{VERSION 4 0 "IBM INTEL NT" "4.0" } {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 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 "Helvetica" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 " " 1 14 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 14 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 1 14 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 1 14 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 1 14 128 0 0 1 0 0 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 "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 "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 Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple P lot" -1 13 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 "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 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 257 31 "Interpolation and Extra polation" }}{PARA 0 "" 0 "" {TEXT 256 26 "Mth 351 Aug 7 2001 Maple 6" }}{PARA 0 "" 0 "" {TEXT 258 16 "Bent E. Petersen" }}{PARA 0 "" 0 "" {TEXT 259 68 "Filename: 351u2001_assignment_05_interpolation_and_extra polation.mws" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 42 "Assignment solutions are due Aug 14, 2001." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 277 11 "Assignment:" } {TEXT -1 515 " This Maple worksheet uses Egyptian cereal imports from \+ 1970 through 1989 to illustrate interpolation and extrapolation. The w orksheet contains a few problems which make up Summer 2001 Mth 351 Ass ignment 5. You can use Maple to do the assignment (recommended) or, if you are reading the PDF version of this document, you can extract the data and do the calculations laboriously with a calculator. Or you ca n write a little program in C or whatever. Above all, think, be creati ve, and write some insightful comments." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 273 15 "Acknowledgment:" }{TEXT 275 1 " " } {TEXT -1 90 "The Egyptian cereal imports data below are from table 3.5 in Sandra Lach Arlinghaus, ed., " }{TEXT 271 34 "Practical Hanbook of Curve Fitting" }{TEXT -1 154 ", CRC Press, Boca Raton 1994. The data \+ in that table are from the World Bank. The World Bank publishes an imm ense amount of world data - see for example, " }{TEXT 272 31 "http://w ww.worldbank.org/data/ " }{TEXT 274 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 350 "Figure 3.6 in the Arlinghaus text show a linear and an exponential fit to the Egyptian data. While the \+ authors of the text have much to say about the art of curve fitting an d predicition the only one I can agree 100 % with is, \"Any of an infi nite number of curves may be used to fit data; the art comes in making reasonable selections.\" Yes, indeed!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 179 "The crazier curve fits below are mi ne (and Maple's) and I take responsibility for them. I particularly li ke the prediction that Egypt in 1992 will (so 'did' from our perspecti ve) " }{TEXT 276 6 "export" }{TEXT -1 67 " more cereal than the world \+ has produced throughout all of history!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "with(stats): with(fit):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "Here's some \"real world\" data, \+ cereal imports, in thousands of tons, by Egypt from 1970 through 1989. First the years" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "Tdata:=[70,71,72,73,74,75,76,77,78,79,80,81,8 2,83,84,85,86,87,88,89];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&TdataG7 6\"#q\"#r\"#s\"#t\"#u\"#v\"#w\"#x\"#y\"#z\"#!)\"#\")\"##)\"#$)\"#%)\"# &)\"#')\"#()\"#))\"#*)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 70 "Now here is the quantity of cereal import ed in the corresponding years" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "Ydata:=[1305,2447,1773,1872 ,3877,4214,4347,4935,5852,5400,6027,7199,6799,8114,8616,8903,8407,9348 ,8500,8542];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&YdataG76\"%08\"%ZC \"%t<\"%s=\"%xQ\"%9U\"%ZV\"%N\\\"%_e\"%+a\"%Fg\"%*>(\"%*z'\"%9\")\"%;' )\"%.*)\"%2%)\"%[$*\"%+&)\"%U&)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "In Maple it is simple to find the l east squares linear fit (this uses the stat library, a standard compon ent of Maple, loaded above)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "eqn1:=leastsquare[\{t,y\},y= m*t+b,\{m,b\}]([Tdata,Ydata]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%e qn1G/%\"yG,&%\"tG#\"&2k\"\"#Q#\"(^_T&\"$!>!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 318 "Because we loaded t he plots library we can save up plots and combine and display them lat er. (This library provides other tools as well.) Note eqn1 is a label \+ for the equation of the line - what we want to plot is the expression \+ in t that forms the right-hand side of the equation. We use the rhs( ) function to do so" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "img1:=plot(rhs(eqn1),t=70..92,color=red): " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "We will want to display the original data points so we first buil t a lsit of points from the original data" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "points:=[seq([Tdata[ k],Ydata[k]],k=1..20)];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'pointsG7 67$\"#q\"%087$\"#r\"%ZC7$\"#s\"%t<7$\"#t\"%s=7$\"#u\"%xQ7$\"#v\"%9U7$ \"#w\"%ZV7$\"#x\"%N\\7$\"#y\"%_e7$\"#z\"%+a7$\"#!)\"%Fg7$\"#\")\"%*>(7 $\"##)\"%*z'7$\"#$)\"%9\")7$\"#%)\"%;')7$\"#&)\"%.*)7$\"#')\"%2%)7$\"# ()\"%[$*7$\"#))\"%+&)7$\"#*)\"%U&)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Now we plot the points (and sav e the plot for later)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "pntplot:=pointplot(points,color=blu e,symbolsize=20,symbol=diamond):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Let's display our two plots to see how well our least squares line fits the data" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "display(\{i mg1,pntplot\});" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'POINTSG687$$\"#q\"\"!$\"%08F)7$$\"#rF)$\"%ZCF)7$$\"#sF)$\"%t(F)7$$\"##)F)$\"%*z'F)7$$\"#$)F)$\"%9\")F)7$$\"#%)F)$\"% ;')F)7$$\"#&)F)$\"%.*)F)7$$\"#')F)$\"%2%)F)7$$\"#()F)$\"%[$*F)7$$\"#)) F)$\"%+&)F)7$$\"#*)F)$\"%U&)F)-%'COLOURG6&%$RGBG$F)F)Fiq$\"*++++\"!\") -%'SYMBOLG6$%(DIAMONDG\"#?-%'CURVESG6$7S7$F'$\"3b)*********4AFir7$$\"3oL$37.y'*3(F]s$\"3SHH[*p'H4@Fir7$$ \"38mmTh8gOrF]s$\"3w*zztF]s$\"3f!)*y4`O=5$Fir7$$\"3+n;a25BmtF]s$\"3E9 ;Nj0N.LFir7$$\"3)***\\P$o`FT(F]s$\"3sTnmP$=U]$Fir7$$\"3_LL$Q(zgguF]s$ \"3w/xxF]s$\"3+x%z&R`5x]F ir7$$\"3y**\\()*y/f#yF]s$\"3E*=o>g_!)G&Fir7$$\"3)HLLVm^\"pyF]s$\"3oe-# GnwZZ&Fir7$$\"3s+]()R.g;zF]s$\"3)e>=FdU'zcFir7$$\"3E+]i*p#yhzF]s$\"3Y_ h)fNBZ(eFir7$$\"3%GL$3_d#*3!)F]s$\"3_=#3!)op#ygFir7$$\"3AL$32zNh#)Fir7$$\"3em;a84fd&)F]s$\"3iz*)Q:P?Z%)Fi r7$$\"3$)****\\$G]Yg)F]s$\"3_Xk?P()Q]')Fir7$$\"39L$3K[H*[')F]s$\"3^hw: 3+dT))Fir7$$\"3++]P0S@&p)F]s$\"3)\\&4oA&49/*Fir7$$\"3!GLLeel/u)F]s$\"3 &=8la<*yO#*Fir7$$\"3Q**\\(ozRyy)F]s$\"3D[j3lNLT%*Fir7$$\"3%ommEzmM$))F ]s$\"3S]*oH7M$Q'*Fir7$$\"3kmm\"f)p7!)))F]s$\"3$e$\\#ok&zR)*Fir7$$\"3*H L3#43SE*)F]s$\"35mZv'*)eR+\"!#87$$\"3_+++Z;#*o*)F]s$\"3nEg&z#yJA5F]`l7 $$\"30MLeD`l_\" *F]s$\"3#z\"*oV2]95\"F]`l7$$\"##*F)$\"3&3Uot%*)3A6F]`l-Ffq6&FhqFjqFiqF iq-%+AXESLABELSG6$Q\"t6\"Q!6\"-%%VIEWG6$;F'Fhal%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "No t too shabby!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "We can use our fitted line to \"predict\" the imports fo r 1992. This kind of extrapolation is open to crticism of course!" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "subs(t=92,rhs(eqn1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"(p>8 #\"$!>" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "That is excessively precise! Let's round it to an integer " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "round(subs(t=92,rhs(eqn1)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"&@7\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 98 "You can also just evaluate the fraction a s a floating point number with 4 or 5 digits of precision" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eval f(subs(t=92,rhs(eqn1)),5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&@7\" \"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "Actually given the data and the wholy unjustified extrap olation we should probably only use 4 digits" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "evalf(subs(t=92,r hs(eqn1)),4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%A6\"\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 151 "U ndoubtedly you are very unhappy about the terminating decimal point (w hich indicates a floating point number here). You can get rid of it by rounding!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "round(evalf(subs(t=92,rhs(eqn1)),4));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"&?7\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 87 "Now it is clear what to do if you wan t only 3 digits of precision and an integer result" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "round(evalf (subs(t=92,rhs(eqn1)),3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"&+7\" " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 445 "Note instead of assigning rhs(eqn1) to a variable (that is, a l abel) I keep using it over an over above. You may think this approach \+ is wasteful because Maple has to evaluate the expression each time. Th is is not the case. Maple remembers the values of expressions it has a lready evaluated, and simply reuses them. Thus you can free your mind \+ to play with the mathematics without having to worry about grubby lowe r order stuff like efficiency." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Co nsider now a least squares cubic polynomial fit to our data" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "e qn3:=leastsquare[\{t,y\},y=a*t^3+b*t^2+c*t+d,\{a,b,c,d\}]([Tdata,Ydata ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/%\"yG,**$)%\"tG\"\"$\" \"\"#!(&)eg#\"(U4Z\"*&#\"'5,;\"$\"RF,)F*\"\"#F,F,*&#\",jP+rb%F/F,F*F,! \"\"#\",FE#p5T\"&&H`F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "i mg3:=plot(rhs(eqn3),t=70..92,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "display(\{img3,pntplot\});" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'POINTSG687$$\"#q\"\"!$\"%08F)7 $$\"#rF)$\"%ZCF)7$$\"#sF)$\"%t(F)7$$\"##)F)$\"%*z'F )7$$\"#$)F)$\"%9\")F)7$$\"#%)F)$\"%;')F)7$$\"#&)F)$\"%.*)F)7$$\"#')F)$ \"%2%)F)7$$\"#()F)$\"%[$*F)7$$\"#))F)$\"%+&)F)7$$\"#*)F)$\"%U&)F)-%'CO LOURG6&%$RGBG$F)F)Fiq$\"*++++\"!\")-%'SYMBOLG6$%(DIAMONDG\"#?-%'CURVES G6$7S7$F'$\"35c@[EhD%\\\"!#97$$\"3ymm;RP&z/(!#;$\"3M&fuGuB\"\\;Fir7$$ \"3oL$37.y'*3(F]s$\"3Q3NM/m(pz\"Fir7$$\"38mmTh8gOrF]s$\"3w-')owF\"o(>F ir7$$\"3)fm;*Hd$Q=(F]s$\"3g'o>%)*)o7<#Fir7$$\"3+L$3%yd#Fir7$$\"3!***\\(Gle&>tF]s$ \"3+5J9w/xxF]s$\"3WU'fp\"\\%yH&Fir7$$\"3y**\\()*y/f#yF]s$\"3Y1M qhdbtbFir7$$\"3)HLLVm^\"pyF]s$\"3tDNR#=#3:eFir7$$\"3s+]()R.g;zF]s$\"3_ 1x:APGwgFir7$$\"3E+]i*p#yhzF]s$\"3N%ef*)p9.K'Fir7$$\"3%GL$3_d#*3!)F]s$ \"3?aI@La(*olFir7$$\"3AL$32z-\"z'Fir7$$\"3-mm\"H59*)4 )F]s$\"3b__b0XuAqFir7$$\"3Om;aZ%=u9)F]s$\"3Cn,;R>v`sFir7$$\"3s+]7A;k*= )F]s$\"3wm;wBa(eW(Fir7$$\"3/nmT2QCN#)F]s$\"3z*))p1oHIk(Fir7$$\"33***** pKbBG)F]s$\"3p;nR/OQMyFir7$$\"3N++vdZWG$)F]s$\"35?>HC\"3%3!)Fir7$$\"3_ **\\(=lQIP)F]s$\"3!\\'p\"[o#Rj\")Fir7$$\"3h++DoDbA%)F]s$\"3jcRBmp))=$) Fir7$$\"3#GLLV-VqY)F]s$\"3H()Q&4V)oU%)Fir7$$\"32++]#3YX^)F]s$\"33b!*y# 3Yrb)Fir7$$\"3em;a84fd&)F]s$\"3oW1qr&=Tk)Fir7$$\"3$)****\\$G]Yg)F]s$\" 3()Fir7$$\"39L$3K[H*[')F]s$\"3WUcB>j)>x)Fir7$$\"3++]P0S@&p)F] s$\"3#*Hq)Fir7$$\"3*HL3#43SE*)F]s$\"3N`(y+E&=8')Fi r7$$\"3_+++Z;#*o*)F]s$\"3'GPU_vth])Fir7$$\"30MLeD`l_\"*F]s$\"3S=k%)**zccxFir7$$\"##*F)$\"3WO Fq7%R(yuFir-Ffq6&FhqFjqFiqFiq-%+AXESLABELSG6$Q\"t6\"Q!6\"-%%VIEWG6$;F' Fgal%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 144 "Nice! Let's use our fitted cubic to \"pr edict\" the imports for 1992. This extrapolation is even more open to \+ criticism than the linear one above!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "round(evalf(subs(t=92,rhs (eqn3)),3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%![(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 277 "If you w ere in charge of formulating policy for Egypt based on predicted cerea l imports, you'd have a headache now. You'd certainly realize that eit her a solid realistic underlying model is needed, or you need to rely \+ on experience and judgment in conjunction with mathematics." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Let's l ook now at a natural cubic spline fit" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "spl:=spline(Tdata,Ydata ,t,cubic):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "I supressed the output here because it is rather long. I t is a piecewise cubic expression in the specified variable t." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "imgspl:=plot(spl,t=70..92,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "display(\{imgspl,pntplot\});" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'POINTSG687$$\"#q\"\"!$\"%08F)7 $$\"#rF)$\"%ZCF)7$$\"#sF)$\"%t(F)7$$\"##)F)$\"%*z'F )7$$\"#$)F)$\"%9\")F)7$$\"#%)F)$\"%;')F)7$$\"#&)F)$\"%.*)F)7$$\"#')F)$ \"%2%)F)7$$\"#()F)$\"%[$*F)7$$\"#))F)$\"%+&)F)7$$\"#*)F)$\"%U&)F)-%'CO LOURG6&%$RGBG$F)F)Fiq$\"*++++\"!\")-%'SYMBOLG6$%(DIAMONDG\"#?-%'CURVES G6$7^u7$F'$\"3O&RS******\\I\"!#97$$\"3ymm;RP&z/(!#;$\"3;va8%*=BO?Fir7$ $\"3A+v=&)e\")oqF]s$\"3'R!\\%[+,5F#Fir7$$\"3oL$37.y'*3(F]s$\"3J@w[ajr\">Fir7$$\"3[*\\7emSt?(F]s$\"3]\\GL2AN:8a\"Fir7$$\"3%y;Hsq&3esF]s$\"3=;/g$pwza\"Fir7$$ \"3CMLLGP`jsF]s$\"3+6'GU[7>c\"Fir7$$\"3k+vV\\<)*osF]s$\"3g)yjG&3_$e\"F ir7$$\"3/n;aq(HWF(F]s$\"3vkj3:e>8;Fir7$$\"3YL$3<@%*pH(F]s$\"3lmX-!*\\e I=Fir7$$\"3!***\\(Gle&>tF]s$\"3bGAXa_P4AFir7$$\"3;M$3-$[*GM(F]s$\"3p4d r*4`Jr#Fir7$$\"3+n;a25BmtF]s$\"35$R$e$z0LC$Fir7$$\"3]L$eaM#\\*Q(F]s$\" 3sE$e]'R'*4PFir7$$\"3)***\\P$o`FT(F]s$\"3#RODlm54.%Fir7$$\"3,$e*)fDZ8O])48%Fir7$$\"3YnTgG3oOuF]s$\"3'*oyI^[;&>%Fir7$$\"3Fe9\" \\hiEW(F]s$\"3#=09\"3=$f@%Fir7$$\"3]](=7SW'[uF]s$\"3([&zsM,HIUFir7$$\" 3sUg_(=EYX(F]s$\"3Z)[3U$[4RUFir7$$\"3_LL$Q(zgguF]s$\"3xRlJ44?VUFir7$$ \"3!)**\\P;Io\"[(F]s$\"3\"oQD+8rFB%Fir7$$\"32mm\"*e!eF](F]s$\"3H*Q+=.3 ?@%Fir7$$\"3/K$3_c$[EvF]s$\"3#Gcr(R/()4UFir7$$\"3W****\\r!4-b(F]s$\"3^ 0omg*yCB%Fir7$$\"3;+++(*>.uvF]s$\"3_eG8>H/xUFir7$$\"3[****\\A\\&yf(F]s $\"31%o\"*[wn/M%Fir7$$\"3G****\\G0xVwF]s$\"3zuow'[_)>XFir7$$\"3)emTvHm ao(F]s$\"3U-#)*[0Y%*z%Fir7$$\"3TLL$)*fY]t(F]s$\"3(Rcl()RR4L&Fir7$$\"3e LLe4r/cxF]s$\"3Wj(Q'e.%Rc&Fir7$$\"3uLLL>w/xxF]s$\"3c)HpFV'p[dFir7$$\"3 MU5llZ:$y(F]s$\"3-6%*f(p*3)y&Fir7$$\"3'4vo>\">E*y(F]s$\"3zNoms3Q>eFir7 $$\"3cfkGe!p`z(F]s$\"3N?u?HQmTeFir7$$\"3vmTg/iZ,yF]s$\"3-uvLyr.aeFir7$ $\"3%R(=#4N$e2yF]s$\"3-.fCFJAceFir7$$\"3c#eRs\\!p8yF]s$\"3-VJK5H=\\eFi r7$$\"3<\"HdNk(z>yF]s$\"3k***e>4CS$eFir7$$\"3y**\\()*y/f#yF]s$\"3_dkbP U&=\"eFir7$$\"3QmT5F#Gv%yF]s$\"3qI>-/FW$p&Fir7$$\"3)HLLVm^\"pyF]s$\"3i *>8#pZ=]bFir7$$\"3i](=K$Q,\")yF]s$\"3avZ[wCyyaFir7$$\"3%o;/@+wG*yF]s$ \"3OH59XP\"GU&Fir7$$\"3mvoa'32))*yF]s$\"3!yry4C`JS&Fir7$$\"32$e*)4)4``S&Fir7$$\"3s]7yCPaLzF]s$\"3)R%)*fV50CaFir7$ $\"3x***\\(>:>RzF]s$\"39)p1S'QN[aFir7$$\"3s+vo4r[]zF]s$\"3rHs&yT4K^&Fi r7$$\"3E+]i*p#yhzF]s$\"3#4az`L!z)f&Fir7$$\"3cmT&eAa`)zF]s$\"3*H0l&\\a< ReFir7$$\"3%GL$3_d#*3!)F]s$\"3E!)GM*yRT:'Fir7$$\"3AL$32z \"Q)oIpz'Fir7$$\"3M+D\"o%fcv!)F]s$\"3C&\\z#yI%\\1(Fir7$$\"3-mm\"H59*)4 )F]s$\"3%)yEHa]>(>(Fir7$$\"3&>z%*f9x\\5)F]s$\"3'**RL,Pi=?(Fir7$$\"3Y;H 2*=S56)F]s$\"3gdoyXlP%>(Fir7$$\"3(4/^@B.r6)F]s$\"3aden(Gvk<(Fir7$$\"3! p;H_FmJ7)F]s$\"3_4j]@!**)\\rFir7$$\"3#fT&QhBHN\")F]s$\"3?w\\$[;$oxqFir 7$$\"3Om;aZ%=u9)F]s$\"3W_7AvBl\"*pFir7$$\"3I+v=TU(z:)F]s$\"3*o=d-Khj\" pFir7$$\"3#GLL[.I&o\")F]s$\"3)*eHGrpL]oFir7$$\"3]]ilJz!Q<)F]s$\"3YkgYr *pO#oFir7$$\"3ym\"z%Ge3z\")F]s$\"3pz;sRQw-oFir7$$\"3/$3-`sjV=)F]s$\"3o PRaSmw)y'Fir7$$\"3s+]7A;k*=)F]s$\"3QT!enVEGy'Fir7$$\"37ekG&*=M&>)F]s$ \"33AwowZy'y'Fir7$$\"3%p\"zWo@/,#)F]s$\"3e!4uz1CH!oFir7$$\"3wv$4;WUn?) 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Again we can make a predici ton for 1992" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "round(evalf(subs(t=92,spl),3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!$/*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 157 "Wow! We have a solid (?) prediciton that Egypt will export cereal in 1992. From the Egyptian point of view thi s prediction is much nice than the ones above. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 151 "Perhaps by now you are n ot prepared to accept anything but a piecewise linear relationship - o ne that makes no assumptions about the data. Let' try it." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "lin: =spline(Tdata,Ydata,t,linear):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "imglin:=plot(lin,t=70..92,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "display(\{imglin,pntplot\});" }}{PARA 13 "" 1 " " {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'POINTSG687$$\"#q\"\"!$\"%08F )7$$\"#rF)$\"%ZCF)7$$\"#sF)$\"%t(F)7$$\"##)F)$\"%*z' F)7$$\"#$)F)$\"%9\")F)7$$\"#%)F)$\"%;')F)7$$\"#&)F)$\"%.*)F)7$$\"#')F) $\"%2%)F)7$$\"#()F)$\"%[$*F)7$$\"#))F)$\"%+&)F)7$$\"#*)F)$\"%U&)F)-%'C OLOURG6&%$RGBG$F)F)Fiq$\"*++++\"!\")-%'SYMBOLG6$%(DIAMONDG\"#?-%'CURVE SG6$7_rF&7$$\"3ymm;RP&z/(!#;$\"3]JLG,)=F \\s7$$\"3qC1*Q'>r*=(Fir$\"3Z#>xLOTB%=F\\s7$$\"3W$eky>)e&>(Fir$\"3y3FRY bt-=F\\s7$$\"3sFir$\"3c6GU-A!>z\"F\\s7$$\"3+L$3r&G(Fir$\"3#3v87([&y&=F\\s7$$\"3 YL$3<@%*pH(Fir$\"3C]7fpU-p=F\\s7$$\"3(fm\"HKkF3tFir$\"3;%*yWnC%z.#F\\s 7$$\"3!***\\(Gle&>tFir$\"3YQP9!*4:kAF\\s7$$\"3;M$3-$[*GM(Fir$\"3r0rnX8 /KFF\\s7$$\"3+n;a25BmtFir$\"3w9/@,<$**>$F\\s7$$\"3]L$eaM#\\*Q(Fir$\"3a )fRk_@jm$F\\s7$$\"3)***\\P$o`FT(Fir$\"3?^P(H\"*z*>RF\\s7$$\"3YnTgG3oOu Fir$\"3UWgR#R91+%F\\s7$$\"3_LL$Q(zgguFir$\"3-M$==()[73%F\\s7$$\"3!)** \\P;Io\"[(Fir$\"3O[P=l.uvFir$\"3R***4gbiCJ%F\\s7$$\"3[****\\A\\&yf (Fir$\"3#)**\\#p/ZTM%F\\s7$$\"3Q****\\DF\"3i(Fir$\"3I&**Rf#)y$pWF\\s7$ $\"3G****\\G0xVwFir$\"3a(**zvqqVg%F\\s7$$\"3)emTvHmao(Fir$\"3m#*\\aH=a \\[F\\s7$$\"3TLL$)*fY]t(Fir$\"3%3nr/KxjD&F\\s7$$\"3eLLe4r/cxFir$\"3Er \"z[*>&*[aF\\s7$$\"3uLLL>w/xxFir$\"3ormGpm_TcF\\s7$$\"3'4vo>\">E*y(Fir $\"33VMvK<``dF\\s7$$\"3vmTg/iZ,yFir$\"3%Q;*=bvKXeF\\s7$$\"3c#eRs\\!p8y Fir$\"3!*)3xC&*=,z&F\\s7$$\"3y**\\()*y/f#yFir$\"3T**\\w\\.\"\\t&F\\s7$ $\"3QmT5F#Gv%yFir$\"39q\"*[8CR0aF\\s7$$\"32$e*)4:>RzFir$\"3G+D$)G3tXcF\\s7$$\"3E+]i *p#yhzFir$\"31]([m_xty&F\\s7$$\"3%GL$3_d#*3!)Fir$\"3\"3m;W\")4;8'F\\s7 $$\"3AL$32z)Fir$\"3PpT&)=Cj)yF\\s7$$\"3N++vdZWG$)Fir $\"370]!Ro#zc#)F\\s7$$\"3_**\\(=lQIP)Fir$\"3U%\\7C.a1[)F\\s7$$\"3h++Do DbA%)Fir$\"3s+v(3(es!o)F\\s7$$\"3sm;H'z(zW%)Fir$\"3k%3d`npXu)F\\s7$$\" 3#GLLV-VqY)Fir$\"3#\\mO)zMT3))F\\s7$$\"3G**\\())y=*y%)Fir$\"3%*[72Cp\\ U))F\\s7$$\"3ulmT`Xz!\\)Fir$\"3KHeIo.ew))F\\s7$$\"3o*\\(oNCt'\\)Fir$\" 3;BJU!4AO*))F\\s7$$\"3hL$ezJqE])Fir$\"3ulm#HAb(*)))F\\s7$$\"3bn\"H-?3' 3&)Fir$\"3')HL'oK..'))F\\s7$$\"32++]#3YX^)Fir$\"3E,+!3V^3$))F\\s7$$\"3 KL3-)\\og`)Fir$\"3djm\")\\-5C()F\\s7$$\"3em;a84fd&)Fir$\"3:LL$)o!\\th) F\\s7$$\"3))\\7.cdNp&)Fir$\"3!***\\3]a**e&)F\\s7$$\"3?L3_)f?6e)Fir$\"3 mmmLJ=k+&)F\\s7$$\"3'[il(>I+(e)Fir$\"3o.D'>-l9Z)F\\s7$$\"3_;/,Ta)Gf)Fi r$\"3TL$)e7#)GU%)F\\s7$$\"3M7Gj^m#ef)Fir$\"3G[7!z!)*pF%)F\\s7$$\"3=3_D iyw)f)Fir$\"3UqT@.968%)F\\s7$$\"3+/w(G24&3*F\\s7$$\"3++]P0S@&p)Fir$\"36`(y0zjHI*F\\s7$$\"3SmTg&zRyr) Fir$\"3enmFL&=n>*F\\s7$$\"3!GLLeel/u)Fir$\"3\\RL8K=&[+*F\\s7$$\"3!o;a8 p_Tw)Fir$\"3DfmJx^)R!))F\\s7$$\"3Q**\\(ozRyy)Fir$\"363+]A&=Jg)F\\s7$$ \"3!y&*[8$3'p!Rva&)F\\s7$$\"3C;H#eaY#*z)Fir$\"3Nq;U\"H*Q 1&)F\\s7$$\"3nuoH?*\\\\!))Fir$\"3W)oCl'*y?])F\\s7$$\"35L3x%H`1\"))Fir$ \"3C]P!QQuW])F\\s7$$\"3(*\\(=P/g?#))Fir$\"3&Q(=O=_E4&)F\\s7$$\"3%ommEz mM$))Fir$\"34,+#H0cS^)F\\s7$$\"3um;H*)ozc))Fir$\"3g+D]$paQ_)F\\s7$$\"3 kmm\"f)p7!)))Fir$\"35+]3MLlL&)F\\s7$$\"3*HL3#43SE*)Fir$\"3g*\\n)R$)3`& )F\\s7$$\"3_+++Z;#*o*)Fir$\"3Y***R<4Z4d)F\\s7$$\"30MLeD`l7CH')F\\s7$$\"3]+]7C')>_\"*Fir$\"3]+D8AM#zk)F\\s7$$\"##* F)$\"%o')F)-Ffq6&FhqFjqFiqFiq-%+AXESLABELSG6$Q\"t6\"Q!6\"-%%VIEWG6$;F' F^fm%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 23 "What about predicition?" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "round(eva lf(subs(t=92,lin),3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%q')" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 293 "P retty reasonable perhaps. Of course one can argue that this prediction makes use only of the last two data points and no use whatsoever of t he earlier history. That may, or may not, be reasonable. We're back to judgment! There's no way around it - simple calculation will not repl ace thought." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 10 "Problem 1:" }{TEXT 262 1 " " }{TEXT 261 0 "" }{TEXT -1 277 "Co nstruct the least squares polynomial of degree 2. Make the 1992 predic tion again. Note if you do this by hand, you will need to solve 3 li near equations in 3 unknowns. Do you think you may be skeptical in t he future about predictions based on trends (rather than models)." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "As our final example let's try an interpolation polynomial" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "poly :=interp(Tdata,Ydata,t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "imgpoly:=plot(poly,t=70..92,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "display(\{imgpoly,pntplot\});" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'POINTSG687$$\"#q\"\"!$\"%08F)7 $$\"#rF)$\"%ZCF)7$$\"#sF)$\"%t(F)7$$\"##)F)$\"%*z'F )7$$\"#$)F)$\"%9\")F)7$$\"#%)F)$\"%;')F)7$$\"#&)F)$\"%.*)F)7$$\"#')F)$ \"%2%)F)7$$\"#()F)$\"%[$*F)7$$\"#))F)$\"%+&)F)7$$\"#*)F)$\"%U&)F)-%'CO LOURG6&%$RGBG$F)F)Fiq$\"*++++\"!\")-%'SYMBOLG6$%(DIAMONDG\"#?-%'CURVES G6$7^`o7$F'$!3+?RFGv!4o'!\"$7$$\"3pm;zM%))>,(!#;$\"3++?b^0*R\\'!\"%7$$ \"3RLLepo(R-(F]s$\"3+?^OBA0jRFir7$$\"33#HKRUva-(F]s$\"3+!ovJehyX*Fir7$ $\"3x]7GyR(p-(F]s$\"3+!3m:0$zvjFir7$$\"3Y4-jKDZGqF]s$\"3+/VZhAX*G\"!\" #7$$\"3tm\"zp3r*HqF]s$\"3+gtG=pcjEFir7$$\"3+C\"G8kp9.(F]s$!3+g4W'35+X' Fir7$$\"3p#3xc>oH.(F]s$!3+!)[Zg5o)z%Fir7$$\"3QTg-]nYMqF]s$!3+SE>-V+;6F ir7$$\"33+]P/`'f.(F]s$\"3+gh^N@zK$*Fir7$$\"3UL3x@&f>/(F]s$\"3+!oBG%3Qv ;Fir7$$\"3ymm;RP&z/(F]s$!3++#z(*GD.r(Fir7$$\"3[v=nv#pJ0(F]s$\"3+gdc[yt yAFir7$$\"3?%3x@\"[QeqF]s$\"3+!3=jw4QV*Fir7$$\"3]\"H#o[.gjqF]s$\"3++#z *[(=$p&)F`s7$$\"3A+v=&)e\")oqF]s$!3++k39zcC8Fet7$$\"3'Q5SMlB92(F]s$!3+ +kW,=Z!Q(F`s7$$\"3%*3Fp@9.uqF]s$\"3+?v]en^\"z\"Fir7$$\"315!>eIN`2(F]s$ !3+gpTt^^3%)Fir7$$\"3e7`%**=Rm2(F]s$!3+k)o3kNK+\"Fet7$$\"37:;2uI%z2(F] s$!3+gDK;ob:XFir7$$\"3C;z>epCzqF]s$\"3++)[z!Q;,;F`s7$$\"3w=UKU3b!3(F]s $\"3++s-reFZ*)Fir7$$\"3I@0XEZ&=3(F]s$!3+o^txVGU9Fet7$$\"3UAod5'eJ3(F]s $!3+G,_(pC_H\"Fet7$$\"3'\\7.Z\\iW3(F]s$!3+Si#3p=?o)Fir7$$\"3gGd&HEqq3( F]s$!3+?ncl)o*zfFir7$$\"3oL$37.y'*3(F]s$!3+S!R+VGXi\"Fir7$$\"3]WtF&QW6 4(F]s$\"3+?.73![>/&Fir7$$\"3MbjMR2h#4(F]s$!3+g`te4^fZFir7$$\"3uk`T$4xS 4(F]s$!3+gT3*Q%p$y$Fir7$$\"3evV[ZMa&4(F]s$\"3+!G@\"H<@[WFir7$$\"3#eRAc :w%)4(F]s$\"3+gFir7$$\"3\\&e(Qv6(F]s$!3+!)3,E2*e:'Fir7$$\"3\"=a)e7^+>rF]s$!3+S=t\")RaOp Fir7$$\"3A^vlm9Z?rF]s$!3+gxB07i<(*Fir7$$\"30ils?y$>7(F]s$!3+![3V%)GDT \"Fet7$$\"3*Gd&zuTSBrF]s$\"3++S%H?X)pXF`s7$$\"3I#ek)G0([7(F]s$!3++++'H n\\H%Fir7$$\"38$fLH)oLErF]s$\"3+S-8@/k,KFir7$$\"3(Rg-qB.y7(F]s$!3++WL: wAIzFir7$$\"3Q8;2\"fp#HrF]s$\"3+g$REO!3^hFir7$$\"3@C19XftIrF]s$\"3+g8* 4^EOM&Fir7$$\"30N'4#*H-A8(F]s$\"3++S]HM.;hF`s7$$\"3YW'yKloO8(F]s$!3+!G &=$z)\\?NFir7$$\"3HbwM2]8NrF]s$\"3++GpZG)pg*F`s7$$\"38mmTh8gOrF]s$\"3+ SagSd#\\l&Fir7$$\"3Sl\"zul0D9(F]s$!3+g\"oejayP'Fir7$$\"35m;a`*4%[rF]s$ !3+#z$>c4,#p\"Fet7$$\"3K#Hdv-'))\\rF]s$\"3++S56l/)R%F`s7$$\"39a)ev\"QG:(F]s$!3++SYE\\sKsFir7$$\"3ymTg\\U JarF]s$\"3++WdDK,?)*F`s7$$\"3g\"z>OK!zbrF]s$!3+W(>&zCA.6Fet7$$\"3U;aj( Rms:(F]s$\"3+?n)e=/9M\"Fir7$$\"3CT5lrCuerF]s$\"3+?>K]Y')eqFir7$$\"31mm mX&=-;(F]s$!3+?6Uo1pS:Fir7$$\"3Kl\"H<%G7mrF]s$!3+?&zj#*4>%zFir7$$\"3-m ;zPr-srF]s$!3+)3$>J3#zp\"Fet7$$\"3qmT&QVJz<(F]s$!3+?BZ_9YH&)Fir7$$\"3) fm;*Hd$Q=(F]s$!3+ghzc;&f^\"Fir7$$\"3+L$3wE(F]s$!3+Sm`Bk$QS(Fir7$$\"3k+vV\\<)*osF]s$\"3+Se6e4?g**Fi r7$$\"3CUNraPMqsF]s$\"3+;K^X-Tg5Fet7$$\"3%Qe*)*fdqrsF]s$\"3+!oz=Bqap&F ir7$$\"3WDcElx1tsF]s$\"3+s%p\")[W4q\"Fet7$$\"3/n;aq(HWF(F]s$!3+3ay'\\S Be#Fet7$$\"3YL$3<@%*pH(F]s$!3+SEblj;)4'Fir7$$\"3!***\\(Gle&>tF]s$\"3+S ]$\\@\\9_\"Fir7$$\"3.n;aTnAJtF]s$!3++W\"QCk)>JFir7$$\"3;M$3-$[*GM(F]s$ !3+kG,$=lL1\"Fet7$$\"3\"zm\"HTLNWtF]s$\"3+?>3S!z!p^Fir7$$\"3m,]P_=\"eM (F]s$!3+/Ju&o=qa#Fet7$$\"3SN$eMOqsM(F]s$!3+S1?lQ5`!*Fir7$$\"3sn;au)G([ tF]s$\"3+g`lcm#zp#Fir7$$\"3!QL3n*ek^tF]s$\"3+?J@QA`M8Fir7$$\"3I,]()=Hc atF]s$!3+g(p?)R!)*4%Fir7$$\"3On;/T*zuN(F]s$\"3+g4o+%G[/%Fir7$$\"3WL$3K 'pRgtF]s$\"3++kS]'z'\\jFir7$$\"3%4+v`)RJjtF]s$\"3++S!zM?S$y!\"&7$$\"3+ n;a25BmtF]s$!3+39!)Q/nw7Fet7$$\"3'4++lnhyP(F]s$!3+g<#3S`G'GFir7$$\"3]L $eaM#\\*Q(F]s$\"3+?ncl)o*zfFir7$$\"3i;zp7+S#R(F]s$!3+SU2!oM!H8Fir7$$\" 3w*\\P*zwI&R(F]s$!3+/2!QJ!e#G\"Fet7$$\"3K\"HdN^hnR(F]s$\"3+K+Mrw\\S9Fe t7$$\"3!H3xrM:#)R(F]s$!3+?n[f=N8#)Fir7$$\"3Yuoz!=p'*R(F]s$\"3++SuV<&3r $F`s7$$\"3-mmT9I7,uF]s$\"3+))y*HF8_;#Fet7$$\"3fdk.[od-uF]s$!3+Se^%G`#) 3$Fir7$$\"3;\\il\"oISS(F]s$!3+_*HolxAB\"Fet7$$\"3sSgF:X[0uF]s$\"3+GDU= 4()G:Fet7$$\"3HKe*)[$QpS(F]s$!3+!3ePCaEf$Fir7$$\"3'Qi:D=#R3uF]s$\"3+!) )e/Zno&RFir7$$\"3V:a8;g%)4uF]s$\"3+gLaA))>+6Fet7$$\"3*p?b(\\)*H6uF]s$ \"3+KkUhPK\"p\"Fet7$$\"3)***\\P$o`FT(F]s$!3+gP$pe(48BFir7$$\"35C=&*H\" \\UT(F]s$!3+?Va\">YRl#Fir7$$\"3!okGldWdT(F]s$\"3+O&49wgJ$=Fet7$$\"3^pa 5B+C(ej#Q\"Fet7$$\"3,$e*)fD)[ G=^\"F`s7$$\"37^+s&f.#HuF]s$\"3+!ofbhn!RgFir7$$\"3CvoHU!*pIuF]s$\"3+!3 1m!zh6JFet7$$\"3O*pt))[%>KuF]s$!3++;'f(>eplFir7$$\"31A0XN**oLuF]s$!3+? 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\\ZtQ&G6Ff\\v7$$\"3[Wt-\\Ma*=*F]s$\"3+3A!>Baj]&Fet7$$\"37QM-Us.\">*F]s $\"3+3!**=;va[(Fet7$$\"3yJ&>].JD>*F]sF_bn7$$\"3UDc,G[-%>*F]sF_g\\l7$$ \"31><,@'=b>*F]s$\"3+Kc'4Wj=^(Fet7$$\"3s7y+9C,(>*F]s$!3+W8%)y#[v\">Ff \\v7$$\"3O1R+2i])>*F]s$\"3+WXG=k?(o#Ff\\v7$$\"##*F)Fa]s-Ffq6&FhqFjqFiq Fiq-%+AXESLABELSG6$Q\"t6\"Q!6\"-%%VIEWG6$;F'Fdh`l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }} }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "W ow!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 10 "P roblem 2:" }{TEXT -1 70 " What happened? A polynomial of degree 19 can not oscillate that much." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 253 "It may help you to know that Maple uses floating \+ point to calculate the points to plot. The default precision is 10 dec imal digits. Let's see what happens if we temporarily increase the pre cision to 30 decimal digits before computing the points to plot." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "Digits:=30:imgpoly:=plot(poly,t=70..92,color=red):Digits:=10:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "display(\{imgpoly,pntplot \});" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'POINT SG687$$\"#q\"\"!$\"%08F)7$$\"#rF)$\"%ZCF)7$$\"#sF)$\"%t(F)7$$\"##)F)$\"%*z'F)7$$\"#$)F)$\"%9\")F)7$$\"#%)F)$\"%;')F)7$$\"# &)F)$\"%.*)F)7$$\"#')F)$\"%2%)F)7$$\"#()F)$\"%[$*F)7$$\"#))F)$\"%+&)F) 7$$\"#*)F)$\"%U&)F)-%'COLOURG6&%$RGBG$F)F)Fiq$\"*++++\"!\")-%'SYMBOLG6 $%(DIAMONDG\"#?-%'CURVESG6$7\\oF&7$$\"?nmmmmmmmm;RP&z/(!#G$!)\"HWv%!\" $7$$\"?LLLLLLLL$37.y'*3(Fir$!(X,$QF\\s7$$\"?nmmmmmmmmTh8gOrFir$\"(cZw* F\\s7$$\"?nmmmmmmmm\"*Hd$Q=(Fir$\"()yvPF\\s7$$\"?LLLLLLLL$3tFir $\"(iTj#F\\s7$$\"?nmmmmmmm;a25BmtFir$\"(vYp$F\\s7$$\"?++++++++]P$o`FT( Fir$\"(5V\"RF\\s7$$\"?LLLLLLLLL$Q(zgguFir$\"(a.,%F\\s7$$\"?nmmmmmmmm\" *e!eF](Fir$\"'WF_v7$$\"?nmmmmm mm;a(Hmao(Fir$\"'d-ZF_v7$$\"?LLLLLLLLL$)*fY]t(Fir$\"'VBaF_v7$$\"?LLLLL LLLLL>w/xxFir$\"';ldF_v7$$\"?++++++++]()*y/f#yFir$\"'b!)eF_v7$$\"?LLLL LLLLLLk;:pyFir$\"'nPbF_v7$$\"?++++++++]()R.g;zFir$\"'K%Q&F_v7$$\"?++++ ++++]i*p#yhzFir$\"')o[&F_v7$$\"?LLLLLLLLL3_d#*3!)Fir$\"'8biF_v7$$\"?LL LLLLLL$32z&[DF^]l7$$\"?nmmmmmmmmm3LCh!*Fir $\"+C@gVZF^]l7$$\"?MLLLLLLL3F\\-[%3*Fir$\"+qA9I))F^]l7$$\"?++++++++]() *=_\"*Fir$\",X*[!))\\%F^]l7$$\"?+++++++v$ 4hzt\"e\"*Fir$\",4LF#R^F^]l7$$\"?+++++++]P4o*[T;*Fir$\",\\Z)>ieF^]l7$$ \"?+++++++D\"y+9C,<*Fir$\",v)\\NxmF^]l7$$\"?++++++++D17$*4w\"*Fir$\",5 W_`f(F^]l7$$\"?+++++++vo/%[u?=*Fir$\",jWhzi)F^]l7$$\"?+++++++]7.c'\\!) =*Fir$\",6yt\")y*F^]l7$$\"?++++++]PM-Us.\">*Fir$\"-@[`0U5F^]l7$$\"?+++ ++++Dc,G[-%>*Fir$\"-gB=.46F^]l7$$\"?++++++]7y+9C,(>*Fir$\"-'\\x[*z6F^] l7$$\"##*F)$\",]C=]D\"F)-Ffq6&FhqFjqFiqFiq-%+AXESLABELSG6$Q\"t6\"Q!6\" -%%VIEWG6$;F'Fafl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "What about the predicit ed cereal import for 1992?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "round(evalf(subs(t=92,poly),3));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\",++++E\"" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Hmm. That is a lot of c ereal." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Let's take a look at the interpolation polynomial, but limit the prec ision." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "poly2:=evalf(sort(poly),10);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&poly2G,J*$)%\"tG\"#>\"\"\"$\"+YeVf7!#>*&$\"+4hW&)=!# ;F*)F(\"#=F*!\"\"*&$\"+!=AlL\"!#8F*)F(\"#sDFLF*)F(FjnF*F**&$\"+G%*eaNFFF*)F(\"\"#F*F4*&$\"+\"G];4 $F@F*F(F*F*$\"+67Ct7F:F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 234 "Here the sort() command arranges the t erms in increasing order by degree. The evalf( ,8) command converts t he coefficients to floating point using 8 decimal digits of precision. Without it we would get an immense exact expression.\n" }}{PARA 0 "" 0 "" {TEXT -1 92 "We have only changed the coefficients a little bit. \+ What happens to our prediction for 1992?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "round(evalf(subs(t=9 2,poly2),3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!>+++++++++++++`%" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 179 " Hmm. We went form a large cereal import to exporting far more cereal t han the entire world has produced throughout history, simply by limit ing the precision of the coefficients. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 10 "Problem 3:" }{TEXT -1 345 " How do \+ you explain that? What is happening? (Mathematics is magic will not do as an explanation!) Has your skeptism increased since problem 1 above ? NASA predicts the positions of its space craft with breath-taking ac curacy over immense distances. How does that kind of prediction differ from the ones discussed here? (Does Newton ring a bell?)" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "I had fun making \+ up this assignment. I hope you enjoy doing it. I especially hope that \+ you will be skeptical of all prediction." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }