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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "# A Maple Session to
 compute the next approximation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 28 "with(plots):with(plottools):" }}{PARA 7 "" 1 "" {TEXT -1 50 "W
arning, the name changecoords has been redefined\n" }}{PARA 7 "" 1 "" 
{TEXT -1 43 "Warning, the name arrow has been redefined\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "f:=x^2-x*y+y^2; # This is the funct
ion on problems #7 & #5 of test 2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"fG,(*$)%\"xG\"\"#\"\"\"F**&F(F*%\"yGF*!\"\"*$)F,F)F*F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "P0:=[1,0]; # This is roughly the Q \+
in #5 of test 2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P0G7$\"\"\"\"\"!
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "x0:=P0[1]; y0:=P0[2]; #
 P0 is a Maple list, x0 is the first element, y0 the 2nd" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#x0G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%#y0G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "z0:=eval(f,
\{x=x0,y=y0\}); #For later use in the 3d plot" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#z0G\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
72 "f_x:=diff(f,x); f_y:=diff(f,y); # This the Maple way to take deriv
atives" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$f_xG,&*&\"\"#\"\"\"%\"xGF
(F(%\"yG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$f_yG,&%\"xG!\"\"*&
\"\"#\"\"\"%\"yGF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "a:=
eval(f_x,\{x=x0,y=y0\}); # This computes f_x(x0,y0)" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"aG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 52 "b:=eval(f_y,\{x=x0,y=y0\}); # This computes f_y(x0,y0)" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"bG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 56 "u:=[-a, -b]; # this is the direction of steepest desc
ent" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG7$!\"#\"\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "PuLine:=[x0-a*t,y0-b*t,t=0..1]; #So
mething you can put into plot" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Pu
LineG7%,&\"\"\"F'*&\"\"#F'%\"tGF'!\"\"F*/F*;\"\"!F'" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 286 "display(\ncontourplot(f,x=-1..2,y=-1..2,
contours=[z0, -1/2,0,1,2,4,8]), #Note contours\nplot(PuLine,color=gree
n), # This is the Puline\ntextplot([x0,y0,\"P0\"],color=green), # The \+
text label P0\narrow([x0,y0],vector(u),.1,.2,.2,color=red) # An arrow,
 just to be fancy\n); # end display command" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 53 "g:=eval(f,\{x=x0-a*t,y=y0-b*t\}); # g(t) = f(x(t),
y(t))" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG,(*$),&\"\"\"F)*&\"\"#F
)%\"tGF)!\"\"F+F)F)*&F(F)F,F)F-*$)F,F+F)F)" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 45 "diff(g,t); # looking for critical points of g" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"&!\"\"*&\"#9\"\"\"%\"tGF(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "expand(diff(g,t)); # force m
aple to expand the powers" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"&!
\"\"*&\"#9\"\"\"%\"tGF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
47 "solve(diff(g,t)=0,t); # find the critical point" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6##\"\"&\"#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
59 "t1:=solve(diff(g,t)=0,t); # A way to pull the first one off" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t1G#\"\"&\"#9" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 80 "# If f was a quadratic instead of a cubic, t0
 would be a fraction and not a surd" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "x1:=eval(x0-a*t,\{t=t1\}); # new approximation's x va
lue" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G#\"\"#\"\"(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "y1:=eval(y0-b*t,\{t=t1\}); # new ap
proximation's y value" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G#\"\"&
\"#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "P1:=[x1,y1]; # new \+
approximation" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P1G7$#\"\"#\"\"(#
\"\"&\"#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "z1:=eval(f,\{x
=x1,y=y1\}); # value of f(P1) for contourplot below" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#z1G#\"\"$\"#G" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 283 "display(\ncontourplot(f,x=0..1.5,y=0..1.5,contours=[
z0, z1,-1/2,0,1,2,4,8]), #Notice z0 z1\nline(P0,P1,color=green), # lin
e seqment P0 to P1\ntextplot([x0,y0,\"P0\"],color=green), # starting p
oint\ntextplot([x1,y1,\"P1\"],color=blue), # next approximation\nscali
ng=constrained\n); #end display" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "23 0
" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }