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{SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 44
"Global Carbon Dioxide Concentrations, Part 2" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 11 "Prepared by" }}{PARA
257 "" 0 "" {TEXT -1 9 "Jim Herod" }}{PARA 258 "" 0 "" {TEXT -1 19 "jh
erod@mail.tds.net" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 205 "This worksheet is continuation of the worksheet Global C
arbon Dioxide Concentrations, Part 1. These two are based on the mater
ial found in Chapter 3: Atmospheric Carbon Dioxide Concentration from \+
the book " }{TEXT 256 72 "Earth Algebra, College algebra with Applicat
ions to Environmental Issues" }{TEXT -1 293 ". This book is authored b
y Christopher Schaufele and Nancy Aumoff of Kennesaw State Collge and \+
University. It is published by Harper Collins College Publishers (1995
). The book has a good collection of problems that catch the attention
of students and make clear the utility of of mathematics." }}{PARA 0
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 244 "In this illustra
tion,we use the author's data and methods to make models for predictin
g the carbon dioxide concentration for the year 2000. Here is the data
they provide. The data gives concentrations in parts per million for \+
four recent years:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 27 " Year Concentration" }}{PARA 0 "" 0 "" {TEXT -1
19 " 1965 319.9" }}{PARA 0 "" 0 "" {TEXT -1 19 " 1970 \+
325.3" }}{PARA 0 "" 0 "" {TEXT -1 19 " 1980 338.5" }}{PARA 0 "
" 0 "" {TEXT -1 19 " 1990 354.0" }}{PARA 0 "" 0 "" {TEXT -1 0
"" }}{PARA 0 "" 0 "" {TEXT -1 147 "We predict the concentration in the
year 2000 by choosing triples of points and finding the quadratic fun
ction that is determined by those points. " }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "The technique for finding quad
ratic functions determined by three points is not complicated. Quadrat
ic functions have the form" }}{PARA 0 "" 0 "" {TEXT -1 35 " \+
" }{XPPEDIT 18 0 "y(x);" "6#-%\"yG6#%\"xG" }
{TEXT -1 4 " = " }{XPPEDIT 18 0 "a*x^2+b*x+c;" "6#,(*&%\"aG\"\"\"*$%
\"xG\"\"#F&F&*&%\"bGF&F(F&F&%\"cGF&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "There are three coeffic
ients to be determined -- " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1
2 ", " }{XPPEDIT 18 0 "b;" "6#%\"bG" }{TEXT -1 6 ", and " }{XPPEDIT
18 0 "c;" "6#%\"cG" }{TEXT -1 85 " . We use three points to set up thr
ee equations with three unknowns and solve these." }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "We make the years the x's
and concentrations the y's." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
63 "xx:=[1965, 1970, 1980, 1990];\nyy:=[319.9, 325.3, 338.5, 354.0];"
}}}{PARA 0 "" 0 "" {TEXT -1 80 "We choose triples of points, find the \+
quadratic functions associated with these." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 17 "i:=1;\nj:=2;\nk:=4;" }}}{PARA 0 "" 0 "" {TEXT -1
29 "Here are the three equations." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 95 "eq1:=yy[i]=a*xx[i]^2+b*xx[i]+c;\neq2:=yy[j]=a*xx[j]^2
+b*xx[j]+c;\neq3:=yy[k]=a*xx[k]^2+b*xx[k]+c;" }}}{PARA 0 "" 0 ""
{TEXT -1 68 "We solve the three equations and assign these values to a
, b, and c." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(\{eq1,e
q2,eq3\},\{a,b,c\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "ass
ign(%);" }}}{PARA 0 "" 0 "" {TEXT -1 68 "Using these values for a, b, \+
and c, we define a quadratic function." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 19 "yq:=x->a*x^2+b*x+c;" }}}{PARA 0 "" 0 "" {TEXT -1 69 "
Here is the estimate for the carbon dioxide content in the year 2000.
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "yq(2000);" }{TEXT -1 0 "
" }}}{PARA 0 "" 0 "" {TEXT -1 75 "We plot the four data points and thi
s quadratic; and, we compute the error." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 114 "with(plots):\nJ:=plot(yq(x),x=1965..2000):\nK:=point
plot(\{[xx[1],yy[1]],[xx[2],yy[2]],[xx[3],yy[3]],[xx[4],yy[4]]\}):" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display(\{J,K\});" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "err:=sum(abs(yq(xx[p])-yy[p]
),p=1..4);" }}}{PARA 0 "" 0 "" {TEXT -1 72 "In order to use a, b, and \+
c again, we clear the values assigned to them." }}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 23 "a:='a';\nb:='b';\nc:='c';" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "As in Part 1, this
worksheet is not designed " }{TEXT 258 4 "just" }{TEXT -1 24 " to giv
e information on " }{XPPEDIT 18 0 "CO[2];" "6#&%#COG6#\"\"#" }{TEXT
-1 214 " build up in the atmosphere, but also to give instruction on M
aple programming. We illustrate here how to write a small program to c
ompute the lines and errors for all the data at one time. First, we i
llustrate a " }{TEXT 259 7 "do loop" }{TEXT -1 134 ". This is a common
programming tool which we illustrate by listing all the triples i, j \+
and k where i, j, and k range between 1 and 4." }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "for i from 1
to 4 do\nfor j from 1 to 4 do\nfor k from 1 to 4 do\nprint( i,j,k );
\nod;\nod;\nod;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 137 "An alternate choice appropriate for this problem is to a
gree that the pair 1, 3,1 and 1,3,1 should not be distinguished. We \+
modify the " }{TEXT 257 7 "do loop" }{TEXT -1 108 " so as not to list \+
both the points 1, 3,1 and 1,3,1. We also do not list any point twice \+
for we always want " }{TEXT 260 5 "three" }{TEXT -1 13 " data points.
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "for i from 1 to 4 do\nfo
r j from i+1 to 4 do\nfor k from j+1 to 4 do\nprint( i,j,k);\nod;\nod;
\nod;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "With the " }{TEXT 261 7
"do loop" }{TEXT -1 105 " tool, we can find all possible quadratic fun
ctions using all possible triples of data points. We make a " }{TEXT
262 7 "do loop" }{TEXT -1 58 " and use the schemes from above. We only
print the errors." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 328 "for i
from 1 to 4 do\nfor j from i+1 to 4 do\nfor k from j+1 to 4 do\neq1:=
yy[i]=a*xx[i]^2+b*xx[i]+c:\neq2:=yy[j]=a*xx[j]^2+b*xx[j]+c:\neq3:=yy[k
]=a*xx[k]^2+b*xx[k]+c:\nsolve(\{eq1,eq2,eq3\},\{a,b,c\}):\nassign(%):
\nyq:=x->a*x^2+b*x+c:\nerr:=sum(abs(yq(xx[p])-yy[p]),p=1..4):\nprint(i
,j,k,` err =`,err);\na:='a': b:='b': c:='c':\nod;\nod;\nod;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 ""
{TEXT 263 10 "Assignment" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 227 "Choose the three points which give the smallest e
rror. Generate the graph of the line associated with these three point
s and the graph of the data. Predict the carbon dioxide concentration \+
in the year 2000 using this quadratic." }}}{PARA 0 "" 0 "" {TEXT -1 0
"" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 8 "Addendum" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 280 "Maple has a routine for \+
computing a quadratic associated with a collection of data. The quadra
tic is also designed to minimize the error. We illustrate the techniqu
es for creating this quadratic. The user can compare the error which t
his routine gives with the one computed above." }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(stats):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "xx:=[1965, 1970, 1980, \+
1990];\nyy:=[319.9, 325.3, 338.5, 354.0];" }}}{PARA 0 "" 0 "" {TEXT
-1 29 "Use the fit routine of Maple." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 47 "fit[leastsquare[[x,y],y=a*x^2+b*x+c]]([xx,yy]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "ym:=unapply(rhs(%),x);" }}}
{PARA 0 "" 0 "" {TEXT -1 72 "Determine the error with the same scheme \+
as above. Compare your results." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 38 "err:=sum(abs(ym(xx[p])-yy[p]),p=1..4);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "1 0" 0
}{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }