{VERSION 5 0 "SUN SPARC SOLARIS" "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 "" -1 256 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 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 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 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 1 }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 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 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 13 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 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 "" 0 260 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 "" 0 261 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 "" 0 262 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 "" 0 263 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 "" 0 264 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 "" 0 265 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 "" 11 266 1 {CSTYLE "" -1 -1 "" 0 1 0 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 "" 11 267 1 {CSTYLE "" -1 -1 "" 0 1 0 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 " " 0 268 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 }} {SECT 0 {PARA 11 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT 256 13 "ACCESS - MATH" }}{PARA 257 "" 0 "" {TEXT -1 9 "July 2007" }}{PARA 256 "" 0 "" {TEXT -1 49 "Notes on Body Mass Index and Actual National \+ Data" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 28 "What is the Body Mass Index?" }}{PARA 0 "" 0 "" {TEXT -1 383 " If you read newspapers and magazines it i s likely that once or twice a year you run across an article about the body mass index (B.M.I.) , and its use in determining health risk fac tors for overweight and underweight people. If you search the interne t for \"body mass index\" you will find many sites which let you compu te your B.M.I., and which tell you a little bit about it. " }}{PARA 0 "" 0 "" {TEXT -1 931 " A person's B.M.I. is computed by dividin g their weight by the square of their height, and then multiplying by \+ a universal constant. If you measure weight in Newtons, i.e. mass in \+ kilograms, and height in meters, this constant is the fine number one. If you use pounds and inches instead, and are good at conversions you will work out that the constant is about 703. Or you might just find a calculator on the internet which confirms this. Since your B.M.I. \+ is supposed to indicate health risks, proponants of the B.M.I index a re claiming that if for two different people, their weights are the s ame multiple (i.e. their common B.M.I.) of their heights-squared, then these people have comparable health risks. Thus, unless height itsel f is a health risk, the B.M.I. hypothesis can be interpreted as a biom etric scaling law that says that on average, weight should scale propo rtionately to the square of height in people." }}{PARA 0 "" 0 "" {TEXT -1 907 " As we have discussed in class, if people were to sc ale equally in all directions (\"self-similar\") when they grew, volum e and hence weight would scale as the cube of height. That particula r power law seems a little high, since adults don't look like uniforml y expanded versions of babies; we seem to get relatively stretched out when we grow taller. One might expect the best predictive power for \+ weight as a function of height to be somewhere between 2 and 3, if one expected a power law at all. If there is a predictive power, and if \+ it is much larger than 2, then one could argue that the body mass inde x might need to be modified to reflect this fact. (In fact, when you f ind body mass index tables, they often explain how and why you should \+ modifiy the acceptable BMI values for children. Is it possible that if a different power law had been used, no such modification would have \+ been needed?)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 612 " You women did a great job collecting height-weight \+ data -with contributions large and small we have 147 of our own data p oints. There was a relative shortage of people between 30 and 50 inch es high, however, so I added 23 data points in that height range - whi ch I stole from the last time I had one of my linear algebra classes d o this project. Each group will use our common data, see if it is co nsistent with a power law relating weights to heights, and decide whet her the B.M.I. power of 2 is a good choice. You may choose to use the \+ data you alone collected, or its union with the 23 points I added." }} {PARA 0 "" 0 "" {TEXT -1 925 " Several years ago I found a nationa l data base at the U.S. Center for Disease Control web site. It conta ined a wide variety of body measurements collected between 1976 and 19 80 (before concerns about obesity), including national median heights \+ and weights for boys and girls, age 2-19. By using only the national \+ medians a lot of the variance has been taken out of the data, compared to what yours will look like. The national data is very consistent wi th a power law, with power = 2.6. When you do a literature and compu ter search into the history of the B.M.I. index and height-weight scal ing laws, as part of your project work for this week, you might discov er that (by centuries) we're not the first people to stumble upon a po wer other than 2, based on empirical data. As far as I know there is \+ no mathematical model to explain any power law for height-weight scali ng in people, despite the empirical evidence!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 31 "How do you test for powe r laws?" }}{PARA 0 "" 0 "" {TEXT -1 209 " Yesterday Meagan explai ned, and some of you seemed to be very familiar with, how to look for \+ power law fits to data points: you look for a best line fit to the co rresponding ln-ln data points. Recall: " }}{PARA 0 "" 0 "" {TEXT -1 148 " Suppose we have a set of \"n\" data points, which you can t hink of as your height-weight data, but which could really be any set \+ of paired data:" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "[[x[1], y[1]], [x[ 2], y[2]], [x[3], y[3]], `...`, [x[n], y[n]]];" "6#7'7$&%\"xG6#\"\"\"& %\"yG6#F(7$&F&6#\"\"#&F*6#F/7$&F&6#\"\"$&F*6#F5%$...G7$&F&6#%\"nG&F*6# F<" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 68 "We want to see if t here is a power m and a proportionality constant " }{XPPEDIT 18 0 "b; " "6#%\"bG" }{TEXT -1 21 " so that the formula" }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "y = b*x^m;" "6#/%\"yG*&%\"bG\"\"\")%\"xG%\"mGF'" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "effectively mirrors the r eal data. " }{TEXT 259 60 "Taking (natural) logarithms of the propose d power law yields" }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "ln(y) = ln(b)+m *ln(x);" "6#/-%#lnG6#%\"yG,&-F%6#%\"bG\"\"\"*&%\"mGF,-F%6#%\"xGF,F," } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 16 "So, if we write " } {XPPEDIT 18 0 "Y = ln(y);" "6#/%\"YG-%#lnG6#%\"yG" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "X = ln(x);" "6#/%\"XG-%#lnG6#%\"xG" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "B = ln(b);" "6#/%\"BG-%#lnG6#%\"bG" }{TEXT -1 59 ", this becomes the equation of a line in the new variables " }{XPPEDIT 18 0 "X;" "6#%\"XG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Y;" "6#%\"YG" }{TEXT -1 1 ":" }}{PARA 262 "" 0 "" {XPPEDIT 18 0 "Y = mX+B;" "6#/%\"Y G,&%#mXG\"\"\"%\"BGF'" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Th us," }{TEXT 262 1 " " }{TEXT 263 207 "in order for there to be a power law for the original data, the ln-ln data should (approximately) sati sfy the equation of a line, and vise verse. If we get a good line fit to the ln-ln data, then the slope " }{XPPEDIT 18 0 "m" "6#%\"mG" } {TEXT 269 75 " of this line is the power relating the original data, a nd the exponential " }{XPPEDIT 18 0 "exp(B);" "6#-%$expG6#%\"BG" } {TEXT 264 8 " of the " }{XPPEDIT 18 0 "Y" "6#%\"YG" }{TEXT 270 43 "-in tercept is the proportionality constant " }{XPPEDIT 18 0 "b" "6#%\"bG " }{TEXT 267 26 " in the original relation " }{XPPEDIT 18 0 "y = b*x^m " "6#/%\"yG*&%\"bG\"\"\")%\"xG%\"mGF'" }{TEXT 268 3 ". " }{TEXT -1 161 "With real data it is not too hard to see if the ln-ln data is wel l approximated by a line, in which case the original data is well-appr oximated by a power law. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 24 "National data example: " }{TEXT -1 162 "I used colons after most of these commands to suppress the out put. If you want to go back and see what each command is doing, repla ce the colons with semicolons." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 257 0 "" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 334 "boyhw:=[[35.9,29.8],[3 8.9,34.1],[41.9,38.8],[44.3,42.8],\n [47.2,48.6],[49.6,54.8],[5 1.4,60.8],[53.6,66.5],\n [55.7,76.8],[57.3,82.3],[59.8,93.8],[6 2.8,106.8],\n [66.0,124.3],[67.3,132.6],[68.4,142.4],[68.9,145. 1],\n [69.6,155.3],[69.6,153.2]]:\n#boy heights (inches) weight s (pounds): Ntl medians for ages 2-19" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 336 "girlhw:=[[35.4,28.0],[38.4,32.6],[41.1,36.8],[43.9,4 1.8],\n [46.6,47.0],[48.9,52.5],[51.4,60.8],[53.1,65.5],\n \+ [55.7,76.1],[58.2,89.0],[61.0,100.1],[62.6,108.1],\n [63.3,1 17.1],[64.2,117.6],[64.3,122.6],[64.2,128.8],\n [64.1,124.5],[6 4.5,126]]:\n#girl heights(inches) weights (pounds): Ntl medians for ag es 2-19 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "boys:=pointpl ot(boyhw,color=blue):\ngirls:=pointplot(girlhw,color=pink):\ndisplay( \{boys,girls\},\n title=`plot of [height,weight], National medians \+ ages 2-19`);\n" }}}{PARA 0 "" 0 "" {TEXT -1 29 "And now for the ln-ln \+ data. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "with(linalg): #o ne of two linear algebra packages in Maple" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 237 "B:=evalm(boyhw): #\"evalm\" stands for evaluate a s a matrix,\n #converts an array of points\n #into a matrix, which will be easiser to manipulate in Maple\nG:=evalm(girlhw):\nBG:=stackm atrix(B,G): #stack the matrices on top of eachother." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 240 "lnBGa:=map(ln,BG): #take ln of th e boy and girl height-weights\nlnBG:=map(evalf,lnBGa): #Get decimal ( floating point) values.\n #This speeds up computations later in the\n #least squares fit - otherwise Maple tries working\n #symbolically. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "lnlnplot:=pointplot(lnB G):\ndisplay(lnlnplot,title=`ln-ln data`);\n" }}}{PARA 0 "" 0 "" {TEXT -1 135 "Notice the ln-ln plot really does seem close to a line! \+ How do we find the best line to fit this data, other than just eye-ba lling it?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 266 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 265 58 "How do I find the \+ best line fit to a collection of points?" }}{PARA 0 "" 0 "" {TEXT -1 249 " Well, any good calculator or mathematical software already h as a canned program to solve this \"linear regression\", or \"least sq uares\" problem. But it's not hard to explain what's going on: Label our data points for which we seek a line fit as" }}{PARA 263 "" 0 "" {XPPEDIT 18 0 "[[X[1], Y[1]], [X[2], Y[2]], [X[3], Y[3]], `...`, [X[n] , Y[n]]];" "6#7'7$&%\"XG6#\"\"\"&%\"YG6#F(7$&F&6#\"\"#&F*6#F/7$&F&6#\" \"$&F*6#F5%$...G7$&F&6#%\"nG&F*6#F<" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "The solution to our problem is the pair " }{XPPEDIT 18 0 "[m, B];" "6#7$%\"mG%\"BG" }{TEXT -1 94 " for which the the sum of the squared vertical distances between the data points and the line " } {XPPEDIT 18 0 "Y = mX+B" "6#/%\"YG,&%#mXG\"\"\"%\"BGF'" }{TEXT -1 35 " is minimized. Precisely, minimize" }}{PARA 265 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "F(m,B) := Sum((m*X[i]+B-Y[i])^2,i = 1 .. n);" "6#>-% \"FG6$%\"mG%\"BG-%$SumG6$*$,(*&F'\"\"\"&%\"XG6#%\"iGF/F/F(F/&%\"YG6#F3 !\"\"\"\"#/F3;F/%\"nG" }}{PARA 0 "" 0 "" {TEXT -1 57 "by setting its d erivatives with respect to the variables " }{XPPEDIT 18 0 "m, B" "6$% \"mG%\"BG" }{TEXT -1 62 " equal to zero. This leads to a system of tw o equations for " }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "B" "6#%\"BG" }{TEXT -1 2 ", " }}{PARA 264 "" 0 "" {XPPEDIT 18 0 "matrix([[Sum(X[i]^2,i = 1 .. n), Sum(X[i],i = 1 .. n)], [Sum(X[i],i = 1 .. n), n]])*matrix([[m], [B]]) = matrix([[Sum(X[i]*Y[ i],i = 1 .. n)], [Sum(Y[i],i = 1 .. n)]]);" "6#/*&-%'matrixG6#7$7$-%$S umG6$*$&%\"XG6#%\"iG\"\"#/F1;\"\"\"%\"nG-F+6$&F/6#F1/F1;F5F67$-F+6$&F/ 6#F1/F1;F5F6F6F5-F&6#7$7#%\"mG7#%\"BGF5-F&6#7$7#-F+6$*&&F/6#F1F5&%\"YG 6#F1F5/F1;F5F67#-F+6$&FU6#F1/F1;F5F6" }{TEXT -1 1 "." }}{PARA 266 "" 1 "" {TEXT -1 109 "This system always has a unique solution, and your \+ calculator solves it to find the linear regression line. " }}{PARA 267 "" 0 "" {TEXT 271 10 "Example 1:" }{TEXT -1 52 " Work out the best -line fit through the three points" }}{PARA 268 "" 0 "" {TEXT -1 19 "[ [0,1],[2,2],[4,1]]" }}{PARA 0 "" 0 "" {TEXT -1 46 "and use Maple softw are to compare your answer." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "Maple 's commands to do this are pretty weird; you'd find them eventually us ing the Help window." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "with (stats): #first load the statistics commands. " }}}{PARA 0 "" 0 "" {TEXT -1 53 "Here's the command: First, name your two variables, " } {XPPEDIT 18 0 "X,Y;" "6$%\"XG%\"YG" }{TEXT -1 29 " in our case. Then \+ list the " }{XPPEDIT 18 0 "X" "6#%\"XG" }{TEXT -1 49 " values, followe d by a list of the corresponding " }{XPPEDIT 18 0 "Y;" "6#%\"YG" } {TEXT -1 8 " values:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "fit[ leastsquare[[X,Y]]]([[0,2,4],[1,2,1]]);\n #so the points are [0,1], [2,2],[4,1].\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 272 11 "Example 2 " }{TEXT -1 86 "Find the linear regression li ne through the national ln(height), ln(weight) data set:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 270 "Xs:=convert(col(lnBG,1),list): #co nvert the first column of\n #the ln-ln data into a list of the \"x's \" The MAPLE least squares\n #command wants to have lists input, not matrix columns, even\n #though it's hard for us to see any differenc e\nYs:=convert(col(lnBG,2),list):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "fit[leastsquare[[X,Y]]]([Xs,Ys]);#this is the bizarre syntax\n #I just introduced you to. \n" }}}{PARA 258 "" 0 "" {TEXT -1 65 "We can paste in the equation of the line and see how well we di d." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "line:=plot(-6.0387033 03+2.593828004*X, X=3.4..4.5,Y=2.7..5.7,\n color=black):\ndisplay(\{ line,lnlnplot\}, title=`least squares fit`);\n" }}}{PARA 0 "" 0 "" {TEXT -1 70 "Finally, we can go back from the least squares line fit t o a power law" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "m:=2.593828 004; #power\nb:=exp(-6.038703303); #proportionality constant\n" }}} {PARA 0 "" 0 "" {TEXT -1 40 "Notice in the command lines below how to " }}{PARA 0 "" 0 "" {TEXT -1 20 " (i) make a title" }}{PARA 0 "" 0 "" {TEXT -1 63 " (ii) get the axes labeled \"h\" and \"w\" for heig ht and weight" }}{PARA 0 "" 0 "" {TEXT -1 241 " (iii) get the displ ay to include appropriate ranges of height and weight - to contain all our data\n (iv) first create a picture of the power function, and \+ then display its graph along with the boys and girls height-weight dat a points." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "powerplot:=plot (b*h^m,h=0..80,w=0..200,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 187 "display(\{powerplot,boys,girls\},title=\n `power la w for national height-weight data`);\n#by calling the variables h and \+ w, and giving their ranges, I\n#get Maple to label the axes as I want \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{MARK "85" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }