Computing Limits using Maclaurin Series
David Dwyer & David Migl
253 Honors Spring 2009
restart:randomize():
with(Maplets[Tools]): with(Maplets[Elements]): with(plots):
StartEngine();
ColorCloseButton:="#9999FF":
ColorNewButton:=magenta:
maroon:="#800000":
burntorange:="#CC5500":
colorbkgd:=gray:
lightcyan:="#DDFFFF":
ColorFooter:=lightcyan:
FontFooter:=Font("helvetica",10):
Formula[1]:=Sum(t^n,n=0..infinity):
Formula[2]:=Sum((-1)^n*t^(n+1)/(n+1),n=0..infinity):
Formula[3]:=Sum(t^n/n!,n=0..infinity):
Formula[4]:=Sum((-1)^n*t^(2*n+1)/(2*n+1)!,n=0..infinity):
Formula[5]:=Sum((-1)^n*t^(2*n)/(2*n)!,n=0..infinity):
Formula[6]:=Sum((-1)^n*t^(2*n+1)/(2*n+1),n=0..infinity):
terms:=[(x-arctan(x))/x^3, (x-x^3/3-arctan(x))/x^5,(cos(x^3)-1+x^6/2)/x^12, (cos(x)-1+x^2/2)/x^4,(sin(x)-x+x^3/6)/(x^5),(sin(x^3)-x^3)/(x^9),(exp(x)-1)/x,(exp(x)-1-x)/(x^2)]:
replace:=[arctan(u),arctan(u),cos(u),cos(u),sin(u),sin(u),e^u,e^u]:
AnswerStep2:=[1/3-x^2/5,-1/5+x^2/7,1/(4!)-x^6/(6!),1/(4!)-x^2/(6!), 1/5!-x^2/7!,-1/3+x^6/5!,1+x/2,1/2+x/3!]:
New_Problem:=proc()
global problem_term,i,OverallAnswer;
local index:
set(reply=NULL):
index:=rand(1..nops(terms));
i:=index():
problem_term:=terms[i]:
Set(MathMLProblem(value)=MathML[Export](L = Limit(problem_term,x=0))):
OverallAnswer:=value(limit(AnswerStep2[i],x=0)):
Clear():
end proc:
CheckStep1:=proc()
global userinput;
userinput:=parse(Get(InputStep1)):
if userinput-replace[i]=0
then
Set(reply(background)=maroon):
Set(reply(foreground)=white):
Set(reply="Step 1 correct: Beat The Hell Outta Step 1"):
else
Set(reply(background)=burntorange):
Set(reply="Step 1 incorrect: T-E-A-M"):
end if:
end proc:
CheckStep2:=proc()
local UserStep2:
UserStep2:=Get(InputStep2::algebraic,corrections=true,update=true):
if simplify(normal(value(AnswerStep2[i]) - value(UserStep2)))=0
or combine(value(AnswerStep2[i]) - value(UserStep2), ln, symbolic)=0
or simplify(expand(value(AnswerStep2[i]) - value(UserStep2)))=0
or simplify(normal(value(AnswerStep2[i]) - value(UserStep2)))=0
or combine(value(AnswerStep2[i]) - value(UserStep2), ln, symbolic)=0
or simplify(expand(value(AnswerStep2[i]) - value(UserStep2)))=0
then
Set(reply(background)=maroon):
Set(reply(foreground)=white):
Set(reply="Step 2 correct: FARMER'S FIGHT!"):
else
Set(reply="Step 2 incorrect: A&M, GIVE US ROOM!"):
Set(reply(background)=burntorange):
end if:
end proc:
ShowStep2:=proc()
Set(InputStep2=AnswerStep2[i]):
Set(reply="Step 2 shown"):
EnableStep3():
Set(reply(background)=yellow):
Set(reply(foreground)=black):
end proc:
CheckStep3:=proc()
local UserStep3:
UserStep3:=Get(InputStep3::algebraic,corrections=true,update=true):
if simplify(normal(value(OverallAnswer) - value(UserStep3)))=0
or combine(value(OverallAnswer) - value(UserStep3), ln, symbolic)=0
or simplify(expand(value(OverallAnswer) - value(UserStep3)))=0
or simplify(normal(value(OverallAnswer) - value(UserStep3)))=0
or combine(value(OverallAnswer) - value(UserStep3), ln, symbolic)=0
or simplify(expand(value(OverallAnswer) - value(UserStep3)))=0
then
Set(reply(background)=maroon):
Set(reply(foreground)=white):
Set(reply="Step 3 correct: WHOOP! GIG 'EM AGGIES!"):
else
Set(reply="Step 3 incorrect: HISS!!!!"):
Set(reply(background)=burntorange):
end if:
end proc:
ShowStep3:=proc()
Set(InputStep3=OverallAnswer):
Set(reply="Step 3 Shown"):
end proc:
EnableStep3:=proc()
Set(MathMLProblem3(value)=MathML[Export](L = Limit(AnswerStep2[i],x=0))):
end proc:
Clear:=proc()
Set(InputStep1(value)=" "):
Set(InputStep2(value)=" "):
Set(InputStep3(value)=" "):
Set(MathMLProblem3(value)=MathML[Export](L = Limit(f(x),x=0))):
Set(reply(value)=" "):
Set(reply(foreground)=black):
Set(reply(background)=white):
end proc:
MaclSerLimit:= Maplet(onstartup = RunWindow(MAIN),
Window[MAIN](title="Limits with Series",
[
[ halign=none,
Button[New]("New Limit",Evaluate(function="New_Problem"), background=ColorNewButton),
HorizontalGlue(),
Button("Quit",background=ColorCloseButton,Shutdown())
],
[ border=true,
caption="Goal: Use a MacLaurin Series to evalute this limit:",
MathMLViewer[MathMLProblem](value=MathML[Export](L = Limit(f(x),x=0))),
HorizontalGlue()
],
[ border=true,
caption="Step 1: Choose a function to replace by its MacLaurin Series:",
DropDownBox[InputStep1](background = white,[" ", e^u, sin(u), cos(u), ln(1+u), arctan(u)]),
Button("Check", Evaluate(function = "CheckStep1")),
Button("Formulas", RunWindow(FORMULAS), background=yellow)
],
BoxColumn( border=true,
caption="Step2: Substitute the series into the limit and simplify:",
[ TextBox( width=60, height=4, background=yellow,
"Cancel the first few terms of the series with other terms in the numerator.\134nCancel what remains with the power of x in the denominator.\134nWrite the FIRST TWO TERMS of what remains:")
],
[ halign=none, inset=0,
"L = lim",
TextField[InputStep2](width=40),
HorizontalGlue(),
Button("Check", Evaluate(function = "CheckStep2")),
Button("Show", Evaluate(function = "ShowStep2"))
]
),
[ border=true,
caption="Step3: Compute the limit:",
MathMLViewer[MathMLProblem3](value=MathML[Export](L = Limit(f(x),x=0)), width=300, height=100),
" = ",
TextField[InputStep3](width=10),
Button("Check", Evaluate(function = "CheckStep3")),
Button("Show", Evaluate(function = "ShowStep3"))
],
[ TextField[reply](width=40, editable=false, background=white)
],
[ halign=none, background=ColorFooter,
Label("Programmers: David Dwyer & David Migl", font=FontFooter),
HorizontalGlue(),
Label("Copyright: P. Yasskin 2009", font=FontFooter)
]
]
),
Window[FORMULAS](title="Formulas",
[ background=colorbkgd, halign=none,
[ background=colorbkgd,
HorizontalGlue(),
"Function",
HorizontalGlue(),
HorizontalGlue(),
HorizontalGlue(),
"Maclaurin Series",
HorizontalGlue(),
HorizontalGlue(),
HorizontalGlue(),
Button[CloseFormulas]("Close", CloseWindow(FORMULAS),background=ColorCloseButton)
],
[ background=colorbkgd,
MathMLViewer(value=MathML[Export](1/(1-t)),width=100,height=60),
"=",
MathMLViewer(value=MathML[Export](Formula[1]),width=200,height=60)
],
[ background=colorbkgd,
MathMLViewer(value=MathML[Export](e^t),width=100,height=60),
"=",
MathMLViewer(value=MathML[Export](Formula[3]),width=200,height=60)
],
[ background=colorbkgd,
MathMLViewer(value=MathML[Export](sin(t)),width=100,height=60),
"=",
MathMLViewer(value=MathML[Export](Formula[4]),width=200,height=60)
],
[ background=colorbkgd,
MathMLViewer(value=MathML[Export](cos(t)),width=100,height=60),
"=",
MathMLViewer(value=MathML[Export](Formula[5]),width=200,height=60)
],
[ background=colorbkgd,
MathMLViewer(value=MathML[Export](ln(1+t)),width=100,height=60),
"=",
MathMLViewer(value=MathML[Export](Formula[2]),width=200,height=60)
],
[ background=colorbkgd,
MathMLViewer(value=MathML[Export](arctan(t)),width=100,height=60),
"=",
MathMLViewer(value=MathML[Export](Formula[6]),width=200,height=60)
]
]
)
):
Maplets[Display]( MaclSerLimit );