{VERSION 3 0 "SUN SPARC SOLARIS" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 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 }{CSTYLE " " 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 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 "Map le 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 "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 6 "Quiz 1" }}{PARA 19 "" 0 " " {TEXT 256 8 "B. Plohr" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 37 "The table below lists the solubility " }{XPPEDIT 18 0 "s;" "6#%\"s G" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 86 "-but ane in anhydrous hydrofluoric acid at high pressures as a function of \+ temperature " }{XPPEDIT 18 0 "T;" "6#%\"TG" }{TEXT -1 33 ". (The data \+ points are listed as " }{XPPEDIT 18 0 "[T, s];" "6#7$%\"TG%\"sG" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "T;" "6#%\"TG" }{TEXT -1 26 " in degree s Farenheit and " }{XPPEDIT 18 0 "s;" "6#%\"sG" }{TEXT -1 44 " in perc entage by weight.) It is known that " }{XPPEDIT 18 0 "s;" "6#%\"sG" } {TEXT -1 13 " varies with " }{XPPEDIT 18 0 "T;" "6#%\"TG" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "s = s0*exp(kappa*T);" "6#/%\"sG*&%#s0G\"\"\"-%$ expG6#*&%&kappaGF'%\"TGF'F'" }{TEXT -1 20 " for some constants " } {XPPEDIT 18 0 "s0;" "6#%#s0G" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "kapp a;" "6#%&kappaG" }{TEXT -1 25 ". Estimate the constants " }{XPPEDIT 18 0 "s0;" "6#%#s0G" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "kappa;" "6#%& kappaG" }{TEXT -1 43 " using a least squares fit to the data for " } {XPPEDIT 18 0 "ln(s);" "6#-%#lnG6#%\"sG" }{TEXT -1 8 " versus " } {XPPEDIT 18 0 "T;" "6#%\"TG" }{TEXT -1 24 ", and draw the graph of " } {XPPEDIT 18 0 "s;" "6#%\"sG" }{TEXT -1 8 " versus " }{XPPEDIT 18 0 "T; " "6#%\"TG" }{TEXT -1 31 " comparing the fit to the data." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "data := [[75,2.2],[103,3.7],[178,6. 9],[237,11.0],[292,20.2]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dataG 7'7$\"#v$\"#A!\"\"7$\"$.\"$\"#PF*7$\"$y\"$\"#pF*7$\"$P#$\"$5\"F*7$\"$# H$\"$-#F*" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 12 "To estimate " }{XPPEDIT 18 0 "s0;" "6#%#s0G" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "kappa;" "6#%&kappaG" }{TEXT -1 20 " , we seek constants " }{XPPEDIT 18 0 "b;" "6#%\"bG" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "kappa;" "6#%&kappaG" }{TEXT -1 11 " such that " } {XPPEDIT 18 0 "ln(s) = b+kappa*T;" "6#/-%#lnG6#%\"sG,&%\"bG\"\"\"*&%&k appaG\"\"\"%\"TGF-F-" }{TEXT -1 19 " fits the data for " }{XPPEDIT 18 0 "ln(s);" "6#-%#lnG6#%\"sG" }{TEXT -1 8 " versus " }{XPPEDIT 18 0 "T; " "6#%\"TG" }{TEXT -1 7 ". Then " }{XPPEDIT 18 0 "b = ln(s0);" "6#/%\" bG-%#lnG6#%#s0G" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "s0 = exp(b) ;" "6#/%#s0G-%$expG6#%\"bG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Define the error function " }{XPPEDIT 18 0 "E;" "6#%\"EG " }{TEXT -1 17 " to be minimized." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "E := (b,kappa) -> sum((b + kappa*data[i][1] - ln(data[i][2]))^2,i= 1..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"EGR6$%\"bG%&kappaG6\"6$% )operatorG%&arrowGF)-%$sumG6$*$),(9$\"\"\"*&9%F4&&%%dataG6#%\"iG6#F4F4 F4-%#lnG6#&F86#\"\"#!\"\"FB\"\"\"/F;;F4\"\"&F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Find the values of " }{XPPEDIT 18 0 "ln_s0;" "6#% &ln_s0G" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "kappa;" "6#%&kappaG" } {TEXT -1 15 " that minimize " }{XPPEDIT 18 0 "E;" "6#%\"EG" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "soln := solve(\{diff(E(b,ka ppa),b)=0,diff(E(b,kappa),kappa)=0\}, \{b,kappa\});" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%solnG<$/%\"bG$\"+c**oe=!#5/%&kappaG$\"+1\\R2'*!#7 " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Here is the answer." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "s0 := evalf(subs(soln,exp(b)));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "kappa := subs(soln,kappa);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#s0G$\"+'[kU?\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&kappaG$\"+1\\R2'*!#7" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }{TEXT -1 30 "Plot the least square fit for " } {XPPEDIT 18 0 "s;" "6#%\"sG" }{TEXT -1 8 " versus " }{XPPEDIT 18 0 "T; " "6#%\"TG" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "data_ plot := plot(data,0..300,style=point):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "fit_plot := plot(s0*exp(kappa*T),T=0..300):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "display(\{data_plot,fit_plot\});" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7S7$\"\"!$\"1+++'[kU?\"!#:7$$\"1+++]i9RlF+$ \"1d^h\"\\[BG\"F+7$$\"1++vVA)GA\"!#9$\"1cQ\"*3nRa8F+7$$\"1++]Peui=F4$ \"1KjNP'p-W\"F+7$$\"1++]i3&o]#F4$\"19]a49@K:F+7$$\"1++voX*y9$F4$\"1N(z B3V&H;F+7$$\"1++vVTAUPF4$\"1Hd/lqHDfS*\\F4$\"1`x+W)*yX>F+7$$\"1++v=$f%GcF4$\"10yx5J2o?F+7$$\" 1,++Dy,\"G'F4$\"1SM#*4#z=?#F+7$$\"1++]7Y'>cj#F+7$$\"1+++D!*oy ()F4$\"1;8;!HG!*z#F+7$$\"1++v$pnsM*F4$\"1[\\Sj+=cHF+7$$\"1++]siL-5!#8$ \"1*f#o]EdaJF+7$$\"1+++!R5'f5Fep$\"1cU')R\">IL$F+7$$\"1+]P/QBE6Fep$\"1 522\"*QL`NF+7$$\"1+++:o?&=\"Fep$\"1l.]T,ZgPF+7$$\"1+]Pa&4*\\7Fep$\"1g$ o:v[;+%F+7$$\"1+]7j=_68Fep$\"18hGe3nXUF+7$$\"1++vVy!eP\"Fep$\"13#H,Jfh ^%F+7$$\"1+](=WU[V\"Fep$\"1;%HO#[qzZF+7$$\"1++DJ#>&)\\\"Fep$\"18)pEQT7 3&F+7$$\"1+]P>:mk:Fep$\"1mCce8h9aF+7$$\"1+]iv&QAi\"Fep$\"1Po!\\,oDs&F+ 7$$\"1++vtLU%o\"Fep$\"1-L#Hzs[2'F+7$$\"1+++bjm[wwsF+7$$\"1++D 6W%)R>Fep$\"15H<\"QQWw(F+7$$\"1+++:K^+?Fep$\"1&*\\Z#*GXI#)F+7$$\"1++]7 ,Hl?Fep$\"18pM)))Q*e()F+7$$\"1+]P4w)R7#Fep$\"10O@bZ2n#*F+7$$\"1++]x%f \")=#Fep$\"1)eK+Q#Rc)*F+7$$\"1+]P/-a[AFep$\"1sKH\"G2X/\"F47$$\"1+](=Yb ;J#Fep$\"1vZ9zI!)46F47$$\"1++]i@OtBFep$\"1M*444'ex6F47$$\"1+]PfL'zV#Fe p$\"1+%35g))HD\"F47$$\"1+++!*>=+DFep$\"1%*HCe45pi#Fep$\"1[y-WYR-:F47$$\"1+++bJ*[o#Fep$\"1N%y ^%GY)e\"F47$$\"1++Dr\"[8v#Fep$\"1.qoBt=$p\"F47$$\"1+++Ijy5GFep$\"1MXVz +p#z\"F47$$\"1+]P/)fT(GFep$\"1KH1')*H_!>F47$$\"1+]i0j\"[$HFep$\"1Z\"z@ Gc&>?F47$$\"$+$F($\"1P<[XU2]@F4-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-F$6%7' 7$$\"#vF($\"1+++++++AF+7$$\"$.\"F($\"1+++++++PF+7$$\"$y\"F($\"1+++++++ pF+7$$\"$P#F($\"#6F(7$$\"$#HF($\"1++++++??F4Fhz-%&STYLEG6#%&POINTG-%+A XESLABELSG6$Q\"T6\"%!G-%%VIEWG6$;F(Fdz%(DEFAULTG" 2 317 317 317 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 -6296 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Alternatively, we can use the statistics package, as follows:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(stats):" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 77 "fit[leastsquare[[T,ln_s]]]([[data[i][1] $i=1..5], [ ln(data[i][2]) $i=1..5]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%%ln_sG ,&$\"+c**oe=!#5\"\"\"%\"TG$\"+1\\R2'*!#7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Here is the answer, which is the same as before." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "s0 := exp(.1858689956);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "kappa := .009607394906;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#s0G$\"+'[kU?\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&kappaG$\"+1\\R2'*!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 11 "" 1 "" {TEXT -1 1 "\n" }}}{MARK "9 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 }