\mnb150ÿ{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fswiss\fprq2 Arial;}{\f4\fmodern\fprq1 Courier New;}{\f5\fswiss\fprq2 Helvetica;}{\f6\froman\fprq2 Times New Roman;}} {\colortbl\red0\green0\blue0;\red0\green128\blue0;\red255\green0\blue0;\red0\green0\blue255;\red128\green0\blue0;\red255\green255\blue255;} \deflang1031\pard\ri4\plain\f4\fs20\cf0\b ________________________________________________________________________________ \par \par Inhalt....: Numerische Integration \par Kategorie.: Unterrichtsmaterial \par Mathematik: Analysis, Numerik \par MuPAD.....: 3.0.0 \par Datum.....: 2003-03-04 \par Autoren...: Alessandro Dell'Aere \par Funktionen: plot, plot::Function2d, plot::Line2d, solve, assign, assume, int \par Funktionen: delete, float \par ________________________________________________________________________________ \par \plain\f3\fs28\cf0 \par \plain\f3\fs36\cf0\b Numerische Integration mit MuPAD \par \plain\f3\fs28\cf0 \par \plain\f3\fs24\cf1 Das folgende Arbeitsblatt basiert auf der Unterrichtseinheit "Numerische Integration" von \par Alexandra Dreiseidler. \par Das Arbeitsblattes bietet eine Einf\'fchrung in die n\'e4herungsweise Berechnung von Integralen mit \par Hilfe der Trapez- und der Simpson-Regel. \par Dabei wird sich herausstellen, dass der Approximationsfehler bei der Simpson-Regel wesentlich \par kleiner ausf\'e4llt, als bei der Trapez-Regel. \par \par \plain\f3\fs28\cf0 \par \plain\f3\fs24\cf4\b Unterrichtliche Voraussetzungen \par \plain\f3\fs24\cf4 \par Eine Unterrichtsreihe mit dem in diesem Notebook beschriebenen Inhalt \par setzt folgende Vorbereitung voraus: \par \par \pard\li500\ri4\plain\f3\fs24\cf4 - Die Idee der Riemannschen Summe \par \par - Die symbolisch Integration einer Quadratischen Funktion \par \par - st\'fcckweise definierte Funktionen \par \par - Die L\'f6sung von Gleichungssystemen \par \pard\ri4\plain\f3\fs28\cf4 \par \par \plain\f3\fs24\cf4\b M\'f6gliche Durchf\'fchrung der Unterrichtseinheit und didaktische Vorbemerkungen \par \plain\f3\fs24\cf4 \par Grunds\'e4tzlich sollte darauf hingewiesen werden, da\'df eine symbolische Integration in der Praxis \par nicht immer m\'f6glich ist. Dazu k\'f6nnen einige Beispiele gegeben werden. So kann man den Sinn \par und Zweck der numerischen Integration begr\'fcnden. \par \par Teilweise ist der Einsatz von MuPAD prinzipiell nicht notwendig. Man kann aber methodisch so \par vorgehen, dass man die komplizierten Ergebnisse der Termumformungen des CAS pr\'e4sentiert \par und dann nach den abgelaufenen Rechenschritten fragt. \par \par Nach Bearbeitung dieses Arbeitsblattes sind \'dcbungen zum Festigen der Erkenntnisse zu \par empfehlen. \par \par \plain\f6\fs22\cf0 \par \plain\f3\fs28\cf0\b Allgemeine \'dcberlegungen: \par \par \plain\f3\fs28\cf0 Zur Berechnung von Fl\'e4chen zwischen Messpunkten (Fahrtenschreiberaufgabe) \par oder Funktionsgraphen und x-Achse \'fcber ein Intervall kann man den Graphen \par durch konstante Funktionen und die Fl\'e4che durch Rechtecksummen, sogenannten \par Riemannschen Summen approximieren. L\'e4\'dft man die Anzahl der dabei benutzten \par Teilintervalle gegen unendlich streben, so erh\'e4lt man das Riemannsche Integral. \par Wir wollen nun versuchen, den Graphen durch solche Funktionen zu approximieren, \par die ihn besser beschreiben mit dem Ziel, dass die sich damit ergebenden Fl\'e4chen \par die tats\'e4chliche Fl\'e4che noch besser ann\'e4hern. Das folgende Problem soll der Aus- \par gangspunkt f\'fcr unsere Betrachtungen sein: \par \par \par \plain\f3\fs28\cf0\b (1) Approximation durch lineare Funktionen (Trapez-Regel): \par \plain\f3\fs28\cf0 \par Wir wollen nun versuchen, die zu integrierende Funktion durch Polygonz\'fcge, d.h. \par durch st\'fcckweise lineare Funktionen anzun\'e4hern. Die Fl\'e4che wird dann durch eine \par Summe von Trapezen approximiert. \par \par Es sei eine auf dem Intervall \plain\f4\fs28\cf3 [\plain\f4\fs28\cf3\i a, b\plain\f4\fs28\cf3 ]\plain\f3\fs28\cf0 zu integrierende Funktion \plain\f4\fs28\cf3\i f\plain\f3\fs28\cf0\i \plain\f3\fs28\cf0 gegeben. \par Das Intervall teilen wir in \plain\f4\fs28\cf3\i n\plain\f3\fs28\cf0 Teile der L\'e4nge \par \par \pard\li3000\ri4\plain\f3\fs28\cf0 {\pict\wmetafile8\picw2126\pich1175\picscalex99\picscaley99\picwgoal1213\pichgoal670 0100090000038F02000008001C0000000000050000000B0200000000050000000C0297044E0803 0000001E00050000000C029F045C08050000000B0200000000030000001E00050000000C02A404 6508050000000B0200000000050000000B0200000000030000001E00050000000C02AD04730805 0000000B0200000000050000000B0200000000050000000B0200000000030000001E0005000000 0C02B2047C08050000000B0200000000050000000B0200000000050000000B0200000000050000 000B0200000000030000001E00050000000C02BB048B08050000000B0200000000050000000B02 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Damit haben wir insbesondere \par \par \pard\li2500\ri4\plain\f3\fs28\cf0 {\pict\wmetafile8\picw1790\pich772\picscalex99\picscaley98\picwgoal1017\pichgoal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}\plain\f3\fs28\cf5 ____\plain\f3\fs28\cf0 {\pict\wmetafile8\picw2152\pich789\picscalex99\picscaley99\picwgoal1225\pichgoal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}\plain\f3\fs28\cf0 \par \pard\ri4\plain\f3\fs28\cf0 Zur Veranschaulichung nehmen wir die folgende Funktion \par \par \pard\li3000\ri4\plain\f3\fs28\cf5 __\plain\f3\fs28\cf0 {\pict\wmetafile8\picw2040\pich862\picscalex99\picscaley98\picwgoal1165\pichgoal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}\plain\f3\fs28\cf0 \par \par \pard\ri4\plain\f3\fs28\cf0 und das Integrationsintervall \plain\f4\fs28\cf3\i [0, 1]\plain\f3\fs28\cf0 , welches wir in 5 gleichlange TeiIintervalle \par aufteilen wollen: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}f := -t^4 + t^2: \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf2 n := 4: \par d := (1 - 0)/n: \par (x[i] := 0 + d*(i - 1)) $i= 1.. n + 1: \par \plain\f4\fs24\cf2 \par \pard\ri4\plain\f3\fs28\cf0 Nun lassen wir sowohl den Graphen der Funktion als auch die Trapeze zeichnen: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}plotf := plot::Function2d(f, t=0..1, Color = RGB::Green): \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf2 plot1 := plot::Line2d([x[1], subs(f, t = x[1])], \par [x[2], subs(f, t = x[2])]): \par plot2 := plot::Line2d([x[2], subs(f, t = x[2])], \par [x[3], subs(f, t = x[3])]): \par plot3 := plot::Line2d([x[3], subs(f, t = x[3])], \par [x[4], subs(f, t = x[4])]): \par plot4 := plot::Line2d([x[4], subs(f, t = x[4])], \par [x[5], subs(f, t = x[5])]): \par plot5 := plot::Line2d([x[2], 0], \par [x[2], subs(f, t = x[2])]): \par plot6 := plot::Line2d([x[3], 0], \par [x[3], subs(f, t = x[3])]): \par plot7 := plot::Line2d([x[4], 0], \par [x[4], subs(f, t = x[4])]): \par \par plot(plotf, plot1, plot2, plot3, plot4, plot5, \par plot6, plot7) \par \par \pard\ri4\plain\f3\fs28\cf0 Die Fl\'e4che des i-ten Trapezes l\'e4\'dft sich berechnen durch \par \par \pard\li2500\ri4\plain\f3\fs28\cf0 {\pict\wmetafile8\picw4558\pich1193\picscalex99\picscaley99\picwgoal2606\pichgoal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}\plain\f3\fs28\cf0 \par \par \pard\ri4\plain\f3\fs28\cf0 Wenden wir diese Formel f\'fcr jedes der Trapeze an und summieren dann alle \par Teilfl\'e4chen auf, so erhalten wir eine N\'e4herung f\'fcr das Integral von \plain\f4\fs28\cf3\i f\plain\f3\fs28\cf0 in den \par Grenzen \plain\f4\fs28\cf3\i 1\plain\f3\fs28\cf0 bis \plain\f4\fs28\cf3\i 2\plain\f3\fs28\cf0 : \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}F := float(sum(d/2*(subs(f, t=x[j]) + subs(f, t=x[j+1])), \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf2 j = 1..n)) \par \pard\ri4\plain\f3\fs28\cf0 Zum Vergleich berechnen wir noch das exakte Integral und berechnen die \par Differenz aus diesem und der N\'e4herung: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}F0 := float(int(f, t=0..1)); \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf2 Fehler1 := abs(F - F0) \par \par \pard\ri4\plain\f3\fs28\cf0 Unsere N\'e4herung weicht also um \par \par \pard\li1500\ri4\plain\f3\fs28\cf0 {\pict\wmetafile8\picw3361\pich872\picscalex99\picscaley99\picwgoal1908\pichgoal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}\plain\f3\fs28\cf0 \par \par \pard\ri4\plain\f3\fs28\cf0 von der exakten L\'f6sung ab. Dieser Fehler wird umso kleiner, je feiner wir das \par Integrationsintervall aufteilen, d.h. je gr\'f6\'dfer \plain\f4\fs28\cf3\i n\plain\f3\fs28\cf0 ist. \par \par \plain\f3\fs28\cf0\b (2) Approximation durch Parabeln (Simpson-Regel): \par \plain\f3\fs28\cf0 \par Es stellt sich nun die Frage, ob wir eine Verbesserung der Werte erhalten, wenn \par wir die Funktionen st\'fcckweise durch Parabeln approximieren. Dazu wollen wir \par zun\'e4chst eine Formel zur Bestimmung der eindeutig bestimmten Parabel durch \par drei gegebene Punkte herleiten. Es seien \plain\f4\fs28\cf3\i x1\plain\f3\fs28\cf0 , \plain\f4\fs28\cf3\i x2\plain\f3\fs28\cf0 und \plain\f4\fs28\cf3\i x3\plain\f3\fs28\cf0 die gegebenen \par St\'fctzstellen und \plain\f4\fs28\cf3\i y1\plain\f3\fs28\cf0 , \plain\f4\fs28\cf3\i y2\plain\f3\fs28\cf0 und \plain\f4\fs28\cf3\i y3\plain\f3\fs28\cf0 die zugeh\'f6rigen Funktionswerte. F\'fcr eine \par einfachere Berechnung k\'f6nnen wir uns die Parabel in Richtung der x-Achse so \par verschoben denken, dass \plain\f4\fs28\cf3\i x2\plain\f3\fs28\cf0 gerade bei \plain\f4\fs28\cf3\i x\plain\f4\fs28\cf3 =\plain\f4\fs28\cf3\i 0\plain\f3\fs28\cf0 liegt. \plain\f4\fs28\cf3\i x1\plain\f3\fs28\cf0 und \plain\f4\fs28\cf3\i x2\plain\f3\fs28\cf0 liegen dann \par symmetrisch zur y-Achse. Die Fl\'e4che unter der verschobenen Parabel ist \par genauso gro\'df wie die der urspr\'fcnglichen Parabel. Wir machen f\'fcr die gesuchte \par Parabel den Ansatz \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}Parabel := a*t^2 + b*t + c \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf2 \par \pard\ri4\plain\f3\fs28\cf0 Wir k\'f6nnen mit Hilfe der St\'fctzstellen und zugeh\'f6rigen Funktionswerten die \par Koeffizienten als L\'f6sung eines Gleichungssystems zu bestimmen: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}delete d: \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf2 Gleichungssystem := \par \{y1 = subs(Parabel, t = -d), \par y2 = subs(Parabel, t = 0), \par y3 = subs(Parabel, t = d)\}; \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}assume(d > 0): \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf2 s := solve(Gleichungssystem,[a,b,c]): \par assign(op(s))\plain\f4\fs24\cf2 \par \pard\ri4\plain\f4\fs24\cf3 \par \plain\f3\fs28\cf0 Damit nimmt die allgemeine Parabel folgende Gestalt an: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}Parabel \par \pard\li600\ri1\fi-300\plain\f4\fs24\cf2 \par \pard\ri4\plain\f3\fs28\cf0 Als n\'e4chstes wollen wir die obige Formel f\'fcr die allgemeine Parabel \'fcbertragen \par auf die beiden Parabeln, welche in den entsprechenden Teilintervallen unsere \par Funktion \plain\f4\fs28\cf3\i f\plain\f3\fs28\cf0 approximieren sollen. Zu diesem Zweck werten wir zun\'e4chst \plain\f4\fs28\cf3\i f\plain\f3\fs28\cf0 in den \par St\'fctzstellen aus: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}f1 := subs(f, t = x[1]): \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf2 f2 := subs(f, t = x[2]): \par f3 := subs(f, t = x[3]): \par f4 := subs(f, t = x[4]): \par f5 := subs(f, t = x[5]): \par \par \pard\ri4\plain\f3\fs28\cf0 Diese Werte sind nun in dem Ausdruck der allgemeinen Parabel einzusetzen. \par Weiterhin ist zu beachten, dass der jeweils mittlere St\'fctzpunkt nicht mehr bei \par \plain\f4\fs28\cf3\i x=0\plain\f3\fs28\cf0 liegt. Deshalb ben\'f6tigen wir noch f\'fcr die erste Parabel die Substitution \par \pard\li3500\ri4\plain\f3\fs28\cf0 {\pict\wmetafile8\picw1866\pich789\picscalex99\picscaley98\picwgoal1064\pichgoal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}\plain\f3\fs28\cf0 \par \pard\ri4\plain\f3\fs28\cf0 und f\'fcr die zweite Parabel die Substitution \par \pard\li3500\ri4\plain\f6\fs22\cf0 {\pict\wmetafile8\picw1866\pich789\picscalex99\picscaley98\picwgoal1064\pichgoal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}\plain\f6\fs22\cf0 \par \pard\ri4\plain\f3\fs28\cf0 Wir bekommen also nun zwei Parabeln mit der Variablen \plain\f4\fs28\cf3\i r\plain\f3\fs28\cf0 : \par \plain\f6\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}Parabel1 := subs(Parabel, y1 = f1, y2 = f2, y3 = f3, \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf2 t = r-x[2], d = 1/n); \par Parabel2 := subs(Parabel, y1 = f3, y2 = f4, y3 = f5, \par t = r-x[4], d = 1/n); \par \plain\f4\fs24\cf2 \par \pard\ri4\plain\f3\fs28\cf0 \par Es ist nun noch zu beachten, dass die beiden Parabeln nur in den jeweiligen \par Teilintervallen ben\'f6tigt werden: \par \plain\f6\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}plotpa1 := plot::Function2d(Parabel1, r = x[1]..x[3]): \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf2 plotpa2 := plot::Function2d(Parabel2, r = x[3]..x[5]): \par \plain\f4\fs24\cf2 \par \pard\ri4\plain\f3\fs28\cf0 Nun wollen wir die aus den beiden Parabeln zusammengesetzte Ersatzfunktion \par zusammen mit der Funktion \plain\f4\fs28\cf3\i f\plain\f3\fs28\cf0 in einem Bild darstellen: \par \plain\f6\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}plot(plotpa1, plotpa2, plot6, plotf) \par \pard\li600\ri1\fi-300\plain\f4\fs24\cf2 \par \pard\ri4\plain\f3\fs28\cf0 Die allgemeine Parabel l\'e4\'dft sich leicht mit MuPAD integrieren und das Ergebnis \par kann dann noch vereinfacht werden: \par \plain\f6\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs24\cf2 {\pntext\f1\'b7\tab}int(Parabel, t= -d..d) \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs24\cf2 {\pntext\f1\'b7\tab}normal(%) \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs24\cf2 {\pntext\f1\'b7\tab}factor(%) \par \pard\ri4\plain\f6\fs22\cf0 \par \plain\f3\fs28\cf0 Diese Formel wenden wir nun auf die beiden Parabeln an: \par \plain\f6\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs24\cf2 {\pntext\f1\'b7\tab}F1 := d/3*(subs(f, t=x[1]) + 4*subs(f, t=x[2]) + subs(f, t=x[3])); \par \pard\li600\ri1\fi-300\plain\f4\fs24\cf2 F2 := d/3*(subs(f, t=x[3]) + 4*subs(f, t=x[4]) + subs(f, t=x[5])); \par \par \par \pard\ri4\plain\f3\fs28\cf0 \par Wir hatten oben die Variable \plain\f4\fs28\cf3\i d\plain\f3\fs28\cf0 aus rechentechnischen Gr\'fcnden gel\'f6scht. Setzen \par wir sie wieder auf den urspr\'fcnglichen Wert und addieren dann die beiden \par Integrale \plain\f4\fs28\cf3\i F1\plain\f3\fs28\cf0 und \plain\f4\fs28\cf3\i F2\plain\f3\fs28\cf0 , so erhalten wir die gesuchte N\'e4herung f\'fcr das Integral \par unserer Ausgangsfunktion \plain\f4\fs28\cf3\i f\plain\f3\fs28\cf0 : \par \plain\f6\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs24\cf2 {\pntext\f1\'b7\tab}d := (1-0)/n: \par \pard\li600\ri1\fi-300\plain\f4\fs24\cf2 F := float(F1 + F2) \par \pard\ri4\plain\f3\fs28\cf0 \par Zum Vergleich berechnen wir auch hierf\'fcr die Differenz aus dem exakten \par Integral und der erhaltenen N\'e4herung: \par \plain\f6\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}Fehler2 := abs(F - F0) \par \pard\ri4\plain\f3\fs28\cf0 \par Abschlie\'dfend \'fcberzeugen wir uns davon, dass in unserem Beispiel der \par Approximationsfehler der Simpson-Regel ungef\'e4hr 20 mal kleiner als der \par Approximationsfehler der Trapezregel ist. \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf2 {\pntext\f1\'b7\tab}("Fehler Trapezregel") = Fehler1; \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf2 ("Fehler Simpsonregel") = Fehler2; \par Fehler1/Fehler2 \par \pard\ri4\plain\f4\fs20\cf0\b ________________________________________________________________________________ \par \plain\f3\fs22\cf0 \par \plain\f3\fs22\cf1\b Anmerkungen:\plain\f3\fs22\cf1 \par \plain\f3\fs20\cf1 \par \plain\f3\fs20\cf1\b 1\plain\f3\fs20\cf1 . Dieses Notebook basiert auf der Ausarbeitung \plain\f3\fs20\cf1\b Numerische Integration \plain\f3\fs20\cf1 von\plain\f3\fs20\cf1\b \par Alexandra Dreiseidler\plain\f3\fs20\cf1 , erschienen in der Reihe \plain\f3\fs20\cf1\b NRW-Learn-Line\plain\f3\fs20\cf1 unter der URL \par \tab \plain\f3\fs20\cf1\b \par \pard\li1000\ri4\plain\f3\fs20\cf1\b http://www.learn-line.nrw.de/angebote/cas/allemat.htm\plain\f3\fs20\cf1 \par \pard\ri4\plain\f3\fs20\cf1 \par \plain\f3\fs20\cf1\b 2\plain\f3\fs20\cf1 . Weitere Anregungen finden Sie in der Buchreihe \plain\f3\fs20\cf2 Mathematik 1 x anders\plain\f3\fs20\cf1 . In dieser Reihe \par wird eine Vielzahl unterschiedlichster mathematischer Probleme mit MuPAD gel\'f6st. Die \par B\'fccher k\'f6nnen unter \plain\f5\fs20\cf3 www.schule.mupad.de/literatur\plain\f3\fs20\cf1 kostenfrei kopiert werden. \par \par \plain\f3\fs20\cf1\b 3\plain\f3\fs20\cf1 . Viele weitere praxisorientierte Aufgaben finden sich unter der Web-Adresse \par \plain\f3\fs20\cf1\b \par \pard\li2500\ri4\plain\f3\fs20\cf1\b http://www.learn-line.nrw.de\plain\f4\fs28\cf2 \par \pard\ri4\plain\f4\fs20\cf0\b ________________________________________________________________________________ \par \plain\f6\fs22\cf0 \par }