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" }}{PARA 0 "" 0 "" {TEXT -1 122 "It can be read into a Maple sessio n by a command similar to the one that follows, where the file path gi ves its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "read \+ \"K:\\\\Maple/procdrs/intg.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 79 "A version of Simpson's rule applicable to possibly unequally spaced data points" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 34 "Suppose that we hav e three points " }{XPPEDIT 18 0 "``(-h,p), ``(0,q)" "6$-%!G6$,$%\"hG! \"\"%\"pG-F$6$\"\"!%\"qG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "``(k,r) " "6#-%!G6$%\"kG%\"rG" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 400 300 300 {PLOTDATA 2 "69-%%TEXTG6%7$$\"$v\"!\"#$\" #N!\"\"Q)y~=~f(x)6\"-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F7F6-%'CURVE SG6$7$7$$\"3!**************R\"!#tQ2(o)=$F?7$$!3rmmm5rI,8F?$\"3?EaEtJK^KF?7$$!3ommmu DgK7F?$\"3P4Ycz?-7LF?7$$!3ALLLzXAk6F?$\"3g(p&QP\"*3qLF?7$$!3ymmmC%H35 \"F?$\"3.Z.G)GO=U$F?7$$!39+++9u=N5F?$\"3+)*R(4!**HtMF?7$$!3'pmmm_.In*! #=$\"3\"*ogw=uDCNF?7$$!35+++g+J'**)F]r$\"3'o&3wh#eFd$F?7$$!3[KLL`wC+$) F]r$\"3$G++4`e-i$F?7$$!3Ommm1b:(o(F]r$\"3QWT(G@!4gOF?7$$!3G******f*ep* pF]r$\"3O*=!GRCo-PF?7$$!3l++++%GRI'F]r$\"3j$=>o+`Iu$F?7$$!33+++S]1OcF] r$\"3oCA;,`ozPF?7$$!3zmmmE6eH]F]r$\"3SH!fgv=5\"QF?7$$!3AMLL$48%3VF]r$ \"3I^'4Uh$)e%QF?7$$!3eKLLt\"*[(p$F]r$\"3mt$*>0UQtQF?7$$!3M+++?%Ro)HF]r $\"3DNy>!*f-.RF?7$$!3INLLtRzdBF]r$\"3)*[\\hpr:FRF?7$$!3w******>9jn;F]r $\"35sv6GnN^RF?7$$!3y******fMV55F]r$\"3-\"GdSs'=sRF?7$$!3'\\KLLLjrC$!# >$\"3%H$Q#**[=;*RF?7$$\"37lmmm/')\\IFdv$\"3Q8]%*4@R2SF?7$$\"3#[LLLL^?% )*Fdv$\"3-M@(*yM=ASF?7$$\"3[KLLtGs*o\"F]r$\"3!)o;Oj^5NSF?7$$\"3p****** R\"yQI#F]r$\"36v+H6W_OF]r$\"3kKBm7-'z0%F?7$$\"3N******RG$GK%F]r$\"3CWNf6ONhSF?7$$\"3 #))******Hr9(\\F]r$\"3u!>$Glz\\iSF?7$$\"3%*)*****>Pn\"p&F]r$\"3w`EmoRI hSF?7$$\"3kimmEw!)QjF]r$\"3sZpM&)*=!eSF?7$$\"3/-+++7wHqF]r$\"3-Ount,?_ SF?7$$\"3)=LLL$y'el(F]r$\"3Jj'H^\"f'[/%F?7$$\"3N+++gxOS$)F]r$\"3n\"))p !e[gMSF?7$$\"3IjmmY)GW)*)F]r$\"3REY\"p\"3\"G-%F?7$$\"3g,++g#ewl*F]r$\" 3?ME$)[bE3SF?7$$\"3immm1jeJ5F?$\"33B6k**R&=*RF?7$$\"3n******\\U\\+6F?$ \"3WM#HRm^B(RF?7$$\"3eLLL*ygo;\"F?$\"3eIC\">.')QO \"F?$\"3oB)G4BDf(QF?7$$\"3#ommEQrZV\"F?$\"36io`:10WQF?7$$\"37LLL&3s\") \\\"F?$\"3XE`,AJT8QF?7$$\"3E+++e/xl:F?$\"3Y " 0 "" {MPLTEXT 1 0 113 "xvalues := [-h,0,k];\nyvalues := [p,q,r];\ninterp(xv alues,yvalues,x):\nint(%,x=-h..k):\nsimplify(%):\nss := factor(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(xvaluesG7%,$%\"hG!\"\"\"\"!%\"kG" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(yvaluesG7%%\"pG%\"qG%\"rG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ssG,$*&*&,&%\"kG\"\"\"%\"hGF*F*,0*& %\"qGF*)F)\"\"#F*!\"\"*&%\"pGF*F/F*F***F0F*F)F*%\"rGF*F+F*F1**F0F*F)F* F.F*F+F*F1**F0F*F)F*F3F*F+F*F1*&F5F*)F+F0F*F**&F.F*F9F*F1F*F**&F+F*F)F *F1#F1\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "This formula is incorporated into the procedure " }{TEXT 0 7 "g ensimp" }{TEXT -1 92 ", given in the next section, to estimate the are a under a curve described by numerical data." }}{PARA 0 "" 0 "" {TEXT -1 21 "Note that, if we let " }{XPPEDIT 18 0 "k = h" "6#/%\"kG%\"hG" } {TEXT -1 59 ", this formula reduces to the usual Simpson's rule formul a." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "subs(k=h,ss);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,(*&%\"qG\"\"\")%\"hG\"\"#F(!\"%*&%\"pGF(F)F(!\"\"* &%\"rGF(F)F(F/F(F*F/#F/\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&% \"hG\"\"\",(%\"qG\"\"%%\"pGF&%\"rGF&F&#F&\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 59 "A procedure for applying Simpson's rule to numerical data: " }{TEXT 0 7 "gensimp" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "gensimp: us age" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 264 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 265 2 " " }{TEXT -1 31 " gensimp( xvalues, yvalues ) " }{TEXT 266 1 "\n" } {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT 23 15 " xvalues \+ - " }{TEXT -1 100 " a list of increasing 1st coordinates of points on a curve, where the total number of values is " }{TEXT 261 3 "odd" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 15 " yvalues - " }{TEXT -1 67 " a corresponding list \+ of 2nd coordinates of points on the curve" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 7 "gensimp" }{TEXT -1 183 " esti mates the total area under a curve passing through data points by appl ying a generalised version of Simpson's rule, which can be applied to \+ possibly unequally spaced data points." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{PARA 0 "" 0 "" {TEXT -1 1 "\004" }}{PARA 0 "" 0 "" {TEXT 261 16 "How to activate:" }{TEXT 262 1 "\n" }{TEXT -1 154 "To make the proced ure active open the subsection, place the cursor anywhere after the pr ompt [ > and press [Enter].\nYou can then close up the subsection." } }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "gensimp: implementation " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 939 "gensimp := proc(x::list,y::list)\n local m,n,h,k,p,q,r,sm,i;\n \+ if not type(x,list(numeric)) then\n error \"the 1st argument, % 1, is invalid .. it should be a list of numerical data\",x;\n end if ;\n if not type(y,list(numeric)) then\n error \"the 2nd argumen t, %1, is invalid .. it should be a list of numerical data\",y;\n en d if;\n n := nops(x);\n m := nops(y);\n if n<>m then\n erro r \"the data lists must have the same length\"\n end if;\n if i rem(n,2)<>1 then\n error \"the data lists must contain an odd num ber of data values\"\n end if;\n\n sm := 0;\n for i to n-2 by 2 do\n h:= x[i+1]-x[i];\n k := x[i+2]-x[i+1];\n if signu m(0,h,0)<>1 or signum(0,k,0)<>1 then\n error \"the values in 1 st argument list must be in strictly increasing\"\n end if; \n \+ p := y[i]; q := y[i+1]; r := y[i+2];\n sm:=sm+(k+h)*((q-p)*k^2 +2*k*h*(r+q+p)+(q-r)*h^2)/(6*h*k);\n end do;\n sm; \nend pro c:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT 0 7 "gensimp" }{TEXT -1 10 ": examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 39 "This exampl e checks that the procedure " }{TEXT 0 7 "gensimp" }{TEXT -1 97 " give s a value which agrees with ordinary integral for data points which li e along the parabola " }{XPPEDIT 18 0 "y=x^2" "6#/%\"yG*$%\"xG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "gensimp([0,3,4,5,8],[0,9.,16,25,64]);\nInt(x^2,x =0..8);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+mmm1 " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)=x^4/10-x^3/8+2*x^2+x+1" "6#/-% \"fG6#%\"xG,,*&F'\"\"%\"#5!\"\"\"\"\"*&F'\"\"$\"\")F,F,*&\"\"#F-*$F'F2 F-F-F'F-F-F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "f := x -> x^4/10-x^3/8+2*x^2 +x+1;\nplot([f(x),[[2,0],[2,f(2)]]],x=0..2.3,color=[red,black]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,,*$)9$\"\"%\"\"\"#F1\"#5*&#F1\"\")F1*$)F/\"\"$F1F1!\"\"*&\"\"#F1)F /F$\"3 ![$=Pi]9b5!#<7$$\"3WmmTN0Vv$*F/$\"3PzK1g(Q76\"F27$$\"3ALL3U^5G9!#=$\"3 %4wb.sxK=\"F27$$\"3VLLe%**=>#>F;$\"3]CX)**R;`E\"F27$$\"3Hm;/OeQ8CF;$\" 3i\"Q&\\s$4kN\"F27$$\"3AL$3-^Q!pGF;$\"3)*[(*)z-d#\\9F27$$\"3$)**\\(=YS 3M$F;$\"3)yV\"R$4$*Qb\"F27$$\"3-L$3_ry(GQF;$\"3!f([h.=?r;F27$$\"3h**\\ PW@::VF;$\"3?.)eDT\\tz\"F27$$\"3_mm;**pW:[F;$\"30y[1$[Mn$>F27$$\"3iKLe zp5c_F;$\"37Dc)y9Dw1#F27$$\"3W++]Zd=_dF;$\"3u(>z*H>A#F27$$\"3u****\\ i9I]iF;$\"3qK_k__4\"R#F27$$\"3m****\\_#G.t'F;$\"3YtczN%*QhDF27$$\"3oK$ 3_cQi;(F;$\"3al1HWR4CFF27$$\"3'fmm\"*3yXo(F;$\"3zgJ?a/mFHF27$$\"3\"emm mlzO7)F;$\"3_$z2qD*y3JF27$$\"3E**\\(o;fWj)F;$\"3.>X.x)R'HLF27$$\"3?lmm \"e&e'3*F;$\"3yh=f$RqV`$F27$$\"3#)**\\(o\"*REe*F;$\"3#G3WUeK\"pPF27$$ \"3$**\\i]4+b+\"F2$\"3#ou7@L0F+%F27$$\"3om;a8gya5F2$\"3[y!>Q>CqD%F27$$ \"3gmT5se/+6F2$\"3V/<2AjG+XF27$$\"3GL$eRuk)[6F2$\"3?P&*=V#4Lx%F27$$\"3 GL3_JQd*>\"F2$\"3Ud*[XAC)o]F27$$\"3\")*\\78C;PC\"F2$\"3)*R.]Qv;O`F27$$ \"3OL$3KD\"R\"H\"F2$\"3ePAE&=)oNcF27$$\"3&)****\\0UkS8F2$\"3Y)QdgQRr&f F27$$\"3#)**\\P5'G))Q\"F2$\"3K\"fRBK3PG'F27$$\"3!)*\\(o*\\\\aV\"F2$\"3 ^U9@Z;L6mF27$$\"3%)**\\i[S@([\"F2$\"3!yULV;b)))pF27$$\"3Ymm;)zEP`\"F2$ \"3@Y!RP!fsStF27$$\"3'****\\i3*Q$e\"F2$\"33tI*)\\\"y*HxF27$$\"3,L3_+0R G;F2$\"3a\"\\Lb9)3&4)F27$$\"3)****\\F$*)ex;F2$\"3p!R@QXt!3&)F27$$\"3[m TNB3)Qs\"F2$\"3+f&R&z'y,\"*)F27$$\"3%)*\\Pu=pAx\"F2$\"3zn8&)f5([M*F27$ $\"3[mm\"zlx&>=F2$\"3c]u/Q\"QWy*F27$$\"3%)*\\(=U_5p=F2$\"3;V4roy/E5!#; 7$$\"3FLLL#>1o\">F2$\"3wsTsz/Zt5F\\x7$$\"35L$eMI(el>F2$\"3T$Q8K#zgB6F \\x7$$\"3ZmTN#[kR,#F2$\"3)3A$ydL,v6F\\x7$$\"31++]&3=%e?F2$\"3\">q5/bmP A\"F\\x7$$\"3_m;HJpO4@F2$\"3CnoEe\"y9G\"F\\x7$$\"3'HLLj=O\\:#F2$\"3;0i 3R/![L\"F\\x7$$\"3))*\\(o;D_.AF2$\"3U#pdW5uMR\"F\\x7$$\"3#**\\7V$e-]AF 2$\"3=m#o3&*Q9X\"F\\x7$$\"3#)*************H#F2$\"3$********\\`d^\"F\\x -%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6$7$7$$\"\"#F)F(7$Fe[l$\"3)**** *********f6F\\x-F[[l6&F][lF)F)F)-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$ ;F($\"#B!\"\"%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 67 "We construct 41 data points along this cu rve with unequally spaced " }{TEXT 272 1 "x" }{TEXT -1 52 " coordinate s between 0 and 2, and use the procedure " }{TEXT 0 7 "gensimp" } {TEXT -1 41 " with this data to estimate the integral " }{XPPEDIT 18 0 "Int(f(x),x=0..2)" "6#-%$IntG6$-%\"fG6#%\"xG/F);\"\"!\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "f := x -> x^4/10-x^3/8+2*x^2+x+1;\nxvalues := [0,seq (i*0.05+0.01*evalf(sin(i)),i=1..39),2.];\nyvalues := map(f,xvalues);\n gensimp(xvalues,yvalues);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6 #%\"xG6\"6$%)operatorG%&arrowGF(,,*&#\"\"\"\"#5F/*$)9$\"\"%F/F/F/*&#F/ \"\")F/*$)F3\"\"$F/F/!\"\"*&\"\"#F/)F3F=F/F/F3F/F/F/F(F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(xvaluesG7K\"\"!$\"+&)4ZTe!#6$\"+V(H44\"!# 5$\"+,?69:F,$\"+](>V#>F,$\"+tv5/CF,$\"+]%e?(HF,$\"+g')plNF,$\"+De$*)4% F,$\"+\\=@TXF,$\"+*)yfX\\F,$\"+z4++aF,$\"+3FMYfF,$\"+/n,UlF,$\"+O21*4( F,$\"+%yG]c(F,$\"+o'47(zF,$\"+^-'QS)F,$\"+v7!\\#*)F,$\"+@x)\\^*F,$\"+` %H\"45!\"*$\"+clOe5FQ$\"+([6**4\"FQ$\"+'zP:9\"FQ$\"+;U%4>\"FQ$\"+#[w'[ 7FQ$\"+&eDwI\"FQ$\"+fPcf8FQ$\"+e!4FS\"FQ$\"+hOOV9FQ$\"+%o>,\\\"FQ$\"+C 'ffa\"FQ$\"+nU^0;FQ$\"+>\"***f;FQ$\"+F3H0YcV=FQ$\"+'ojH!>FQ$\"+azjf>FQ$\"\"#F&" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(yvaluesG7K\"\"\"$\"+:b@l5!\"*$\"+*RZF8\"F)$\"+m0)o> \"F)$\"+1&QdE\"F)$\"+(e-YN\"F)$\"+VrOr9F)$\"+OO!og\"F)$\"+!\\L,u\"F)$ \"+_(>\"f=F)$\"+Y/gu>F)$\"+h.-7@F)$\"++Y.)G#F)$\"+b)zM\\#F)$\"+ck^)p#F )$\"+D/tzGF)$\"+sm)\\/$F)$\"+'*=dGKF)$\"+/>:gMF)$\"+HX[OPF)$\"+e61@SF) $\"+\"fFfF%F)$\"+'3]&*\\%F)$\"+11iJZF)$\"+A\\m<]F)$\"+GS<6F\\p$\"++++g6F\\p" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+OJLt%*!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Compare this value with the analytical value for the i ntegral." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "f := x -> x^4/10-x^3/8+2*x^2+x+1;\nInt(f(x),x=0..2); \nvalue(%);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#% \"xG6\"6$%)operatorG%&arrowGF(,,*&#\"\"\"\"#5F/*$)9$\"\"%F/F/F/*&#F/\" \")F/*$)F3\"\"$F/F/!\"\"*&\"\"#F/)F3F=F/F/F3F/F/F/F(F(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$IntG6$,,*&#\"\"\"\"#5F)*$)%\"xG\"\"%F)F)F)*&# F)\"\")F)*$)F-\"\"$F)F)!\"\"*&\"\"#F))F-F7F)F)F-F)F)F)/F-;\"\"!F7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##\"%@9\"$]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+LLLt%*!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 63 "\"Indef inite\" numerical integration via parabolic interpolation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 40 "A gain suppose that we have three points " }{XPPEDIT 18 0 "``(-h,p)" "6# -%!G6$,$%\"hG!\"\"%\"pG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "``(0,q)" "6# -%!G6$\"\"!%\"qG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "``(k,r)" "6#-%!G 6$%\"kG%\"rG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 29 "This time we need the actual " }{TEXT 261 12 "coefficients" }{TEXT -1 1 " " } {TEXT 284 1 "a" }{TEXT -1 2 ", " }{TEXT 285 1 "b" }{TEXT -1 5 " and " }{TEXT 286 1 "c" }{TEXT -1 29 " of the quadratic polynomial " } {XPPEDIT 18 0 "y=a*x^2+b*x+c" "6#/%\"yG,(*&%\"aG\"\"\"*$%\"xG\"\"#F(F( *&%\"bGF(F*F(F(%\"cGF(" }{TEXT -1 39 " which fits through these three \+ points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "xvalues := [-h,0,k];\nyvalues := [p,q,r];\npx := int erp(xvalues,yvalues,x):\na = simplify(coeff(px,x,2));\nb = simplify(co eff(px,x,1));\nc = simplify(coeff(px,x,0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(xvaluesG7%,$%\"hG!\"\"\"\"!%\"kG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%(yvaluesG7%%\"pG%\"qG%\"rG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"aG,$*&,**&%\"rG\"\"\"%\"hGF*!\"\"*&%\"qGF*%\"kGF*F* *&F.F*F+F*F**&%\"pGF*F/F*F,F**(F+F*,&F/F*F+F*F*F/F*F,F," }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%\"bG,$*&,**&%\"rG\"\"\")%\"hG\"\"#F*!\"\"*&%\" qGF*)%\"kGF-F*F.*&F0F*F+F*F**&%\"pGF*F1F*F*F**(F,F*,&F2F*F,F*F*F2F*F.F ." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG%\"qG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "The coefficient of " }{XPPEDIT 18 0 " x^2" "6#*$% \"xG\"\"#" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "a = ((r-q)*h+(p-q)*k)/(h *(k+h)*k)" "6#/%\"aG*&,&*&,&%\"rG\"\"\"%\"qG!\"\"F*%\"hGF*F**&,&%\"pGF *F+F,F*%\"kGF*F*F**(F-F*,&F1F*F-F*F*F1F*F," }{TEXT -1 21 ", the coeffi cient of " }{TEXT 263 1 "x" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "b = ((r -q)*h^2+(q-p)*k^2)/(h*(k+h)*k)" "6#/%\"bG*&,&*&,&%\"rG\"\"\"%\"qG!\"\" F**$%\"hG\"\"#F*F**&,&F+F*%\"pGF,F**$%\"kGF/F*F*F**(F.F*,&F4F*F.F*F*F4 F*F," }{TEXT -1 34 ", and the constant coefficient is " }{XPPEDIT 18 0 "c = q" "6#/%\"cG%\"qG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "We can obtain any integral of the \+ form " }{XPPEDIT 18 0 "Int(a*x^2+b*x+c,x = 0 .. z);" "6#-%$IntG6$,(*& %\"aG\"\"\"*$%\"xG\"\"#F)F)*&%\"bGF)F+F)F)%\"cGF)/F+;\"\"!%\"zG" } {TEXT -1 30 " by evaluating the function " }{XPPEDIT 18 0 "g(z) = a* z^3/3+b*z^2/2+c*z;" "6#/-%\"gG6#%\"zG,(*(%\"aG\"\"\"*$F'\"\"$F+F-!\"\" F+*(%\"bGF+*$F'\"\"#F+F2F.F+*&%\"cGF+F'F+F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Given a set of data consisting of " }{TEXT 274 1 "x " }{TEXT -1 8 " values " }{XPPEDIT 18 0 "x[1],x[2],` . . . `,x[n]" "6& &%\"xG6#\"\"\"&F$6#\"\"#%(~.~.~.~G&F$6#%\"nG" }{TEXT -1 20 ", and corr esponding " }{TEXT 273 1 "y" }{TEXT -1 8 " values " }{XPPEDIT 18 0 "y[ 1],y[2],` . . . `,y[n]" "6&&%\"yG6#\"\"\"&F$6#\"\"#%(~.~.~.~G&F$6#%\"n G" }{TEXT -1 8 ", where " }{TEXT 275 1 "n" }{TEXT -1 4 " is " }{TEXT 261 3 "odd" }{TEXT -1 16 ", the procedure " }{TEXT 0 10 "simpinterp" } {TEXT -1 50 " constructs a numerical procedure which estimates " } {XPPEDIT 18 0 "Int(f(x),x = 0 .. z);" "6#-%$IntG6$-%\"fG6#%\"xG/F);\" \"!%\"zG" }{TEXT -1 5 " for " }{TEXT 276 1 "z" }{TEXT -1 17 " in the i nterval " }{XPPEDIT 18 0 "x[1] <= z;" "6#1&%\"xG6#\"\"\"%\"zG" } {XPPEDIT 18 0 "``<=x[n]" "6#1%!G&%\"xG6#%\"nG" }{TEXT -1 2 ". 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_mFg_mFi_m-F`_m6%7$F^[mFe_mFg_mFi_m-F`_m6%7$Fa\\mFe_mQ\"zFh_mFi_m-F`_m 6%7$$\"$2%Ff[mFe_mFg_mFi_m-F`_m6%7$FdzFe_mFg_mFi_m-F`_m6%7$$\"\"&Ff[m$ F`[mF`[mQ\"1Fh_m-Fj_m6$F\\`mFj\\m-F`_m6%7$$\"$4$Ff[mFgamQ$i-1Fh_mFiam- F`_m6%7$$\"#OF`[mFgamQ\"iFh_mFiam-F`_m6%Fh`mFi`mFiam-F`_m6%7$$\"$D%Ff[ mFgamQ$i+1Fh_mFiam-F`_m6%7$$\"#hF`[mFgamQ\"nFh_mFiam-F`_m6&7$$\"$&\\Ff [m$\"# " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 68 "A proce dure to perform \"indefinite integration\" for numerical data: " } {TEXT 0 10 "simpinterp" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "gensimp: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 268 18 "Call ing Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 269 2 " " }{TEXT -1 31 " g ensimp( xvalues, yvalues ) " }{TEXT 270 1 "\n" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 4 " \+ " }}{PARA 0 "" 0 "" {TEXT 23 15 " xvalues - " }{TEXT -1 100 " \+ a list of increasing 1st coordinates of points on a curve, where the total number of values is " }{TEXT 261 3 "odd" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 15 " yvalue s - " }{TEXT -1 67 " a corresponding list of 2nd coordinates of \+ points on the curve" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 43 "Given a s et of data consisting of x values " }{XPPEDIT 18 0 "x[1],x[2],` . . . \+ `,x[n]" "6&&%\"xG6#\"\"\"&F$6#\"\"#%(~.~.~.~G&F$6#%\"nG" }{TEXT -1 29 ", and corresponding y values " }{XPPEDIT 18 0 "y[1],y[2],` . . . `,y[ n]" "6&&%\"yG6#\"\"\"&F$6#\"\"#%(~.~.~.~G&F$6#%\"nG" }{TEXT -1 13 ", w here n is " }{TEXT 261 3 "odd" }{TEXT -1 16 ", the procedure " }{TEXT 0 10 "simpinterp" }{TEXT -1 50 " constructs a numerical procedure whic h estimates " }{XPPEDIT 18 0 "Int(f(x),x = 0 .. z);" "6#-%$IntG6$-%\"f G6#%\"xG/F);\"\"!%\"zG" }{TEXT -1 23 " for z in the interval " } {XPPEDIT 18 0 "x[1] <= z;" "6#1&%\"xG6#\"\"\"%\"zG" }{XPPEDIT 18 0 "`` <=x[n]" "6#1%!G&%\"xG6#%\"nG" }{TEXT -1 1 "." }}}{PARA 0 "" 0 "" {TEXT -1 1 "\004" }}{PARA 0 "" 0 "" {TEXT 261 16 "How to activate:" } {TEXT 262 1 "\n" }{TEXT -1 154 "To make the procedure active open the \+ subsection, place the cursor anywhere after the prompt [ > and press \+ [Enter].\nYou can then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "simpinterp: implementation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2896 "simpinterp := proc(x::list,y::list)\n local m,n,h,k,p,q,r,sm,i,a,b,c,d,e,f,loc simp,data,\n saveDigits,xi,xip,xip2;\nlocsimp :=\nproc(x_simp::realc ons)\n local a,b,c,xk,xm,sm,jF,jM,jS,n,h,k,u,j,\n data,xx,va l,saveDigits;\n options `Copyright 2003 by Peter Stone`;\n \n da ta := _data;\n\n saveDigits := Digits;\n Digits := Digits+length(D igits)+1;\n\n xx := evalf(x_simp);\n n := nops(data);\n if (data [1,1]data[n,1] or xxdata[1,1])) then\n error \"indepen dent variable is outside the interpolation interval %1 to %2\",evalf(d ata[n,1]),evalf(data[1,1]);\n end if;\n \n # Peform a binary sea rch for the interval containing x.\n n := nops(data);\n jF := 0;\n jS := n+1;\n\n if data[1,1]1 d o\n jM := trunc((jF+jS)/2);\n if xx>=data[jM,1] then jF \+ := jM else jS := jM end if;\n end do;\n if jM = n then jF := n-1; jS := n end if;\n else\n while jS-jF>1 do\n jM := \+ trunc((jF+jS)/2);\n if xx<=data[jM,1] then jF := jM else jS := \+ jM end if;\n end do;\n if jM = n then jF := n-1; jS := n end if;\n end if;\n \n # Get the data needed from the list.\n xk \+ := data[jF,1];\n xm := data[jF,2];\n sm := data[jF,3];\n a := da ta[jF,4];\n b := data[jF,5];\n c := data[jF,6];\n u := xx-xm;\n \+ val := sm+((a*u+b)*u+c)*u;\n Digits := saveDigits;\n evalf(val); \nend proc: # of locsimp\n \n # start of main procedure\n if not \+ type(x,list(numeric)) then\n error \"the 1st argument, %1, is inv alid .. it should be a list of numerical data\",x;\n end if;\n if \+ not type(y,list(numeric)) then\n error \"the 2nd argument, %1, is invalid .. it should be a list of numerical data\",y;\n end if;\n \+ n := nops(x);\n m := nops(y);\n if n<>m then\n error \"the d ata lists must have the same length\"\n end if;\n if irem(n,2)< >1 then\n error \"the data lists must contain an odd number of da ta values\"\n end if;\n\n sm := 0;\n m := (n-1)/2;\n data := N ULL;\n\n saveDigits := Digits;\n Digits := Digits+length(Digits)+1 ;\n\n for i to n-2 by 2 do\n xi := evalf(x[i]);\n xip := \+ evalf(x[i+1]);\n xip2 := evalf(x[i+2]);\n h := xip-xi;\n \+ k := xip2-xip;\n if h*k<=0 then\n error \"values in 1st argument list must be in strictly increasing or strictly decreasing\" \n end if; \n p := evalf(y[i]);\n q := evalf(y[i+1]);\n r := evalf(y[i+2]);\n d := h*(k+h)*k;\n e := r-q;\n \+ f := p-q;\n a := (e*h+f*k)/(3*d);\n b := (e*h^2-f*k^2)/(2 *d);\n sm := sm+((a*h-b)*h+q)*h;\n data := data,[xi,xip,sm,a ,b,q];\n sm := sm+((a*k+b)*k+q)*k;\n end do;\n data := data,[ xip2];\n subs(_data=[data],eval(locsimp)); \nend proc:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 10 "simpinterp" }{TEXT -1 10 ": examples" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 39 "This example c hecks that the procedure " }{TEXT 0 10 "simpinterp" }{TEXT -1 61 " con structs a numerical procedure equivalent to the function " }{XPPEDIT 18 0 "f(x) = x^3/3;" "6#/-%\"fG6#%\"xG*&F'\"\"$F)!\"\"" }{TEXT -1 53 " when given data points which lie along the parabola " }{XPPEDIT 18 0 "y=x^2" "6#/%\"yG*$%\"xG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "gn := simpin terp([0,3,4,5,8],[0,9,16,25,64]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#gnGf*6#'%'x_simpG%)realconsG64%\"aG%\"bG%\"cG%#xkG%#xmG%#smG%#jFG%#j MG%#jSG%\"nG%\"hG%\"kG%\"uG%\"jG%%dataG%#xxG%$valG%+saveDigitsG6#%>Cop yright~2003~by~Peter~StoneG6\"C6>827%7($\"\"!FF$\"\"$FF$\".++++++*!#7$ \".LLLLLL$!#8$\".++++++$FK$\"\"*FF7($\"\"%FF$\"\"&FF$\".mmmmm;%!#6FL$ \".++++++&FK$\"#DFF7#$\"\")FF>85%'DigitsG>F^o,(F^o\"\"\"-%'lengthG6#F^ oFaoFaoFao>83-%&evalfG6#9$>8--%%nopsG6#FB@&32&FB6$FaoFao&FB6$F\\pFao52 FepFfo2FfoFcpY6%Q^oindependent~variable~is~outside~the~interpolation~i nterval~%1~to~%2F?-Fho6#Fcp-Fho6#Fep32FepFcp52FfoFep2FcpFfoY6%F\\qF_qF ]q>F\\pF]p>8*FF>8,,&F\\pFaoFaoFao@%FbpC$?(F?FaoFaoF?2Fao,&F\\rFaoFjq! \"\"C$>8+-%&truncG6#,&*&#Fao\"\"#FaoFjqFaoFao*&F\\sFaoF\\rFaoFao@%1&FB 6$FfrFaoFfo>FjqFfr>F\\rFfr@$/FfrF\\pC$>Fjq,&F\\pFaoFaoFcr>F\\rF\\pC$?( F?FaoFaoF?FarC$>FfrFgr@%1FfoFas>FjqFfr>F\\rFfr@$FfsC$>FjqFis>F\\rF\\p> 8'&FB6$FjqFao>8(&FB6$FjqF]s>8)&FB6$FjqFH>8$&FB6$FjqFU>8%&FB6$FjqFW>8&& FB6$Fjq\"\"'>80,&FfoFaoF\\uFcr>84,&F`uFao*&,&*&,&*&FduFaoFavFaoFaoFhuF aoFaoFavFaoFaoF\\vFaoFaoFavFaoFao>F^oF]o-Fho6#FdvF?F?F?" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Here is one numeri cal example." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "xx := 5.663966958;\ngn(xx);\nf := x -> x^3/3;\nf(x x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"+ep'Rm&!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+krwcg!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&#\"\"\"\"\"$F/*$)9$ F0F/F/F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+jrwcg!\")" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "The curve for the numerical procedure coincides with that for " }{XPPEDIT 18 0 "y=x^3/3" "6#/%\"yG*&%\"xG\"\"$F'!\"\"" }{TEXT -1 28 " as far as the e ye can tell." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "f := x -> x^3/3;\nplot(['gn(x)',x^3/3],x=0..8,colo r=[red,green],thickness=[1,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&#\"\"\"\"\"$F/*$)9$F0F/F/F/ F(F(F(" }}{PARA 13 "" 1 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q7#Fjal7$$\"3oKLL=2VsTF[^l$\"3tt13;jG@CFjal7$$\"3f*****\\`pfK%F[^l$\"3 9n'fleT&)p#Fjal7$$\"3!HLLLm&z\"\\%F[^l$\"3Z`\\&)*[;4-$Fjal7$$\"3s***** *z-6jYF[^l$\"3A+L)>?:*zLFjal7$$\"3<******4#32$[F[^l$\"3!f*GH=`gdPFjal7 $$\"3O*****\\#y'G*\\F[^l$\"3i$[L5lh)[TFjal7$$\"3G******H%=H<&F[^l$\"3` +W$*za39YFjal7$$\"35mmm1>qM`F[^l$\"3-JmG`Rog]Fjal7$$\"3%)*******HSu]&F [^l$\"3[Nz\\,2PobFjal7$$\"3'HLL$ep'Rm&F[^l$\"3%f;\"zkrwcgFjal7$$\"3')* *****R>4NeF[^l$\"3Gz*=wH)\\AmFjal7$$\"3#emm;@2h*fF[^l$\"39]UJ_]*f=(Fja l7$$\"3]*****\\c9W;'F[^l$\"3P=@\"G()f#3yFjal7$$\"3Lmmmmd'*GjF[^l$\"3s \\L%y^%R]%)Fjal7$$\"3j*****\\iN7]'F[^l$\"3W)[aZ<)Qf\"*Fjal7$$\"3aLLLt> :nmF[^l$\"3A8RT&G+(y)*Fjal7$$\"35LLL.a#o$oF[^l$\"3?^[8:rAl5!#:7$$\"3am mm^Q40qF[^l$\"3u3LSP6$e9\"Fihl7$$\"3y******z]rfrF[^l$\"3*[^Jl#HRB7Fihl 7$$\"3gmmmc%GpL(F[^l$\"3vdLlEA];8Fihl7$$\"3/LLL8-V&\\(F[^l$\"3/hJ!41\" o.9Fihl7$$\"3=+++XhUkwF[^l$\"3?u,T@Ay+:Fihl7$$\"3=+++:oyf\"Fihl7$Ffz$\"3dmmmmmm1 " 0 "" {MPLTEXT 1 0 361 "randomize():\nDigits := 10:\neps := Float(3,-Digi ts);\nfor i from 1 to 10000 do\n xx := rand()/Float(1,12)*8;\n sxx := evalf(evalf(f(xx),15));\n pxx := gn(xx);\n e := abs(sxx-pxx); \n if e>=eps then \n printf(\" trial no. %d, x = %.10e,\\n\", i,xx);\n printf(\" f(x) = %.10e, gn(x) = %.10e, error = %.2e\\n \\n\",sxx,pxx,e);\n end if;\nend do:\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epsG$\"\"$!#5" }}{PARA 6 "" 1 "" {TEXT -1 39 " \+ trial no. 539, x = 2.9799297120e+00," }}{PARA 6 "" 1 "" {TEXT -1 69 " f(x) = 8.8205731630e+00, gn(x) = 8.8205731620e+00, error = 1.00e-09 " }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 " tri al no. 5411, x = 5.2678818120e+00," }}{PARA 6 "" 1 "" {TEXT -1 69 " \+ f(x) = 4.8728923090e+01, gn(x) = 4.8728923080e+01, error = 1.00e-08" } }{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 " trial \+ no. 7615, x = 7.9246638870e+00," }}{PARA 6 "" 1 "" {TEXT -1 69 " f(x ) = 1.6589041710e+02, gn(x) = 1.6589041720e+02, error = 1.00e-07" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 " trial n o. 7715, x = 3.5160726570e+00," }}{PARA 6 "" 1 "" {TEXT -1 69 " f(x) = 1.4489462250e+01, gn(x) = 1.4489462260e+01, error = 1.00e-08" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 86 "The code for the numerical procdure can b e viewed by executing the following commands." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "interface(ve rboseproc=2):\neval(gn);\ninterface(verboseproc=0):" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "It is also contained in the next subsection." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 287 28 "Code for numerical procedure" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1468 "proc(z::realcons)\nlocal a ,b,c,xk,xm,sm,jF,jM,jS,n,h,k,u,j,data,xx,val,saveDigits;\noption `Copy right 2001 by Peter Stone`;\n data := [[0.,3.,9.000000000000,.33333 33333333,3.000000000000,9.],\n [4.,5.,41.66666666666,.333333333 3333,5.000000000000,25.],[8.]];\n saveDigits := Digits;\n Digits := Digits+length(Digits)+1;\n xx := evalf(z);\n n := nops(data) ;\n if (data[1,1]data[n,1] or xxdata[1,1])) then\n ERROR(`inde pendent variable is outside the interpolation interval`,evalf(data[n,1 ]),`to`,evalf(data[1,1]));\n end if;\n n := nops(data);\n jF \+ := 0;\n jS := n+1;\n if data[1,1] " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 60 "We set up 101 un equally spaced data points along the curve " }{XPPEDIT 18 0 "y= exp(- x/2)" "6#/%\"yG-%$expG6#,$*&%\"xG\"\"\"\"\"#!\"\"F-" }{TEXT -1 25 ", w ith x between 0 and 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "f := x -> exp(-x/2);\nxvalues := [ 0,seq(i*0.01+0.005*evalf(sin(i)),i=1..99),1.]:\nyvalues := map(f,xvalu es):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operat orG%&arrowGF(-%$expG6#,$9$#!\"\"\"\"#F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Some sample " }{TEXT 293 1 "x" }{TEXT -1 14 " values . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "xvalues[1],xvalues[2],xvalu es[3];\nxvalues[50],xvalues[51],xvalues[52];\nxvalues[99],xvalues[100] ,xvalues[101];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!$\"+#\\N2U\"!#6 $\"+8([YX#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"+nBJ_[!#5$\"+d7)o) \\F%$\"+f9^L^F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"+14Lr(*!#5$\"+e' R+&)*F%$\"\"\"\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 7 "We use " }{TEXT 0 10 "simpinterp" }{TEXT -1 164 " to con struct a numerical procedure which uses parabolic interpolation over e ach consecutive pair of intervals to estimate the integral under the \+ \"numerical curve\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "gn := simpinterp(xvalues,yvalues);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "We can co mpare this numerical procedure with the analytical integral:" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(-t/2),t=0..x) = 2-2 *exp(-x/2)" "6#/-%$IntG6$-%$expG6#,$*&%\"tG\"\"\"\"\"#!\"\"F//F,;\"\"! %\"xG,&F.F-*&F.F--F(6#,$*&F3F-F.F/F/F-F/" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "h := \+ x -> 2-2*exp(-x/2);\nplot(['gn(x)',h(x)],x=0..1,thickness=[1,2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,&\"\"#\"\"\"*&F-F.-%$expG6#,$9$#!\"\"F-F.F6F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 319 181 181 {PLOTDATA 2 "6&-%'CURVESG6%7S7$$\"\"!F)F(7$ $\"+;arz@!#6$\"+j0)y;#F-7$$\"+XTFwSF-$\"+7\\,NSF-7$$\"+\"z_\"4iF-$\"+% feP6'F-7$$\"+S&phN)F-$\"+)e6S=)F-7$$\"+*=)H\\5!#5$\"+_wCA5FB7$$\"+[!3u C\"FB$\"+uOI47FB7$$\"+J$RDX\"FB$\"+jm/,9FB7$$\"+)R'ok;FB$\"+%o*G(f\"FB 7$$\"+1J:w=FB$\"+gE%3z\"FB7$$\"+3En$4#FB$\"+U>\"y)>FB7$$\"+/RE&G#FB$\" +Eq`f@FB7$$\"+D.&4]#FB$\"+!f+4N#FB7$$\"+vB_o#GU$FB7$$\"+$Q*o]RFB$\"+6I' \\e$FB7$$\"+\"=lj;%FB$\"+*GG5w$FB7$$\"+V&RY2aFB$\"+F`5QZFB7$$\"+yXu9cFB$\"+yZY &*[FB7$$\"+\\y))GeFB$\"+r(Gj0&FB7$$\"+i_QQgFB$\"+=[/7_FB7$$\"+!y%3TiFB $\"+bM;h`FB7$$\"+O![hY'FB$\"+ZK(\\_&FB7$$\"+#Qx$omFB$\"+>&*fqcFB7$$\"+ u.I%)oFB$\"+t(pW#eFB7$$\"+(pe*zqFB$\"+k;ZifFB7$$\"+C\\'QH(FB$\"+\"Q2=6 'FB7$$\"+8S8&\\(FB$\"+z'p3D'FB7$$\"+0#=bq(FB$\"+D;u%R'FB7$$\"+2s?6zFB$ \"+tr%R`'FB7$$\"+IXaE\")FB$\"+Fm:ymFB7$$\"+l*RRL)FB$\"+2te:oFB7$$\"+`< .Y&)FB$\"+iQmapFB7$$\"+8tOc()FB$\"+k)R64(FB7$$\"+\\Qk\\*)FB$\"+9!)G:sF B7$$\"+p0;r\"*FB$\"+(43hN(FB7$$\"+lxGp$*FB$\"+tYu![(FB7$$\"+!oK0e*FB$ \"+Z-G7wFB7$$\"+<5s#y*FB$\"+#)>)ot(FB7$$\"\"\"F)$\"+1oQpyFB-%'COLOURG6 &%$RGBG$\"#5!\"\"F(F(-%*THICKNESSG6#Fcz-F$6%7SF'7$$\"3emmm;arz@!#>$\"3 !3$y$=b!)y;#Ff[l7$$\"3[LL$e9ui2%Ff[l$\"3M.*[1!\\,NSFf[l7$$\"3nmmm\"z_ \"4iFf[l$\"3jmIq*eeP6'Ff[l7$$\"3[mmmT&phN)Ff[l$\"3O(fozd6S=)Ff[l7$$\"3 CLLe*=)H\\5!#=$\"3t`PN^wCA5F[]l7$$\"3gmm\"z/3uC\"F[]l$\"3X6*)HsOI47F[] l7$$\"3%)***\\7LRDX\"F[]l$\"3*=]%)=mY5S\"F[]l7$$\"3]mm\"zR'ok;F[]l$\"3 i@'4?o*G(f\"F[]l7$$\"3w***\\i5`h(=F[]l$\"3m+\"y)>F[]l7$$\"3qmm;/RE&G#F[]l$\"3#\\N&pCq`f@F[]l7$$ \"3\")*****\\K]4]#F[]l$\"3#eR_se+4N#F[]l7$$\"3$******\\PAvr#F[]l$\"3'f ST&RY)4a#F[]l7$$\"3)******\\nHi#HF[]l$\"3W(f4'>&GAs#F[]l7$$\"3jmm\"z*e v:JF[]l$\"39&>Vns%=&)GF[]l7$$\"3?LLL347TLF[]l$\"3]gG()ei&p2$F[]l7$$\"3 ,LLLLY.KNF[]l$\"305tKd(HxB$F[]l7$$\"3w***\\7o7Tv$F[]l$\"3Oa!o$=o#GU$F[ ]l7$$\"3'GLLLQ*o]RF[]l$\"3I;I%)4I'\\e$F[]l7$$\"3A++D\"=lj;%F[]l$\"3gsY <)GG5w$F[]l7$$\"31++vV&RY2aF[]l$\"34h&GlK0\"QZF[]l7$$\"39mm;zXu9cF[]l$\"3#[[ptxka *[F[]l7$$\"3l******\\y))GeF[]l$\"3Aam%3xGj0&F[]l7$$\"3'*)***\\i_QQgF[] l$\"3Hzh6<[/7_F[]l7$$\"3@***\\7y%3TiF[]l$\"3_;tt`M;h`F[]l7$$\"35****\\ P![hY'F[]l$\"3Hi<*pCt\\_&F[]l7$$\"3kKLL$Qx$omF[]l$\"3W^n:=&*fqcF[]l7$$ \"3!)*****\\P+V)oF[]l$\"3)pn7:xpW#eF[]l7$$\"3?mm\"zpe*zqF[]l$\"39$zjRm rC'fF[]l7$$\"3%)*****\\#\\'QH(F[]l$\"3HC:j!Q2=6'F[]l7$$\"3GKLe9S8&\\(F []l$\"3YNAqx'p3D'F[]l7$$\"3R***\\i?=bq(F[]l$\"3S[h!\\iTZR'F[]l7$$\"3\" HLL$3s?6zF[]l$\"3*[o'esr%R`'F[]l7$$\"3a***\\7`Wl7)F[]l$\"3mll#eic\"ymF []l7$$\"3#pmmm'*RRL)F[]l$\"3tuqE2te:oF[]l7$$\"3Qmm;a<.Y&)F[]l$\"3=)ot(F[]l7$Fbz$\"3_Jtu0oQpyF[]l-Fgz6&FizF( FjzF(-F^[l6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!F\\[m-%%VIEWG6$;F(Fbz%(DEFAU LTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Here is the absolute error curve." }}{PARA 0 "" 0 "" {TEXT -1 67 "Note that the original data values were only computed to 10 digits ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "evalf(plot(h(x)-'gn(x)',x=0..1,color=blue),15);" }} {PARA 13 "" 1 "" {GLPLOT2D 360 176 176 {PLOTDATA 2 "6&-%'CURVESG6#7iam 7$$\"\"!F)$!&&GH!#>7$$\"0m;a8ABO\"!#<$!'2ZgF07$$\"0KL3FWYs#F0$!(p)p@F0 7$$\"0)*\\iSmp3%F0$!(_@N%F07$$\"0lm;a)G\\aF0$!(J>&oF07$$\"0I$3x1h6oF0$ !(xbT*F07$$\"0(**\\7G$R<)F0$!)ou#=\"F07$$\"0j;z%\\DO&*F0$!)wz\"R\"F07$ $\"0LL$3x&)*3\"!#;$!(Zgb\"FT7$$\"0+v=#**3E7FT$!(Mom\"FT7$$F/FT$!(%>?FT$!(AXU\"FT7$ $\"0mmmT:(z@FT$!(,s8\"FT7$$\"0dRsL]#)H#FT$!(X5/\"FT7$$\"0\\7yD&y;CFT$! 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)p;H2)4ZV\"*F[r$\"3#=P\"He\"pFv'F07$$\"3CLL$3dg6<*F[r$\"3zmMs#>s(3oF07 $$\"3))*\\7.ZEf>*F[r$\"3ugmfLt5$y'F07$$\"3^m;zpBp?#*F[r$\"3g'QBR3tUo'F 07$$\"39L3Fp#eaC*F[r$\"3-+11$*F[r$\"3[fD\\;rDTbF07$$\"3ImmmmxGp$*F[r$\"3c.L%HZgW _%F07$$\"3sK$eRA5\\Z*F[r$\"3a^&)RA=;e9F07$$\"3A++D\"oK0e*F[r$!3eri$)>& H1^#F07$$\"3C+++]oi\"o*F[r$!3C\\Qxb4*p^'F07$$\"3A++v=5s#y*F[r$!37Uv?'H eAs*F07$$\"3v]iS\"*3))4)*F[r$!3tiFC@d*p-\"FF7$$\"3;+D1k2/P)*F[r$!3%3+m ?IR@1\"FF7$$\"3e\\(=nj+U')*F[r$!3K'pT*o**yt5FF7$$\"35+]P40O\"*)*F[r$!3 C\"R6?%Htd5FF7$$\"31+voa-oX**F[r$!3<%*flp@wP#*F07$$\"\"\"F)$!3mjJ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 90 "Use the following commands to construct some unequally sp aced data points along the curve " }{XPPEDIT 18 0 "y=2*x/(1+x^2)" "6#/ %\"yG*(\"\"#\"\"\"%\"xGF',&F'F'*$F(F&F'!\"\"" }{TEXT -1 5 " for " } {TEXT 288 1 "x" }{TEXT -1 17 " between 0 and 1." }}{PARA 0 "" 0 "" {TEXT -1 42 "Use this data together with the procedure " }{TEXT 0 10 " simpinterp" }{TEXT -1 166 " to construct a numerical procedure which p rovides indefinite integration over the interval from 0 to 1 by parabo lic interpolation over successive pairs of intervals." }}{PARA 0 "" 0 "" {TEXT -1 100 "Compare this numerical procedure with the analytical \+ function which gives the \"indefinite integral\" " }{XPPEDIT 18 0 "Int (2*x/(1+x^2),x=0..z)" "6#-%$IntG6$*(\"\"#\"\"\"%\"xGF(,&F(F(*$F)F'F(! \"\"/F);\"\"!%\"zG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "__ ___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "f := x -> 2*x/(1+x^2);\nxv alues := [0,seq(i*0.02+0.01*evalf(sin(i)),i=1..49),1.]:\nyvalues := ma p(f,xvalues):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 38 "Use the data below with the procedure " }{TEXT 0 10 "si mpinterp" }{TEXT -1 102 " to construct a numerical procedure which pro vides indefinite integration for the associated function " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 93 " over the interval from 0 to 1 by parabolic interpolation over successive pairs of inter vals." }}{PARA 0 "" 0 "" {TEXT -1 37 "________________________________ _____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1331 "xvalues := [0, .2999394741e-1, .5995160856e-1, .898 3684340e-1, .1196138030, .1492470720, .1787018075, .2079438734, .23693 99681, .2656577467, .2940659340, .3221344306, .3498344082, .3771383968 , .4040203606, .4304557641, .4564216271, .4818965700, .5068608469, .53 12963698, .5551867220, .5785171613, .6012746141, .6234476611, .6450265 143, .6660029866, .6863704521, .7061238050, .7252594083, .7437750389, \+ .7616698292, .7789442057, .7955998215, .8116394921, .8270671222, .8418 876403, .8561069244, .8697317333, .8827696366, .8952289454, .907118644 0, .9184483249, .9292281235, .9394686568, .9491809620, .9583764424, .9 670668090, .9752640312, .9829802866, .9902279146, 1.]:\nyvalues := [1, 1.014886174, 1.029539513, 1.043952510, 1.058118048, 1.072029418, 1.08 5680343, 1.099064999, 1.112178029, 1.125014554, 1.137570189, 1.1498410 46, 1.161823742, 1.173515401, 1.184913651, 1.196016624, 1.206822948, 1 .217331742, 1.227542605, 1.237455603, 1.247071258, 1.256390529, 1.2654 14799, 1.274145856, 1.282585870, 1.290737381, 1.298603270, 1.306186742 , 1.313491305, 1.320520745, 1.327279107, 1.333770672, 1.339999934, 1.3 45971579, 1.351690468, 1.357161612, 1.362390151, 1.367381342, 1.372140 531, 1.376673144, 1.380984665, 1.385080620, 1.388966567, 1.392648074, \+ 1.396130711, 1.399420038, 1.402521590, 1.405440867, 1.408183329, 1.410 754378, 1.414213562]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_______________________________ ______" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 13 "The velocity " }{XPPEDIT 18 0 "v = v(x);" "6#/%\"vG-F$6#%\"xG" }{TEXT -1 45 " metres per sec of a \+ particle, at a distance " }{TEXT 271 1 "x" }{TEXT -1 74 " meters from \+ a fixed point O in its path, is given by the following table." }} {PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "matrix([[x, 0, 10, 2 0, 30, 40, 50, 60], [v(x), 47, 58, 64, 65, 61, 52, 38]]);" "6#-%'matri xG6#7$7*%\"xG\"\"!\"#5\"#?\"#I\"#S\"#]\"#g7*-%\"vG6#F(\"#Z\"#e\"#k\"#l \"#h\"#_\"#Q" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 56 "(a) Show that the time taken to arrive at a distance of " }{TEXT 289 1 "s" } {TEXT -1 27 " meters from O is given by " }{XPPEDIT 18 0 "Int(1/v(x),x =0..s)" "6#-%$IntG6$*&\"\"\"F'-%\"vG6#%\"xG!\"\"/F+;\"\"!%\"sG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 49 "(b) Construct a numerica l procedure for the time " }{TEXT 290 1 "t" }{TEXT -1 31 " as a functi on of the distance " }{TEXT 291 1 "x" }{TEXT -1 21 ", and plot its gra ph." }}{PARA 0 "" 0 "" {TEXT -1 55 "(c) Estimate the time taken to tra verse the 60 meteres." }}{PARA 0 "" 0 "" {TEXT -1 37 "________________ _____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_______________________________ ______" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 24 "Code for drawing picture" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 21 "Code fo r 1st picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 577 "p1:=plot(-x^2/4+x/4+4,x=-1.5..1.7,tickmarks=[0, 0]):\np2:=plot([[[1.4,0],[1.4,3.86]],[[-1,0],[-1,3.5]]],color=black): \np3:=plot([[[-1,3.5]],[[0,4]],[[1.4,3.86]],[[-1,3.5]],[[0,4]],\n[[1.4 ,3.86]],[[-1,3.5]],[[0,4]],[[1.4,3.86]]],style=point,color=blue,\nsymb ol=[circle$2,diamond$2,cross$1]): \nt1:=plots[textplot]([[-1.2,3.7,`(- h,p)`],\n[-0.22,4.2,`(0,q)`],[1.45,4.1,`(k,r)`]],color=blue):\nt2:=plo ts[textplot]([[1.8,-0.1,`x`],[-0.1,4.8,`y`],\n[1.4,-0.2,`k`],[-1,-0.2, `-h`]]):\nt3:=plots[textplot]([1.75,3.5,`y = f(x)`],color=red):\nplots [display](\{p1,p2,p3,t1,t2,t3\},view=[-2..2,-.2..5]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 258 "" 0 "" {TEXT -1 21 "Code for 2nd picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 853 "i := 'i': n := 'n':\np1:=plot(x^2/50+x/10+1/2,x=0..6,y=0..1.75,color=re d):\np2:=plots[polygonplot]([[3,0],[0,0],op(op(1,op(1,plot(x^2/50+x/10 +1/2,\n x=0..3.5)))),[3.5,0]],color=COLOR(RGB,.93,.93,.95)):\np3:=pl ots[polygonplot]([[3.5,0],[3.5,1.095],[3.8,1.1688],[3.8,0]],\n colo r=COLOR(RGB,.8,.8,.95)):\np4:=plot([[[-.5,0],[6.5,0]],[[3,0],[3,.98]], [[4,0],[4,1.22]],\n [[6,0],[6,1.82]]],color=[black$3,navy$2]):\n t1:=plots[textplot]([[-.05,-0.07,`x`],[2.95,-0.07,`x`],[3.5,-0.07,`x`] ,\n [3.8,-0.07,`z`],[4.07,-0.07,`x`],[6,-0.07,`x`]],font=[HELVETICA ,10]):\nt2:=plots[textplot]([[.05,-0.1,`1`],[3.09,-0.1,`i-1`],[3.6,-0. 1,`i`],\n [3.8,-0.07,`z`],[4.25,-0.1,`i+1`],[6.1,-0.1,`n`]],font=[H ELVETICA,8]):\nt3:=plots[textplot]([4.95,1.7,`y = f(x)`],color=red,fon t=[HELVETICA,10]):\nplots[display]([p1,p2,p3,p4,t1,t2,t3],view=[-.5..6 .5,-.2..1.82],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }