Andrew Garia & Kelley StoreyMath 252- Section 202Finding Taylor Polynomialsrestart:with(Maplets[Tools]): with(Maplets[Elements]): with(RandomTools):
StartEngine():randomize():lightorange:="#FFB300":
lightcyan:="#DDFFFF":ColorFooter:=lightcyan:
FontFooter:=Font("helvetica",10):alias(e=exp(1)):Funcs:=[exp(x), ln(x), ln(1+x), sin(x), cos(x), x^6, sin(2*x)]:centers:=[
[0,1,2,-1,-2,3,-3,4,-4],
[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],
[0,1,2,3,4,-1,-2,-3,-4],
[0, Pi, Pi/2, Pi/3, Pi/4, Pi/6]
]:
randFuncs:=rand(1..nops(Funcs)):problem:=proc()
global func, func1, func2, func3, func4, func5, funca, func1a, func2a, func3a, func4a, func5a, val_a;
local whichFunc;
whichFunc:=randFuncs();
func:=Funcs[whichFunc];
val_a:=Generate(choose(centers[whichFunc]));
func1:=diff(func,x);
func2:=diff(func1,x);
func3:=diff(func2,x);
func4:=diff(func3,x);
func5:=diff(func4,x);
funca:=eval(func, x=val_a);
func1a:=eval(func1, x=val_a);
func2a:=eval(func2, x=val_a);
func3a:=eval(func3, x=val_a);
func4a:=eval(func4, x=val_a);
func5a:=eval(func5, x=val_a);
Set(MMLfn(value)=MathML[Export](f(x)=func));
Set(aequals=val_a);
Set(poly0="");
Set(poly1="");
Set(poly2="");
Set(poly3="");
Set(poly4="");
Set(poly5="");
Set(poly0a="");
Set(poly1a="");
Set(poly2a="");
Set(poly3a="");
Set(poly4a="");
Set(poly5a="");
Set(polyc0="");
Set(polyc1="");
Set(polyc2="");
Set(polyc3="");
Set(polyc4="");
Set(polyc5="");
Set(TF0="", TF0(background)=white);
Set(TF1="", TF1(background)=white);
Set(TF2="", TF2(background)=white);
Set(TF3="", TF3(background)=white);
Set(TF4="", TF4(background)=white);
Set(TF5="", TF5(background)=white);
Set(TF6="", TF6(background)=white);
Set(TFpoly="", TFpoly(background)=turquoise);
Set(reply="", reply(background)=white);
end proc:check0:=proc()
global func, funca;
local user_func, user_funca, user_c0;
if Get(poly0)="" or Get(poly0a)="" or Get(polyc0)="" then
Set(TF0="Blank");
return;
end if;
user_func:=Get(poly0::anything):
if simplify(func - user_func)<>0 then
Set(TF0="Incorrect", TF0(background)=red):
Set(reply="The function is incorrect. Try again.", reply(background)=red):
return;
end if;
user_funca:=Get(poly0a::anything):
if simplify(funca - user_funca)<>0 then
Set(TF0="Incorrect", TF0(background)=red):
Set(reply="Check the value of the function.", reply(background)=red):
return;
end if;
user_c0:=Get(polyc0::anything):
if simplify(funca - user_c0)<>0 then
Set(TF0="Incorrect", TF0(background)=red):
Set(reply="The coefficient is incorrect.", reply(background)=red):
return;
end if;
Set(TF0="Correct", TF0(background)=green):
Set(reply="Great job! Now on to the next step.", reply(background)=green):
end proc:check1:=proc()
global func1, func1a;
local user_func1, user_func1a, user_c1;
if Get(poly1)="" or Get(poly1a)="" or Get(polyc1)="" then
Set(TF1="Blank");
return;
end if;
user_func1:=Get(poly1::anything):
if simplify(func1 - user_func1)<>0 then
Set(TF1="Incorrect", TF1(background)=red):
Set(reply="Check your derivative.", reply(background)=red):
return;
end if;
user_func1a:=Get(poly1a::anything):
if simplify(func1a - user_func1a)<>0 then
Set(TF1="Incorrect", TF1(background)=red):
Set(reply="The value of the derivative is wrong. Try again!", reply(background)=red):
return;
end if;
user_c1:=Get(polyc1::anything):
if simplify(func1a - user_c1)<>0 then
Set(TF1="Incorrect", TF1(background)=red):
Set(reply="Wrong coefficient. Give it another shot!", reply(background)=red):
return;
end if;
Set(TF1="Correct", TF1(background)=green):
Set(reply="Awesome! Keep going!", reply(background)=green):
end proc:check2:=proc()
global func2, func2a;
local user_func2, user_func2a, user_c2;
if Get(poly2)="" or Get(poly2a)="" or Get(polyc2)="" then
Set(TF2="Blank");
return;
end if;
user_func2:=Get(poly2::anything):
if simplify(func2 - user_func2)<>0 then
Set(TF2="Incorrect", TF2(background)=red):
Set(reply="Try computing that second derivative again.", reply(background)=red):
return;
end if;
user_func2a:=Get(poly2a::anything):
if simplify(func2a - user_func2a)<>0 then
Set(TF2="Incorrect", TF2(background)=red):
Set(reply="The value of the function is incorrect.", reply(background)=red):
return;
end if;
user_c2:=Get(polyc2::anything):
if simplify((func2a/2) - user_c2)<>0 then
Set(TF2="Incorrect", TF2(background)=red):
Set(reply="Try recomputing the coefficient.", reply(background)=red):
return;
end if;
Set(TF2="Correct", TF2(background)=green):
Set(reply="You're the master!", reply(background)=green):
end proc:check3:=proc()
global func3, func3a;
local user_func3, user_func3a, user_c3;
if Get(poly3)="" or Get(poly3a)="" or Get(polyc3)="" then
Set(TF3="Blank");
return;
end if;
user_func3:=Get(poly3::anything):
if simplify(func3 - user_func3)<>0 then
Set(TF3="Incorrect", TF3(background)=red):
Set(reply="The derivative is incorrect. Try recomputing it.", reply(background)=red):
return;
end if;
user_func3a:=Get(poly3a::anything):
if simplify(func3a - user_func3a)<>0 then
Set(TF3="Incorrect", TF3(background)=red):
Set(reply="Check that value at a.", reply(background)=red):
return;
end if;
user_c3:=Get(polyc3::anything):
if simplify((func3a/6) - user_c3)<>0 then
Set(TF3="Incorrect", TF3(background)=red):
Set(reply="Your coefficient is incorrect. Try it again", reply(background)=red):
return;
end if;
Set(TF3="Correct", TF3(background)=green):
Set(reply="Correct! Now on to the next problem.", reply(background)=green):
end proc:check4:=proc()
global func4, func4a;
local user_func4, user_func4a, user_c4;
if Get(poly4)="" or Get(poly4a)="" or Get(polyc4)="" then
Set(TF4="Blank");
return;
end if;
user_func4:=Get(poly4::anything):
if simplify(func4 - user_func4)<>0 then
Set(TF4="Incorrect", TF4(background)=red):
Set(reply="Sorry! Try solving for the derivative again.", reply(background)=red):
return;
end if;
user_func4a:=Get(poly4a::anything):
if simplify(func4a - user_func4a)<>0 then
Set(TF4="Incorrect", TF4(background)=red):
Set(reply="The value of the function is incorrect. Give it another try.", reply(background)=red):
return;
end if;
user_c4:=Get(polyc4::anything):
if simplify((func4a/24) - user_c4)<>0 then
Set(TF4="Incorrect", TF4(background)=red):
Set(reply="Close, but you have the wrong coefficient. Try one more time!", reply(background)=red):
return;
end if;
Set(TF4="Correct", TF4(background)=green):
Set(reply="Great Job! You're almost done now.", reply(background)=green):
end proc:check5:=proc()
global func5, func5a;
local user_func5, user_func5a,user_c5;
if Get(poly5)="" or Get(poly5a)="" or Get(polyc5)="" then
Set(TF5="Blank");
return;
end if;
user_func5:=Get(poly5::anything):
if simplify(func5 - user_func5)<>0 then
Set(TF5="Incorrect", TF5(background)=red):
Set(reply="You have the wrong 5th derivative. See if you can get it right this time!", reply(background)=red):
return;
end if;
user_func5a:=Get(poly5a::anything):
if simplify(func5a - user_func5a)<>0 then
Set(TF5="Incorrect", TF5(background)=red):
Set(reply="Try finding the value of the function again.", reply(background)=red):
return;
end if;
user_c5:=Get(polyc5::anything):
if simplify((func5a/120) - user_c5)<>0 then
Set(TF5="Incorrect", TF5(background)=red):
Set(reply="Give that coefficient one more try!", reply(background)=red):
return;
end if;
Set(TF5="Correct", TF5(background)=green):
Set(reply="Oh snap! You've really got it now!", reply(background)=green):
end proc:check6:=proc()
global funca, func1a, func2a, func3a, func4a, func5a;
local user_func6;
if Get(TFpoly)="" then
Set(TF6="Blank");
return;
end if;
user_func6:=Get(TFpoly::anything):
if simplify(funca+func1a*(x-val_a)+(func2a/2)*(x-val_a)^2+(func3a/6)*(x-val_a)^3+(func4a/24)*(x-val_a)^4 +(func5a/120)*(x-val_a)^5- user_func6)<>0 then
Set(TF6="Incorrect", TF6(background)=red):
Set(reply="Try again! You're so close!", reply(background)=red):
return;
end if;
Set(TF6="Correct", TF6(background)=green):
Set(reply="Fantastic! You have found the Taylor polynomial!", reply(background)=green):
end proc:show0:=proc()
global func, funca;
Set(poly0=func):
Set(poly0a=funca):
Set(polyc0=funca):
Set(reply="See if you can get it yourself next time.", reply(background)=yellow):
end proc:show1:=proc()
global func1, func1a;
Set(poly1=func1):
Set(poly1a=func1a):
Set(polyc1=func1a):
Set(reply="I helped you out on this one, but next time try it yourself.", reply(background)=yellow):
end proc:show2:=proc()
global func2, func2a;
Set(poly2=func2):
Set(poly2a=func2a):
Set(polyc2=(func2a/2)):
Set(reply="This one we can just call practice. Try the next one on your own.", reply(background)=yellow):
end proc:show3:=proc()
global func3, funca3;
Set(poly3=func3):
Set(poly3a=func3a):
Set(polyc3=(func3a/6)):
Set(reply="Alright, let's see what you can do by yourself next time.", reply(background)=yellow):
end proc:show4:=proc()
global func4, func4a;
Set(poly4=func4):
Set(poly4a=func4a):
Set(polyc4=(func4a/24)):
Set(reply="Try the next one without my help.", reply(background)=yellow):
end proc:show5:=proc()
global func5, func5a;
Set(poly5=func5):
Set(poly5a=func5a):
Set(polyc5=(func5a/120)):
Set(reply="See if you can get the next one on your own.", reply(background)=yellow):
end proc:show6:=proc()
global funca, func1a, func2a, func3a, func4a, func5a;
Set(TFpoly=funca+func1a*(x-val_a)+(func2a/2)*(x-val_a)^2+(func3a/6)*(x-val_a)^3+(func4a/24)*(x-val_a)^4 +(func5a/120)*(x-val_a)^5):
Set(reply="Next time see if you can get the polynomial by yourself.", reply(background)=yellow):
end proc:FindTaylorPoly:=Maplet(onstartup = RunWindow(MAIN),
Window[MAIN](title="Finding Taylor Polynomials",
[ halign=none, inset=0,
[ Button("New Function", Evaluate(function="problem"), background=magenta),
HorizontalGlue(),
Button("Quit", Shutdown(), background=pink)
],
[ border=true,
caption="Find the 5th degree Taylor polynomial for",
MathMLViewer[MMLfn](value = MathML[Export]('f'(x)=`...`), width=300, height=60),
"at a =",
TextField[aequals](width=7, editable=false, background=white)
],
BoxColumn( border=true,
caption="Enter f(x) and its first 5 derivatives, their values at x=a and their corresponding coefficients in the Taylor polynomial.",
[ "f (x) = ",
TextBox[poly0](width=30, height=2),
"f (a) = ",
TextField[poly0a](width=10),
"c0 =",
TextField[polyc0](width=10),
HorizontalGlue(),
Button("Check", Evaluate(function = "check0"), background=green),
TextField[TF0](width=6, editable=false, background=white),
Button("Show", Evaluate(function = "show0"), background=lightorange)
],
[ "f '(x) = ",
TextBox[poly1](width=30, height=2),
"f '(a) = ",
TextField[poly1a](width=10),
"c1 =",
TextField[polyc1](width=10),
HorizontalGlue(),
Button("Check", Evaluate(function = "check1"), background=green),
TextField[TF1](width=6, editable=false, background=white),
Button("Show", Evaluate(function = "show1"), background=lightorange)
],
[ "f ''(x) = ",
TextBox[poly2](width=30, height=2),
"f ''(a) = ",
TextField[poly2a](width=10),
"c2 =",
TextField[polyc2](width=10),
HorizontalGlue(),
Button("Check", Evaluate(function = "check2"), background=green),
TextField[TF2](width=6, editable=false, background=white),
Button("Show", Evaluate(function = "show2"), background=lightorange)
],
[ "f '''(x) =" ,
TextBox[poly3](width=30, height=2),
"f '''(a) = ",
TextField[poly3a](width=10),
"c3 =",
TextField[polyc3](width=10),
HorizontalGlue(),
Button("Check", Evaluate(function = "check3"), background=green),
TextField[TF3](width=6, editable=false, background=white),
Button("Show", Evaluate(function = "show3"), background=lightorange)
],
[ "f^4(x) =",
TextBox[poly4](width=30, height=2),
"f^4(a) =",
TextField[poly4a](width=10),
"c4 =",
TextField[polyc4](width=10),
HorizontalGlue(),
Button("Check", Evaluate(function = "check4"), background=green),
TextField[TF4](width=6, editable=false, background=white),
Button("Show", Evaluate(function = "show4"), background=lightorange)
],
[ "f^5(x) =",
TextBox[poly5](width=30, height=2),
"f^5(a) =",
TextField[poly5a](width=10),
"c5 =",
TextField[polyc5](width=10),
HorizontalGlue(),
Button("Check", Evaluate(function = "check5"), background=green),
TextField[TF5](width=6, editable=false, background=white),
Button("Show", Evaluate(function = "show5"), background=lightorange)
]
),
BoxColumn( border=true,
caption="Write out the Taylor polynomial.",
[ TextField[TFpoly](width=75, background=turquoise),
HorizontalGlue(),
Button("Check", Evaluate(function = "check6"), background=green),
TextField[TF6](width=6, editable=false, background=white),
Button("Show", Evaluate(function = "show6"), background=lightorange)
]
),
[ TextField[reply](width=40, background=white)
],
[ halign=none, background=ColorFooter,
Label("Programmers: Andrew Garia & Kelley Storey", font=FontFooter),
HorizontalGlue(),
Label("Copyright: P. Yasskin 2008", font=FontFooter)
]
]
)
):Maplets[Display]( FindTaylorPoly );