Aaron Rice & Amy PhillipsUsing Maclaurin Series to Compute Limitsrestart;with(Maplets[Tools]):with(Maplets[Elements]):StartEngine():randomize():lightorange:="#FFB300":
lightcyan:="#DDFFFF":ColorFooter:=lightcyan:
FontFooter:=Font("helvetica",10):Formula1:=Sum(t^n/n!,n=0..infinity):
Formula1t:=1+t+t^2/2+t^3/3!+`...`:
Formula2:=Sum((-1)^n*t^(2*n+1)/(2*n+1)!,n=0..infinity):
Formula2t:=t-t^3/3!+t^5/5!-t^7/7!+`...`:
Formula3:=Sum((-1)^n*t^(2*n)/(2*n)!,n=0..infinity):
Formula3t:=1-t^2/2+t^4/4!-t^6/6!+`...`:fns:=[exp(t), sin(t), cos(t)]:
randfn:=rand(1..nops(fns)):
as:=[-2,-1,1,2]:
randa:=rand(1..nops(as)):
ns:=[1,2,3]:
randn:=rand(1..nops(ns)):ask:=proc()
global fns,randfn,as,randa,ns,randn,Npx,L;
local Nf,ft,a,n,fx,px,i,Nx,degN,Dx,Lim;
Nf:=randfn():
ft:=fns[Nf]:
a:=as[randa()]:
n:=ns[randn()]:
fx:=eval(ft,t=a*x^n):
px:=convert(taylor(fx,x,10),polynom):
for i from 1 to 10 while nops(px)>2 do
px:=convert(taylor(fx,x,10-i),polynom):
end do:
Nx:=fx-px:
Npx:=convert(taylor(Nx,x,24),polynom):
for i from 1 to 24 while nops(Npx)>2 do
Npx:=convert(taylor(Nx,x,24-i),polynom):
end do:
degN:=-degree(eval(Npx,x=1/t)):
Dx:=x^degN:
Lim:='L' = Limit(Nx/Dx,x=0);
L:=limit(Nx/Dx,x=0);
Set(func= MathML[Export](Lim));
Set(answer1="");
Set(answer2="");
Set(reply ="");
end proc:hint:=proc()
Set(reply ="Use 3 or 4 terms of one of these formulas to rewrite the numerator.");
end proc:check1:=proc()
global Npx;
local answer;
answer:=Get(answer1::polynom, corrections = true, update=true):
if simplify(answer - Npx)=0
then Set(reply ="Way to go, on to part 2!");
else Set(reply="You might want to go back over you calculations");
end if;
end proc:show1:=proc()
global Npx;
Set(answer1=Npx);
Set(reply ="For a novel idea, you could always do it yourself.");
end proc:check2:=proc()
global L;
local answer;
answer:=Get(answer2::realcons, corrections = true, update=true):
if simplify(answer - L)=0
then Set(reply ="Congrats, you did it!");
else Set(reply ="FAIL, retry.");
end if;
end proc:show2:=proc()
global L;
Set(answer2=L);
Set(reply ="Next time see if you can work it by yourself.");
end proc:MaclaurinLimit:=Maplet(onstartup=RunWindow(MAIN),
Window[MAIN](title="Using Maclaurin Series to Compute Limits",
[ halign=none,
[ Button("New Limit",Evaluate(function="ask"), background=magenta),
HorizontalGlue(),
Button("Quit",Shutdown(), background=pink)
],
[ "Use a Maclaurin Series to compute the Limit",
MathMLViewer[func](value=MathML[Export](" "), width=250, height=75)
],
[ border=true,
caption="Find the first 2 terms of the Maclaurin series for the numerator in the limit:",
"N = ",
TextField[answer1](width=30),
HorizontalGlue(),
Button("Formulas", onclick=AFormula, background=lightorange),
Button("Check",Evaluate(function="check1"), background=green),
Button("Show",Evaluate(function="show1"), background=lightorange)
],
[ border=true,
caption="Compute the Limit:",
"L = ",
TextField[answer2](width=30),
HorizontalGlue(),
Button("Check",Evaluate(function="check2"), background=green),
Button("Show",Evaluate(function="show2"), background=lightorange)
],
[ TextBox[reply](width=60,height=2,editable=false)
],
[ halign=none, background=ColorFooter,
Label("Programmers: Aaron Rice & Amy Phillips", font=FontFooter),
HorizontalGlue(),
Label("Copyright: P. Yasskin 2008", font=FontFooter)
]
]),
Window[FORMULAS](title="Formulas",
[ [ "Function",
HorizontalGlue(),
"Maclaurin Series",
HorizontalGlue(),
HorizontalGlue(),
HorizontalGlue(),
HorizontalGlue(),
HorizontalGlue(),
Button[CloseFormulas]("Close", CloseWindow(FORMULAS),background=pink)
],
[ MathMLViewer(value=MathML[Export](e^t),width=60,height=60),
"=",
MathMLViewer(value=Formula1,width=160,height=60),
"=",
MathMLViewer(value=Formula1t,width=240,height=60)
],
[ MathMLViewer(value=MathML[Export](sin(t)),width=60,height=60),
"=",
MathMLViewer(value=Formula2,width=160,height=60),
"=",
MathMLViewer(value=Formula2t,width=240,height=60)
],
[ MathMLViewer(value=MathML[Export](cos(t)),width=60,height=60),
"=",
MathMLViewer(value=Formula3,width=160,height=60),
"=",
MathMLViewer(value=Formula3t,width=240,height=60)
]
]),
Action[AFormula](
RunWindow(FORMULAS),
Evaluate(function="hint")
)
):Maplets[Display](MaclaurinLimit);