Derek OudieImproper Integralsrestart:with(Maplets[Tools]): with(Maplets[Elements]): with(plots): with(plottools):
StartEngine();randomize():lightorange:="#FFB300":
lightcyan:="#DDFFFF":ColorFooter:=lightcyan:
FontFooter:=Font("helvetica",10):rand22:=rand(-2..2):
rand14:=rand(1..4):
rand02:=rand(0..2):ps:=[1,2,3,4,1/2,1/3,2/3]:
randp:=rand(1..nops(ps)):ask:=proc()
global ImpType, lbound, ubound, p, grand, intplot, IntValue;
local q, dbound;
ImpType:=rand14();
p:=ps[randp()];
q:=rand02();
if ImpType=1 then
lbound:=rand22();
ubound:=lbound+rand14();
if lbound=0 then
grand:=(-1)^rand02()*(x-rand14())^q/(x-lbound)^p;
else
grand:=(-1)^rand02()*x^q/(x-lbound)^p;
end if
elif ImpType=2 then
lbound:=rand22();
ubound:=lbound+rand14();
if ubound=0 then
grand:=(-1)^rand02()*(x-rand14())^q/(x-ubound)^p;
else
grand:=(-1)^rand02()*x^q/(x-ubound)^p;
end if
elif ImpType=3 then
lbound:=-infinity;
ubound:=rand22();
dbound:=ubound+rand14();
grand:=(-1)^rand02()*1/(x-dbound)^p;
elif ImpType=4 then
lbound:=rand22();
ubound:=infinity;
dbound:=lbound-rand14();
grand:=(-1)^rand02()*1/(x-dbound)^p;
end if;
IntValue:=int(grand, x=lbound..ubound);
if lbound=-infinity then
if IntValue<0 then
intplot:=plot(grand,x=-20..ubound,y=-1..1,color=green,discont=true);
elif IntValue>0 then
intplot:=plot(grand,x=-20..ubound,y=-1..1,color=green,discont=true);
else
intplot:=plot(grand,x=-20..ubound,y=-10..10,color=green,discont=true);
end if;
elif ubound=infinity then
if IntValue<0 then
intplot:=plot(grand,x=lbound..20,y=-1..1,color=green,discont=true);
elif IntValue>0 then
intplot:=plot(grand,x=lbound..20,y=-1..1,color=green,discont=true);
else
intplot:=plot(grand,x=lbound..20,y=-10..10,color=green,discont=true);
end if;
else
intplot:=plot(grand,x=lbound..ubound,y=-10..10,color=green,discont=true);
end if;
Set(IntPlot=plot(0,x=-5..5,y=-10..10,color=black));
Set(INT=Int(grand,x=lbound..ubound));
Set(CON=false);
Set(DEFV(enabled)=false);
Set(TFDEFV="", TFDEFV(enabled)=false);
Set(DIV=false);
Set(INF=false, INF(enabled)=false);
Set(NIN=false, NIN(enabled)=false);
Set(NEI=false, NEI(enabled)=false);
Set(reply="");
end proc:plotit:=proc()
global intplot;
Set(IntPlot=intplot);
end proc:cod:=proc()
if Get(CON) then
Set(DEFV(enabled)=true);
Set(TFDEFV="", TFDEFV(enabled)=true);
Set(INF=false, INF(enabled)=false);
Set(NIN=false, NIN(enabled)=false);
Set(NEI=false, NEI(enabled)=false);
end if;
if Get(DIV) then
Set(DEFV(enabled)=false);
Set(TFDEFV="", TFDEFV(enabled)=false);
Set(INF=false, INF(enabled)=true);
Set(NIN=false, NIN(enabled)=true);
Set(NEI=false, NEI(enabled)=true);
end if;
end proc:check:=proc()
global IntValue;
local userIntValue;
if IntValue = infinity then
if Get(DIV) and Get(INF) then
Set(reply="Nice work kiddo.");
else
Set(reply="You need more practice. Try again.");
end if;
elif IntValue = -infinity then
if Get(DIV) and Get(NIN) then
Set(reply="Way to go.");
else
Set(reply="Not quite. Look at the antiderivative at each endpoint.");
end if;
else
userIntValue:=Get(TFDEFV::realcons);
if Get(CON) and IntValue=userIntValue then
Set(reply="Smooth sailing.");
else
Set(reply="Not this time.");
end if;
end if;
end proc:JSFHshow:=proc()
global IntValue;
if IntValue = infinity then
Set(DIV=true);
Set(INF=true,INF(enabled)=true);
Set(NIN(enabled)=true);
Set(NEI(enabled)=true);
Set(DEFV(enabled)=false);
Set(TFDEFV(enabled)=false);
elif IntValue = -infinity then
Set(DIV=true);
Set(NEI(enabled)=true);
Set(NIN=true,NIN(enabled)=true);
Set(INF(enabled)=true);
Set(DEFV(enabled)=false);
Set(TFDEFV(enabled)=false);
else
Set(CON=true);
Set(DEFV(enabled)=true);
Set(TFDEFV=IntValue,TFDEFV(enabled)=true);
end if;
Set(reply="Next time try it yourself.")
end proc:ImproperIntegrals:=Maplet(onstartup=RunWindow(MAIN),
Window[MAIN](title="Improper Integrals",
[ halign=none, valign=none, inset=0, spacing=0,
[ halign=none,
Button("New Problem", Evaluate(function="ask"), background=magenta),
HorizontalGlue(),
Button("Quit",Shutdown(), background=pink)
],
[ halign=center, inset=0, spacing=0,
[ valign=none, inset=0, spacing=0, border=true,
caption="Problem Statement",
[ halign=none,
"Determine if the following improper integral is convergent or divergent.",
HorizontalGlue()
],
[ halign=center,
MathMLViewer[INT](width=200, height=60, background=white)
],
[ halign=none,
"If convergent, compute it.",
HorizontalGlue()
],
[ halign=none,
"If divergent, determine if it is infinity, negative infinity, or neither.",
HorizontalGlue()
]
],
[ border=true,
caption="Plot of the Integrand",
Plotter[IntPlot](width=200, height=200),
Button("Plot", Evaluate(function="plotit"), background=magenta)
]
],
[ halign=none, border=true,
caption="Determine Convergence and Value or Divergence and Type:",
[
[ halign=none,
RadioButton[CON]("Convergent", value=false, group=COD),
HorizontalGlue(),
Label[DEFV] ("Value = ", enabled=false),
TextField[TFDEFV](width=22, enabled=false),
HorizontalGlue()
],
[ halign=none,
RadioButton[DIV]("Divergent", value=false, group=COD),
HorizontalGlue(),
RadioButton[INF]("Infinity", value=false, enabled=false, group=TDIV),
RadioButton[NIN]("-Infinity", value=false, enabled=false, group=TDIV),
RadioButton[NEI]("Neither", value=false, enabled=false, group=TDIV),
HorizontalGlue()
]
],
Button("Check", Evaluate(function="check"), background=green),
Button("Show", Evaluate(function="show"), background=lightorange)
],
[ halign=none,
TextField[reply](width=40)
],
[ halign=none, background=ColorFooter,
Label("Programmer: Derek Oudie", font=FontFooter),
HorizontalGlue(),
Label("Copyright: P. Yasskin 2008", font=FontFooter)
]
]
),
ButtonGroup[COD](onchange=Evaluate(function="cod")),
ButtonGroup[TDIV]()
):Maplets[Display]( ImproperIntegrals );