{VERSION 5 0 "IBM INTEL NT" "5.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 13 "Project \+
Three" }}{PARA 256 "" 0 "" {TEXT -1 27 "Maxima, Minima, and Saddles" }
}{PARA 256 "" 0 "" {TEXT -1 56 "Authors: Sammuel Robles Jr. , Brandon \+
Johns , Bar Nguyen" }}{PARA 256 "" 0 "" {TEXT -1 27 "Math 253 Section \+
201 Honors" }}{PARA 0 "" 0 "" {TEXT -1 379 "     This maplet is design
ed to test ones knowledge in computing maximum, minimum, and saddle po
int values with their corresponding coordinates.  Our maplet will rand
omly generate a three dimensional function for which the user is expec
ted to calculate the various points and check them with Maple.  To see
 our maplet, click on the triple exclamation mark icon on the menu abo
ve." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "Project 3" }}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 8 "Start up" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
103 "with(Maplets): with(Maplets[Tools]): with(Maplets[Elements]): wit
h(plots): with(student): with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "StartEngine();" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
6 "Maplet" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2567 "MaxMin:= Mapl
et(onstartup=RunWindow(MAIN),\nWindow[MAIN](title=\"Maxima and Minima \+
and Saddles\",\n  [ [ \"Find the critical points for the following fun
ction and classify each as a maximum, minimum or saddle point.\"],\n  \+
  [ Button(\"Show Function\", Evaluate(function=\"show\")),\n      Mat
hMLViewer[showf]('width'=300, 'height'=50)\n    ],\n    [ Button(\"Sho
w Me the Graph\", 'onclick'='Plot_It'),\n      \"    \",\n      Button
(\"Help\", RunWindow(Help)),\n      \"    \"\n    ],\n    [ \"Enter an
swers in exact form, e.g. [1+sqrt(3), 5], or as decimals, e.g. [2.732,
 5.], with at least four significant figures.\"\n    ],\n    [ \"Local
 minimum  \", TextField[min](\"[    ,    ]\"),\n      [Button(\"Check
\", Evaluate(function=\"check_min\"))],\n      [Button(\"Show Answer\"
, Evaluate(function=\"ans1\"))]\n    ],\n    [ \"Local maximum\", Text
Field[max](\"[    ,    ]\"),\n      [Button(\"Check\", Evaluate(functi
on=\"check_max\"))],\n      [Button(\"Show Answer\", Evaluate(function
=\"ans2\"))]\n    ],\n    [ \"Saddle Point     \", TextField[saddle](
\"[    ,    ]\"),\n      [Button(\"Check\", Evaluate(function=\"check_
saddle\"))],\n      [Button(\"Show Answer\", Evaluate(function=\"ans3
\"))]\n    ],\n    [ TextField[reply1](width=50,editable=false),\n    \+
  Button(\"Done\", Shutdown())\n    ]\n  ]  \n),\nWindow[PlotIt](title
=\"The Plot\",\n  [ [ MathMLViewer[showfplot]('width'=300, 'height'=50
)\n    ],\n    [ [ [ \"Enter your intervals (in the form a..b):\"],\n \+
       [ \"x-Interval:\",\n          TextField['interv1']('width'=15)]
,\n        [ \"y-Interval:\",\n          TextField['interv2']('width'=
15)],\n        [ \"z-Interval:\",\n          TextField['interv3']('wid
th'=15)],\n        [ Button(\"Plot It\", Evaluate(function=\"plot_grap
h\")),\n          \"   \",\n          Button(\"Close\", CloseWindow(Pl
otIt))]\n      ],\n      [ Plotter[myplot](width=300,height=300)\n    \+
  ]\n    ]\n  ]\n),\nWindow[Help](title=\"HELP\",\n  [ [\"How to Deter
mine Maxima, Minima, and Saddle Points:\"],\n    \"\",\n    TextBox(\"
Step1:  Compute the first partial derivatives, fx and fy, equate both \+
of these to zero and solve for the critical points.\", width=47, heigh
t=3),\n    TextBox(\"Step 2: Compute the second partial derivatives, f
xx, fyy and fxy, and compute D=fxx*fyy-fxy^2.\", width=47, height=2),
\n    TextBox(\"Step 3: For each critical point evaluate fxx and D. If
 D>0 and fxx>0, then the critical point is a local minimum. If D>0 and
 fxx<0 then the critical point is a local maximum.  If D<0 then the cr
itical point is a saddle point. Otherwise the Second Derivative Test f
ails.\", width=47, height=6)\n  ]\n),\nAction[Plot_It](RunWindow(PlotI
t), Evaluate(function=\"show_f_plot\")\n)\n):" }}}}{SECT 0 {PARA 4 "" 
0 "" {TEXT -1 10 "Procedures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
952 "show:=proc()\nlocal rand4,rand6, r,s,x0, a,c,d, h, eqs, sol, fxx,
fyy,fxy,Df, i, sadcount;\nglobal f, maximum, minimum, saddle1, saddle2
;\nrand4:=rand(1..4);\nrand6:=rand(1..6);\nr:=rand4();\ns:=rand4();\nx
0:=rand6();\nwhile x0<s  do \n  x0:=rand6();\nend do;\na:=r^2;\nc:=-3*
r^2*x0^2;\nd:=3*sqrt(x0^2-s^2);\nf:=a*x^3+3*x*y^2+c*x+d*y^2;\nSet('sho
wf' = MathML[Export]('f'=f)):\nh:=unapply(f,x,y);\neqs:=equate(grad(f,
[x,y]),[0,0]);\nsol:=solve(eqs,\{x,y\});\nfxx:=D[1,1](h)(x,y);\nfyy:=D
[2,2](h)(x,y);\nfxy:=D[1,2](h)(x,y);\nDf:=fxx*fyy-fxy^2;\nsadcount:=1;
\nfor i from 1 to 4 do \n  if evalf(subs(sol[i],Df))>0 and evalf(subs(
sol[i],fxx))>0\n    then minimum:=subs(sol[i],[x,y]);\n  elif evalf(su
bs(sol[i],Df))>0 and evalf(subs(sol[i],fxx))<0\n    then maximum:=subs
(sol[i],[x,y]);\n  elif evalf(subs(sol[i],Df))<0\n    then if sadcount
=1 \n           then saddle1:=subs(sol[i],[x,y]); sadcount:=2\n       \+
    else saddle2:=subs(sol[i],[x,y]);\n         end if;\n  end if;\nen
d do\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "show_f_p
lot:=proc()\nglobal f;\nSet('showfplot' = MathML[Export]('f'=f))\nend \+
proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 301 "plot_graph:=proc(
)\nlocal p1,userinterv1,userinterv2,userinterv3;\nglobal f;\nuserinter
v1:=Get(thismaplet,interv1::range):\nuserinterv2:=Get(thismaplet,inter
v2::range):\nuserinterv3:=Get(thismaplet,interv3::range):\nSet('myplot
'=plot3d(f, x=userinterv1, y=userinterv2, view=userinterv3, axes=box))
\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 286 "check_min:=
proc()\nlocal usermin;\nglobal minimum;\nusermin:= Get(thismaplet, min
::[realcons,realcons]):\nif evalf[4](usermin[1]-minimum[1])=0. and eva
lf[4](usermin[2]-minimum[2])=0.\n  then Set(reply1=\"He Shoots, HE SCO
RES!!.\")\n  else Set(reply1=(\"Not quite, try again.\"))\nend if;\nen
d proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "check_max:=proc
()\nlocal usermax;\nglobal maximum;\nusermax:= Get(thismaplet, max::[r
ealcons,realcons]):\nif evalf[4](usermax[1]-maximum[1])=0. and evalf[4
](usermax[2]-maximum[2])=0.\n  then Set(reply1=\"Thats one point for y
ou!\")\n  else Set(reply1=\"Think about it some more.\")\nend if;\nend
 proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 440 "check_saddle:=pr
oc()\nlocal usersaddle;\nglobal saddle1, saddle2;\nusersaddle:= Get(th
ismaplet,saddle::[realcons,realcons]):\nif (evalf[4](usersaddle[1]-sad
dle1[1])=0. and\n    evalf[4](usersaddle[2]-saddle1[2])=0.)\nor (evalf
[4](usersaddle[1]-saddle2[1])=0. and\n    evalf[4](usersaddle[2]-saddl
e2[2])=0.)\n  then Set(reply1=\"Great Job, go ahead and try another fu
nction!\")\n  else Set(reply1=\"Don't give up just yet, try again!\")
\nend if;\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "an
s1:=proc()\nglobal minimum;\nSet(reply1=cat(\"The correct minimum is \+
\", convert(minimum,string)));\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 107 "ans2:=proc()\nglobal maximum;\nSet(reply1=cat(\"Th
e correct maximum is \", convert(maximum,string)));\nend proc:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "ans3:=proc()\nglobal saddle
1, saddle2;\nSet(reply1=cat(\"The correct saddles are \", convert(sadd
le1,string), \" and \", convert(saddle2,string)));\nend proc:" }}}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Display(MaxMin);" }}}{EXCHG 
}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 1 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 
2 33 1 1 }