Susan Vanderzyl, Rebecca Hollkamp
Math 253- Section 200
Taylor Polynomials
restart:
with(Maplets[Tools]): with(Maplets[Elements]): with(plots): with(plottools): with(RandomTools):
StartEngine();
randomize():
alias(e=exp(1)):
Funcs:=[exp(x), ln(x), ln(1+x), sin(x), cos(x)]:
centers:=[
[0,1,2,-1],
[1,2,3,e, e^2, 1/2],
[0, 1,2,3,e, 1/2, -1/2],
[0, Pi, Pi/2, Pi/3, Pi/4, Pi/6],
[0, Pi, Pi/2, Pi/3, Pi/4, Pi/6]
]:
xinterval:=[
-2..3,
0..10,
-1..10,
-Pi/2..2*Pi,
-Pi/2..2*Pi
]:
yinterval:=[
-2..25,
-1..3,
-1..4,
-1.5..1.5,
-1.5..1.5
]:
randFuncs:=rand(1..nops(Funcs)):
problem:=proc()
global func, val_a, x_int, y_int, taylor_1, taylor_2, taylor_3, plotfunc, plotcenter;
local whichFunc;
whichFunc:=randFuncs();
func:=Funcs[whichFunc];
val_a:=Generate(choose(centers[whichFunc]));
x_int:=xinterval[whichFunc];
y_int:=yinterval[whichFunc];
Set(MMLfn(value)=MathML[Export](f(x)=func));
Set(aequals=val_a);
Set(poly1="");
Set(poly2="");
Set(poly3="");
Set(TF1="", TF1(background)=white);
Set(TF2="", TF2(background)=white);
Set(TF3="", TF3(background)=white);
taylor_1:=convert(taylor(func,x=val_a,2),polynom);
taylor_2:=convert(taylor(func,x=val_a,3),polynom);
taylor_3:=convert(taylor(func,x=val_a,4),polynom);
plotfunc:=plot(func, x=x_int, y=y_int, color=blue);
plotcenter:=plot([[val_a,eval(func,x=val_a)]], x=x_int, y=y_int, style=point, symbol=circle, symbolsize=20, color=red);
Set(PolyPlot=display(plotfunc,plotcenter));
Set(reply="", reply(background)=white);
end proc:
check1:=proc()
global rand_a, taylor_1;
local user_1;
if Get(poly1)="" then
Set(TF1="Blank");
return;
end if;
user_1:=Get(poly1::anything):
if simplify(taylor_1 - user_1)= 0 then
Set(TF1="Correct", TF1(background)=green):
else
Set(TF1="Incorrect", TF1(background)=red):
end if;
end proc:
show1:=proc()
global taylor_1;
Set(poly1=taylor_1):
end proc:
hint1:=proc()
Set(reply(value)=MathML[Export](f(a)+`f '`(a)*(x-a)));
end proc:
plot1:=proc()
global Funcs, func, val_a, taylor_1, taylor_2, taylor_3, plot_t1;
plot_t1:=plot(taylor_1, x=x_int, y=y_int, color=green);
Set(PolyPlot=display(plotfunc,plotcenter,plot_t1));
end proc:
check2:=proc()
global rand_a, taylor_2;
local user_2;
if Get(poly2)="" then
Set(TF2="Blank");
return;
end if;
user_2:=Get(poly2::anything):
if simplify(taylor_2 - user_2)= 0 then
Set(TF2="Correct", TF2(background)=green):
else
Set(TF2="Incorrect", TF2(background)=red):
end if;
end proc:
show2:=proc()
global taylor_2;
Set(poly2=taylor_2):
end proc:
hint2:=proc()
Set(reply(value)=MathML[Export](f(a)+`f '`(a)*(x-a)+`f ''`(a)/2*(x-a)^2));
end proc:
plot2:=proc()
global Funcs, func, val_a, taylor_1, taylor_2, taylor_3, plot_t2;
plot_t2:=plot(taylor_2, x=x_int, y=y_int, color=orange);
Set(PolyPlot=display(plotfunc,plotcenter,plot_t2));
end proc:
check3:=proc()
global rand_a, taylor_3;
local user_3;
if Get(poly3)=""then
Set(TF3="Blank");
return;
end if;
user_3:=Get(poly3::anything):
if simplify(taylor_3 - user_3=0) then
Set(TF3="Correct", TF3(background)=green):
else
Set(TF3="Incorrect", TF2(background)=red):
end if;
end proc:
show3:=proc()
global taylor_3;
Set(poly3=taylor_3):
end proc:
hint3:=proc()
Set(reply(value)=MathML[Export](f(a)+`f '`(a)*(x-a)+`f ''`(a)/2*(x-a)^2+`f '''`(a)/6*(x-a)^3));
end proc:
plot3:=proc()
global Funcs, func, val_a, taylor_1, taylor_2, taylor_3, plot_t3;
plot_t3:=plot(taylor_3, x=x_int, y=y_int, color=violet);
Set(PolyPlot=display(plotfunc,plotcenter,plot_t3));
end proc:
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
showall:=proc()
global taylor_1, taylor_2, taylor_3;
Set(poly1=taylor_1):
Set(poly2=taylor_2):
Set(poly3=taylor_3):
end proc:
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
plotall:=proc()
global Funcs, func, val_a, taylor_1, taylor_2, taylor_3, plot_t1, plot_t2, plot_t3, plotfunc, plotcenter;
plot_t1:=plot(taylor_1, x=x_int, y=y_int, color=green);
plot_t2:=plot(taylor_2, x=x_int, y=y_int, color=orange);
plot_t3:=plot(taylor_3, x=x_int, y=y_int, color=violet);
Set(PolyPlot=display(plotfunc, plotcenter, plot_t1, plot_t2, plot_t3));
end proc:
TaylorP:=Maplet(onstartup = RunWindow(MAIN),
Font[helv10]("helvetica",10),
Window[MAIN](title="Taylor Polynomials",
[ halign=none, inset=0,
[ inset=0, [ [ Button("New Function", Evaluate(function="problem")),
HorizontalGlue(),
Button("Quit", Shutdown())
],
BoxColumn( border=true,
caption="Consider the Function:",
MathMLViewer[MMLfn](value = MathML[Export](f(x)), width=300)
),
[ border=true,
caption="Goal:",
"Find the Taylor polynomial approximations at a =",
TextField[aequals](width=7, editable=false, background=white),
HorizontalGlue()
]
],
[ border=true,
caption="Plot",
Plotter[PolyPlot](width=300, height=200)
]
],
[ border=true,
caption="1st Order:",
"T1 =",
TextBox[poly1](width=40, height=2),
HorizontalGlue(),
Button("Hint", Evaluate(function = "hint1")),
Button("Check", Evaluate(function = "check1")),
TextField[TF1](width=6, editable=false, background=white),
Button("Show", Evaluate(function = "show1")),
Button("Plot", Evaluate(function = "plot1"))
],
[ border=true,
caption="2nd Order:",
"T2 =",
TextBox[poly2](width=40, height=2),
HorizontalGlue(),
Button("Hint", Evaluate(function = "hint2")),
Button("Check", Evaluate(function = "check2")),
TextField[TF2](width=6, editable=false, background=white),
Button("Show", Evaluate(function = "show2")),
Button("Plot", Evaluate(function = "plot2"))
],
[ border=true,
caption="3rd Order:",
"T3 =",
TextBox[poly3](width=40, height=2),
HorizontalGlue(),
Button("Hint", Evaluate(function = "hint3")),
Button("Check", Evaluate(function = "check3")),
TextField[TF3](width=6, editable=false, background=white),
Button("Show", Evaluate(function = "show3")),
Button("Plot", Evaluate(function = "plot3"))
],
[ [ Button("Show All", Evaluate(function = "showall")),
Button("Plot All", Evaluate(function = "plotall"))
],
MathMLViewer[reply](width=500, height=50, background=white)
],
BoxRow( background="#DDFFFF", inset=0, spacing=0,
Label("Programmers: Susan Vanderzyl and Rebecca Hollkamp", font=helv10),
HorizontalGlue(),
Label("\302\251 Copyright: P. Yasskin 2008", font=helv10)
)
]
)
):
Maplets[Display]( TaylorP );