Simpson's Error Formula MapletTroy Marschang and Kathleen KarikaMath 152-201Wednesday November 21, 2007
restart:with(Maplets[Tools]): with(Maplets[Elements]): with(plots): with(plottools):
StartEngine();randomize():lightorange:="#FFB300":great:=[" Bullseye. "," Right on the mark. "," Great. "," That's absolutely right. "," Awesome. You're unstoppable. "," Correct. "," WOW, nailed that one. "," You got it! "," YES! Right on. "," Way to go. "," Don't stop now. "," I knew you could do it! "," Perfect. You're unstoppable. "," Radical Dude. "," Power house. "," Sweet! Keep on Rockin'. "," You rock! "," You got those mad skills. "," Awesome! "," Wowzers! That was great! "," Awesome man! "," Good job! You really know what you are doing. "," You're awesome kid. "," Tearing it up. "," Right on the money. "," High Five!!!!! "," Cool Beans. "," What a nerd! "," You're so money. "," You're a genious. "," You hit the nail on the head. "," Excellent! "," If I had a cookie, I'd give it to you. "," You're a winner. "," That's a winner. "," Whoop! Way to go. "," That's the way, ah ha ah ha, I like it. "," Oh yeah! "," You're the bomb! "," That was fantastic! "," Oh Snap! "]:randgreat:=rand(1..nops(great)):nextquest:=["Keep it up.","Keep going.","On to the next integral.","Find another error bound.","Bound another error term."]:randnext:=rand(1..nops(nextquest)):sorry:=[" Sorry, that's not right. "," Nope, but don't give up. "," Not quite, it's a little off. "," Bummer, thought you would get it this time. "," Close, but no cigar. "," Slightly off. "," If you need help, please ask a human. "," I don't know where you went wrong. "," Sorry, killer problem. "," No, but don't blow it off. "," D'oh! "," That's whack. "," Yikes! Really close. You'll get it next time. "," Thought you would get it this time. "," Nice try, but your answer is wrong. "," Ask your neighbor for help. "," Brutal kid. "," That won't fly. "," So close and yet so far. "," So, that was just practice. "," Ooooh...close. "," A swing and a miss. "," Better luck next time. "," Oops. "," Tilt. Play again. "," I know you can do better. "," Sorry Charlie. "," Better luck next time. "," Tisk, tisk. It's a tough one. "," It takes a lot of wrongs to make a right. "," Even Einstein didn't get everything right. "," Oh man, you were so close. "," Today's not your lucky day. "," Don't quit your day job. "," You're colder than a polar bear's toenail. "]:randsorry:=rand(1..nops(sorry)):tryagain:=["Try the hints.","Fix your answer and check it again.","Give it another try.","Try again.","Please try again."]:randtry:=rand(1..nops(tryagain)):showit:=[" OK That's just practice. But try doing it yourself."," That's fine for practice. Try doing it yourself."," Think about it and enter an answer yourself."," Next time, try the hints."," Next time, compute an answer yourself."," So that was practice. Try doing it yourself."]:randshow:=rand(1..nops(showit)):randstart:=rand(0..5):randlength:=rand(1..4):rand15:=rand(1..5):Funcs:=[exp(x^2), sqrt(x), exp(-x), ln(x+1), exp(x), 1/(x+2), x^7+x^3, ln(3*x), sqrt(x+4), x^5+5, (1/24)*x^4-(1/360)*x^6, (5/12)*x^4-(1/360)*(x-3)^6, sin(3*x)
];randFuncs:=rand(1..nops(Funcs)):NewFunction:=proc()
global Funcs, func, __a, __b, __n, F4, K, E;
func:=Funcs[randFuncs()];
__a:=randstart();
__b:=__a+randlength();
__n:=10*rand15();
Set(MMLintegral(value)=MathML[Export](Int(func,x=__a..__b)));
Set(TFn=__n);
F4:=diff(func,x,x,x,x);
K:=maximize(abs(F4),x=__a..__b);
E:=K*(__b-__a)^5/(180*__n^4);
Set(TFF4="");
Set(TFK="");
Set(TFE="");
Set(PF4=plot(0, x=__a..__b, color=black));
Set(reply="", reply(background)=white);
end proc:Check_F4:=proc()
global Funcs, func, __a, __b, __n, F4, K, E;
local useranswer;
useranswer:=Get(TFF4::anything, corrections=true, update=true):
if simplify(F4-useranswer)=0 then
Set(reply=nextquest[randnext()],
reply(background)=green, reply(foreground)=black)
else
Set(reply=sorry[randsorry()],
reply(background)=red, reply(foreground)=white)
end if;
end proc:Show_F4:=proc()
global Funcs, func, __a, __b, __n, F4, K, E;
Set(TFF4=F4);
end proc:Plot_F4:=proc()
global Funcs, func, __a, __b, __n, F4, K, E;
Set(PF4=plot(abs(F4), x=__a..__b, color=green));
end proc:Check_K:=proc()
global Funcs, func, __a, __b, __n, F4, K, E;
local useranswer;
useranswer:=Get(TFK::realcons, corrections=true, update=true):
if evalf(useranswer-K)>=0 then
Set(reply=nextquest[randnext()],
reply(background)=green, reply(foreground)=black)
else
Set(reply=sorry[randsorry()],
reply(background)=red, reply(foreground)=white)
end if;
end proc:Show_K:=proc()
global Funcs, func, __a, __b, __n, F4, K, E;
Set(TFK=K);
end proc:Plot_K:=proc()
global Funcs, func, __a, __b, __n, F4, K, E;
Set(PF4=plot([abs(F4),K], x=__a..__b, color=[green,blue]));
end proc:Check_E:=proc()
global Funcs, func, __a, __b, __n, F4, K, E;
local useranswer;
useranswer:=Get(TFE::anything, corrections=true, update=true):
if evalf(useranswer-E)>=0 then
Set(reply=great[randgreat()],
reply(background)=green, reply(foreground)=black)
else
Set(reply=sorry[randsorry()],
reply(background)=red, reply(foreground)=white)
end if;
end proc:Show_E:=proc()
global Funcs, func, __a, __b, __n, F4, K, E;
Set(TFE=E);
end proc:SimpsonError:=Maplet(onstartup=RunWindow(MAIN), Font[helv10]("helvetica",10),
Window[MAIN]( title="Simpson's Error Formula", defaultbutton=BHint,
[ background=wheat,
[ halign=none, background=wheat,
Button[Bnew]("New Integral", onclick=Evaluate(function=NewFunction),
background=magenta),
HorizontalGlue(),
Button[BHint]("Hint", onclick=RunWindow(HINT), background=lightorange),
Button("Quit", Shutdown(), background=pink)
],
BoxColumn( halign=none, valign=none, inset=0, spacing=0, background=wheat, border=true,
caption="Goal: To Use Simpson's Error Formula:",
[ background=wheat,
"Suppose you use Simpson's Rule to approximate the integral:",
HorizontalGlue()
],
[ background=wheat,
MathMLViewer[MMLintegral](value=MathML[Export](Int(f(x),x=a..b)), height=100, width=500),
"using n =",
TextField[TFn](width=5),
"intervals."
],
[ background=wheat,
"Use Simpson's Error Formula to bound the error in the approximation.",
HorizontalGlue()
]
),
[ halign=none, inset=0, spacing=0, background=wheat,
[ halign=none, background=wheat, border=true,
caption="Calculate the Error",
[ background=wheat,
"Find the fourth derivative of the integrand:",
HorizontalGlue()
],
[ halign=none, background=wheat,
"f''''(x) =",
TextField[TFF4](width=20),
HorizontalGlue(),
Button[BCheckF4]("Check", Evaluate(function=Check_F4), background=green),
Button[BShowF4]("Show", Evaluate(function=Show_F4), background=lightorange),
Button[BPlotF4]("Plot", Evaluate(function=Plot_F4), background=tan)
],
[ background=wheat,
"Find the maximum value of the absolute value of the fourth derivative:",
HorizontalGlue()
],
[ halign=none, background=wheat,
"K =",
TextField[TFK](width=20),
HorizontalGlue(),
Button[BCheckK]("Check", Evaluate(function=Check_K), background=green),
Button[BShowK]("Show", Evaluate(function=Show_K), background=lightorange),
Button[BPlotK]("Plot", Evaluate(function=Plot_K), background=tan)
],
[ background=wheat,
"Find a bound on the Error in the Simpson's Rule Approximation:",
HorizontalGlue()
],
[ halign=none, background=wheat,
"| E | <",
TextField[TFE](width=20),
HorizontalGlue(),
Button[BCheckE]("Check", Evaluate(function=Check_E), background=green),
Button[BShowE]("Show", Evaluate(function=Show_E), background=lightorange),
HorizontalGlue(),
" "
]
],
[ halign=none, background=wheat, border=true,
caption="Plot of | f''''(x) |:",
Plotter[PF4](height=200,width=200)
]
],
[ background=wheat,
TextField[reply](background=white, width=50)
],
[ background=cyan, inset=0, spacing=0,
Label("Programmers: Troy Marschang and Kathleen Karika", font=helv10),
HorizontalGlue(),
Label("@ Copyright P. Yasskin 2007", font=helv10)
]
]
),
Window[HINT]( title="Hint",
[ valign=none, inset=0, spacing=0, background=wheat,
[ background=wheat,
MathMLViewer(value=('abs(E)' < 'K'('b'-'a')^5/(180*'n'^4)), width=200, height=60)
],
[ background=wheat,
"where",
MathMLViewer(value=('K' >= 'abs(`f''''`(x))'), width=150, height=40)
],
[ background=wheat,
"for all",
MathMLViewer(value=('x' in ['a','b']), width=150, height=40)
],
Button("Close", CloseWindow(HINT), background=pink)
]
)
):Maplets[Display](SimpsonError);