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{SECT 0 {PARA 256 "" 0 "" {TEXT 256 13 "Curve Fitting" }}{PARA 256 "" 
0 "" {TEXT -1 27 "(Prelude to Simpson's Rule)" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 257 12 "by Greg Crow" }}{PARA 
258 "" 0 "" {TEXT 258 22 "(please report errors)" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 328 "You are all aware that y
ou can find a line that goes through two points.  The goal of this lab
oratory is to try and extend this idea to fit a quadratic through thre
e points, and a cubic through four points.  In so doing, we will be ab
le to approximate continuous functions with polynomials.  We begin wit
h two points and a line." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT 261 9 "Example 1" }{TEXT -1 57 ":  Find the equation of t
he line through (1,4) and (2,7)." }}{PARA 0 "" 0 "" {TEXT 262 10 "Solu
tion 1" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 47 "Store the coordinates of the points as follows:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "x:=array([1,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG-%'VECTO
RG6#7$\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "y:=arr
ay([4,7]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG-%'VECTORG6#7$\"\"
%\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
38 "The points can be displayed by typing " }{MPLTEXT 1 0 12 "[x[1],y[
1]];" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "[x[1],y[1]];\n[x[2],
y[2]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"\"\"\"%" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#7$\"\"#\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 30 "The generic form of a line is " }
{XPPEDIT 18 0 "y=mx+b" "/%\"yG,&%#mxG\"\"\"%\"bGF&" }{TEXT -1 32 ", so
 let's call such a function " }{XPPEDIT 18 0 "f" "I\"fG6\"" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "f:=x->a*x+b;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"f
G:6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&%\"aG\"\"\"9$F/F/%\"bGF/F(F(" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "We will
 attempt to find values for a and b that satisfy the requirements that
 " }{XPPEDIT 18 0 "f(1)=4" "/-%\"fG6#\"\"\"\"\"%" }{TEXT -1 5 " and " 
}{XPPEDIT 18 0 "f(2)=7" "/-%\"fG6#\"\"#\"\"(" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f(
x[1])=y[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"aG\"\"\"%\"bGF&\"
\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f(x[2])=y[2];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"aG\"\"#%\"bG\"\"\"\"\"(" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Maple wil
l happily find the solution if one exists using the " }{MPLTEXT 1 0 5 
"solve" }{TEXT -1 9 " command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "solve(\{f(x[1])=y[1], f(x[2]
)=y[2]\}, \{a,b\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"bG\"\"\"/
%\"aG\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 111 "If on your second or third try this gives an error, re-execute
 the definition of f(x) above and then try again." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(\",f(x)
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"$\"\"\"F&" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The following c
ommand makes a function out of the " }{XPPEDIT 18 0 "3x+1" ",&*&\"\"$
\"\"\"%\"xGF%F%F%F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f := unapply(\",x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"xG6\"6$%)operatorG%&arrowG
F(,&9$\"\"$\"\"\"F/F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 46 "Check to see if the result is what you wanted." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
15 "[x[1],f(x[1])];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"\"\"\"%" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "[x[2],f(x[2])];" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7$\"\"#\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 51 "Plot the graph of f(x) and verify th
at all is well." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 259 9 "Example 2" }{TEXT -1 68 ": Find the equation of the parab
ola through (1,4), (2,7), and (3,6)." }}{PARA 0 "" 0 "" {TEXT 260 10 "
Solution 2" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "x:=array([1,2,3]):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "y:=array([4,7,6]):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f:=x->a*x^2+b*x+c;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"fG:6#%\"xG6\"6$%)operatorG%&arrowGF(,(*&%\"aG
\"\"\"9$\"\"#F/*&%\"bGF/F0F/F/%\"cGF/F(F(" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 59 "solve(\{f(x[1])=y[1], f(x[2])=y[2], f(x[3])=y[3]\},
 \{a,b,c\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"cG!\"$/%\"bG\"\"
*/%\"aG!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(\",f(x)
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$%\"xG\"\"#!\"#F%\"\"*!\"$\"
\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f := unapply(\",x);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"xG6\"6$%)operatorG%&ar
rowGF(,(*$9$\"\"#!\"#F.\"\"*!\"$\"\"\"F(F(" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Plot the graph of f(x) and veri
fy that it is what was needed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Question 1:  Fi
nd the equation of the line that approximately fits " }{XPPEDIT 18 0 "
f(x) = sin(x)" "/-%\"fG6#%\"xG-%$sinG6#F&" }{TEXT -1 61 " at the origi
n by using the example above with the points at " }{XPPEDIT 18 0 "x=0
" "/%\"xG\"\"!" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "x=Pi/6" "/%\"xG*&
%#PiG\"\"\"\"\"'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 82 "Question 2:  Plot the line you genera
ted in question 1 and the graph of sin(x) on " }{XPPEDIT 18 0 "[-Pi..P
i]" "7#;,$%#PiG!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Quest
ion 3: Find the equation of a parabola that approximately fits  " }
{XPPEDIT 18 0 "f(x)=cos(x)" "/-%\"fG6#%\"xG-%$cosG6#F&" }{TEXT -1 63 "
 at the origin by using the example above with the points at x=" }
{XPPEDIT 18 0 "-Pi/4" ",$*&%#PiG\"\"\"\"\"%!\"\"F'" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "x=0" "/%\"xG\"\"!" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 
"x=Pi/4" "/%\"xG*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Question 4: Find the
 equation of the cubic polynomial that  fits " }{XPPEDIT 18 0 "f(x)=si
n(x)" "/-%\"fG6#%\"xG-%$sinG6#F&" }{TEXT -1 54 " at the origin.by  usi
ng the points in question 3 and " }{XPPEDIT 18 0 "x=Pi/2" "/%\"xG*&%#P
iG\"\"\"\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 88 "Question 5: Will this work for higher ord
er polynomials?  Take a stab at it with sin(x)." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{MARK "61 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }