Making Piecewise Functions ContinuousChris PatrickMATH 152-201restart:with(Maplets[Tools]): with(Maplets[Elements]): with(plots):
StartEngine();randomize():lightorange:="#FFB300":rights:= ["Bullseye!", "Great!", "That's absolutely right!", "You're unstoppable!", "Correct!", "You nailed that one!", "YES!", "Way to go!", "I knew you could do it!", "Perfecto!", "You're awesome!", "That's amazing!", "You're a math whiz!", "Whoop!", "Nice job!", "That's right!", "That's the way!", "I like the sound of that!", "Ding! That's correct!", "You're a genius!", "Right on the mark!", "Brick...house!", "Oh yeah!", "Wonderful!", "Excellent!", "You got it!", "Can I get a what what!", "Holla!", "You're incredible!"]: rightrand := rand(1..nops(rights)):wrong:=["Try again.", "Nope. I'm sorry.", "C'mon, you can do it.", "You can do better.", "You're close.", "Look closer.", "Put on your thinking cap.", "Better luck next time.", "Keep trying.", "Never give up.", "Try again. I have faith in you.", "Try harder."]:wrongrand := rand(1..nops(wrong)):final:=["You know your stuff! Move on to the next one!", "You're doing so awesome! Keep on keepin' on!", "Wow! That's awesome! Don't stop now!", "You got all of them right! Way to go! Go on to the next problem!", "You're doing great! Keep on truckin'!", "Oh yeah! Try the next function!"]:finalrand := rand(1..nops(final)):hint:=["Now try it yourself.", "There it is. Now figure it out.", "The hint is great, but try it yourself.", "That was practice. Try it again.", "Don't settle for the hint. Work it out."]:hintrand := rand(1..nops(hint)):constanthintC:=["You got one of the limits right. :)", "Check one of the limits.", "One of your limits is a little iffy."]:constCrand := rand(1..nops(constanthintC)):gen1:=rand(-3..3):
gen2:=rand(1..2):
gen3:=rand(1..3):ask:=proc()
global correctanswerf1, correctanswerf2, correctanswerC, fp;
local a, b, c ,d, p, q, g, h, i, whichequal, fmml;
a:=gen1();
b:=RandomTools[Generate](choose({-1,-2,1,2}));
c:=gen1();
d:=gen1();
while (c=0 and d=0) do
c:=gen1();
d:=gen1();
end do;
p:=gen3();
q:=gen3();
i:=gen1();
while(i=0) do
i:=gen1();
end do;
g:=a*x^p+C*b*x;
h:=c*x^q+d*x;
if gen2()=2 then g,h:=h,g end if;
correctanswerf1:=limit(g,x=i,'left');
correctanswerf2:=limit(h,x=i,'right');
correctanswerC:=solve(correctanswerf1=correctanswerf2,C);
whichequal:=gen2();
fp:=`if`(whichequal=1,
piecewise(x<=i,g,x>i,h),
piecewise(x<i,g,x>=i,h)
);
fmml:=`if`(whichequal=1,
piecewise(x<=i,f[1](x),x>i,f[2](x)),
piecewise(x<i,f[1](x),x>=i,f[2](x))
);
Set(MMLfunction(value)=MathML[Export](f(x) = fmml));
Set(MMLf1(value)=MathML[Export](f[1](x) = g));
Set(MMLf2(value)=MathML[Export](f[2](x) = h));
Set(MMLf3(value)=MathML[Export](Limit(f[1],x=i,'left')));
Set(MMLf4(value)=MathML[Export](Limit(f[2],x=i,'right')));
Set(answer1=""):
Set(answer2=""):
Set(responsef1f2="", responsef1f2(background)=white);
Set(answerC=""):
Set(responseC="", responseC(background)=white);
Set(reply="", reply(background)=white);
end proc:checkf1f2:=proc()
global correctanswer1,correctanswer2;
local useranswerf1, useranswerf2, constantcheck;
useranswerf1:=Get(answer1::algebraic, corrections=true, update=true):
useranswerf2:=Get(answer2::algebraic, corrections=true, update=true):
if ((useranswerf1=correctanswerf1) and (useranswerf2=correctanswerf2)) then
Set(responsef1f2=" Correct ", responsef1f2(background)=green);
Set(reply=rights[rightrand()], reply(background)=green);
elif (useranswerf1=correctanswerf1 or evalb(useranswerf2=correctanswerf2)) then
Set(responsef1f2=" Warning ", responsef1f2(background)=lightorange);
Set(reply=constanthintC[constCrand()], reply(background)=lightorange);
else
Set(responsef1f2="Incorrect", responsef1f2(background)=red);
Set(reply=wrong[wrongrand()], reply(background)=red);
end if;
end proc:showf1f2:=proc()
global correctanswerf1, correctanswerf2;
Set(answer1=correctanswerf1);
Set(answer2=correctanswerf2);
Set(responsef1f2=" Shown ", responsef1f2(background)=lightorange);
Set(reply=hint[hintrand()], reply(background)=lightorange);
end proc:plotC:=proc()
global fp;
local useranswerC, p1;
useranswerC:=Get(answerC::realcons, corrections=true, update=true):
p1:=plot(eval(fp,C=useranswerC), x=-4..4, discont=true, tickmarks=[9,default]);
Set(PlotC=p1);
end proc:checkC:=proc()
global correctanswerC;
local useranswerC;
useranswerC:=Get(answerC::realcons, corrections=true, update=true):
if useranswerC=correctanswerC then
Set(responseC=" Correct ", responseC(background)=green);
Set(reply=final[finalrand()], reply(background)=green);
else
Set(responseC="Incorrect", responseC(background)=red);
Set(reply=wrong[wrongrand()], reply(background)=red);
end if;
end proc:showC:=proc()
global correctanswerC;
Set(answerC=correctanswerC);
Set(responseC=" Shown ", responseC(background)=lightorange);
Set(reply=hint[hintrand()], reply(background)=lightorange);
end proc:PiecewiseCont:=Maplet(onstartup = RunWindow(MAIN), Font[helv10]("helvetica",10),
Window[MAIN](title="Making Piecewise Functions Continuous",
[ halign=none,
[ halign=none,
Button("New Function", Evaluate(function = "ask"), background=magenta),
HorizontalGlue(),
Button("Quit", Shutdown(), background=pink)
],
[ halign=none, border=true,
caption="Goal: Find the value of C so that the following function is continuous:",
MathMLViewer[MMLfunction](value=
MathML[Export](f(x) = piecewise(x<=i,f[1](x),x>i,f[2](x))),
width=170, height=120),
"where",
[ valign=none, inset=0,
MathMLViewer[MMLf1](value=
MathML[Export](f[1](x) = f[1]), width=160, height=40),
MathMLViewer[MMLf2](value=
MathML[Export](f[2](x) = f[2]), width=160, height=40)
]
],
[ halign=none, border=true,
caption="Compute the Limits from the Left and Right.",
MathMLViewer[MMLf3](value=
MathML[Export](Limit(f[1](x),x=1,'left')), width=75, height=45),
"=",
TextField[answer1](width=5, background=turquoise),
" ",
MathMLViewer[MMLf4](value=
MathML[Export](Limit(f[2](x),x=1,'right')), width=75, height=45),
"=",
TextField[answer2](width=5, background=turquoise),
Button("Check", Evaluate(function = "checkf1f2"), background=green),
TextField[responsef1f2](width=10, editable=false, background=white),
Button("Show", Evaluate(function = "showf1f2"), background=lightorange)
],
[ halign=none, inset=0, border=true,
caption="Equate and Solve for C. Plot it to try your C.",
HorizontalGlue(),
"C =", TextField[answerC](width=5, background=turquoise),
HorizontalGlue(),
Button("Plot", onclick=AP1, background=tan),
Button("Check", Evaluate(function = "checkC"), background=green),
TextField[responseC](width=10, editable=false, background=white),
Button("Show", Evaluate(function = "showC"), background=lightorange)
],
[ halign=none,
TextField[reply](width=40, editable=false, background=white)
],
[ inset=0, spacing=0, background="#DDFFFF",
Label("Programmer: Chris Patrick", font=helv10),
HorizontalGlue(),
Label("Copyright 2007, P. Yasskin", font=helv10)
]
]
),
Window[PLOT](title="Plot with your C:",
[ [Plotter[PlotC](width=200, height=200)
],
[ Button("Close", CloseWindow(PLOT), background=pink)
]
]
),
Action[AP1](RunWindow(PLOT), Evaluate(function = "plotC"))
):Maplets[Display]( PiecewiseCont );