{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Helvetica" 1 14 128 0 0 1 0 0 2 0 0 0 0 0 0 1 } {CSTYLE "" -1 257 "" 0 24 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Helvetica" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Time s" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Title" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }1 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "T imes" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Title" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 261 1 {CSTYLE "" -1 -1 "C ourier" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 263 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 264 1 {CSTYLE "" -1 -1 "Couri er" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 2" -1 265 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 257 30 "MLC Lab Visit - Lab 09 \+ - Maple" }}{PARA 0 "" 0 "" {TEXT 256 46 "Mth 355 (a.k.a. Mth 399) Mar \+ 12, 2003 Maple 7" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 16 "Bent E. Petersen" }{TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT 260 0 "" }} {PARA 264 "" 0 "" {TEXT 261 22 "petersen@math.orst.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT 259 194 "There are 2 extr a credit problems below. Problem solutions are due Mar 17, 2003. Email your solutions to me as Maple worksheet attachments. Your worksheet m ust execute correctly for full credit." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 80 "In this week's lab we investigate par t of Maple's pedagogical package, student. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 "Double Integrals" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 174 "Maple has some limited support for double and triple integrals. Note these integrals are ret urned unevaluated. If an exact evaluation is possible, value() will gi ve it to us." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "g:=(x,y)->x^3*y^2-4*x^2*y+2*x*y+y^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowG F),**&)9$\"\"$\"\"\")9%\"\"#F2F2*(\"\"%F2)F0F5F2F4F2!\"\"*(F5F2F4F2F0F 2F2*$)F4F1F2F2F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Dou bleint(g(x,y),x=-1..3,y=2..4): %=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-F%6$,**&)%\"xG\"\"$\"\"\")%\"yG\"\"#F.F.*(\" \"%F.)F,F1F.F0F.!\"\"*(F1F.F0F.F,F.F.*$)F0F-F.F./F,;F5F-/F0;F1F3#\"%78 F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 144 "Some variable limits of integration can be handled. Jsut keep \+ in mind Maple takes the first specified range as the range for the inn er integral." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Doubleint(g(x,y),x=-1..y,y=2..4): %=value(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-F%6$,**&)%\"xG\"\"$\"\"\") %\"yG\"\"#F.F.*(\"\"%F.)F,F1F.F0F.!\"\"*(F1F.F0F.F,F.F.*$)F0F-F.F./F,; F5F0/F0;F1F3#\"&_:#\"#N" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 88 "Here's an example where the limits for y depend on x and so must be specified first." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Doubleint(g (x,y),y=x..4,x=-1..3): %=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%$IntG6$-F%6$,**&)%\"xG\"\"$\"\"\")%\"yG\"\"#F.F.*(\"\"%F.)F,F1F.F0F .!\"\"*(F1F.F0F.F,F.F.*$)F0F-F.F./F0;F,F3/F,;F5F-#\"&nH%\"$0\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 60 "int2:=Doubleint(x^2+y^2,y=-sqrt(1-x^2)..sqrt(1-x^2) ,x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%int2G-%$IntG6$-F&6$,&* $)%\"xG\"\"#\"\"\"F/*$)%\"yGF.F/F//F2;,$*$-%%sqrtG6#,&F/F/F+!\"\"F/F;F 6/F-;\"\"!F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "changevar( \{x=r*cos(t),y=r*sin(t)\}, int2, [r,t]);" }}{PARA 7 "" 1 "" {TEXT -1 51 "Warning, Computation of new ranges not implemented\n" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$*&)%\"rG\"\"#\"\"\"-%$absG6#F*F,F* %\"tG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "Maybe in the future ... but for now we need to compute th e limits of integration ourselves." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "Doubleint(integrand(change var(\{x=r*cos(t),y=r*sin(t)\},int2,[r,t])),r=0..1, t=-Pi/2..Pi/2); val ue(%);" }}{PARA 7 "" 1 "" {TEXT -1 51 "Warning, Computation of new ran ges not implemented\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6 $*&)%\"rG\"\"#\"\"\"-%$absG6#F*F,/F*;\"\"!F,/%\"tG;,$%#PiG#!\"\"F+,$F7 #F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\"%" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "To turn off the warning we need to remove the limits of integration in \+ int2. Here's an example" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Doubleint(x^3+y^3,y=0..sqrt(1-x^2), x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$,&*$)%\"xG\" \"$\"\"\"F-*$)%\"yGF,F-F-/F0;\"\"!*$-%%sqrtG6#,&F-F-*$)F+\"\"#F-!\"\"F -/F+;F3F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "int3z:=Doublei nt(x^3+y^3,x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&int3zG-%$IntG6$ -F&6$,&*$)%\"xG\"\"$\"\"\"F/*$)%\"yGF.F/F/F-F2" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 56 "int3c:=changevar(\{x=r*cos(t),y=r*sin(t)\}, in t3z, [r,t]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&int3cG-%$IntG6$-F&6 $,(*()%\"rG\"\"$\"\"\"-%$absG6#F-F/)-%$sinG6#%\"tGF.F/F/**F,F/F0F/-%$c osGF6F/)F4\"\"#F/!\"\"*(F,F/F0F/F9F/F/F-F7" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 61 "int3:=Doubleint(integrand(int3c),r=0..1,t=0..Pi/2); value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%int3G-%$IntG6$-F&6$,( *()%\"rG\"\"$\"\"\"-%$absG6#F-F/)-%$sinG6#%\"tGF.F/F/**F,F/F0F/-%$cosG F6F/)F4\"\"#F/!\"\"*(F,F/F0F/F9F/F//F-;\"\"!F//F7;FA,$%#PiG#F/F<" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"%\"#:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Nice, but a more complete implementation would be nicer." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 14 "Line Integrals" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Ma ple can compute some line integrals. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "Lineint(f(x,y),x=2*cos( t),y=2*sin(t),t=0..Pi); simplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%\"fG6$,$-%$cosG6#%\"tG\"\"#,$-%$sinGF-F/\"\"\"-%%sqrtG 6#,&*$)-%%diffG6$F*F.F/F3F3*$)-F;6$F0F.F/F3F3F3/F.;\"\"!%#PiG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$-%\"fG6$,$-%$cosG6#%\"tG\" \"#,$-%$sinGF-F//F.;\"\"!%#PiGF/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Lineint(x^4-y^4,x=2*cos(t),y=2*sin(t),t=0..Pi/4); sim plify(%); value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&,&*$ )-%$cosG6#%\"tG\"\"%\"\"\"\"#;*&F0F/)-%$sinGF,F.F/!\"\"F/-%%sqrtG6#,&* $)-%%diffG6$,$F*\"\"#F-F@F/F/*$)-F=6$,$F3F@F-F@F/F/F//F-;\"\"!,$%#PiG# F/F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$,&!\"\"\"\"\"*&\"\" #F))-%$cosG6#%\"tGF+F)F)/F0;\"\"!,$%#PiG#F)\"\"%\"#K" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "Tangent line to a Graph" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 242 "M aple has a routine to plot the graph of a function and a tangent to th e graph. This routine is rather limited, so you will want to compute a nd plot the tangent yourself. Of course, if you are teaching calculus, this routine may fit the bill." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "showtangent(x^2*sin(x),x=2); " }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$ 7S7$$!#5\"\"!$!3z:ngiQS.?!#;7$$!3!pmmm\"p0k&*!#<$!39Ws0F'4u\">F-7$$!3u KL$3(ftMI9F-7$$!3Kmm;/siqmF1$!3 O3xz)*4lY8F-7$$!3Q****\\(y$pZiF1$!3wLm;`IAj7F-7$$!3jKLL$yaE\"eF1$!3#=/ ([crSx6F-7$$!3s%HaF1$!31\"4\")zY?=5\"F-7$$!3]******\\$*4)*\\F1$ !33s=Qfxs;5F-7$$!3o******\\_&\\c%F1$!33bgX>l&GJ*F17$$!3%)******\\1aZTF 1$!3'G`KR\\j%*[)F17$$!3Imm;/#)[oPF1$!3U.cHWPuTxF17$$!3%HLLL=exJ$F1$!37 PZ(4kKE&oF17$$!3lKLLL2$f$HF1$!3%z`m:\"*Q%*4'F17$$!3%)****\\PYx\"\\#F1$ !3cgCA[bHB_F17$$!3gLLLL7i)4#F1$!3gDK)[@gxW%F17$$!3o)***\\P'psm\"F1$!3# )H?ZQ](of$F17$$!3?****\\74_c7F1$!3-,CrL6j'y#F17$$!3M:LL$3x%z#)!#=$!3zs HE>mAT>F17$$!3()HLL3s$QM%Fer$!35t4Fo8)[;\"F17$$!3]^omm;zr)*!#?$!3![\"o vW8)[F$Fer7$$\"3fVLLezw5VFer$\"3'4tnN!4GBaFer7$$\"3-.++v$Q#\\\")Fer$\" 3%H7b5n0&*H\"F17$$\"3%\\LL$e\"*[H7F1$\"3(o+np5+#[*zN&*F17$$\"3X,+]7k.6aF1$\"3HfY%)=3eO5F-7$$\"3#emmmT9C#eF1$\" 3q=w2T$Hx6\"F-7$$\"33****\\i!*3`iF1$\"3B->SBVo-7F-7$$\"3;NLLL*zym'F1$ \"3'fM>&og]%G\"F-7$$\"3'eLL$3N1#4(F1$\"3_I%y1i!=o8F-7$$\"3,pm;HYt7vF1$ \"3an#*H0B;^9F-7$$\"37-+++xG**yF1$\"3&QW@`&QTF:F-7$$\"3gpmmT6KU$)F1$\" 3pj)Hmt1[h\"F-7$$\"3qNLLLbdQ()F1$\"3c:-\\r>(Hp\"F-7$$\"3[++]i`1h\"*F1$ \"3_.j%\\T7jx\"F-7$$\"3A-+]P?Wl&*F1$\"3)HvML&)zg&=F-7$$\"#5F*$\"3P)f@' f3!=%>F--%'COLOURG6&%$RGBG$Fhz!\"\"$F*F*Fa[l-F$6$7_r7$F($\"33*p$*)36@S aF-7$$!3Gmm;HU,\"*)*F1$\"3sw\\poV#yR%F-7$$!3cKLLe%G?y*F1$\"3InY'o()3iM $F-7$$!3i++](oUIn*F1$\"3#*QD3;!o\"*H#F-7$F/$\"3'z`&[Ch))p7F-7$$!3QL$3- )*G#p%*F1$\"37-1()*)eW%)RF17$$!3#)***\\P/,WP*F1$!3rT/(fo?_U%F17$$!3Im; H2Jdz#*F1$!3k))fjqm(fC\"F-7$F5$!3..l!))>A)F1$!38?a=88b2j F-7$$!3YmT5l^]o\")F1$!3OggEHL7XjF-7$$!3?++Dc[3:\")F1$!3m)*GUR6DijF-7$$ !3%R$eRZXmh!)F1$!3=\"***z&e(QfjF-7$$!3nn;aQUC3!)F1$!3>$y-q]TqL'F-7$$!3 _+voHR#[&zF1$!3U1p#*>OiF-7$$!3`****\\iPH.xF1$!3# z\"4#R&)Go'eF-7$FI$!3Um&*H#y'f$H&F-7$$!31ML$3i_+I(F1$!3Wo**=/'*=KXF-7$ FN$!37_%4I2I>l$F-7$$!3#RLL3FuF)oF1$!3)4(G'3')\\Kn#F-7$FS$!3#3It&>4?\"o \"F-7$$!3SKL$e\\g\"fkF1$!3uVJ'zc$)RI(F17$FX$\"3Uun'>rt]Q\"F17$$!3+mmT& Gu,.'F1$\"3h#)GP>dR-\"*F17$Fgn$\"3kbxf%yi<`\"F-7$$!3'*)***\\([j5i&F1$ \"3-%oqtjF-7$F\\o$\"3I;Z>65#>A#F-7$$!31***\\7)*G;K&F1$\"3\\T]oxPXA BF-7$$!3$GLL3x&y8_F1$\"3'H$o)3JfPQ#F-7$$!3s**\\ilT')f^F1$\"3jIwmdWG+CF -7$$!3hmmTgD%f5&F1$\"3%o`s^agxS#F-7$$!3iK$3_&4-_]F1$\"3*H\"))Rsy\\1CF- 7$Fao$\"3O(3Cy&4$oR#F-7$$!3g*******HF:y%F1$\"3qSs-G\"Q3G#F-7$Ffo$\"3I# Qy'yUFh?F-7$F[p$\"3wN]rx)3IX\"F-7$F`p$\"3G+>[&zw5L)F17$Fep$\"3mF^=I58H >F17$$!3!GLL$eW%o7$F1$!3B_^BC=!>W\"Fer7$Fjp$!3!*=ptd8Fg&H#F1$!3e0T=8\\8XR F17$Fdq$!3N7@Qg#=[!QF17$$!39mmTNa%H)=F1$!3WG.NT2:uLF17$Fiq$!3e@Ox\">io w#F17$F^r$!3n#\\;1'R^,:F17$Fcr$!3K4]AZ`**[]Fer7$Fir$!3da*3]z75%z!#>7$F ^s$!3*yp!f'))G,i*!#C7$Fds$\"3==Y5&GxZw(Fbjl7$Fis$\"3\"ej%4U][K[Fer7$F^ t$\"3/UGuo)[WU\"F17$Fct$\"3S))oS:6$yt#F17$$\"3Q++]7JFn=F1$\"301lkI)*eM LF17$Fht$\"313bZR\\ef1%R)3#F-7$$\"37MLeR*)**)y% F1$!3/k$HZ9CnG#F-7$F[w$!35sa$Q**[ZR#F-7$$\"3GLL3xS'G/&F1$!3'pW[;8RaS#F -7$$\"3y*****\\7ga4&F1$!3ra]dl#*=3CF-7$$\"3Gmm\"HY]o-CF -7$$\"3nLL$3A_1?&F1$!35;nh)oL')Q#F-7$$\"3cnmm;V%eI&F1$!3X_L=Jp$QL#F-7$ F`w$!34!)pdNZ&>C#F-7$$\"3jLLe9as;cF1$!3)>r'=(*eH]>F-7$Few$!3!*G0hmzM2: F-7$$\"3WKLeRm<\"F17$$\"37nm\"z \\%[gkF1$\"3!z087Qz8O(F17$F_x$\"3')*y6'44]o;F-7$$\"3_NL$3sr*zoF1$\"3Wl $4&z66gEF-7$Fdx$\"3aJ@rQ43ROF-7$$\"3W-+vo!*R-tF1$\"3*GPQ\"*3t;a%F-7$Fi x$\"3enbNHum=`F-7$$\"3dNLek6,1xF1$\"3E'z&3c%pL(eF-7$F^y$\"3Gs(y^grMB'F -7$$\"3-O$3-)omazF1$\"3[D\\?ahi&H'F-7$$\"3*)pmTgg/5!)F1$\"3,SX%R%o5QjF -7$$\"3w.]iS_Ul!)F1$\"35k:p6HBgjF-7$$\"3(eLL3U/37)F1$\"3Gy/_.9QhjF-7$$ \"3'zmT5g$=w\")F1$\"3-voPi3*4M'F-7$$\"3&=+]7yi:B)F1$\"3k/+!)e[c)H'F-7$ $\"3sN$e9'>%pG)F1$\"3goAgZ`nLiF-7$Fcy$\"3)>U=U$e'f9'F-7$$\"3l-+]P$[/a) F1$\"3!Q_yli!yTcF-7$Fhy$\"3$HJR\\Np$Q[F-7$$\"3*=+D1*z>W))F1$\"3'>%>)oE G/H%F-7$$\"33om\"zW?)\\*)F1$\"3]&)>pjc%Hm$F-7$$\"3GM$3_!HWb!*F1$\"33(* *y$)4+-'HF-7$F]z$\"3;=E([WZw=#F-7$$\"3#4+]7`f@E*F1$\"3w@Xp+R#*)Q\"F-7$ $\"3M,+++PDj$*F1$\"3k6h#*G`]VNF17$Fbz $!3eX-BhN!GG\"F-7$$\"3w+]7G:3u'*F1$!3$\\:9T%[34BF-7$$\"35,+v=5s#y*F1$! 3YPLj]8*GN$F-7$$\"3W,]P40O\"*)*F1$!3`O4j.%e6S%F-7$Fgz$!33*p$*)36@SaF-- F\\[l6&F^[lFa[lF_[lFa[l-%+AXESLABELSG6$Q\"x6\"Q!Fjim-%%VIEWG6$;F(Fgz%( DEFAULTG-F$6$7$7$$\"\"#F*Fa[l7$Fejm$\"+2(*=PO!\"*-F\\[l6&F^[lF_[lF_[lF a[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 "Trapezoidal and Simpson's Rule s" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "The student package implements both the (compound) trapezoidal ru le and the (compound) Simpson's rule for numerical quadrature." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "trapezoid(f(x),x=a..b,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$* (,&%\"bG\"\"\"%\"aG!\"\"F'%\"nGF),(-%\"fG6#F(F'*&\"\"#F'-%$SumG6$-F-6# ,&F(F'*(%\"iGF'F%F'F*F)F'/F8;F',&F*F'F'F)F'F'-F-6#F&F'F'#F'F0" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "simpson(f(x),x=a..b,n);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,&%\"bG\"\"\"%\"aG!\"\"F'%\"nGF),* -%\"fG6#F(F'-F-6#F&F'*&\"\"%F'-%$SumG6$-F-6#,&F(F'*(,&%\"iG\"\"#F'F)F' F%F'F*F)F'/F;;F',$F*#F'F " 0 "" {MPLTEXT 1 0 27 "trapezoid(f (x),x=a..b,2*n);" }}{PARA 8 "" 1 "" {TEXT -1 67 "Error, (in trapezoid) usage: trapezoid( f(x), x=a...b, iterations)\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Let's write our own \+ (compound) trapezoidal rule - call it T." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "T:=proc(f,R,n)" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "local a, b, sm, h, k;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " a:=lhs(R);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " b:=rhs(R);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " h:=(b-a)/n; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "sm:=f(a)/2;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "for k from 1 to n-1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " sm:=sm+f(a+k*h);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " sm:=sm+f(b)/2;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "sm:=h*sm;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Let's check it." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "T(exp,1..3,12): evalf(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ZQuS " 0 "" {MPLTEXT 1 0 38 "trapezoid(exp(x),x=1..3,12): evalf(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+[QuS " 0 "" {MPLTEXT 1 0 44 "S:=(f,R,n)->(4/3)*T(f,R,n)-(1/3)*T( f,R,n/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SGf*6%%\"fG%\"RG%\"nG 6\"6$%)operatorG%&arrowGF*,&-%\"TG6%9$9%9&#\"\"%\"\"$*&#\"\"\"F7F:-F06 %F2F3,$F4#F:\"\"#F:!\"\"F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "S(exp,1..3,12): evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+IHtO " 0 "" {MPLTEXT 1 0 36 "simpson(exp (x),x=1..3,12): evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+IHtO< !\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "Perhaps you suspect that S(r,R,n) is just Simpson's rule. Tha t is correct!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Further extrapolation is possible:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "B:=(f,R,n)- >(16/15)*S(f,R,n)-(1/15)*S(f,R,n/2); # n divisible by 4" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"BGf*6%%\"fG%\"RG%\"nG6\"6$%)operatorG%&arrow GF*,&-%\"SG6%9$9%9&#\"#;\"#:*&#\"\"\"F7F:-F06%F2F3,$F4#F:\"\"#F:!\"\"F *F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "B(exp,1..3,12): ev alf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'eDnt\"!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "int(exp,1..3): evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4bsO " 0 "" {MPLTEXT 1 0 28 "Rom[0]:=(f,R,n)-> T(f,R,2^n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$RomG6#\"\"!f*6%%\"f G%\"RG%\"nG6\"6$%)operatorG%&arrowGF--%\"TG6%9$9%)\"\"#9&F-F-F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Rom[1]:=(f,R,n)->(4/3)*Rom[0 ](f,R,n)-(1/3)*Rom[0](f,R,n-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&% $RomG6#\"\"\"f*6%%\"fG%\"RG%\"nG6\"6$%)operatorG%&arrowGF-,&-&F%6#\"\" !6%9$9%9&#\"\"%\"\"$*&#F'F " 0 "" {MPLTEXT 1 0 62 "Rom[2]:=(f,R,n)->(64/63)*Rom [1](f,R,n)-(1/63)*Rom[1](f,R,n-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%$RomG6#\"\"#f*6%%\"fG%\"RG%\"nG6\"6$%)operatorG%&arrowGF-,&-&F%6#\" \"\"6%9$9%9&#\"#k\"#j*&#F5F " 0 "" {MPLTEXT 1 0 65 "Rom[3]:=(f,R,n)->(256/255)*R om[2](f,R,n)-(1/255)*Rom[2](f,R,n-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$RomG6#\"\"$f*6%%\"fG%\"RG%\"nG6\"6$%)operatorG%&arrowGF-,&-&F%6 #\"\"#6%9$9%9&#\"$c#\"$b#*&#\"\"\"F " 0 "" {MPLTEXT 1 0 68 "Rom[4]:=(f,R,n)->(1024/ 1023)*Rom[3](f,R,n)-(1/1023)*Rom[3](f,R,n-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$RomG6#\"\"%f*6%%\"fG%\"RG%\"nG6\"6$%)operatorG%&arr owGF-,&-&F%6#\"\"$6%9$9%9&#\"%C5\"%B5*&#\"\"\"F \+ " 0 "" {MPLTEXT 1 0 29 "Rom[0](exp,1..2,0): evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+k*oO0&!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Rom[1](exp,1..2,1): evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+M!\\Bn%!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Rom[2](exp,1..2,2): evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+$[^3n%!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Rom[3] (exp,1..2,3): evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+8)y2n%! \"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Rom[4](exp,1..2,4): \+ evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+]XxqY!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "The actua l integral is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "Int(exp,1..2): evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+qUxqY!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 "Integration by Parts" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 " Thiswould have been handy when you studied calculus." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "intparts( Int(exp(k*x)*cos(w*x),x),cos(w*x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,&*(-%$cosG6#*&%\"wG\"\"\"%\"xGF*F*%\"kG!\"\"-%$expG6#*&F,F*F+F*F*F*- %$IntG6$,$**-%$sinGF'F*F)F*F,F-F.F*F-F+F-" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 44 "intparts(Int(exp(k*x)*cos(w*x),x),exp(k*x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(-%$expG6#*&%\"kG\"\"\"%\"xGF*F*%\" wG!\"\"-%$sinG6#*&F,F*F+F*F*F*-%$IntG6$**F)F*F%F*F,F-F.F*F+F-" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "T he second expression here is pretty useless, but the first one is what we need. Surely you remember all of this from calculus!" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} }{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Completion of the Square" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "Ma ple can complete the completion of the square, but we have to do some \+ of the work." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "completesquare(8*x^2-x,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$),&%\"xG\"\"\"#F(\"#;!\"\"\"\"#F(\"\")#F(\"#KF+" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "expr1:=completesquare(x^2-3 *x*y+16*y^2-7*y,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&expr1G,(*$), &%\"xG\"\"\"*&#\"\"$\"\"#F*%\"yGF*!\"\"F.F*F**&#\"#b\"\"%F*)F/F.F*F**& \"\"(F*F/F*F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "(x-(3/2)*y )^2 + completesquare(expr1-(x-(3/2)*y)^2,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$),&%\"xG\"\"\"*&#\"\"$\"\"#F(%\"yGF(!\"\"F,F(F(*&# \"#b\"\"%F(),&F-F(#\"#9F1F.F,F(F(#\"#\\F1F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 8 "Problems " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 262 9 "Problem 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 304 "Construct your own quadrature rule with the following specific ations. (a) It will have 9 nodes located at k/8, k=0..8, (b) It will give the exact integral on the interval [0,1] (up to floating point p recision) for functions of the form p(x)*exp(-x^2) where p(x) is a polynomial of degree at most 8." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 30 "Your method will have the form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "G :=f->sum(A[k]*f((k-1)/8),k=1..9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"GGf*6#%\"fG6\"6$%)operatorG%&arrowGF(-%$sumG6$*&&%\"AG6#%\"kG\"\"\" -9$6#,&F3#F4\"\")#F4F:!\"\"F4/F3;F4\"\"*F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 208 "Maple can actually \+ evaluate the integrals here (in terms of the error function erf) so yo u could find an exact symbolic solution if you desire. You may grow si gnificantly older though while waiting for Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 263 9 "Problem 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 262 "Let romberg(f,R,n) be Rom[n](f,R,n) as defined above. Write a procedure which computes romberg(f,R,n) for any n. N ote: recursion would be cool here if not very practical. If you do use recursion be sure to also use \"option remember\" to help speed thing s up." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 6 0" 11 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }