DERIVE for Windows version 5.05 DfW file saved on 12 Dec 2002 -CENTER_OF_CURVATURE(y, x):=CENTER_OF_CURVATURE_AUX([x, y], DIF(y, x), DIF(y, x, 2)) CENTER_OF_CURVATURE_AUX(v, d1, d2):=v + [-d1, 1](1 + d1^2)/d2 CIRCLE_AUX(v0, r, ):=[v01 + rCOS(), v02 + rSIN()] CURVATURE(y, x):=CURVATURE_AUX(DIF(y, x), DIF(y, x, 2)) CURVATURE_AUX(d1, d2):=d2/(1 + d1^2)^(3/2) IMP_CENTER_OF_CURVATURE(u, x, y):=CENTER_OF_CURVATURE_AUX([x, y], IMP_DIF(u, x, y, 1), IMP_DIF(u, x, y, 2)) IMP_CURVATURE(u, x, y):=CURVATURE_AUX(IMP_DIF(u, x, y, 1), IMP_DIF(u, x, y, 2)) IMP_DIF(u, x, y, default1):=IMP_DIF_AUX(- DIF(u, x)/DIF(u, y), x, y, default1) IMP_DIF_AUX(d1, x, y, n):=ITERATE(DIF(dk, x) + d1DIF(dk, y), dk, d1, n - 1) IMP_OSCULATING_CIRCLE(u, x, y, x0, y0, ):=CIRCLE_AUX(LIM(IMP_CENTER_OF_CURVATURE(u, x, y), [x, y], [x0, y0]), LIM(IMP_CURVATURE(u, x, y), [x, y], [x0, y0])^(-1), x, ) IMP_PERPENDICULAR(u, x, y, x0, y0):=LIM(LINE_AUX(x_, y_, - LIM(IMP_DIF(u, x, y, 1), [x, y], [x_, y_])^(-1), x), [x_, y_], [x0, y0]) IMP_TANGENT(u, x, y, x0, y0):=LIM(LINE_AUX(x_, y_, LIM(IMP_DIF(u, x, y, 1), [x, y], [x_, y_]), x), [x_, y_], [x0, y0]) LINE_AUX(x0, y0, d1, x):=y0 + d1(x - x0) NORMAL_LINE(u, v, v0, t):=v0 + tLIM(GRAD(u, v), v, v0) OSCULATING_CIRCLE(y, x, ):=CIRCLE_AUX(CENTER_OF_CURVATURE(y, x), 1/CURVATURE(y, x), ) PARA_CENTER_OF_CURVATURE(v, t):=CENTER_OF_CURVATURE_AUX(v, PARA_DIF(v, t, 1), PARA_DIF(v, t, 2)) PARA_CURVATURE(v, t):=CURVATURE_AUX(PARA_DIF(v, t, 1), PARA_DIF(v, t, 2)) PARA_DIF(v, t, default1):=ITERATE(DIF(dk, t)/DIF(v1, t), dk, v2, default1) PARA_OSCULATING_CIRCLE(v, t, t0, ):=CIRCLE_AUX(LIM(PARA_CENTER_OF_CURVATURE(v, t), t, t0), 1/LIM(PARA_CURVATURE(v, t), t, t0), ) PARA_PERPENDICULAR(v, t, t0, x):=LIM(LINE_AUX(LIM(v1, t, t_), LIM(v2, t, t_), - 1/LIM(PARA_DIF(v, t, 1), t, t_), x), t_, t0, 0) PARA_TANGENT(v, t, t0, x):=LIM(LINE_AUX(LIM(v1, t, t_), LIM(v2, t, t_), LIM(PARA_DIF(v, t, 1), t, t_), x), t_, t0, 0) PERPENDICULAR(y, x, x0):=LIM(LINE_AUX(x_, LIM(y, x, x_), - 1/LIM(DIF(y, x), x, x_), x), x_, x0, 0) POLAR_CENTER_OF_CURVATURE(r, ):=PARA_CENTER_OF_CURVATURE([rCOS(), rSIN()], ) POLAR_CURVATURE(r, ):=PARA_CURVATURE([rCOS(), rSIN()], ) POLAR_DIF(r, , default1):=PARA_DIF([rCOS(), rSIN()], , default1) POLAR_OSCULATING_CIRCLE(r, , 0, ):=PARA_OSCULATING_CIRCLE([rCOS(), rSIN()], , 0, ) POLAR_PERPENDICULAR(r, , 0, x):=PARA_PERPENDICULAR([rCOS(), rSIN()], , 0, x) POLAR_TANGENT(r, , 0, x):=PARA_TANGENT([rCOS(), rSIN()], , 0, x) TANGENT(y, x, x0):=LIM(LINE_AUX(x_, LIM(y, x, x_), LIM(DIF(y, x), x, x_), x), x_, x0, 0) TANGENT_PLANE(u, v, v0):=LIM(GRAD(u, v), v, v0) (v - v0) f(x):=x g(x):=SIN(x) d1:= d2:= default1:=1 hCross:=APPROX(6653225806451613/500000000000000) t0:= v0:= vCross:=APPROX(1032258064516129/500000000000000) x0:= y0:= := := 0:= := CTextObj y{\rtf1\ansi\ansicpg1252\deff0\deflang1030{\fonttbl{\f0\froman\fprq2\fcharset0 Times New Roman;}} {\colortbl ;\red0\green0\blue0;} \viewkind4\uc1\pard\cf1\b\f0\fs36 Taylorudvikling og approksimerende polynomier.\b0\fs24 \par \par Det approksimerende f\'f8rstegradspolynomium til en funktion f er det polynomium, hvis graf er tangenten til grafen for f i i punktet (x0, f(x0)). \par } {\rtf1\ansi\ansicpg1252\deff0\deflang1030{\fonttbl{\f0\froman\fprq2\fcharset0 Times New Roman;}} {\colortbl ;\red0\green0\blue0;} \viewkind4\uc1\pard\cf1\f0\fs24 Vi definerer en funktion f: \par } CExpnObj8User f(x):=SQRT(x){\rtf1\ansi\ansicpg1252\deff0\deflang1030{\fonttbl{\f0\froman\fprq2\fcharset0 Times New Roman;}{\f1\fnil\fcharset2 DfW5 Printer;}} {\colortbl ;\red0\green0\blue0;} \viewkind4\uc1\pard\cf1\f0\fs24 Tangenten i (2, f(2)) = (2 , \f1\'8b\f0 2) findes med TANGENT: \par } 8UserTANGENT(f(x),x,2)8 Simp(User){Gz?SQRT(2)*x/4+SQRT(2)/2#6{\rtf1\ansi\ansicpg1252\deff0\deflang1030{\fonttbl{\f0\froman\fprq2\fcharset0 Times New Roman;}} {\colortbl ;\red0\green0\blue0;} \viewkind4\uc1\pard\cf1\f0\fs24 (Tast Ctrl + Enter. S\'e5 kommer resultatet med eksakte v\'e6rdier.) \par } CPlotObjB^ C2DPlotView CExplicitPlotUz&@$@ vH'BTYBU&@$@ vH7B@TYB *$@ w?B@0BBxy??B!@s9?11??BM46(4){\rtf1\ansi\ansicpg1252\deff0\deflang1030{\fonttbl{\f0\froman\fprq2\fcharset0 Times New Roman;}} {\colortbl ;\red0\green0\blue0;} \viewkind4\uc1\pard\cf1\f0\fs24 Imidlertid kan denne tangent ogs\'e5 findes med den indbyggede facilitet, "Taylor Series", der ligger under Calculus i menubj\'e6lken \'f8verst. \par \par Pr\'f8v at markere f i #1. \par V\'e6lg Calculus, Taylor Series. V\'e6lg x-v\'e6rdien 2 (Expansion point), og orden: 1 \par } 8 Taylor(#1,x)TAYLOR(f(x):=SQRT(x),x,2,1)8Simp(Taylor(#1,x))Q?SQRT(2)*x/4+SQRT(2)/2(/{\rtf1\ansi\ansicpg1252\deff0\deflang1030{\fonttbl{\f0\froman\fprq2\fcharset0 Times New Roman;}} {\colortbl ;\red0\green0\blue0;} \viewkind4\uc1\pard\cf1\f0\fs24 Det giver samme resultat. \par Pr\'f8v nu at s\'e6tte ordenen op. Beregn Taylorudviklingen, og tegn resultatet ind p\'e5 grafen. \par } 4m'{\rtf1\ansi\ansicpg1252\deff0\deflang1030{\fonttbl{\f0\froman\fprq2\fcharset0 Times New Roman;}} {\colortbl ;\red0\green0\blue0;} \viewkind4\uc1\pard\cf1\f0\fs24 Hvilken betydning har ordenen? \par Hvad kan man mon kalde en n'te ordens Taylorudvikling. (Sammenlign med overskriften) \par } y{\rtf1\ansi\ansicpg1252\deff0\deflang1030{\fonttbl{\f0\froman\fprq2\fcharset0 Times New Roman;}} {\colortbl ;\red0\green0\blue0;} \viewkind4\uc1\pard\cf1\f0\fs24 \par } 3({\rtf1\ansi\ansicpg1252\deff0\deflang1030{\fonttbl{\f0\froman\fprq2\fcharset0 Times New Roman;}} {\colortbl ;\red0\green0\blue0;} \viewkind4\uc1\pard\cf1\b\f0\fs28 Ekstra-opgave.\b0\fs24 \par S\'e6t vinkelm\'e5let til at v\'e6re radianer. Det g\'f8res i Declare, Simplification Settings. \par Tegn nu grafen for funktionen g(x):=SIN(x) \par Find Taylorudviklingen med orden 1, 2, 3, ....... \par Tegn funktionerne ind p\'e5 grafen. \par Lav en tabel, der viser sammenh\'e6ngen mellem ordenen og antal toppe, der approksimeres "godt". \par } 8?KUser Angle:=Radian8WcUser g(x):=SIN(x)8o{UserTAYLOR(g(x),x,0,1) Approx(#8') x8User TAYLOR(g(x),x,0,2) Approx(#10) x8User TAYLOR(g(x),x,0,3)8User TAYLOR(g(x),x,0,3)(/ Simp(#13){Gz?x-x^3/6;N{\rtf1\ansi\ansicpg1252\deff0\deflang1030{\fonttbl{\f0\froman\fprq2\fcharset0 Times New Roman;}} {\colortbl ;\red0\green0\blue0;} \viewkind4\uc1\pard\cf1\f0\fs24 En tabel laves f.eks. s\'e5ledes: \par } 8ZfUser"TABLE[1,0;2,0;3,0]"r{\rtf1\ansi\ansicpg1252\deff0\deflang1030{\fonttbl{\f0\froman\fprq2\fcharset0 Times New Roman;}} {\colortbl ;\red0\green0\blue0;} \viewkind4\uc1\pard\cf1\f0\fs24 eller: \par } 8User"TABLE[[1,0],[2,0],[3,0]]"{\rtf1\ansi\ansicpg1252\deff0\deflang1030{\fonttbl{\f0\froman\fprq2\fcharset0 Times New Roman;}} {\colortbl ;\red0\green0\blue0;} \viewkind4\uc1\pard\cf1\f0\fs24 N\'e5r du har fundet flere punkter (spring lidt i ordenerne), kan du rette i tabellen ved at markere kommandoen og taste Enter. Lav dine tilf\'f8jelser, og tast Enter igen. \par \par Hvor stor skal ordenen v\'e6re for at polymomiet "stemmer overens" med grafen for g over 10 toppe i alt? \par }