Continuous functions of one variable

ContFunc

2-(1.8)^2

Show[Plot[2-x^2,{x,-2,2}],Graphics[{Red,Line[{{0,1},{1,1},{1,0}}],Line[{{0,1},{1,1}}],Brown,Line[{{-.5,0},{-.5,1.75},{0,1.75}}],Green,Line[{{-1.8,0},{-1.8,2-(1.8)^2},{0,2-(1.8)^2}}],
Thick,Purple,Line[{{Sqrt[2],0},{0,0}}],
Black,Inset["(1,1)",{1+.15,1+.15}],Inset["(0.5,1.75)",{-.5+-.3,1.75+.1}],Inset["(-1.8,-1.24)",{2-(1.8)^2-.05,2-(1.8)^2-.15}],Inset["((Sqrt[2]),0)",{Sqrt[2]+.3,.15}],Inset["(0,0)",{.24,-.15}]}
],AspectRatio->1,ImageSize->275
]

uppar[t_]:=2-t^2

Manipulate[
Show[
Plot[uppar[x],{x,-2,2}],
Graphics[{Red,Line[{{t,0},{t,uppar[t]},{0,uppar[t]}}]}
],
AspectRatio->1,ImageSize->175
],{t,-2,2}]

upparmovie=Table[Show[
Plot[uppar[x],{x,-2,2}],
Graphics[{Red,Line[{{t,0},{t,uppar[t]},{0,uppar[t]}}]}
],
AspectRatio->1,ImageSize->175
],{t,-2,2,.1}];

Export["W:\\www\\MM\\Mathematica\\Function Representations\\upparmovie.gif",upparmovie]

ContFunc2

foolyou[x_]:=.0002((x^3-10)/(3 Exp[-x]+1))^6

Plot[foolyou[x],{x,-1,3},PlotRange->{{-1,3},{0,3}},AspectRatio->3/4]

10^(1/3)//N

Solve[x^3-10==0,x]

-(((-10))^(1/3))//N

(((-1))^(2/3)) (10^(1/3))//N

foolyou'[x]

foolyou''[x]

NSolve[foolyou'[x]==0,x,Reals]

foolyou''[1.1648251336938016`]

Discontinuous function

Remove[dfunc]

dfunc[x_]:=2-x^2/;x>0;dfunc[x_]:=2-x^2/;x<-1;dfunc[x_]:=1-x^2/;x>-1&&x<0;

Plot[dfunc[x],{x,-2,2},AspectRatio->1,Exclusions->{x=0,x=-1 },ImageSize->150]

Polar Plot

PolarPlot[t,{t,0,2Pi},ImageSize->200,PlotRange->{{-4,7},{-5,2}},AspectRatio->7/11]

ToPolarCoordinates[{t,1}]

FromPolarCoordinates[{(Sqrt[1+(t^2)]),ArcTan[t,1]}]

ToPolarCoordinates[{x,y}]

FromPolarCoordinates[{r,t}]

PolarPlot[1,{t,0,2Pi},ImageSize->150]

PolarPlot[Sin[t],{t,0,2Pi},ImageSize->150]

PolarPlot[Sin[2t],{t,0,2Pi},PlotRange->{{-1,1},{-1,1}},ImageSize->150]

PolarPlot[Cos[2t],{t,0,2Pi},ImageSize->150]

PolarPlot[Sin[3t],{t,0,2Pi},PlotRange->{{-1,1},{-1,1}},ImageSize->150]

Functions from R to R x R

Unit Circle

Bare unit circle

ParametricPlot[{Cos[t],Sin[t]},{t,0,2Pi},ImageSize->150]

Show[{ParametricPlot[{Cos[t],Sin[t]},{t,0,2Pi},PlotRange->{-1.3,1.3},ImageSize->200],Graphics[{PointSize[Large],Black,Point[{Cos[5 Pi/3],Sin[5 Pi/3]}],Text["5π/3",{Cos[5Pi/3]+.15,Sin[5 Pi/3]-.15}],Point[{Cos[-Pi/2],Sin[-Pi/2]}],Text["3π/2",{Cos[3Pi/2]-.2,Sin[3Pi/2]-.2}],Point[{Cos[Pi],Sin[Pi]}],Text["π",{Cos[Pi]-.1,Sin[Pi]+.1}],Point[{Cos[Pi/4],Sin[Pi/4]}],Text["π/4",{Cos[Pi/4]+.15,Sin[Pi/4]+.15}]}]}]

Sqrt[3]/2//N

Manipulate[
Show[ParametricPlot[{Cos[t],Sin[t]},{t,0,2Pi},PlotRange->{-1.3,1.3},ImageSize->200],Graphics[{PointSize[Large],Black,Point[{Cos[t],Sin[t]}]}],ImageSize->150],{{t,0},0,10 Pi}]

Animate[
Show[
ParametricPlot[{Cos[t],Sin[t]},{t,0,8Pi},PlotRange->{{-1.3,2.3},{-1.3,1.3}},ImageSize->200],Graphics[{PointSize[Large],Black,Point[{Cos[t],Sin[t]}],Text[t,{2,.5}]}],ImageSize->150],{{t,0},0,10 Pi},AnimationRunning->False]

outstring[t_]:=StringJoin["t=",ToString[NumberForm[N[(t/Pi)],{3,3}]],"π"]

outstring[1.3]

Table[Show[
ParametricPlot[{Cos[t],Sin[t]},{t,0,2Pi},PlotRange->{{-1.3,2.5},{-1.3,1.3}},ImageSize->200],Graphics[{PointSize[Large],Black,Point[{Cos[t],Sin[t]}],Text[NumberForm["t="N[(t/Pi)],{3,3}]"π",{2,.5}]}],ImageSize->250,Axes->False],{t,0,2Pi,Pi/30}];

circlemovie=Table[Show[
ParametricPlot[{Cos[t],Sin[t]},{t,0,2Pi},PlotRange->{{-1.3,2.5},{-1.3,1.3}},ImageSize->200],Graphics[{PointSize[Large],Black,Point[{Cos[t],Sin[t]}],Text[outstring[t],{2,.5}]}],ImageSize->250,Axes->False],{t,0,2Pi,Pi/30}];

ListAnimate[circlemovie, AnimationRunTime->1,AnimationRunning->False]

Export["W:\www\MM\Mathematica\Function Representations\circlemovie.gif",circlemovie]

Unit circle with universal cover

ParametricPlot3D[{{Cos[t],Sin[t],0},{Cos[t],Sin[t],.2t}},{t,0,10Pi},ImageSize->200]

Show fibers

Show[ParametricPlot3D[{{Cos[t],Sin[t],0},{Cos[t],Sin[t],.2t}},{t,0,8Pi}],Graphics3D[{PointSize[Large],Black,Point[{Cos[5 Pi/3],Sin[5 Pi/3],0}],Point[{Cos[3 Pi/2],Sin[3 Pi/2],0}],Point[{Cos[Pi],Sin[Pi],0}],Point[{Cos[Pi/4],Sin[Pi/4],0}],Point[{Cos[5 Pi/3],Sin[5 Pi/3],.2 (5Pi/3)}],Point[{Cos[3 Pi/2],Sin[3 Pi/2],.2 (3 Pi/2)}],Point[{Cos[Pi],Sin[Pi],.2  Pi}],Point[{Cos[Pi/4],Sin[Pi/4],.2(Pi/4)}],Point[{Cos[5 Pi/3],Sin[5 Pi/3],.2(5 Pi/3+2 Pi)}],Point[{Cos[3 Pi/2],Sin[3 Pi/2],.2(3 Pi/2+2 Pi)}],Point[{Cos[Pi],Sin[Pi],.2 (Pi+2 Pi)}],Point[{Cos[Pi/4],Sin[Pi/4],.2(Pi/4+2 Pi)}],Point[{Cos[5 Pi/3],Sin[5 Pi/3],.2(5 Pi/3+4Pi)}],Point[{Cos[3 Pi/2],Sin[3 Pi/2],.2(3 Pi/2+4 Pi)}],Point[{Cos[Pi],Sin[Pi],.2(Pi+4 Pi)}],Point[{Cos[Pi/4],Sin[Pi/4],.2(Pi/4+4 Pi)}],Point[{Cos[5 Pi/3],Sin[5 Pi/3],.2(5 Pi/3+6Pi)}],Point[{Cos[3 Pi/2],Sin[3 Pi/2],.2(3 Pi/2+6 Pi)}],Point[{Cos[Pi],Sin[Pi],.2(Pi+6 Pi)}],Point[{Cos[Pi/4],Sin[Pi/4],.2(Pi/4+6 Pi)}]
}],Boxed->True
]

Manipulate[Show[ParametricPlot3D[{{Cos[t],Sin[t],0},{Cos[t],Sin[t],.2t}},{t,0,10Pi},ImageSize->170],Graphics3D[{PointSize[Large],Black,Point[{Cos[t],Sin[t],0}],Point[{Cos[t],Sin[t],.2t}]
}]
],{t,0,10 Pi}]

covermovie=Table[Show[
ParametricPlot3D[{{Cos[t],Sin[t],0},{Cos[t],Sin[t],.2t}},{t,0,8Pi},Axes->False],Graphics3D[{PointSize[Large],Black,Point[{Cos[t],Sin[t],0}],Point[{Cos[t],Sin[t],.2t}],Text[outstring[t],{2,.5,1}]}],ImageSize->250],{t,0,8Pi,Pi/30}];

ListAnimate[covermovie, AnimationRunTime->1,AnimationRunning->False]

Export["W:\www\MM\Mathematica\Function Representations\covermovie.gif",covermovie]

Spiral

ParametricPlot[{2 t Cos[t],2 t Sin[t]},{t,0,20 Pi}]

Spiral with universal cover

ParametricPlot3D[{2 t Cos[t],2 t Sin[t],8t},{t,0,20 Pi},ImageSize->200]

ParametricPlot3D[{{2 t Cos[t],2 t Sin[t],0},{2 t Cos[t],2 t Sin[t],8t}},{t,0,20 Pi},ImageSize->200]

Figure eight

PolarPlot[Sqrt[4Cos[2t]],{t,0,2 Pi},ImageSize->300]

ParametricPlot[{x=Cos[t],y=Sin[2t]},{t,0,2Pi},ImageSize->150]

Figure 8 with universal cover

ParametricPlot3D[{{x=Cos[t],y=Sin[2t],0},{x=Cos[t],y=Sin[2t],.1t}},{t,0,10Pi},ImageSize->250,AxesLabel->{"x","y","z"}]

Manipulate[
ParametricPlot3D[{{x=Cos[t],y=Sin[2t],0},{x=Cos[t],y=Sin[2t],.1t}},{t,0,10Pi},ImageSize->150,AxesLabel->{X,Y,Z},ViewPoint->{a,b,c}],{{a,-1},-5,5},{{b,4.3},-5,5},{{c,1.92},-5,5}]

covermovie=Table[Show[
ParametricPlot3D[{{Cos[t],Sin[t],0},{Cos[t],Sin[t],.2t}},{t,0,8Pi},Axes->False],Graphics3D[{PointSize[Large],Black,Point[{Cos[t],Sin[t],0}],Point[{Cos[t],Sin[t],.2t}],Text[outstring[t],{2,.5,1}]}],ImageSize->250],{t,0,8Pi,Pi/30}];

Fig8movie=Table[Show[
ParametricPlot3D[{{Cos[t],Sin[2t],0},{Cos[t],Sin[2t],.1t}},{t,0,8Pi},AxesLabel->{X,Y,Z},ViewPoint->{-2.65,-5,1.12}],Graphics3D[{PointSize[Large],Black,Point[{Cos[t],Sin[2t],0}],Point[{Cos[t],Sin[2t],.1t}]}],ImageSize->250],{t,0,8Pi,Pi/30}];

ListAnimate[Fig8movie, AnimationRunTime->3,AnimationRunning->True]