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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 20 "Fitting a Polynomial" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 22 "Linear Algebra \+
Project" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{SECT 0 {PARA 5 "" 0 "" {TEXT 281 11 "Objective: " }}{PARA 0 "" 
0 "" {TEXT -1 72 " To learn how systems of linear equations are used t
o fit a polynomial ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 142 " Maple will help with solving the systems of equations
. It will also help us verify that the polynomial found passes through
 the given points." }}}{PARA 5 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT 282 16 "Solved Example
 :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 129 " F
ind a polynomial of degree 2 whose graph goes through points (1,2), (2
,6), and (3,4).  Plot the polynomial and the three points" }}{PARA 0 "
" 0 "" {TEXT -1 15 " in one window." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 9 "
Solution:" }}{PARA 0 "" 0 "" {TEXT -1 45 "We are looking for a polynom
ial of the form  " }{TEXT 258 5 " p(x)" }{TEXT -1 3 " = " }{TEXT 263 
1 "a" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT 
-1 1 " " }{TEXT 259 10 "+ bx + c, " }{TEXT -1 13 "  such that  " }
{TEXT 260 1 "p" }{TEXT 264 3 "(1)" }{TEXT 265 3 " = " }{TEXT 266 2 "2,
" }{TEXT 269 3 "  p" }{TEXT 267 8 "(2) = 6," }{TEXT 268 2 "  " }{TEXT 
270 3 "and" }{TEXT 271 3 "  p" }{TEXT 272 8 "(3) = 4." }{TEXT 273 1 " \+
" }{TEXT -1 18 " These conditions " }}{PARA 0 "" 0 "" {TEXT -1 67 "gen
erate a system of  three linear equations with three unknowns,  " }
{TEXT 274 1 "a" }{TEXT -1 2 ", " }{TEXT 275 1 "b" }{TEXT -1 6 ", and \+
" }{TEXT 276 1 "c" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "p:=x->a*x^2+b*x+c;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(*
&%\"aG\"\"\")9$\"\"#F/F/*&%\"bGF/F1F/F/%\"cGF/F(F(F(" }}}{PARA 11 "" 
1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "eqn1:=p
(1)=2; eqn2:=p(2)=6; eqn3:=p(3)=4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%%eqn1G/,(%\"aG\"\"\"%\"bGF(%\"cGF(\"\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%eqn2G/,(%\"aG\"\"%*&\"\"#\"\"\"%\"bGF+F+%\"cGF+\"\"'
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/,(%\"aG\"\"**&\"\"$\"\"\"
%\"bGF+F+%\"cGF+\"\"%" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "solve(\{eqn1, eqn2, eqn3\}, \{a,b,c
\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"cG!\")/%\"bG\"#8/%\"aG!
\"$" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "
So the polynomial  we are looking for is  " }{TEXT 261 6 "p(x) =" }
{TEXT -1 3 " -3" }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 1 " \+
" }{TEXT 262 2 "+ " }{TEXT 277 2 "13" }{TEXT 278 2 "x " }{TEXT 279 3 "
- 8" }{TEXT 280 1 "." }{TEXT -1 59 "  We will graph  the polynomial an
d the three given points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "p:='p';" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"pGF$" }}}{PARA 0 "" 0 "" {TEXT -1 43 "The last comm
and clears the definition of  " }{TEXT 256 1 "p" }{TEXT -1 34 "  so th
at we can reuse the letter " }{TEXT 257 1 "p" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "p:
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\"6$%)operatorG%&arrowGF(,(*$)9$\"\"#\"\"\"!\"$*&\"#8F1F/F1F1\"\")!\"
\"F(F(F(" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 25 "plot1:=plot(p(x),x=0..4):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 73 "plot2:=plot(\{[1,2],[2,6],[3,4]\}, style=point, \+
symbol=circle,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name cha
ngecoords has been redefined\n" }}}{PARA 12 "" 1 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "display([plot1,plot2]);" }}
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rve 1" "Curve 2" }}}}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 76 "_______________________________________________________
_____________________" }}{PARA 257 "" 0 "" {TEXT -1 10 "Assignment" }}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "Problem:" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }{TEXT 256 1 " " }{TEXT -1 157 "Find a third degree polynomial
 that passes through the points (1,2), (2,4), (3, -1), and (4,0).  Plo
t the polynomial and the three points in the same window." }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 75 "_______________________________________________________
____________________" }}{PARA 0 "" 0 "" {TEXT -1 177 "MSIP Grant #P120
A80089-98: \"Three Urban Calculus Reform Programs: Adopting the Best,
\" 1998-2001; MSEIP Grant #P120AA010031:  \"Four Colleges: Calculus + \+
Enhancements\", 2001-2004 " }}}{MARK "11 22 0" 0 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }