xi7 := 1.65: xi8 := 1.97:\nfor i from \+ 1 to 8 do c||i := f(xi||i) od:\ndisplay( [\nplot(f(x), x = 0..2, color =red, thickness=5),\nplot([[0, 0], [0, a1], [0.25, a1], [0.25, 0], [0. 25, a2], [0.5, a2], [0.5, 0], [0.5, a3], [0.75, a3], [0.75, 0], [0.75, a4], [1, a4], [1, 0], [1, a5], [1.25, a5], [1.25, 0], [1.25, a6], [1. 5, a6], [1.5, 0], [1.5, a7], [1.75, a7], [1.75, 0], [1.75, a8], [2, a8 ], [2, 0]], style = line, color=green, thickness=3), \nplot([[0, a1], \+ [0, b1], [0.25, b1], [0.25, a2], [0.25, b2], [0.5, b2], [0.5,a3], [0.5 , b3], [0.75, b3], [0.75, a4], [0.75, b4], [1, b4], [1, a5], [1, b5], \+ [1.25, b5], [1.25, a6], [1.25, b6], [1.5, b6], [1.5, a7], [1.5, b7], [ 1.75, b7], [1.75, a8], [1.75, b8], [2, b8], [2, a8]], style = line, co lor=blue, thickness=3),\nseq(plot([[(i-1)/4, c||i], [i/4, c||i]], colo r=cyan, thickness=3), i = 1..8),\nseq(plottools[disk]([xi||i, c||i], 0 .02, color=black), i = 1..8),\nseq(plottools[disk]([xi||i, 0], 0.02, c olor=red), i = 1..8) ], scaling=constrained);" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 310 "Um zu sehen, warum die Ableitung des Integrals glei ch der urspr\374nglichen Funktion ist, beachte man, da\337 in den folg enden Zeichnungen die rote Fl\344che jeweils gleich F(x+h)-F(x) ist, w obei h ihre Breite bezeichnet. Das schwarz umrandete Rechteck soll die selbe Breite und dieselbe Fl\344che haben, hat also die H\366he " } {XPPEDIT 18 0 "(F(x+h)-F(x))/h;" "6#*&,&-%\"FG6#,&%\"xG\"\"\"%\"hGF*F* -F&6#F)!\"\"F*F+F." }{TEXT -1 6 ". F\374r " }{XPPEDIT 18 0 "proc (h) o ptions operator, arrow; 0 end;" "6#R6#%\"hG7\"6$%)operatorG%&arrowG6\" \"\"!F*F*F*" }{TEXT -1 22 " geht das gegen F'(x)." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 414 "Fx := seq(plottools[polygon]([[i /100,0],[i/100,f(i/100)], [(i+1)/100,f((i+1)/100)],[(i+1)/100,0]], sty le=patchnogrid, color=green),i=0..175):\nKurve := plot(f(x), x=0..2.1, thickness=5, color=blue):\nt1 := textplot([0.85,0.9,\"F(x)\"], color= black,\nfont=[HELVETICA,24]):\nt2 := textplot([1.9,1.3, \"F(x+h)-F(x) \"], color=red,\nfont=[HELVETICA,24]):\nfor i from 0 to 23 do ff[i] := 100*int(f(x), x=1.76..2-i/100)/(24-i) od: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 310 "display([seq(\ndisplay([Fx, Kurve, t1, t2,\nplot([[1 .76,ff[i]], [2-i/100,ff[i]], [2-i/100,0], [1.76,0],[1.76,ff[i]]], colo r=black, thickness=5),\nseq(plottools[polygon]([[j/100,0],[j/100,f(j/1 00)],[(j+1)/100,f((j+1)/100)],[(j+1)/100,0]], style=patchnogrid, color =red),j=176..199-i)]),\ni=0..23)],\ninsequence=true);" }}}}{MARK "1 0 \+ 0" 21 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }