{VERSION 2 3 "IBM INTEL NT" "2.3" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 
0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }
{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 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 }0 0 
0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" 
-1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 }1 0 0 -1 -1 -1 0 0 
0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 
1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Error" 
7 8 1 {CSTYLE "" -1 -1 "" 0 1 255 0 255 1 0 0 0 0 0 0 0 0 0 }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 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }
{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 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 "Maple Plot" 0 13 1 {CSTYLE "
" -1 -1 "" 0 1 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 "" 11 256 1 {CSTYLE "" -1 -1 "" 0 1 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 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Question #1, part a and b.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "u:=vector(3);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%&arrayG6$;\"\"\"\"\"$7\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "F:=u->vector([u[1]^2+u[2]^2+u[3]^4-
257,u[3]^2-u[1]*u[2],u[1]-3*u[2]-u[3]]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"FG:6#%\"uG6\"6$%)operatorG%&arrowGF(-%'vectorG6#7%,**$&9$6#
\"\"\"\"\"#F5*$&F36#F6F6F5*$&F36#\"\"$\"\"%F5!$d#F5,&*$F;F6F5*&F2F5F8F
5!\"\",(F2F5F8!\"$F;FCF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
11 "F([x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%,**$%
\"xG\"\"#\"\"\"*$%\"yGF*F+*$%\"zG\"\"%F+!$d#F+,&*$F/F*F+*&F)F+F-F+!\"
\",(F)F+F-!\"$F/F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "newto
nroots(F,[1,0,-1],10,0.001);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VEC
TORG6#7%$\"+'G9dB*!\")$\"+9dG9PF)$!+dG92>F)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'VECTORG6#7%$\"+%Q3&pY!\")$\"+3aKL?F)$!+RyYI9F)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+dg\"\\Y#!\")$\"+sFXz
6F)$!+fAWt5F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+DU'
>T\"!\")$\"+R*=\"*R(!\"*$!+mXrx!)F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#-%'VECTORG6#7%$\"+[5p]!*!\"*$\"+nf([1&F)$!+`o$R9'F)" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+?k<Wl!\"*$\"+Tc+#z$F)$!+.0%=$[F)" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+6\")p%R&!\"*$\"+\\
IRoJF)$!+M5[5TF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+
C*\\D1&!\"*$\"+f!eA)HF)$!+`UA%)QF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%'VECTORG6#7%$\"+&pPQ.&!\"*$\"+0/#f'HF)$!+>N#R'QF)" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+;biL]!\"*$\"+G&)zlHF)$!+p+xjQF)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+;biL]!\"*$\"+G&)zlHF
)$!+p+xjQF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "newtonroots(
F,[1,10,10],10,0.001);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#
7%$\"+vn&GA'!\"*$!+6Qp_X!#5$\"+>\\m)e(F)" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'VECTORG6#7%$\"+'*>\"*QB!\")$\"+93E(*e!\"*$\"+9vL(p&F," }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+z)*4(R\"!\")$\"+Zy*f
;$!\"*$\"+a_+tWF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"
+9!>gy*!\"*$\"+zAlz>F)$\"+w@1ZQF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-
%'VECTORG6#7%$\"++\"zpd)!\"*$\"+:\"=.j\"F)$\"+bZ-'o$F)" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+(oPxY)!\"*$\"+]m'pf\"F)$\"+Ox$on
$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+S-)oY)!\"*$\"
+(Q)o'f\"F)$\"+y]\"on$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG
6#7%$\"+S-)oY)!\"*$\"+(Q)o'f\"F)$\"+y]\"on$F)" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 36 "newtonroots(F,[10,10,-10],10,0.001);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+LndB%*!\"*$\"+Sg9acF)$!+'Q
h)QvF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+KCp=u!\"*$
\"+0!GUR%F)$!+%e\"*Rw&F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG
6#7%$\"+_%\\R(f!\"*$\"+xCXCNF)$!+')zS*f%F)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'VECTORG6#7%$\"+PE4J_!\"*$\"+!\\EF3$F)$!+Jo3<SF)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+$>0U/&!\"*$\"+ZH0sHF
)$!+[O&>(QF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+\"\\
dO.&!\"*$\"+7u\"e'HF)$!+YZzjQF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'
VECTORG6#7%$\"+0biL]!\"*$\"+A&)zlHF)$!+h+xjQF)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'VECTORG6#7%$\"+0biL]!\"*$\"+A&)zlHF)$!+h+xjQF)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "newtonroots(F,[5,10,10],10,0
.001);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+=60o(*!\"*
$\"+fKYNt!#5$\"+T@TnvF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6
#7%$\"+[=`\"Q\"!\")$\"+MG/)p#!\"*$\"+u**=@dF," }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'VECTORG6#7%$\"+GyE]5!\")$\"+M8+-?!\"*$\"+yUn'\\%F," 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+kU<6*)!\"*$\"+0*RO
o\"F)$\"+[XDgQF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+
nl2%\\)!\"*$\"+IF'>g\"F)$\"+v$)=)o$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#-%'VECTORG6#7%$\"+C2*pY)!\"*$\"+t'4nf\"F)$\"+0<'on$F)" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+!>!)oY)!\"*$\"+q$)o'f\"F)$\"+z
]\"on$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+!>!)oY)!
\"*$\"+q$)o'f\"F)$\"+z]\"on$F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 37 "newtonroots(F,[-5,-10,-10],10,0.001);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'VECTORG6#7%$!+=60o(*!\"*$!+fKYNt!#5$!+T@TnvF)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$!+[=`\"Q\"!\")$!+NG/)p#
!\"*$!+u**=@dF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$!+Gy
E]5!\")$!+M8+-?!\"*$!+yUn'\\%F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'
VECTORG6#7%$!+kU<6*)!\"*$!+0*ROo\"F)$!+[XDgQF)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'VECTORG6#7%$!+nl2%\\)!\"*$!+JF'>g\"F)$!+v$)=)o$F)" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$!+D2*pY)!\"*$!+t'4nf\"
F)$!+0<'on$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$!+!>!)
oY)!\"*$!+q$)o'f\"F)$!+z]\"on$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%
'VECTORG6#7%$!+!>!)oY)!\"*$!+q$)o'f\"F)$!+z]\"on$F)" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 34 "newtonroots(F,[-1,5,-5],10,0.001);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+'*=7ZI!\"*$\"+cd:UCF
)$!+s`MzUF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"++qL;a
!\"*$\"+0N^8JF)$!+:N?CRF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTOR
G6#7%$\"+lc4Y]!\"*$\"+6>UqHF)$!+n+<lQF)" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'VECTORG6#7%$\"+j&QO.&!\"*$\"+`J!e'HF)$!+(*3xjQF)" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+/biL]!\"*$\"+A&)zlHF)$!+h+xjQ
F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+/biL]!\"*$\"+A
&)zlHF)$!+h+xjQF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "Idea: we keep on hitting the same
 solutions, thus, we have found all the solutions that are possible vi
a Newton's Method" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "F([-8.
466880190,-1.596688370,-3.676815079]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'VECTORG6#7%$\"\"\"!\"(\"\"!$!\"\"!\"*" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 43 "F([-5.033625505,-2.965798522,3.863770061]);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"\"\"!\"(\"\"!F*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Note: if we take the negation of a
 solution, we have another solution. " }}}{EXCHG {PARA 256 "> " 1 "" 
{MPLTEXT 1 0 43 "F([8.466880190, 1.596688370, 3.676815079]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"\"\"!\"(\"\"!$F(!\"*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Part b, " }}{PARA 0 "" 0 "" {TEXT 
-1 405 "we can not be certain that we have every solution. Since newto
n's method can not find every solution, we are limited in this fashion
. However, through exploration and many different starting points, we \+
can conclude we have discovered all the solutions we can via Newtons M
ethod. Since solutions can be hidden in the nature of the surface, and
 we can not plot the curves/surfaces, we are at a disadvantage." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Question #2." }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 46 "Why this function has a global min and max in " }
{XPPEDIT 18 0 "R^3" "*$%\"RG\"\"$" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 13 "v:=vector(3);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"vG-%&arrayG6$;\"\"\"\"\"$7\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 93 "f:=v->(100*v[3]^3*v[1]*v[2]+20*v[1]*v[1]+80*v[2]*v[2]
)*exp(-(v[1]*v[1]+v[2]*v[2]+v[3]*v[3]));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"fG:6#%\"vG6\"6$%)operatorG%&arrowGF(*&,(*(&9$6#\"\"$F2&F06#
\"\"\"F5&F06#\"\"#F5\"$+\"*$F3F8\"#?*$F6F8\"#!)F5-%$expG6#,(F:!\"\"F<F
B*$F/F8FBF5F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "f([x,y,z
]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,(*(%\"zG\"\"$%\"xG\"\"\"%\"y
GF)\"$+\"*$F(\"\"#\"#?*$F*F-\"#!)F)-%$expG6#,(F,!\"\"F/F5*$F&F-F5F)" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 457 "As x,y,z grow larger, the expon
ential term approaches 0 extremely rapidly. Much more rapidly than the
 other term. Thus the exponential term \"dominates\" the other term an
d brings the function to 0 as x,y,z grow large. Thus this function has
 a global maximum because it does not get infinately larger as x,y,z g
et large. This function might have a global minimum, but it would be a
 critical point. And thus we could solve for it by setting the gradien
t to 0. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "part b. find values.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "newtoncritpts(f,[1,0,0]
,10,0.001);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'VECTORG6#7%$\"\"\"\"
\"!F)F)$\"+C))edt!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7
%$\"\"\"\"\"!F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "newton
critpts(f,[-1,0,0],10,0.001);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'VE
CTORG6#7%$!\"\"\"\"!F)F)$\"+C))edt!\"*" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'VECTORG6#7%$!\"\"\"\"!F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 75 "Here are the points we found: (mostly by accident starting at t
hese points)" }}{PARA 0 "" 0 "" {TEXT -1 58 "[1,0,0] [-1,0,0] [0,1,0] \+
[0,-1,0] [0,0,1] [0,0,-1] [0,0,0]" }}{PARA 0 "" 0 "" {TEXT -1 85 "They
 are mostly saddle points, but two of them are maximums. The origin is
 a minimum." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Grad(f,[0,1,
0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%\"\"!F'F'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "H2:=evalf(Hess(f,[0,1,0]));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H2G-%'MATRIXG6#7%7%$!+%H`XT%!\"
)\"\"!F-7%F-$!+7U@x6!\"(F-7%F-F-$!+f52')eF," }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 28 "definite(H2,'positive_def');" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "
definite(H2,'negative_def');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%true
G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Thus [0,1,0] is a maximum...
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Grad(f,[0,0,0]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%\"\"!F'F'" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "H2:=evalf(Hess(f,[0,-1,0]));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H2G-%'MATRIXG6#7%7%$!+%H`XT%!\")\"
\"!F-7%F-$!+7U@x6!\"(F-7%F-F-$!+f52')eF," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "definite(H2,'positive_def');" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "
definite(H2,'negative_def');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%true
G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Values at these points:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalf(f([0,-1,0]));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Ib.VH!\")" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 18 "evalf(f([0,1,0]));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"+Ib.VH!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Note: the \+
same, this leads me to the conclusion that this graph is symmetric....
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 215 "The points [1,0,0] [-1,0,0] \+
[0,0,1] [0,0,-1] are not negative definite, positive definite, not sem
i-negative or semi-positive, therefore they are saddle points. Also, t
he point [0,0,0] is only positive semi-definite" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 27 "H2:=evalf(Hess(f,[0,0,0]));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#H2G-%'MATRIXG6#7%7%$\"#S\"\"!F,F,7%F,$\"$g\"F,F,7%
F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "definite(H2,'posi
tive_semidef');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 49 "The rest are false. Thus the origin is a \+
minimum." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "evalf(f([0,0,0]
));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 41 "Here are some examples of newtons method:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "newtoncritpts(f,[1,0,0],10,0.0001);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'VECTORG6#7%$\"\"\"\"\"!F)F)$\"+
C))edt!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"\"\"\"
\"!F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "We found out all of th
e values by guessing. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 179 "Thus I
 would say, there are no other solutions. Any other values larger than
 1 or so in the inital approxmation leads to a divergant case. So our \+
global maximums and minimums are:" }}{PARA 0 "" 0 "" {TEXT -1 60 "Max:
 [0,1,0] and [0,-1,0] value at these points: 29.43035530" }}{PARA 0 "
" 0 "" {TEXT -1 36 "Min: [0,0,0] value at this point:  0" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 135 "Notes: anything that is in form [a,b,c] \+
where a b and c are not 0 diverges. anything in the form [0,0,a] or [0
,a,0] or [a,0,0] either: " }}{PARA 0 "" 0 "" {TEXT -1 38 "1.) converge
s to the origin in a < 1, " }}{PARA 0 "" 0 "" {TEXT -1 26 "2.) IS a so
lution is a = 1" }}{PARA 0 "" 0 "" {TEXT -1 21 "3.) diverges if a > 1
" }}{PARA 0 "" 0 "" {TEXT -1 71 "This seems to be the case. Any value,
 even small ones in a,b,c diverge:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "newtoncritpts(f,[0.1,0.1,0.1],10,0.00001);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$-%'VECTORG6#7%$!*0(>&R\"!#5$!)f:WYF)$\"+i[mL
BF)$\"+W`h(R&!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'VECTORG6#7%$!*<
+JW#!#6$!*\"R:pR!#7$\"+[Bu,w!#5$\"+s@E&z*!#9" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$-%'VECTORG6#7%$!*2Z?E&!#7$\"+(R(=SC!#8$\"+h2\"Rd)!#5$\"
+'3&ye5!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'VECTORG6#7%$!+FqMHE!#
8$\"+(G6NJ\"F)$\"+WWCE*)!#5$\"+F56#Q\"!#;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$-%'VECTORG6#7%$!+=CEW8!#8$\"+J+F@n!#9$\"+\"\\$R/\"*!#5$
\"+r+9)y\"!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'VECTORG6#7%$!+Iwn1
o!#9$\"+P(QLS$F)$\"+(HWP>*!#5$\"+S27yA!#=" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$-%'VECTORG6#7%$!+S$4gU$!#9$\"+rY+8<F)$\"+BdXQ#*!#5$\"+6
/AwG!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'VECTORG6#7%$!+AH%)=<!#9$
\"+5Y@%f)!#:$\"+Kg\"3E*!#5$\"+aGp8O!#?" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$-%'VECTORG6#7%$!+)yE!4')!#:$\"+%R8XI%F)$\"+nn*>F*!#5$\"+yb!)GX!#
@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'VECTORG6#7%$!+;PC3V!#:$\"+e=7a
@F)$\"+1sex#*!#5$\"+.+Poc!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%FAIL
G$\"(R/f&!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$!+;PC3V
!#:$\"+e=7a@F)$\"+1sex#*!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "Th
is this leads me to believe there is only 7 solutions, 2 global maximu
ms and 1 global minimum." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Quest
ion #3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 368 "data:=[[13,254.07],[15,252.
10],[23,250.07],[31,249.03],[33,247.10],[34,246.13],[37,246.07],[42,24
6.03],[42,244.10],[43,242.10],[44,241.10],[45,241.07],[54,239.10],[54,
238.00],[57,237.03],[58,234.08],[62,234.07],[64,234.02],[65,233.10],[6
6,231.05],[67,231.02],[75,231.00],[75,229.07],[79,229.00],[80,228.93],
[81,228.88],[81,228.67],[81,227.55],[85,226.53],[93,224.65]];" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%dataG7@7$\"#8$\"&2a#!\"#7$\"#:$\"&5
_#F*7$\"#B$\"&2]#F*7$\"#J$\"&.\\#F*7$\"#L$\"&5Z#F*7$\"#M$\"&8Y#F*7$\"#
P$\"&2Y#F*7$\"#U$\"&.Y#F*7$FD$\"&5W#F*7$\"#V$\"&5U#F*7$\"#W$\"&5T#F*7$
\"#X$\"&2T#F*7$\"#a$\"&5R#F*7$FW$\"&+Q#F*7$\"#d$\"&.P#F*7$\"#e$\"&3M#F
*7$\"#i$\"&2M#F*7$\"#k$\"&-M#F*7$\"#l$\"&5L#F*7$\"#m$\"&0J#F*7$\"#n$\"
&-J#F*7$\"#v$\"&+J#F*7$Fdp$\"&2H#F*7$\"#z$\"&+H#F*7$\"#!)$\"&$*G#F*7$
\"#\")$\"&))G#F*7$Fcq$\"&nG#F*7$Fcq$\"&bF#F*7$\"#&)$\"&`E#F*7$\"#$*$\"
&lC#F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Least square linear cur
ves:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 228 "A:=matrix(30,2,[[1
,13],[1,15],[1,23],[1,31],[1,33],[1,34],[1,37],[1,42],[1,42],[1,43],[1
,44],[1,45],[1,54],[1,54],[1,57],[1,58],[1,62],[1,64],[1,65],[1,66],[1
,67],[1,75],[1,75],[1,79],[1,80],[1,81],[1,81],[1,81],[1,85],[1,93]]);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7@7$\"\"\"\"#87$
F*\"#:7$F*\"#B7$F*\"#J7$F*\"#L7$F*\"#M7$F*\"#P7$F*\"#UF87$F*\"#V7$F*\"
#W7$F*\"#X7$F*\"#aF@7$F*\"#d7$F*\"#e7$F*\"#i7$F*\"#k7$F*\"#l7$F*\"#m7$
F*\"#n7$F*\"#vFP7$F*\"#z7$F*\"#!)7$F*\"#\")FVFV7$F*\"#&)7$F*\"#$*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 228 "Y:=matrix(30,1,[254.07,252.
10,250.07,249.03,247.10,246.13,246.07,246.03,244.10,242.10,241.10,241.
07,239.10,238.00,237.03,234.08,234.07,234.02,233.10,231.05,231.02,231.
00,229.07,229.00,228.93,228.88,228.67,227.55,226.53,224.65]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG-%'MATRIXG6#7@7#$\"&2a#!\"#7#$\"
&5_#F,7#$\"&2]#F,7#$\"&.\\#F,7#$\"&5Z#F,7#$\"&8Y#F,7#$\"&2Y#F,7#$\"&.Y
#F,7#$\"&5W#F,7#$\"&5U#F,7#$\"&5T#F,7#$\"&2T#F,7#$\"&5R#F,7#$\"&+Q#F,7
#$\"&.P#F,7#$\"&3M#F,7#$\"&2M#F,7#$\"&-M#F,7#$\"&5L#F,7#$\"&0J#F,7#$\"
&-J#F,7#$\"&+J#F,7#$\"&2H#F,7#$\"&+H#F,7#$\"&$*G#F,7#$\"&))G#F,7#$\"&n
G#F,7#$\"&bF#F,7#$\"&`E#F,7#$\"&lC#F," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "B:=matrix(2,1,[a,b]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"BG-%'MATRIXG6#7$7#%\"aG7#%\"bG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "l:=evalm(transpose(A)&*A);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"lG-%'MATRIXG6#7$7$\"#I\"%z;7$F+\"'Rx5" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "m:=evalm(transpose(A)&*Y);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG-%'MATRIXG6#7$7#$\"'sCr!\"#7#$\"
)L1MRF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "linsolve(l,m);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7$7#$\"+9-&>f#!\"(7#$!+A
g3yQ!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Least squares linear a
pproximation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "lin:=x->25
9.1950214-.3878086022*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$linG:6#
%\"xG6\"6$%)operatorG%&arrowGF(,&$\"+9-&>f#!\"(\"\"\"9$$!+Ag3yQ!#5F(F(
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Now for the least squares qua
dratic..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 374 "A:=matrix(30,
3,[[1,13,169],[1,15,225],[1,23,529],[1,31,961],[1,33,1089],[1,34,1156]
,[1,37,1369],[1,42,1764],[1,42,1764],[1,43,1849],[1,44,1936],[1,45,202
5],[1,54,2916],[1,54,2916],[1,57,3249],[1,58,3364],[1,62,3844],[1,64,4
096],[1,65,4225],[1,66,4356],[1,67,4489],[1,75,5625],[1,75,5625],[1,79
,6241],[1,80,6400],[1,81,6561],[1,81,6561],[1,81,6561],[1,85,7225],[1,
93,8649]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7@7%
\"\"\"\"#8\"$p\"7%F*\"#:\"$D#7%F*\"#B\"$H&7%F*\"#J\"$h*7%F*\"#L\"%*3\"
7%F*\"#M\"%c67%F*\"#P\"%p87%F*\"#U\"%k<F?7%F*\"#V\"%\\=7%F*\"#W\"%O>7%
F*\"#X\"%D?7%F*\"#a\"%;HFK7%F*\"#d\"%\\K7%F*\"#e\"%kL7%F*\"#i\"%WQ7%F*
\"#k\"%'4%7%F*\"#l\"%DU7%F*\"#m\"%cV7%F*\"#n\"%*[%7%F*\"#v\"%DcF]o7%F*
\"#z\"%Ti7%F*\"#!)\"%+k7%F*\"#\")\"%hlFfoFfo7%F*\"#&)\"%Ds7%F*\"#$*\"%
\\')" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "We can use our old defini
tions of Y to recompute..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
26 "l:=evalm(transpose(A)&*A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"
lG-%'MATRIXG6#7%7%\"#I\"%z;\"'Rx57%F+F,\"(*f(\\(7%F,F.\"*Ve\"*\\&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "m:=evalm(transpose(A)&*Y);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG-%'MATRIXG6#7%7#$\"'sCr!\"#7#$
\")L1MRF,7#$\"+*\\@@]#F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 
"linsolve(l,m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7#$\"
+idT/E!\"(7#$!+d]vPW!#57#$\"+Az\"3D&!#8" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 54 "And thus our least squares quadratic approximation is:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "quad:=x->260.4415762-.4437
755057*x+.5250817922e-3*x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%qua
dG:6#%\"xG6\"6$%)operatorG%&arrowGF(,($\"+idT/E!\"(\"\"\"9$$!+d]vPW!#5
*$F1\"\"#$\"+Az\"3D&!#8F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
61 "p1:=plot(data,x=13..100,y=200..260,style=line,symbol=circle):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "p2:=plot(lin(x),x=13..100,y=
200..260,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "p3
:=plot(quad(x),x=13..100,y=200..260,color=green):" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "display(p1,p
2,p3);" }}{PARA 13 "" 1 "" {INLPLOT "6'-%'CURVESG6&7@7$$\"#8\"\"!$\"1+
++++qSD!#87$$\"#:F*$\"1++++++@DF-7$$\"#BF*$\"1+++++q+DF-7$$\"#JF*$\"1+
++++I!\\#F-7$$\"#LF*$\"1++++++rCF-7$$\"#MF*$\"1+++++IhCF-7$$\"#PF*$\"1
+++++qgCF-7$$\"#UF*$\"1+++++IgCF-7$FM$\"1++++++TCF-7$$\"#VF*$\"1++++++
@CF-7$$\"#WF*$\"1++++++6CF-7$$\"#XF*$\"1+++++q5CF-7$$\"#aF*$\"1++++++
\"R#F-7$F^o$\"$Q#F*7$$\"#dF*$\"1+++++IqBF-7$$\"#eF*$\"1+++++!3M#F-7$$
\"#iF*$\"1+++++qSBF-7$$\"#kF*$\"1+++++?SBF-7$$\"#lF*$\"1++++++JBF-7$$
\"#mF*$\"1+++++]5BF-7$$\"#nF*$\"1+++++?5BF-7$$\"#vF*$\"$J#F*7$Fiq$\"1+
++++q!H#F-7$$\"#zF*$\"$H#F*7$$\"#!)F*$\"1+++++I*G#F-7$$\"#\")F*$\"1+++
++!))G#F-7$F[s$\"1+++++q'G#F-7$F[s$\"1+++++]vAF-7$$\"#&)F*$\"1+++++IlA
F-7$$\"#$*F*$\"1+++++]YAF--%'COLOURG6&%$RGBG$\"#5!\"\"F*F*-%'SYMBOLG6#
%'CIRCLEG-%&STYLEG6#%%LINEG-F$6$7S7$F($\"1+Sr&4N:a#F-7$$\"1++DT_j*[\"!
#9$\"1(HIz(3=MDF-7$$\"1+vo]eja;Fgu$\"1([fB,#yFDF-7$$\"1+](GH'>S=Fgu$\"
1_')y=ee?DF-7$$\"1+]7]n)p-#Fgu$\"15c<C>M8DF-7$$\"1+v$\\U*)G@#Fgu$\"1#=
H&eC81DF-7$$\"1+vo,]C&Q#Fgu$\"1.\"**4O[%*\\#F-7$$\"1+v==#4Pc#Fgu$\"1XZ
;lt_#\\#F-7$$\"1+v=mrF[FFgu$\"1lAPh'p`[#F-7$$\"1+vV-KDKHFgu$\"1CmU7\\B
yCF-7$$\"1++Dp^\\@JFgu$\"15R;Yf*3Z#F-7$$\"1+]i'fz\")G$Fgu$\"1I]o!yJWY#
F-7$$\"1,+v#yEeZ$Fgu$\"1S#*QhY:dCF-7$$\"1++DmWCkOFgu$\"1]Cahw%)\\CF-7$
$\"1++D<)>e%QFgu$\"1AfB8g!GW#F-7$$\"1+v=Jwq5SFgu$\"1I;(>_6kV#F-7$$\"1+
+D!>vn?%Fgu$\"1]*pL&y!)GCF-7$$\"1,++J,(GP%Fgu$\"1w%*o[lOACF-7$$\"1+voK
!ygc%Fgu$\"1L21!yt[T#F-7$$\"1****\\j(*4PZFgu$\"16NA59C3CF-7$$\"1+vo2xt
C\\Fgu$\"12!Q$\\Y'4S#F-7$$\"1+D1.MT.^Fgu$\"1e4<_a.%R#F-7$$\"1+](ouU)*G
&Fgu$\"1fw%=c0oQ#F-7$$\"1+vV\"3V5Y&Fgu$\"1uLgli;!Q#F-7$$\"1+]iqdqXcFgu
$\"1Whm()[+tBF-7$$\"1+v=1%=v$eFgu$\"1lzlGiclBF-7$$\"1+DJp=\\/gFgu$\"1l
S7a34fBF-7$$\"1+](QyF[='Fgu$\"1r(GAF(4_BF-7$$\"1++]HC8rjFgu$\"1^%3y@s[
M#F-7$$\"1+]Py^R`lFgu$\"1<5i6R!yL#F-7$$\"1+vofPuHnFgu$\"1U\"R>'\\'4L#F
-7$$\"1+]i#z[b#pFgu$\"1?kKu9PBBF-7$$\"1****\\B$)[,rFgu$\"1%QszQ[lJ#F-7
$$\"1++DE8M*G(Fgu$\"1H3$pGj#4BF-7$$\"1+v=nScfuFgu$\"1m#3E!>m-BF-7$$\"1
++v%[ick(Fgu$\"17'*)e%[W&H#F-7$$\"1*\\(o#fm2#yFgu$\"1Sd&z:a')G#F-7$$\"
1+vVR3!Q+)Fgu$\"1bqTKfb\"G#F-7$$\"1***\\7F]F=)Fgu$\"1'[:&>hhuAF-7$$\"1
+v=UP4q$)Fgu$\"1Hfby2NnAF-7$$\"1+++rx_]&)Fgu$\"1@cq\"R`.E#F-7$$\"1**\\
7Ew/N()Fgu$\"1.i*Hb(>`AF-7$$\"1,voj&R!=*)Fgu$\"10UCo45YAF-7$$\"1++]\\,
>'3*Fgu$\"1**)zQ%*z&RAF-7$$\"1+]i'p4*y#*Fgu$\"1$=hS61@B#F-7$$\"1+++d.G
^%*Fgu$\"1^^dJ9UDAF-7$$\"1+voKM1N'*Fgu$\"18h!e;%H=AF-7$$\"1+DJ'Gn4\")*
Fgu$\"1Al/jCZ6AF-7$$\"$+\"F*$\"1++!=hTT?#F--F`t6&FbtF*F*$\"*++++\"!\")
-F$6$7S7$F($\"1=y[MBhZDF-7$Feu$\"1GL:iXZRDF-7$F[v$\"1Nw=bYUKDF-7$F`v$
\"1Nq=a/`CDF-7$Fev$\"1>RdX/i;DF-7$Fjv$\"1\\#f>T%y3DF-7$F_w$\"1\"zl$G=b
,DF-7$Fdw$\"1BDr\"y&4%\\#F-7$Fiw$\"18:B5*>k[#F-7$F^x$\"11didU!)yCF-7$F
cx$\"1)*>e3x+rCF-7$Fhx$\"1\\8nb;<kCF-7$F]y$\"1l`a\"z5lX#F-7$Fby$\"1\"R
Qqnb)[CF-7$Fgy$\"1gbfJQ^TCF-7$F\\z$\"1![5#Hn([V#F-7$Faz$\"1I$>Mt@qU#F-
7$Ffz$\"1ci22\"*R?CF-7$F[[l$\"1VV]p=t7CF-7$F`[l$\"1f`0tx(fS#F-7$Fe[l$
\"1IZ#ez-')R#F-7$Fj[l$\"1J,ySWh\"R#F-7$F_\\l$\"1K()yl&eVQ#F-7$Fd\\l$\"
1QPLcvsxBF-7$Fi\\l$\"1d-PA'41P#F-7$F^]l$\"1ux-2SDjBF-7$Fc]l$\"1;;syB)o
N#F-7$Fh]l$\"1GLqKP.]BF-7$F]^l$\"1>P!zF%*HM#F-7$Fb^l$\"1/io<G9OBF-7$Fg
^l$\"1ou6%)oaHBF-7$F\\_l$\"1_&Rp[hAK#F-7$Fa_l$\"1L)*Gw&\\dJ#F-7$Ff_l$
\"10$pbgK)3BF-7$F[`l$\"136:2of-BF-7$F``l$\"1([&)4D9eH#F-7$Fe`l$\"1_zg/
cY*G#F-7$Fj`l$\"1Ct!)fQ'GG#F-7$F_al$\"1++ogMWwAF-7$Fdal$\"1\"fu-$zvpAF
-7$Fial$\"1L^59QNjAF-7$F^bl$\"1*p7+/SoD#F-7$Fcbl$\"1L!**e^:/D#F-7$Fhbl
$\"1tU#)[IaWAF-7$F]cl$\"1>*pl0\\yB#F-7$Fbcl$\"1+j$R\"\\*=B#F-7$Fgcl$\"
1\">k_#4eDAF-7$F\\dl$\"11'\\X&3d>AF-7$Fadl$\"1++_N%[J@#F--F`t6&FbtF*Fg
dlF*-%+AXESLABELSG6$%\"xG%\"yG-%%VIEWG6$;F(Fadl;$\"$+#F*$\"$g#F*" 2 
496 496 496 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 
10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 434 294 
0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "The year 2000 is re
presented by the value '100' in our example. thus evaluate the quadrat
ic at 100:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "quad(100);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+N%[J@#!\"(" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 70 "The value of the one mile run will be 221.31 seconds
 in the year 2000." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Question #4
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 59 "data:=[[-5,127],[-3,151],[-1,379],[1,426],[3,
460],[5,421]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dataG7(7$!\"&\"$F
\"7$!\"$\"$^\"7$!\"\"\"$z$7$\"\"\"\"$E%7$\"\"$\"$g%7$\"\"&\"$@%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "v:=vector(3);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"vG-%&arrayG6$;\"\"\"\"\"$7\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 187 "f:=v->vector([127-v[1]-v[2]*exp(v[
3]*(-5)),151-v[1]-v[2]*exp(v[3]*(-3)),379-v[1]-v[2]*exp(v[3]*(-1)),426
-v[1]-v[2]*exp(v[3]*(1)),460-v[1]-v[2]*exp(v[3]*(3)),421-v[1]-v[2]*exp
(v[3]*(5))]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"vG6\"6$%)
operatorG%&arrowGF(-%'vectorG6#7(,(\"$F\"\"\"\"&9$6#F2!\"\"*&&F46#\"\"
#F2-%$expG6#,$&F46#\"\"$!\"&F2F6,(\"$^\"F2F3F6*&F8F2-F<6#,$F?!\"$F2F6,
(\"$z$F2F3F6*&F8F2-F<6#,$F?F6F2F6,(\"$E%F2F3F6*&F8F2-F<6#F?F2F6,(\"$g%
F2F3F6*&F8F2-F<6#,$F?FAF2F6,(\"$@%F2F3F6*&F8F2-F<6#,$F?\"\"&F2F6F(F(" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "g:=v->(f(v)[1])^2 + (f(v)
[2])^2 + (f(v)[3])^2 + (f(v)[4])^2 + (f(v)[5])^2 + (f(v)[6])^2;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG:6#%\"vG6\"6$%)operatorG%&arrowG
F(,.*$&-%\"fG6#9$6#\"\"\"\"\"#F4*$&F/6#F5F5F4*$&F/6#\"\"$F5F4*$&F/6#\"
\"%F5F4*$&F/6#\"\"&F5F4*$&F/6#\"\"'F5F4F(F(" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 5 "g(v);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,.*$,(\"$F
\"\"\"\"&%\"vG6#F'!\"\"*&&F)6#\"\"#F'-%$expG6#,$&F)6#\"\"$!\"&F'F+F/F'
*$,(\"$^\"F'F(F+*&F-F'-F16#,$F4!\"$F'F+F/F'*$,(\"$z$F'F(F+*&F-F'-F16#,
$F4F+F'F+F/F'*$,(\"$E%F'F(F+*&F-F'-F16#F4F'F+F/F'*$,(\"$g%F'F(F+*&F-F'
-F16#,$F4F6F'F+F/F'*$,(\"$@%F'F(F+*&F-F'-F16#,$F4\"\"&F'F+F/F'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "h:=v->vector([(-v[1]-v[2]*e
xp(-5*v[3])+127)^2+(-v[1]-v[2]*exp(-3*v[3])+151)^2+(-v[1]-v[2]*exp(-v[
3])+379)^2+(-v[1]-v[2]*exp(v[3])+426)^2+(-v[1]-v[2]*exp(3*v[3])+460)^2
+(-v[1]-v[2]*exp(5*v[3])+421)^2]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#
>%\"hG:6#%\"vG6\"6$%)operatorG%&arrowGF(-%'vectorG6#7#,.*$,(\"$F\"\"\"
\"&9$6#F4!\"\"*&&F66#\"\"#F4-%$expG6#,$&F66#\"\"$!\"&F4F8F<F4*$,(\"$^
\"F4F5F8*&F:F4-F>6#,$FA!\"$F4F8F<F4*$,(\"$z$F4F5F8*&F:F4-F>6#,$FAF8F4F
8F<F4*$,(\"$E%F4F5F8*&F:F4-F>6#FAF4F8F<F4*$,(\"$g%F4F5F8*&F:F4-F>6#,$F
AFCF4F8F<F4*$,(\"$@%F4F5F8*&F:F4-F>6#,$FA\"\"&F4F8F<F4F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "g([a,b,c]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,.*$,(\"$F\"\"\"\"%\"aG!\"\"*&%\"bGF'-%$expG6#,$%\"cG!
\"&F'F)\"\"#F'*$,(\"$^\"F'F(F)*&F+F'-F-6#,$F0!\"$F'F)F2F'*$,(\"$z$F'F(
F)*&F+F'-F-6#,$F0F)F'F)F2F'*$,(\"$E%F'F(F)*&F+F'-F-6#F0F'F)F2F'*$,(\"$
g%F'F(F)*&F+F'-F-6#,$F0\"\"$F'F)F2F'*$,(\"$@%F'F(F)*&F+F'-F-6#,$F0\"\"
&F'F)F2F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Some educated guesse
s:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(g([420,0.001,1]
));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+r9E:;!\"%" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 31 "evalf(Grad(g,[420,0.00001,1]));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+KM+76!\"'$!+7*R\\(=F)$!+iu
Jlk!#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Idea: compute the gradi
ent, then set it equal to 0. Then work it out more." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 17 "gr:=grad(g(v),v);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%#grG-%'VECTORG6#7%,2!%GR\"\"\"&%\"vG6#F+\"#7*&&F-6#\"
\"#F+-%$expG6#,$&F-6#\"\"$!\"&F+F3*&F1F+-F56#,$F8!\"$F+F3*&F1F+-F56#,$
F8!\"\"F+F3*&F1F+-F56#F8F+F3*&F1F+-F56#,$F8F:F+F3*&F1F+-F56#,$F8\"\"&F
+F3,.*&,(\"$F\"F+F,FEF0FEF+F4F+!\"#*&,(\"$^\"F+F,FEF<FEF+F=F+FV*&,(\"$
z$F+F,FEFAFEF+FBF+FV*&,(\"$E%F+F,FEFFFEF+FGF+FV*&,(\"$g%F+F,FEFIFEF+FJ
F+FV*&,(\"$@%F+F,FEFMFEF+FNF+FV,.*(FTF+F1F+F4F+\"#5*(FXF+F1F+F=F+\"\"'
*(FenF+F1F+FBF+F3*(FhnF+F1F+FGF+FV*(F[oF+F1F+FJF+!\"'*(F^oF+F1F+FNF+!#
5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "fsolve(\{gr[1]=0,gr[2]
=0,gr[3]=0\},\{v[1],v[2],v[3]\},\{v[1]=300..1000,v[2]=-1..1,v[3]=-500.
.500\});" }}{PARA 8 "" 1 "" {TEXT -1 42 "Error, (in fsolve/gensys) did
 not converge" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Ok, didn't work.
 Lets try newtons method instead." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 678 "t:=v->vector([-3928+12*v[1]+2*v[2]*exp(-5*v[3])+2*v[
2]*exp(-3*v[3])+2*v[2]*exp(-v[3])+2*v[2]*exp(v[3])+2*v[2]*exp(3*v[3])+
2*v[2]*exp(5*v[3]), -2*(127-v[1]-v[2]*exp(-5*v[3]))*exp(-5*v[3])-2*(15
1-v[1]-v[2]*exp(-3*v[3]))*exp(-3*v[3])-2*(379-v[1]-v[2]*exp(-v[3]))*ex
p(-v[3])-2*(426-v[1]-v[2]*exp(v[3]))*exp(v[3])-2*(460-v[1]-v[2]*exp(3*
v[3]))*exp(3*v[3])-2*(421-v[1]-v[2]*exp(5*v[3]))*exp(5*v[3]), 10*(127-
v[1]-v[2]*exp(-5*v[3]))*v[2]*exp(-5*v[3])+6*(151-v[1]-v[2]*exp(-3*v[3]
))*v[2]*exp(-3*v[3])+2*(379-v[1]-v[2]*exp(-v[3]))*v[2]*exp(-v[3])-2*(4
26-v[1]-v[2]*exp(v[3]))*v[2]*exp(v[3])-6*(460-v[1]-v[2]*exp(3*v[3]))*v
[2]*exp(3*v[3])-10*(421-v[1]-v[2]*exp(5*v[3]))*v[2]*exp(5*v[3])]);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"tG:6#%\"vG6\"6$%)operatorG%&arrowG
F(-%'vectorG6#7%,2!%GR\"\"\"&9$6#F2\"#7*&&F46#\"\"#F2-%$expG6#,$&F46#
\"\"$!\"&F2F:*&F8F2-F<6#,$F?!\"$F2F:*&F8F2-F<6#,$F?!\"\"F2F:*&F8F2-F<6
#F?F2F:*&F8F2-F<6#,$F?FAF2F:*&F8F2-F<6#,$F?\"\"&F2F:,.*&,(\"$F\"F2F3FL
F7FLF2F;F2!\"#*&,(\"$^\"F2F3FLFCFLF2FDF2Fgn*&,(\"$z$F2F3FLFHFLF2FIF2Fg
n*&,(\"$E%F2F3FLFMFLF2FNF2Fgn*&,(\"$g%F2F3FLFPFLF2FQF2Fgn*&,(\"$@%F2F3
FLFTFLF2FUF2Fgn,.*(FenF2F8F2F;F2\"#5*(FinF2F8F2FDF2\"\"'*(F\\oF2F8F2FI
F2F:*(F_oF2F8F2FNF2Fgn*(FboF2F8F2FQF2!\"'*(FeoF2F8F2FUF2!#5F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "newtonroots(t,[346.1041099, \+
-3.413518858, .6965486345],100,0.001);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'VECTORG6#7%$\"+%Qg0P$!\"($!+#o<%yN!\"*$\"+?mQga!#5" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+BHR>L!\"($!+&*e$>x$!\"*$\"
+$p.im$!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+:)oFH$
!\"($!+$e-U\\$!\"*$\"+w;aZ;!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'V
ECTORG6#7%$\"+69JvK!\"($!+e6W(4\"!\"*$\"*-?.w\"!#5" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+4Q9tK!\"($\")tb#\\\"!\"*$!)M@b7!#6
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+6LLtK!\"($\"'?NA
!#6$!&+S*!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+LLLt
K!\"($\"#E!#:$!)BtpM!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6
#7%$\"+LLLtK!\"($\"#E!#:$!)BtpM!#=" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 68 "l:=x->327.3333333+(.171e-1210^(-12))*exp((-.2715400e-
1010^(-10))*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"lG:6#%\"xG6\"6$
%)operatorG%&arrowGF(,&$\"+LLLtK!\"(\"\"\"-%$expG6#,$9$$!+iiS)e%\"&'45
$\"+Z*H(*f\"\"&?X\"F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "
l:=x->327.3333333+(0.171*(10^(-12)))*exp((-.2715400*(10^(-10)))*x);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"lG:6#%\"xG6\"6$%)operatorG%&arrow
GF(,&$\"+LLLtK!\"(\"\"\"-%$expG6#,$9$$!+++S:F!#?$\"++++5<!#AF(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "p1:=plot(data,x=-6..6,y=100.
.500,style=point):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "p2:=p
lot(l(x),x=-6..6,y=100..500):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 17 "display(\{p1,p2\});" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$
7S7$$!\"'\"\"!$\"1-++LLLtK!#87$$!1+++]TVQd!#:F+7$$!1++]-r%3^&F1F+7$$!1
+++l;!\\D&F1F+7$$!1+++b'fs*\\F1F+7$$!1++]s@%3u%F1F+7$$!1++]U.6.XF1F+7$
$!1++]-G&pD%F1F+7$$!1++]AjP-SF1F+7$$!1++]sih[PF1F+7$$!1+++qGf([$F1F+7$
$!1+++:LodKF1F+7$$!1+++5'f))*HF1F+7$$!1+++]J(*QFF1F+7$$!1+++!RC&)[#F1F
+7$$!1++]AH4hAF1F+7$$!1+++5\\l!*>F1F+7$$!1+++S%e:w\"F1F+7$$!1++]#yk]\\
\"F1F+7$$!1+++SF<f7F1F+7$$!1++]#yh.+\"F1F+7$$!1(****\\ZD\"Rv!#;F+7$$!1
'*****\\ion\\FhoF+7$$!1'****\\K-jg#FhoF+7$$!16,++]2Bf!#=F+7$$\"1*****
\\xgke#FhoF+7$$\"1)****\\-V&*)[FhoF+7$$\"1.++]\\$pP(FhoF+7$$\"1'******
>am%**FhoF+7$$\"1+++:B1Y7F1F+7$$\"1++]P<I*[\"F1F+7$$\"1+++XwPf<F1F+7$$
\"1******fG0-?F1F+7$$\"1+++]/;hAF1F+7$$\"1****\\P/&f\\#F1F+7$$\"1,++5z
j_FF1F+7$$\"1****\\<3;%*HF1F+7$$\"1++]Z=iYKF1F+7$$\"1******\\'[M\\$F1F
+7$$\"1****\\PM&=v$F1F+7$$\"1,++gzs+SF1F+7$$\"1+++0\"Q_D%F1F+7$$\"1,+]
x2k2XF1F+7$$\"1,++?EdRZF1F+7$$\"1+++&o#R0]F1F+7$$\"1+++?`9V_F1F+7$$\"1
++]<#Rm\\&F1F+7$$\"1++]A_ERdF1F+7$$\"\"'F*F+-%'COLOURG6&%$RGBG$\"#5!\"
\"F*F*-F$6%7(7$$!\"&F*$\"$F\"F*7$$!\"$F*$\"$^\"F*7$$FauF*$\"$z$F*7$$\"
\"\"F*$\"$E%F*7$$\"\"$F*$\"$g%F*7$$\"\"&F*$\"$@%F*F[u-%&STYLEG6#%&POIN
TG-%+AXESLABELSG6$%\"xG%\"yG-%%VIEWG6$;F(Fit;$\"$+\"F*$\"$+&F*" 2 496 
496 496 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 
10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 145 -20656 0 
0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 727 "Even though this is \+
not a minimum, i do not know if we can get the equations to converge w
ithin our lifetimes. The solutions we have found are saddle points. Ne
ither maximums nor minimums. We computed over 600 seconds of time for \+
this question #4. The equations are extremely complex and possibly do \+
not have a minimal solution that can be computed with any known numeri
cal methods. However it seems like a should be near 400 or so, and b b
e a negative fraction, and c be some other value. However, when plugge
d into the newton roots routine, we derive a curve which looks like th
e one obtained above. This brings us to the conclusion that there is n
o minimum computable via newton root or other numerical computation me
thods. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "104 \+
0 0" 727 }{VIEWOPTS 1 1 0 1 1 1803 }