\mnb150ÿ{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fmodern\fprq1 Courier New;}{\f4\fmodern\fprq1\fcharset1 Courier New;}{\f5\fswiss\fprq2\fcharset1 Arial;}{\f6\fswiss\fprq2 Arial;}{\f7\fmodern\fprq2 COURIER NEW;}{\f8\froman\fcharset1 Times New Roman;}} {\colortbl\red0\green0\blue0;\red0\green128\blue0;\red0\green0\blue255;\red128\green0\blue0;\red255\green0\blue0;\red255\green255\blue255;} \deflang1031\pard\ri4\plain\f3\fs20\cf0 ________________________________________________________________________________ \par \par Inhalt....: Volumen einer Weinflasche als Rotationsk\'f6rper \par Kategorie.: Unterrichtsmaterial \par Mathematik: Analysis \par MuPAD.....: 3.1.1 \par Datum.....: 2005-04-04 \par Autoren...: Robert Krell \par Funktionen: \plain\f4\fs20\cf0 -->, piecewise, solve, plot, plot::XRotate, int, float \par Freitext..: \plain\f3\fs20\cf0 Rotationsk\'f6rper, Umgekehrte Kurvendiskussion ("Steckbrief-Aufgabe") \par \plain\f4\fs20\cf0 Freitext..: \plain\f3\fs20\cf0 http://www.r-krell.de \par ________________________________________________________________________________ \par \plain\f3\fs20\cf0\b \par \plain\f5\fs40\cf0\b \'c4u\'dferes Volumen einer kleinen Weinflasche \par \par \plain\f5\fs24\cf1 Als praktisches Beispiel f\'fcr einen Rotationsk\'f6rper wurde eine Weinflasche vermessen und \par ihre Form durch den Graph einer abschnittsweise definierten Funktion nachgebildet, deren \par St\'fccke aus Polynomen 0. bzw. 3. Grades bestehen. Das f\'fcr die (um die x-Achse) rotierende \par Funktion berechnete Volumen wurde mit dem im Tauchversuch ermittelten echten Volumen \par der Flasche verglichen. \plain\f3\fs20\cf0\b \par ________________________________________________________________________________ \par \plain\f6\fs24\cf2\b \par \par \plain\f6\fs24\cf3\b\ul Vorbemerkungen: \par \plain\f6\fs24\cf3 \par \plain\f5\fs24\cf3 1. Dieses MuPAD-Notebook kann (weil die Weinflasche nicht mitgeliefert wird) nicht als \par fertiges Blatt mit in den Unterricht genommen werden, sondern soll dazu anregen, selbst \par verf\'fcgbare K\'f6rper in \'e4hnlicher Weise zu behandeln oder von der Sch\'fclerinnen und \par Sch\'fclern behandeln zu lassen -- am besten in Gruppen, um unterschiedliche Messungen \par und Ans\'e4tze sp\'e4ter miteinander vergleichen zu k\'f6nnen. \par \plain\f5\fs28\cf3\b \par \plain\f6\fs24\cf3 2. Zu dieser Aufgabe wurde ich inspiriert durch die in einer MuPAD-Fortbildung sowie \par im Buch bzw. im Band 4 von "\plain\f6\fs24\cf4 Mathematik 1 x anders\plain\f6\fs24\cf3 " (S. 80 ff) \par [\plain\lang1033\f7\fs20\cf2 http://schule.mupad.de/literatur/m1xa/band-4.pdf\plain\f6\fs24\cf3 ] bzw. in \par \plain\lang1033\f7\fs20\cf2 http://schule.mupad.de/material/notebooks/Mensch_aergere_dich_nicht.html\plain\f6\fs24\cf3 \par gezeigte Volumenbestimmung einer Spielfigur. Allerdings gefiel mir nicht, dass eine einfache \par Figur durch ein Polynom 6. Grades angen\'e4hert wurde, das ja bekanntlich bis zu 5 Extrema \par hat und deshalb viel zu wellig ist. Tats\'e4chlich hat die Spielfigur am unteren Ende auch eine \par deutliche Einschn\'fcrung, weicht also erheblich von der realen Spielfigur ab. Die Volumen- \par berechnung mit der erkennbar schlechten N\'e4herung ist wenig sinnvoll. (Auf der Webseite \par \plain\lang1033\f7\fs20\cf2 http://www.learn-line.nrw.de/angebote/cas/barzel1.htm \plain\lang1033\f6\fs24\cf3 von Frau Barzel, auf die \par die Aufgabenstellung -- aber nicht die in der MuPAD "Beispiell\'f6sung" -- zur\'fcckgeht, findet \par man Hinweise auf Alternativen). \par \par Deshalb habe ich eine K\'f6rper gesucht (und mit der Weinflasche gefunden), der sich \par sinnvoll und einigerma\'dfen genau mit einem Polynomansatz mit niedrigem Grad ann\'e4hern \par l\'e4sst. Wenn Sie \'c4hnliches in Ihrem Unterricht durchf\'fchren wollen, achten Sie auf eine \par geeignete Flaschen- bzw. K\'f6rperform und vergessen Sie nicht die R\'fcckf\'fchrung des \par Modells in die Wirklichkeit mit einer Betrachtung m\'f6glicher Fehler -- sonst verkommt \par 'Anwendungsorientierung' zur beliebigen Dekoration!\plain\f6\fs24\cf3\b \par \plain\f5\fs40\cf0\b \par \'c4u\'dferes Volumen einer kleinen Weinflasche \par \plain\f8\fs22\cf0 Robert Krell (www.r-krell.de) \par \par \pard\li1500\ri4\plain\f8\fs22\cf0 {\pict\wmetafile8\picw12144\pich8479\picscalex99\picscaley99\picwgoal6932\pichgoal4842 010009000003BF7301000200B06A01000000050000000B0200000000050000000C021F21702F03 0000001E00050000000C025D21C42F050000000B0200000000030000001E00050000000C027621 ED2F050000000B0200000000050000000B0200000000030000001E00050000000C02B621423005 0000000B0200000000050000000B0200000000050000000B0200000000030000001E0005000000 0C02CF216A30050000000B0200000000050000000B0200000000050000000B0200000000050000 000B0200000000030000001E00050000000C020F22C130050000000B0200000000050000000B02 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80008000800000808000C0C0C000C0DCC000A6CAF000FFFBF000A0A0A40080808000FF00000000 FF0000FFFF00000000FF00FF00FF0000FFFF00FFFFFF00040000003402010004000000F0010000 040000002701FFFF030000000000 }\plain\f8\fs22\cf0 \par \pard\ri4\plain\f8\fs22\cf0 \par \plain\f8\fs28\cf0 Die Weinflasche ist 19 cm hoch (inkl. Verschluss). Sie wird horizontal gelegt, x-Achse ist die \par Symmetrie-/Rotationsachse. Der Ursprung wurde auf Sch\'fclervorschlag in die Mitte der Flasche, \par also 9,5 cm \'fcber den (bzw. rechts vom) Flaschenboden verlegt. \par \par Die Flasche besteht im Wesentlichen aus einem Zylinder mit r = 3.3 cm (f\'fcr -9,5 <= x <= -3,5), \par einer durch die Funktion g zu beschreibenden Verj\'fcngungszone (-3,5 <= x <= 3,5) und wieder \par einem (nahezu) zylindrischen Hals mit r=1.1 (f\'fcr 3,5 <= x <= 9,5). \par \par Die Funktion g wird als kubisches Polynom angesetzt, damit eine erkannte Wendestelle auf- \par treten kann \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf4 {\pntext\f1\'b7\tab}g := x --> a*x^3 + b*x^2 + c*x + d \par \pard\li600\ri1\fi-300\plain\f3\fs28\cf4 \par \pard\ri4\plain\f8\fs28\cf0 Die \'dcbergangsbedingungen zu den Zylindern f\'fchren zu folgenden 4 Gleichungen bzw. damit \par zu den Koeffizienten \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf4 {\pntext\f1\'b7\tab}solve(\{g(-3.5) = 3.3, \par \pard\li600\ri1\fi-300\plain\f3\fs28\cf4 g( 3.5) = 1.1, \par g'(-3.5) = 0, \par g'( 3.5) = 0\}, \par \{a, b, c, d\}) \par \par \pard\ri4\plain\f8\fs28\cf0 Im Unterricht war allerdings die Funktion g auf Sch\'fclervorschlag in einem um 2,2 nach oben \par verschobenen Koord.-system berechnet worden, weil das verschobene g (hier g1) dann punkt- \par symmetrisch ist und einem einfacheren Ansatz gehorcht: Es m\'fcssen nur noch 2 Koeffizienten \par bestimmt werden. (Alle Rechnungen im Unterricht erfolgten von Hand ohne MuPAD!) \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf4 {\pntext\f1\'b7\tab}g1 := x --> a1*x^3 + c1*x \par \pard\li600\ri1\fi-300\plain\f3\fs28\cf4 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf4 {\pntext\f1\'b7\tab}solve(\{g1(-3.5) = 1.1, \par \pard\li600\ri1\fi-300\plain\f3\fs28\cf4 g1'(-3.5) = 0\}, \par \{a1, c1\}) \par \par \pard\ri4\plain\f8\fs28\cf0 Im Folgenden wird mit etwas st\'e4rker gerundeten Koeffizienten weiter gerechnet \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf4 {\pntext\f1\'b7\tab}a1 := 0.01283: \par \pard\li600\ri1\fi-300\plain\f3\fs28\cf4 c1 := -0.4714: \par \pard\ri4\plain\f8\fs28\cf0 \par Und durch Zur\'fcckverschieben erh\'e4lt man f\'fcr g (in der neuen Form als g2 bezeichnet, aber \par identisch mit dem oben berechneten g) \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf4 {\pntext\f1\'b7\tab}g2 := x --> g1(x) + 2.2 \par \pard\li600\ri1\fi-300\plain\f3\fs28\cf4 \par \pard\ri4\plain\f8\fs28\cf0 Zur Kontrolle werden die Anschlusswerte \'fcberpr\'fcft; weitere Werte sollen mit der Schieblehre \par an der Flasche nachgepr\'fcft werden, um die Realit\'e4tsn\'e4he des Ansatzes zu pr\'fcfen (es werden \par nat\'fcrlich unn\'f6tig viele bzw. irref\'fchrend viele Kommastellen angegeben, \'fcber die hier gesprochen \par werden muss -- s.u.!): \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf4 {\pntext\f1\'b7\tab}g2(-3.5); g2(0); g2(3.5); g2'(-3.5); g2'(0); \par \pard\li600\ri1\fi-300\plain\f3\fs28\cf4 g2'(3.5); g2(-2); g2(-1); g2(1); g2(2); \par \par \pard\ri4\plain\f8\fs28\cf0 Insgesamt wird die Silhouette der Flasche also durch eine abschnittsweise definierte Funktion \par f beschrieben mit zwei Zylindern bzw. mit zwei konstanten Radien und dem Mittelst\'fcck g \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf4 {\pntext\f1\'b7\tab}f2 := x --> piecewise([x < -3.5, 3.3], \par \pard\li600\ri1\fi-300\plain\f3\fs28\cf4 [x >= -3.5 and x < 3.5, g2(x)], \par [x>=3.5, 1.1]) \par \par \pard\ri4\plain\f8\fs28\cf0 Der Graph bzw. der durch f bestimmte Rotationsk\'f6rper sieht tats\'e4chlich flaschenartig aus und \par ermutigt daher zum Weiterrechnen mit diesem Ansatz: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf4 {\pntext\f1\'b7\tab}plot(plot::XRotate(f2(x), x = -9.5 .. 9.5), \par \pard\li600\ri1\fi-300\plain\f3\fs28\cf4 Scaling = Constrained): \par \pard\ri4\plain\f8\fs28\cf0 Durch Integration von f (von Hand: mit st\'fcckweise gebildeten Stammfunktionen bzw. 3 Teil- \par integralen) in den Grenzen von -9,5 <= x <= 9,5 erh\'e4lt man das (\'fcbertrieben bzw. schein-genaue) \par Ergebnis \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf4 {\pntext\f1\'b7\tab}AussenVolumen := int(PI*(f2(x))^2, x=-9.5..9.5); \par \pard\li600\ri1\fi-300\plain\f3\fs28\cf4 float(AussenVolumen) \par \par \pard\ri4\plain\f8\fs28\cf0 Die Festlegung auf zwei oder h\'f6chstens drei signifikante Stellen w\'e4re besser (da alle Ma\'dfe auch \par nur mit zwei Stellen Genauigkeit aufgenommen wurden, mehr Exakthaeit aufgrund des Ansatzes \par also gar nicht zu erwarten ist), wobei allerdings die von MuPAD erg\'e4nzte .0 etwas st\'f6rt: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf4 {\pntext\f1\'b7\tab}DIGITS := 3 : \par \pard\li600\ri1\fi-300\plain\f3\fs28\cf4 float(AussenVolumen); \par delete DIGITS: \par \par \pard\ri4\plain\f8\fs28\cf0 Ob dies stimmt bzw. wie gut die N\'e4herung der Flaschenform durch f ist, musste ein Tauchversuch \par der Flasche zeigen, wobei das verdr\'e4ngte Wasser gemessen wird! \'dcber m\'f6gliche Ungenauigkeiten \par und Fehlerquellen hinsichtlich der N\'e4herung als auch der ersten (Durchmesser-)Messungen an der \par Flasche oder m\'f6glicher Fehler beim Tauchversuchs muss gesprochen werden! \par \par Beim Tauchversuch verdr\'e4ngte die Flasche 350 ml ! Das ist ein Fehler < 1% , der allerdings nicht \par dar\'fcber hinweg t\'e4uschen darf, dass allein aufgrund der zweistelligen Messungen mit ca. 10% Fehler \par gerechnet werden muss und dar\'fcber hinaus einige Fehlerquellen existieren: unklare Zylinderl\'e4nge, \par ungenaue Messung (wird der untere Zylinder mir r=3,2cm berechnet, treten schon 6% Fehler auf!), \par Flasche ist nicht wirklich rund, N\'e4herung durch g bzw. g2 entspricht nicht genau der Wirklichkeit \par (der Glasbl\'e4ser hatte sicher kein Polynom als Vorlage), der \'dcbergang vom Boden zum unteren \par Zylinderteil hat keine scharfe Kante, sondern ist mit etwa r=0,5 cm abgerundet, usw.! \par \plain\f3\fs22\cf0 \par \plain\f3\fs20\cf0 ________________________________________________________________________________ \par \plain\f6\fs24\cf2\b \par \plain\f6\fs24\cf1\b Anmerkungen: \par \plain\f6\fs24\cf2\b \par \plain\f6\fs24\cf1 1. In den Vorbemerkungen -- auf die hier extra verwiesen wird -- wurden auch bereits \par \plain\f6\fs24\cf5 __\plain\f6\fs24\cf1 einige Webseiten mit weiterf\'fchrenden Inhalten angegeben. \par \par 2. F\'fcr Rotationsk\'f6rper mit Spline-Ansatz (der sinnvollerweise auch f\'fcr die Mensch- \par \plain\f6\fs24\cf5 __\plain\f6\fs24\cf1 \'c4rgere-Dich-Nicht-Figur durchgef\'fchrt werden sollte) gibt es im MuPAD-Material \par \plain\f6\fs24\cf5 __\plain\f6\fs24\cf1 schon Beispiele, u.a. "\plain\f6\fs24\cf4 Rotationsk\'f6rper, Polynom- und Splineinterpolation\plain\f6\fs24\cf1 " auf \par \plain\f3\fs20\cf5 __\plain\f3\fs20\cf2 http://schule.mupad.de/material/notebooks/SchneckeSpline.html\plain\f6\fs24\cf1 bzw. -\plain\f3\fs20\cf2 .mnb \par \plain\f6\fs24\cf1 \par 3. F\'fcr realistische Anwendungsorientierung setzt sich der Verein MUED e.V. ein:\plain\f3\fs20\cf2 \par http://www.mued.de/ \par \plain\f6\fs24\cf1 \par 4. Der Autor dieses Beitrags unterh\'e4lt eine Webseite zur Mathematik, Physik, \plain\f6\fs24\cf5 \par __\plain\f6\fs24\cf1 Informatik und Fotovoltaik unter\plain\f3\fs20\cf2 http://www.r-krell.de \par \plain\f3\fs20\cf0 ________________________________________________________________________________ \par \plain\f8\fs22\cf0 \par }