{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Helvetica" 1 9 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 1 10 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{PSTYLE " Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "R 3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 0 12 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 0 10 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 3" -1 258 1 {CSTYLE "" -1 -1 "Courier" 0 14 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 4" -1 259 1 {CSTYLE "" -1 -1 "Courier" 0 14 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 5" -1 260 1 {CSTYLE "" -1 -1 "Courier" 0 14 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 6 " -1 261 1 {CSTYLE "" -1 -1 "Courier" 0 14 0 0 0 0 2 2 2 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 7" -1 262 1 {CSTYLE "" -1 -1 "Courier" 0 14 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 8" -1 263 1 {CSTYLE "" -1 -1 "Courier" 0 14 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 26 "Problem 3: Data Regressio n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 207 "The problem at hand is the fitting of experimental data \+ to equations used to represent the vapor pressure. We begin with a lis t of the temperatures (in Celcius) at which the vapor pressure has bee n measured." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "T(C) := [-36 .7,-19.6,-11.5,-2.6,7.6,15.4,26.1,42.2,60.6,80.1];" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>-%\"TG6#%\"CG7,$!$n$!\"\"$!$'>F+$!$:\"F+$!#EF+$\"#wF +$\"$a\"F+$\"$h#F+$\"$A%F+$\"$1'F+$\"$,)F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "Note the perhaps unconventional symbol used to denote th e list of temperatures. As far as Maple is concerned " }{XPPEDIT 18 0 "T(C)" "-%\"TG6#%\"CG" }{TEXT -1 26 " here is a name just like " } {XPPEDIT 18 0 "T" "I\"TG6\"" }{TEXT -1 166 " would be, and can be assi gned to any valid Maple object; in this case a list of temperature val ues. The corresponding values of the vapor pressure (in mm Hg) is next " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "P(mmHg) := [1,5,10,20,4 0,60,100,200,400,760];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"PG6#%%m mHgG7,\"\"\"\"\"&\"#5\"#?\"#S\"#g\"$+\"\"$+#\"$+%\"$g(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "A quick check to see that the number of v alues of temperature and pressure are equal." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "nops(P(mmHg)), nops(T(C));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#5F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "We plot \+ the data to see its general form" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "datapoints := [seq([T(C)[k],P(mmHg)[k]],k=1..nops(T(C )))];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+datapointsG7,7$$!$n$!\"\" \"\"\"7$$!$'>F)\"\"&7$$!$:\"F)\"#57$$!#EF)\"#?7$$\"#wF)\"#S7$$\"$a\"F) \"#g7$$\"$h#F)\"$+\"7$$\"$A%F)\"$+#7$$\"$1'F)\"$+%7$$\"$,)F)\"$g(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "dataplot:=plot(datapoints,T= -60..100,style=point,symbol=box): dataplot;" }}{PARA 13 "" 1 "" {INLPLOT "6'-%'CURVESG6$7,7$$!1++++++qO!#9$\"\"\"\"\"!7$$!1++++++g>F*$ \"\"&F-7$$!1++++++]6F*$\"#5F-7$$!1+++++++E!#:$\"#?F-7$$\"1+++++++wF;$ \"#SF-7$$\"1++++++S:F*$\"#gF-7$$\"1++++++5EF*$\"$+\"F-7$$\"1++++++?UF* $\"$+#F-7$$\"1++++++ggF*$\"$+%F-7$$\"1************4!)F*$\"$g(F--%'COLO URG6&%$RGBG$F7!\"\"F-F--%'SYMBOLG6#%$BOXG-%+AXESLABELSG6$%\"TG%!G-%&ST YLEG6#%&POINTG-%%VIEWG6$;$!#gF-FK%(DEFAULTG" 2 217 188 188 5 3 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10020 10061 10055 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 196 117 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "p2:=logplot([seq([Tlist[k],Plist[k] ],k=1..nops(Tlist))],style=point,symbol=box):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "We continue by attempting to fit simple polynomials t o the data. We define a function to give us a polynomial in " } {XPPEDIT 18 0 "T" "I\"TG6\"" }{TEXT -1 15 " of any order. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "p := n->sum(a[k]*T^k,k=0..n);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG:6#%\"nG6\"6$%)operatorG%&arrowG F(-%$sumG6$*&&%\"aG6#%\"kG\"\"\")%\"TGF3F4/F3;\"\"!9$F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "A quadratic, for example, is given by" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "p(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(&%\"aG6#\"\"!\"\"\"*&&F%6#F(F(%\"TGF(F(*&&F%6#\"\"#F( F,F0F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "and a cubic is" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "p(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*&%\"aG6#\"\"!\"\"\"*&&F%6#F(F(%\"TGF(F(*&&F%6#\"\"#F( F,F0F(*&&F%6#\"\"$F(F,F4F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "Ma ple includes a least squares fitting procedure in its statistical pack age. The package is loaded as follows" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(stats):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 " We fit a quadratic to the data as follows" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "n:=2; expr := P=p(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG/%\"PG,(& %\"aG6#\"\"!\"\"\"*&&F)6#F,F,%\"TGF,F,*&&F)6#\"\"#F,F0F4F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "eqn[n]:=fit[leastsquare[[T,P], expr , \{seq(a[k],k=0..n)\}]]([T(C),P(mmHg)]);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "n:=2; expr := P=p(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG/%\"PG,(& %\"aG6#\"\"!\"\"\"*&&F)6#F,F,%\"TGF,F,*&&F)6#\"\"#F,F0F4F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "eqn[n]:=fit[leastsquare[[T,P], expr , \{seq(a[k],k=0..n)\}]]([T(C),P(mmHg)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$eqnG6#\"\"#/%\"PG,($!+6ak?e!#5\"\"\"%\"TG$\"+_z9n?! \"**$F/F'$\"+`qD:')!#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "We can \+ fit higher order polynomials just as easily" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "n:=3; expr := P=p(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG /%\"PG,*&%\"aG6#\"\"!\"\"\"*&&F)6#F,F,%\"TGF,F,*&&F)6#\"\"#F,F0F4F,*&& F)6#\"\"$F,F0F8F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "eqn[n] :=fit[leastsquare[[T,P], expr, \{seq(a[k],k=0..n)\}]]([T(C),P(mmHg)]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$eqnG6#\"\"$/%\"PG,*$\"+(*e$fW# !\")\"\"\"%\"TG$\"+v$*4)>\"!\"**$F/\"\"#$\"+ns![%R!#6*$F/F'$\"+vB6\\u! #8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "n:=4; expr := P=p(n); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG/%\"PG,,&%\"aG6#\"\"!\"\"\"*&&F)6#F,F,%\"TGF,F, *&&F)6#\"\"#F,F0F4F,*&&F)6#\"\"$F,F0F8F,*&&F)6#\"\"%F,F0F " 0 "" {MPLTEXT 1 0 74 "eqn[n]:=fit[leastsquare[[T,P ], expr, \{seq(a[k],k=0..n)\}]]([T(C),P(mmHg)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$eqnG6#\"\"%/%\"PG,,$\"+Nd(yY#!\")\"\"\"%\"TG$\"+9e> 1;!\"**$F/\"\"#$\"+'fHWg$!#6*$F/\"\"$$\"+tf@JT!#8*$F/F'$\"+Hx9jR!#:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "n:=5; expr := P=p(n);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG/%\"PG,.&%\"aG6#\"\"!\"\"\"*&&F)6#F,F,%\"TGF,F, *&&F)6#\"\"#F,F0F4F,*&&F)6#\"\"$F,F0F8F,*&&F)6#\"\"%F,F0F " 0 "" {MPLTEXT 1 0 74 "eqn[n]:=fit[ leastsquare[[T,P], expr, \{seq(a[k],k=0..n)\}]]([T(C),P(mmHg)]);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>&%$eqnG6#\"\"&/%\"PG,.$\"+:iUvC!\")\" \"\"%\"TG$\"+Wk,4;!\"**$F/\"\"#$\"+#*H`gN!#6*$F/\"\"$$\"+!Q#yHT!#8*$F/ \"\"%$\"+]'egA%!#:*$F/F'$!+=o-0D!#=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "We can compare these expressions graphically" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "n:='n':\npolyplot:=plot([seq(rhs(eqn[n]), n=2..5)],T=-40..100,color=[red,black,blue,green]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plots[display](\{polyplot,dataplot\});" 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which we observe that the red line (the quadratic) \+ is clearly poorer than the others; all of which seem more or less equa lly good. 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We define two new variables " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "v1:= Tau =1/(T+273.15): v1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$TauG*$,&%\"TG\"\"\"$\"&:t#!\"#F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "v2:=rho=log10(P): v2;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%$rhoG-%&log10G6#%\"PG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "and substitute into the CC equation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "CC2:=subs(rhs(v2)=lhs(v2),rhs(v1)=lhs(v1) ,CCeqn): CC2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$rhoG,&%\"AG\"\"\"* &%\"BGF'%$TauGF'!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Thus, th is equation is linear in " }{XPPEDIT 18 0 "1/T(K)" "*&\"\"\"F#-%\"TG6# %\"KG!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "log10(P)" "-%&log10G6# %\"PG" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "The nex t step is to transform the data, beginning with the vapor pressures" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "lnpdata := evalf(map(log10 ,P(mmHg)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(lnpdataG7,\"\"!$\"+V +q*)p!#5$\"+++++5!\"*$\"+'**H5I\"F,$\"+\"**f?g\"F,$\"+]7:y " 0 " " {MPLTEXT 1 0 50 "ToKelvin:=x->x+273.15;\nT(K) := map(ToKelvin,T(C)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)ToKelvinG:6#%\"xG6\"6$%)operato rG%&arrowGF(,&9$\"\"\"$\"&:t#!\"#F.F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"TG6#%\"KG7,$\"&XO#!\"#$\"&b`#F+$\"&lh#F+$\"&bq#F+$\"&v!GF+$ \"&b)GF+$\"&D*HF+$\"&N:$F+$\"&vL$F+$\"&D`$F+" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 43 "We must take the reciprocal of these values" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "r := x-> 1/x;\nTover := map(r,T(K)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG:6#%\"xG6\"6$%)operatorG%&a rrowGF(*$9$!\"\"F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&ToverG7,$\" 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