{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 "" 0 12 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 271 "" 1 18 0 0 0 0 0 1 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 0 12 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }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 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Headi ng 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 0 0 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 1 }3 3 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 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 1 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }} {SECT 0 {PARA 18 "" 0 "" {TEXT 256 37 "Modeling Data by Polynomial Fun ctions" }}{PARA 4 "" 0 "" {TEXT -1 20 "Precalculus Project " }}{PARA 0 "" 0 "" {TEXT -1 135 " \+ \+ " }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 257 11 "Objectives:" }}{PARA 0 "" 0 "" {TEXT -1 99 "1. To perform mode ling data by graphing and fitting linear, quadratic and cubic functio ns to data " }}{PARA 0 "" 0 "" {TEXT -1 82 "2. To use Maple statistica l package for curve fitting by Least Squares to data. " }}{PARA 0 " " 0 "" {TEXT -1 100 " (The method of Least Squares is discussed in de tail Statistics and Calculus. Here we concentrate " }}{PARA 0 "" 0 " " {TEXT -1 86 "on the curve fitting and practical conclusions and not on details of computation.) " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 258 15 "Solved Example:" }}{PARA 0 "" 0 "" {TEXT -1 71 "T he amounts y of the merchandise trade balance of the United States in \+ " }}{PARA 0 "" 0 "" {TEXT -1 106 " billions of dollars for the years 1986 to 1993 were as follows. (1986, -152.7), (1987, -152.1), \+ " }}{PARA 0 "" 0 "" {TEXT -1 103 " (1988, -118.6), (1989, -10 9.6), (1990, -101.7), (1991, -65.4), (1992, -85.4), (1993, -115.8)." } }{PARA 0 "" 0 "" {TEXT -1 109 " a) Let t be the time in year s, with t = 0 corresponding to 1990. Use the regression capabilities \+ " }}{PARA 0 "" 0 "" {TEXT -1 109 " to find 1) a linear 2 ) a quadratic and 3) a cubic model for the data. What is the domain \+ of " }}{PARA 0 "" 0 "" {TEXT -1 26 " the models?" }} {PARA 0 "" 0 "" {TEXT -1 39 " b) Graph the data and models." }}{PARA 0 "" 0 "" {TEXT -1 100 " c) For which year does the \+ model most accurately estimate the actual data? During which " }} {PARA 0 "" 0 "" {TEXT -1 41 " year is it least accurate? " }}{PARA 0 "" 0 "" {TEXT -1 86 " d) Why would economists be \+ concerned if this model remained in the future? " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 259 8 "Solution" }{TEXT 273 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 39 "The problem is solved in several steps." }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 260 15 "1. Scatter plot" }}{PARA 0 "" 0 "" {TEXT -1 101 "A scatter plot is just plotting the points and not \+ connecting them. By putting a colon (:) instead " }}{PARA 0 "" 0 "" {TEXT -1 60 "of semicolon (;) we suppress the output for later displa y. " }}{PARA 0 "" 0 "" {TEXT -1 92 "Since 1990 is t = 0, 1986 becomes \+ t = -4 and 1993 becomes t = 3. We prepare a set of pairs " }}{PARA 0 "" 0 "" {TEXT -1 98 "with t as the first coordinates, and correspondin g y as second coordinates. We can plot this set " }}{PARA 0 "" 0 "" {TEXT -1 20 "as a scatter plot. " }}{PARA 0 "" 0 "" {TEXT -1 27 "Prep are the list of pairs. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 " datalist := [ [-4, -152.7], [-3 \+ , -152.1], [-2, -118.6], [-1, -109.6], [0, -101.7], [1, -65.4], [2, -8 5.4],[3, -115.8] ];\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)datalistG7 *7$!\"%$!%F:!\"\"7$!\"$$!%@:F*7$!\"#$!%'=\"F*7$F*$!%'4\"F*7$\"\"!$!%<5 F*7$\"\"\"$!$a'F*7$\"\"#$!$a)F*7$\"\"$$!%e6F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Define and store a scatte rplot. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "PointPlot := plot(datalist, t = -4..3, y = -153..-60 , style = POINT, symbol = CIRCLE, title = `Plot of points`, color = re d):" }}}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 261 21 "2. Prepare two lists" }}{PARA 0 "" 0 " " {TEXT -1 64 " One list is of t coordinates and the other of y coordi nates. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Tvalues := [-4, -3, -2, -1, 0, 1, 2, 3]; \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(Tva luesG7*!\"%!\"$!\"#!\"\"\"\"!\"\"\"\"\"#\"\"$" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Yvalues := [ -152.7, -152.1, -118.6, -109.6, -101.7, -65.4, -85.4, -115.8];\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(YvaluesG7*$!%F:!\"\"$!%@:F($!%'=\"F ($!%'4\"F($!%<5F($!$a'F($!$a)F($!%e6F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 5 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 262 38 "3. Fit polynomial models to the lists" }} {PARA 0 "" 0 "" {TEXT -1 67 " First define linear, quadratic and cubic curves to fit the data. " }}{PARA 0 "" 0 "" {TEXT -1 39 "Then use Ma ple's curvefitting package. " }}{PARA 0 "" 0 "" {TEXT -1 94 "In the fo llowing after defining linear_model we use Maple's Copy/Paste feature \+ to avoid extra " }}{PARA 0 "" 0 "" {TEXT -1 56 "inputting. We make ap propriate changes after pasting. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 54 " For these two lists we fit i) a linear model y = a" }{TEXT 263 1 "t" }{TEXT -1 7 " + b, " }}{PARA 0 "" 0 " " {TEXT -1 67 " ii) a quadratic model y = a" }{XPPEDIT 18 0 "t^2;" "6#*$%\"tG\"\"#" }{TEXT -1 4 " + b " }{TEXT 264 1 "t" }{TEXT -1 5 " + c " }}{PARA 0 "" 0 "" {TEXT -1 64 " iii) a cubic model: y = a" } {XPPEDIT 18 0 "t^3;" "6#*$%\"tG\"\"$" }{TEXT -1 4 " + b" }{XPPEDIT 18 0 "t^2;" "6#*$%\"tG\"\"#" }{TEXT -1 4 " + c" }{TEXT 265 2 "t " }{TEXT -1 4 "+ d." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "linear_model : = a*t + b;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-linear_modelG,&*&% \"aG\"\"\"%\"tGF(F(%\"bGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "quadratic_model := a*t^2 + b*t + c;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0quadratic_modelG,(*&%\"aG\"\"\")%\"tG\"\"#F(F(*&%\"b GF(F*F(F(%\"cGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "cubic_m odel := a*t^3 + b*t^2 + c*t + d;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%,cubic_modelG,**&%\"aG\"\"\")%\"tG\"\"$F(F(*&%\"bGF()F*\"\"#F(F(*&% \"cGF(F*F(F(%\"dGF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 5 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 266 9 "4. Apply " }{TEXT 274 8 "Maple V " }{TEXT 275 45 "statistical and g raphing packages to the data" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "We activate the graphing and stati stical packages." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "with(pl ots): with(stats):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "lin ear_fit:= fit[leastsquare[ [t,y], y = linear_model, \{a, b, c, d\}]]([ Tvalues, Yvalues]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+linear_fit G/%\"yG,&*&$\"+9dGR!*!\"*\"\"\"%\"tGF,F,$\"+r&G93\"!\"(!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "quadratic_fit:= fit[leastsqu are[ [t,y], y = quadratic_model, \{a, b, c, d\}]]([Tvalues, Yvalues]); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.quadratic_fitG/%\"yG,(*&$\"+9 dyEI!\"*\"\"\")%\"tG\"\"#F,!\"\"*&$\"+++]7gF+F,F.F,F,$\"+dG*3I*!\")F0 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "cubic_fit:= fit[leastsq uare[ [t,y], y = cubic_model, \{a, b, c, d\}]]([Tvalues, Yvalues]);\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*cubic_fitG/%\"yG,**&$\"+:::!>\" !\"*\"\"\")%\"tG\"\"$F,!\"\"*&$\"+()H,7[F+F,)F.\"\"#F,F0*&$\"+)yyGh\"! \")F,F.F,F,$\"+vYKl()F9F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 267 15 "5. Draw graphs" }} {PARA 0 "" 0 "" {TEXT -1 85 "We plot the scatter plot and curves. To \+ make them stand out choose different styles " }}{PARA 0 "" 0 "" {TEXT -1 30 " and colors for graphing." }}{PARA 0 "" 0 "" {TEXT -1 20 " Define eq_fit as an " }{TEXT 268 13 "implicit plot" }{TEXT -1 66 " bec ause y and x both exist in the equation. Suppress the output." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "linear_plot := implicitplot( linear_fit, t = -4..3, y = -153..-60, color = red, style = line):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "quadratic_plot := implicit plot(quadratic_fit, t = -4..3, y = -153..-60, color = blue, style = l ine):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "cubic_plot := impl icitplot(cubic_fit, t = -4..3, y = -153..-60, color = green, style = line):" }}}{PARA 0 "" 0 "" {TEXT -1 67 " Now we display all the previ ous graphs with one set of axes. 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ip7$7$Fd]m$!3W,\")y#)yoB%*FF7$$\"367K?7Yi4CFcoFfjl7$7$$\"3c7K?7Yi4CFco Ffjl7$$\"3u0<7X,$\\P#Fco$!333u%*\\+bh#*FF7$7$$\"3C,]hm>LgAFcoF^\\mFg[q 7$7$$\"3!3+:m'>LgAFcoF^\\m7$Fh\\m$!31ncM)y(*fs)FF7$7$F__m$!3(fZRvIVL. \"F-7$$\"3sw31$4#pXEFco$!3-+++++?45F-7$7$F\\]qFdfl7$$\"33C*>%evG%f#Fco $!3%[v+>/#)\\#**FF7$7$$\"3?,=Y&\\)>JDFcoF\\hlFb]q7$Fh]qF][q7$7$Fg`m$!3 +,+)*344Z6F-7$$\"31i%4FUF>F[am-%+AXESLABELSG6%Q\"t6\"Q\"yFd`q-%%FONT G6#%(DEFAULTG-%&TITLEG6#%J~Data~points~and~least~squares~curves~fitG-% %VIEWG6$;F(FM;Fdam$!#gF*" 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" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "The green plot or cu bic plot includes more points than any other plots. So we choose the \+ cubic " }}{PARA 0 "" 0 "" {TEXT -1 42 "model as the best model to repr esent data." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 269 19 " 6. Answer questions" }}{PARA 0 "" 0 "" {TEXT -1 29 " a) The domain is [-4, 3]. " }}{PARA 0 "" 0 "" {TEXT -1 99 " b) Maple draws the graph with position of x-axis through y = -152.7 and not through the origin . " }}{PARA 0 "" 0 "" {TEXT -1 32 " Read the graph carefully." }}{PARA 0 "" 0 "" {TEXT -1 98 " c) The data is best fit by the cubic \+ curve for the years 1986, 1989, 1992, 1993, least accurate " }}{PARA 0 "" 0 "" {TEXT -1 30 " for 1990 and 1991. " }}{PARA 0 "" 0 "" {TEXT -1 96 " d) The values on the y axis are negative. The curve \+ is decreasing from 1992 means the debt is " }}{PARA 0 "" 0 "" {TEXT -1 99 " increasing. The graph shows that the debt will keep in creasing and will never decrease. " }}{PARA 0 "" 0 "" {TEXT -1 98 " \+ The economists would be concerned if the model remains in the fu ture, because the debt has " }}{PARA 0 "" 0 "" {TEXT -1 42 " sta rted increasing from 1992. " }}}}{PARA 0 "" 0 "" {TEXT -1 92 "___ ______________________________________________________________________ ___________________" }}{PARA 0 "" 0 "" {TEXT -1 75 " \+ " }{TEXT 270 0 "" }{TEXT 271 10 "ASSIGNMENT" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 272 8 "Problem:" }}{PARA 0 "" 0 "" {TEXT -1 96 "The amounts \+ y of the Federal budget deficit over a 15 year period in billions of d ollars are as " }}{PARA 0 "" 0 "" {TEXT -1 107 " follows. \+ (1977, 53.6), (1978, 59.2), (1979, 40.2), (1980, 73.8), (1981, 78.9), \+ (1982, 127.9), " }}{PARA 0 "" 0 "" {TEXT -1 100 " (1983, 207. 8), (1984, 185.3), (1985, 212.3), (1986, 221.2), (1987, 149.7), (1988, 155.1), " }}{PARA 0 "" 0 "" {TEXT -1 54 " (1989, 153.4), (19 90, 220.4), (1991, 320.1)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 " (a) Let t be the time in years, with t \+ = 0 corresponding to 1984. Use the regression capabilities " }}{PARA 0 "" 0 "" {TEXT -1 93 " to find an appropriate model for the data. What is the domain of the model? " }}{PARA 0 "" 0 "" {TEXT -1 67 " Try linear model: y = ax + b, quadratic mo del: y = a" }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 4 " + b" }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 6 " + c, " }}{PARA 0 "" 0 "" {TEXT -1 39 " cubic model: y = a" }{XPPEDIT 18 0 " x^3;" "6#*$%\"xG\"\"$" }{TEXT -1 4 " + b" }{XPPEDIT 18 0 "x^2;" "6#*$% \"xG\"\"#" }{TEXT -1 4 " + c" }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 32 " + d, or a quadric model: y = a" }{XPPEDIT 18 0 "x^4;" "6#*$%\"xG \"\"%" }{TEXT -1 4 " + b" }{XPPEDIT 18 0 "x^3;" "6#*$%\"xG\"\"$" } {TEXT -1 4 " + c" }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 4 " + d" }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 8 " + e. " }}{PARA 0 " " 0 "" {TEXT -1 42 " Modify the solved example. " }} {PARA 0 "" 0 "" {TEXT -1 96 " (b) Graph the data and model. F ind the best fitting model among the above four models." }}{PARA 0 "" 0 "" {TEXT -1 69 " Answer the following questions for th e best fit model:" }}{PARA 0 "" 0 "" {TEXT -1 100 " (c) For w hich year does the model most accurately estimate the actual data? Du ring which " }}{PARA 0 "" 0 "" {TEXT -1 42 " year is it least accurate?" }}{PARA 0 "" 0 "" {TEXT -1 93 " (d) By exte nding the curve on the hard copy, estimate the budget deficit for 1992 . " }}}{PARA 0 "" 0 "" {TEXT -1 91 "__________________________________ _________________________________________________________" }}{PARA 0 " " 0 "" {TEXT -1 79 "MSEIP Grant #P120AA010031: \"Four Colleges: Calcu lus + Enhancements\", 2001-2004" }}}{MARK "5 10 7 0" 99 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }