\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;}} {\colortbl\red0\green0\blue0;\red0\green0\blue255;\red0\green128\blue0;\red255\green0\blue0;} \deflang1031\pard\ri4\plain\f4\fs20\cf0\b ________________________________________________________________________________ \par \par \plain\f4\fs20\cf0 Inhalt....: Berechnung des Differenzenquotienten \par Kategorie.: Handwerkskasten \par Mathematik: Analysis \par MuPAD.....: 3.0.0 \par Datum.....: 2002-02-06 \par Autoren...: Kai Gehrs \par Funktionen: limit \par \plain\f4\fs20\cf0\b ________________________________________________________________________________ \par \plain\f3\fs36\cf0\b \par \plain\f3\fs40\cf0\b Elementare MuPAD-Funktionen: \par Berechnung des Differenzenquotienten \plain\f3\fs24\cf2 \par \par Gew\'f6hnlich wird die Differenziation als Grenzwertproze\'df eingef\'fchrt. Wie man mit MuPAD auch \par Grenzwerte des Differenzenquotienten berechnen ka\plain\f3\fs24\cf0 nn, sehen wir im folgenden\plain\f3\fs24\cf2 . \plain\f3\fs28\cf0 \par \par \plain\f3\fs28 Die Definition der Ableitung einer Funktion f an einer Stelle x_0 ist gegeben \par als Grenzwert des Differenzenquotienten: \par {\pict\wmetafile8\picw4600\pich1338\picscalex99\picscaley99\picwgoal2629\pichgoal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}\plain\f3\fs28 \par Wir k\'f6nnen mit MuPAD Grenzwerte mit Hilfe der Funktion \plain\f3\fs28\cf3 limit\plain\f3\fs28 berechnen. \par \par \plain\f3\fs28\cf0 Sie erh\'e4lt stets \plain\f3\fs28\cf2 zwei Argumente\plain\f3\fs28\cf0 : \par \par \plain\f3\fs28\cf3 1. Argument: \plain\f3\fs28\cf0 Die \plain\f3\fs28\cf2 Funktion\plain\f3\fs28\cf0 , deren Grenzwert wir betrachten m\'f6chten. \par \par \plain\f3\fs28\cf3 2. Argument: \plain\f3\fs28\cf0 Eine \plain\f3\fs28\cf2 Gleichung\plain\f3\fs28\cf0 , die angibt, f\'fcr welchen Parameter welcher \par Grenzwert gebildet werden soll. \par \par Wir betrachten ein ganz einfaches Beispiel: Die Normalparabel.\plain\f3\fs28 \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}f:= x -> x^2\plain\f3\fs28\cf3 \par \pard\ri4\plain\f3\fs28 \par Der Differenzenquotient an einer beliebigen Stelle x_0 ist also gegeben durch \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}Dquotient_f:= (f(x_0 + h) - f(x_0)) / h\plain\f3\fs28\cf3 \par \pard\ri4\plain\f3\fs28 \par Jetzt erhalten wir die Ableitung durch Grenzwertbildung f\'fcr h gegen 0. \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}limit(Dquotient_f, h = 0)\plain\f3\fs28\cf3 \par \pard\ri4\plain\f3\fs28 \par Auch kompliziertere Ableit\plain\f3\fs28\cf0 ungen lassen sich auf diese \plain\f3\fs28 Weise ermitteln. \par Sei dazu g die Funktion sin(x): \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}g:= x -> sin(x): \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf3 Dquotient_g:= (g(x_0 + h) - g(x_0)) / h \par \pard\ri4\plain\f3\fs28 \par \par Grenzwertbildung liefert das erwartete Ergebnis: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf3 {\pntext\f1\'b7\tab}limit(Dquotient_g, h = 0)\plain\f3\fs28\cf3 \par \pard\ri4\plain\f3\fs28 \par \plain\f4\fs20\cf0\b ________________________________________________________________________________ \par \plain\f3\fs22\cf0 \par \plain\f3\fs22\cf1\b Aufgaben:\plain\f3\fs22\cf1 \par \plain\f3\fs20\cf1 1. Berechnen Sie mit Hilfe der Funktion \plain\f3\fs20\cf3 limit\plain\f3\fs20\cf1 wie oben die Ableitungen der folgenden Funktionen als \par Grenzwert des Differenzenquotienten an einer beliebigen Stelle x_0: \par (a) f:= 4*x^2 - 2*x \par (b) g:= 6*x^3 - 3*x \par (c) h:= cos(x) \plain\f3\fs20\cf2 \par \plain\f3\fs20\cf1 (d) l:= 3*exp(x) - 1 \par \plain\f4\fs20\cf0\b _______________________________________________________________________________ \par \plain\f3\fs22\cf0 \par \plain\f3\fs22\cf2\b Anmerkungen:\plain\f3\fs22\cf2 \par \plain\f3\fs20\cf2\b 1\plain\f3\fs20\cf2 . Weitere Anregungen finden Sie in der Buchreihe \plain\f3\fs20\cf3 Mathematik 1 x anders\plain\f3\fs20\cf2 . In dieser Reihe \par wird eine Vielzahl unterschiedlichster mathematischer Probleme mit MuPAD gel\'f6st. Die \par B\'fccher k\'f6nnen unter \plain\f5\fs20\cf1 www.schule.mupad.de/literatur\plain\f3\fs20\cf2 kostenfrei kopiert werden. \par \plain\f3\fs20\cf1 \par \plain\f4\fs20\cf0\b _______________________________________________________________________________ \par \par \par }