DERIVE for Windows version 5.05 DfW file saved on 02 May 2002 0ARCS(w, z, r0, rm, m, 0, n, n, r, ):=GRID(RE_IM(LIM(w, z, ^()r)), , 0, n, n, r, r0, rm, m) AreaBetweenCurves(u, v, x, a, b, y):=[u, v, a x b (v - y)(y - u) > 0] AreaOverCurve(u, x, a, b, y):=[u, a x b u < y < 0] AreaUnderCurve(u, x, a, b, y):=[u, a x b 0 < y < u] BINORMAL(v, t):=SIGN(CROSS(DIF(v, t), DIF(v, t, 2))) CONE(, , z):=[zSIN()COS(), zSIN()SIN(), z] COPROJECTION(v):=VECTOR(VECTOR(u_n_, u_, v), n_, DIM(v1)) CYLINDER(r, , z):=[rCOS(), rSIN(), z] GRID(u, s, s0, sm, m, t, t0, tn, n):=VECTOR(VECTOR(LIM(u, [s, t], [s0 + j(sm - s0)/m, t0 + k(tn - t0)/n]), j, 0, m), k, 0, n) HORIZONTALS(w, z, z00, zmn, m, n, x, y):=GRID(RE_IM(LIM(w, z, x + y)), x, RE(z00), RE(zmn), m, y, IM(z00), IM(zmn), n) ISOMETRIC(v):=[v2 - v1, v3 - (v1 + v2)/2] ISOMETRICS(v, s, s0, sm, m, t, t0, tn, n):=GRID(ISOMETRIC(v), s, s0, sm, m, t, t0, tn, n) NORMAL_VECTOR(v, t):=SIGN(DIF(v, t, 2)) PlotInt(u, x, a, b, y):=IF(a < b, [u, a x b 0 < y < u, a x b u < y < 0], [u, b x a u < y < 0, b x a 0 < y < u]) RAYS(w, z, z00, zmn, m, n, x, y):=GRID([x, y; [x, y], [x, y] + RE_IM(LIM(w, z, x + y))], x, RE(z00), RE(zmn), m, y, IM(z00), IM(zmn), n) RE_IM(w):=[RE(w), IM(w)] ROTATE_X():=[1, 0, 0; 0, COS(), - SIN(); 0, SIN(), COS()] ROTATE_Y():=[COS(), 0, SIN(); 0, 1, 0; - SIN(), 0, COS()] ROTATE_Z():=[COS(), - SIN(), 0; SIN(), COS(), 0; 0, 0, 1] SPACE_TUBE(v, t, r, ):=v + r(SIN()NORMAL_VECTOR(v, t) + COS()BINORMAL(v, t)) SPHERE(r, , ):=r[SIN()COS(), SIN()SIN(), COS()] TORUS(rc, rs, , ):=SPACE_TUBE(rc[COS(), SIN(), 0], , rs, ) f(x):=1/x hjresum(a, b, n):=(f(a + i(b - a)/n)(b - a)/n, i, 1, n) plothjresum(x, a, b, n):=a < x < b 0 < y < (f(a + i(b - a)/n)CHI(a + (i - 1)(b - a)/n, x, a + i(b - a)/n), i, 1, n) plotvenstresum(x, a, b, n):=a < x < b 0 < y < (f(a + (i - 1)(b - a)/n)CHI(a + (i - 1)(b - a)/n, x, a + i(b - a)/n), i, 1, n) trapezsum(a, b, n):=(hjresum(a, b, n) + venstresum(a, b, n))/2 venstresum(a, b, n):=(f(a + (i - 1)(b - a)/n)(b - a)/n, i, 1, n) axes:=[-t_, - t_/2; t_, - t_/2; 0, t_] hCross:=APPROX(2990365612645829/2500000000000000) r0:= rc:= rm:= rs:= s0:= sm:= t0:= tn:= vCross:=APPROX(25709053343344007/100000000000000000) z00:= zmn:= := := 0:= n:= := := := CTextObj Z{\rtf1\ansi\ansicpg1252\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fnil\fcharset2 Times New Roman;}{\f4\fmodern\fcharset1 DfW5 Printer;}{\f5\fmodern\fcharset2 DfW5 Printer;}{\f6\froman\fcharset1 Times New Roman;}} {\colortbl\red0\green0\blue0;} \deflang1030\pard\plain\f3\fs28\ul H\'f8jresummer, venstresummer, trapezsummer og grafer. \par \plain\f3\fs24 \par Tegn en graf der forbinder de enkelte punkter med en ret linie. \par Her f\'e5r du et eksempel \par } CExpnObj8frUser f(x):=SQRT(x)~{\rtf1\ansi\ansicpg1252\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fnil\fcharset2 Times New Roman;}{\f4\fmodern\fcharset1 DfW5 Printer;}{\f5\fmodern\fcharset2 DfW5 Printer;}{\f6\froman\fcharset1 Times New Roman;}} {\colortbl\red0\green0\blue0;} \deflang1030\pard\plain\f3\fs24 Vi vil nu lave en tabel for f(x) \par \par } 8 UserVECTOR([x,f(x)],x,0,4,1){\rtf1\ansi\ansicpg1252\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fnil\fcharset2 Times New Roman;}{\f4\fmodern\fcharset1 DfW5 Printer;}{\f5\fmodern\fcharset2 DfW5 Printer;}{\f6\froman\fcharset1 Times New Roman;}} {\colortbl\red0\green0\blue0;} \deflang1030\pard\plain\f3\fs24 Dette betyder at tabellen starter i med x = 0 og slutter med x = 4 og x stiger med 1 \par } 0SSimp(#2)Mb`?+[[0,0],[1,1],[2,SQRT(2)],[3,SQRT(3)],[4,2]]_#{\rtf1\ansi\ansicpg1252\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fnil\fcharset2 Times New Roman;}{\f4\fmodern\fcharset1 DfW5 Printer;}{\f5\fmodern\fcharset2 DfW5 Printer;}{\f6\froman\fcharset1 Times New Roman;}} {\colortbl\red0\green0\blue0;} \deflang1030\pard\plain\f3\fs24 Vi kan indtegne disse punkter ved blot at markere dem. Hvis vi vil forbinde punkterne med rette linier s\'e5 g\'e5 over i plot-vinduet tryk options, display, points,og ved connect tryk yes. \par } CPlotObjC C2DPlotViewCPointListPlot   ? ? @R$A,A @-A9A @ @xy??V9&@@11??BM:/6(fs"H?{\rtf1\ansi\ansicpg1252\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fnil\fcharset2 Times New Roman;}{\f4\fmodern\fcharset1 DfW5 Printer;}{\f5\fmodern\fcharset2 DfW5 Printer;}{\f6\froman\fcharset1 Times New Roman;}} {\colortbl\red0\green0\blue0;} \deflang1030\pard\plain\f3\fs24 Du kan ogs\'e5 tegne en punktgraf, hvor punkterne er givet. For at skrive tabellen op skal du prikke p\'e5 matrixen til venstre for lighedstegnet (parantes med ni prikker i ) og her skrive 5 r\'e6kker og 2 s\'f8jler. \par } 8TUser [[1,2],[2,5],[4,6],[8,10],[9,9]]{\rtf1\ansi\ansicpg1252\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fnil\fcharset2 Times New Roman;}{\f4\fmodern\fcharset1 DfW5 Printer;}{\f5\fmodern\fcharset2 DfW5 Printer;}{\f6\froman\fcharset1 Times New Roman;}} {\colortbl\red0\green0\blue0;} \deflang1030\pard\plain\f3\fs24 Grafen tegnes blot ved at markere punkterne og trykke p\'e5 tegn ligesom vi plejer. \par } Cp z ? @ @ @ @ @  @ $@ "@ "@xy ??@@Cy 5@$I$I@11??BM:/6(fs|{\rtf1\ansi\ansicpg1252\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fnil\fcharset2 Times New Roman;}{\f4\fmodern\fcharset1 DfW5 Printer;}{\f5\fmodern\fcharset2 DfW5 Printer;}{\f6\froman\fcharset1 Times New Roman;}} {\colortbl\red0\green0\blue0;} \deflang1030\pard\plain\f3\fs24 Du kan nu hente mit program til at finde h\'f8jresummer venstresummer og trapezsummer. \par Skriv load, utility file se stien nedenunder \par } 8xUser*LOAD("C:\DfW5\Math\hjreogvenstresum.mth")t {\rtf1\ansi\ansicpg1252\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fnil\fcharset2 Times New Roman;}{\f4\fmodern\fcharset1 DfW5 Printer;}{\f5\fmodern\fcharset2 DfW5 Printer;}{\f6\froman\fcharset1 Times New Roman;}} {\colortbl\red0\green0\blue0;} \deflang1030\pard\plain\f3\fs24 Dette skal st\'e5 p\'e5 dit ark, f\'f8r du kan bruge mine programmer. \par Du kan se selve programmerne, hvis du g\'e5r ind og \'e5bner det, som du plejer at \'e5bne en derivefil. Hvis du blot vil se funktionsordrene s\'e5 g\'e5 op under declare, function definition og ved pilen findes venstresum, h\'f8jresum og trapezsum, plotvenstresum og ploth\'f8jresum. \par \par Opgave 1 \par \plain\f6\fs24 Betragt punktm\'e6ngderne \plain\f4\fs24 \par M1=\{(x,y)|3x4\'8f0y\'8bx\} \par M2=\{(x,y)|0\plain\f5\fs24 \plain\f4\fs24 x\plain\f5\fs24 \plain\f4\fs24 1\plain\f5\fs24 \'8f\plain\f4\fs24 0\plain\f5\fs24 \plain\f4\fs24 y\plain\f5\fs24 \plain\f4\fs24 x^3\} \par M3=[(x,y)|0\plain\f5\fs24 \plain\f4\fs24 x\plain\f5\fs24 \plain\f4\fs24 1\plain\f5\fs24 \'8f\plain\f4\fs24 0\plain\f5\fs24 \plain\f4\fs24 y\plain\f5\fs24 \plain\f4\fs24 sin(x)\} \par M4=\{(x,y)|1\plain\f5\fs24 \plain\f4\fs24 x\plain\f5\fs24 \plain\f4\fs24 2\plain\f5\fs24 \'8f\plain\f4\fs24 0\plain\f5\fs24 \plain\f4\fs24 y\plain\f5\fs24 \plain\f4\fs24 1/x\} \par \plain\f6\fs24 Find h\'f8jresummen, venstresummen og trapezsummen for hver af ovenst\'e5ende punktm\'e6ngder ved at inddele intervallet p\'e5 x-aksen i 10 lige store dele. Du skal ogs\'e5 tegne graferne, hvor man kan se de tre summer. \par Gentag beregningerne, nu blot med 1000 indelinger. \par Find for hver af de fire punktm\'e6ngder de eksakte v\'e6rdier for arealet ved at finde integralet. \par \par \par Opgave 2 \par Betragt igen punktm\'e6ngden M2 fra opgave 1. \par Du skal nu finde h\'f8jresum, venstresum og trapezsum ved at inddele intervallet p\'e5 x-aksen i n lige store dele. \par Find nu arealet ved at lade n g\'e5 mod \plain\f5\fs24 \endash \plain\f6\fs24 i alle tre tilf\'e6lde . \par Sammenlign disse gr\'e6nsev\'e6rdier med arealet bestemt ved hj\'e6lp af integralet. \par Det kan ikke altid lade sig g\'f8re at bestemme arealet p\'e5 denne m\'e5de, pr\'f8v selv efter ved at beregne arealet af M1 p\'e5 samme m\'e5de som ovenfor. \par \plain\f4\fs24 \par \plain\f6\fs24 \par }