SetDirectory[NotebookDirectory[]]

th1=0.007;th0=0.0025; ps1=0.015; ps0=0.01

Bijections

figff1=Plot[{(x^2),(Sqrt[x]),x},{x,0,10},PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]},{RGBColor[0.5,0,0],Thickness[th1]},{RGBColor[0.5,0.5,0.5],Thickness[th0]}},PlotRange->{{-0.2,5},{-0.2,5}},AspectRatio->Automatic]

(* Export["figff1.png",figff1] *)

figff1a=Plot[{x^3,Sign[x]Abs[x]^(1/3),x},{x,-10,10},Exclusions->None,PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]},{RGBColor[0.5,0,0],Thickness[th1]},{RGBColor[0.5,0.5,0.5],Thickness[th0]}},PlotRange->{{-3.1,3.1},{-3.1,3.1}},AspectRatio->Automatic]

(* Export["figff1a.png",figff1a] *)

figff2=Plot[{Exp[x],Log[x],x},{x,-4,10},PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]},{RGBColor[0.5,0,0],Thickness[th1]},{RGBColor[0.5,0.5,0.5],Thickness[th0]}},PlotRange->{{-3.1,5},{-3.1,5}},AspectRatio->Automatic]

(* Export["figff2.png",figff2] *)

ff3=Plot[{Sin[x],x},{x,-10,10},PlotStyle->{{RGBColor[0.5,0.5,1],Thickness[th0]},{RGBColor[0.5,0.5,.5],Thickness[th0]}}];
ff3a=Plot[Sin[x],{x,-Pi/2,Pi/2},PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]}}];
fi3a=Plot[ArcSin[x],{x,-1,1},PlotStyle->{{RGBColor[0.5,0,0],Thickness[th1]}}];
figff3=Show[ff3,ff3a,fi3a,Epilog->{{PointSize[ps1],RGBColor[0,0,.5],Point[{Pi/2,1}],Point[{-Pi/2,-1}]},{PointSize[ps1],RGBColor[0.5,0,0],Point[{1,Pi/2}],Point[{-1,-Pi/2}]}},PlotRange->{{-4,4},{-((3 )/4)Pi,((3 )/4)Pi}},Ticks->{Union[Range[-3/2 Pi,3/2 Pi,(Pi/2)],{-1,1}],Union[Range[-3 Pi,3 Pi,(Pi/2)],{-1,1}]},AspectRatio->Automatic]

(*  Export["figff3.png",figff3] *)

ff4=Plot[{Cos[x],x},{x,-10,10},PlotStyle->{{RGBColor[0.5,0.5,1],Thickness[th0]},{RGBColor[0.5,0.5,.5],Thickness[th0]}}];
ff4a=Plot[Cos[x],{x,0,Pi},PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]}}];
fi4a=Plot[ArcCos[x],{x,-1,1},PlotStyle->{{RGBColor[0.5,0,0],Thickness[th1]}}];
figff4=Show[ff4,ff4a,fi4a,Epilog->{{PointSize[ps1],RGBColor[0,0,.5],Point[{0,1}],Point[{Pi,-1}]},{PointSize[ps1],RGBColor[0.5,0,0],Point[{-1,Pi}],Point[{1,0}]}},PlotRange->{{-3,5},{-1.4,Pi+.5}},Ticks->{Union[Range[-3/2 Pi,3/2 Pi,(Pi/2)],{-1,1}],Union[Range[-3 Pi,3 Pi,(Pi/2)],{-1,1}]},AspectRatio->Automatic]

(* Export["figff4.png",figff4] *)

ff5=Plot[{Tan[x],x},{x,-10,10},Exclusions->Range[-3 Pi,3 Pi,Pi/2],PlotStyle->{{RGBColor[0.5,0.5,1],Thickness[th0]},{RGBColor[0.5,0.5,.5],Thickness[th0]}}];
ff5a=Plot[Tan[x],{x,-Pi/2,Pi/2},PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]}}];
fi5a=Plot[ArcTan[x],{x,-10,10},PlotStyle->{{RGBColor[0.5,0,0],Thickness[th1]}}];
figff5=Show[ff5,ff5a,fi5a,PlotRange->{{-5,5},{-5,5}},Ticks->{Union[Range[-3/2 Pi,3/2 Pi,(Pi/2)]],Union[Range[-3 Pi,3 Pi ,Pi/2]]},AspectRatio->Automatic]

(*  Export["figff5.png",figff5] *)

figW0a=Plot[{x Exp[x],ProductLog[x]},{x,-1,10},PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]},{RGBColor[0.5,0,0],Thickness[th1]}},PlotRange->{{-2.1,5},{-2.1,5}},AspectRatio->Automatic];

figW0b=Plot[{x Exp[x],x},{x,-10,10},Exclusions->Range[-3 Pi,3 Pi,Pi/2],PlotStyle->{{RGBColor[0.5,0.5,1],Thickness[th0]},{RGBColor[0.5,0.5,.5],Thickness[th0]}}];

figW0=Show[figW0b,figW0a,Epilog->{{PointSize[ps1],RGBColor[0,0,.5],Point[{-1,-1/E}]},{PointSize[ps1],RGBColor[0.5,0,0],Point[{-1/E,-1}]},{Thickness[0.001],Line[{{-1/E,0},{-1/E,0.05}}],Line[{{0,-1/E},{0.05,-1/E}}]},{Text[-1/E,{-1/E,.25}],Text[-1/E,{.22,-1/E}]}},PlotRange->{{-2.1,4.7},{-2.1,4.7}},AspectRatio->Automatic,Ticks->{Range[-10,10,1],Range[-10,10,1]}]

(* Export["figW0.png",figW0] *)

Solve[x Exp[x]==-1/4,x]

figWn1a=Plot[{x Exp[x]},{x,-10,-1},PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]}},PlotRange->{{-5.1,2.1},{-5.1,2.1}},AspectRatio->Automatic];

figWn1aa=Plot[{ProductLog[-1,x]},{x,-1/E,0},PlotStyle->{{RGBColor[0.5,0,0],Thickness[th1]}},PlotRange->{{-5.1,2.1},{-5.1,2.1}},AspectRatio->Automatic];

figWn1b=Plot[{x Exp[x],x},{x,-10,10}, PlotStyle->{{RGBColor[0.5,0.5,1],Thickness[th0]},{RGBColor[0.5,0.5,.5],Thickness[th0]}}];

figWn1=Show[figWn1a,figWn1aa,figWn1b,Epilog->{{PointSize[ps1],RGBColor[0,0,.5],Point[{-1,-1/E}]},{PointSize[ps1],RGBColor[0.5,0,0],Point[{-1/E,-1}]},{PointSize[ps0],White,Point[{-1,-1/E}],Point[{-1/E,-1}]},{Thickness[0.001],Line[{{-1/E,0},{-1/E,0.05}}],Line[{{0,-1/E},{0.05,-1/E}}]},{Text[-1/E,{-1/E,.25}],Text[-1/E,{.22,-1/E}]}},PlotRange->{{-4.7,2.1},{-4.7,2.1}},AspectRatio->Automatic,Ticks-> {Range[-10,10,1],Range[-10,10,1]}]

(* Export["figWn1.png",figWn1] *)

figExf= Plot[{((-x)/(Exp[-x]-1)),If[x≤1,x+ProductLog[-1,-x Exp[-x]],x+ProductLog[0,-x Exp[-x]]],x},{x,-15,15},PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]},{RGBColor[0.5,0,0],Thickness[th1]},{RGBColor[0.5,0.5,.5],Thickness[th0]}},PlotRange->{{-4.9,4.9},{-4.9,4.9}},AspectRatio->Automatic,Ticks->{Range[-10,10,1],Range[-10,10,1]}]

(* Export["figExf.png",figExf] *)

figExga= Plot[{((Exp[x]-1)/x),x},{x,-15,15},PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]},{RGBColor[0.5,0.5,.5],Thickness[th0]}},PlotRange->{{-4.9,4.9},{-4.9,4.9}},AspectRatio->Automatic,Ticks->{Range[-10,10,1],Range[-10,10,1]}];

figExgb= Plot[{If[x≤1,-(1/x)-ProductLog[0,-(1/x) Exp[-1/x]],-(1/x)-ProductLog[-1,-(1/x) Exp[-1/x]]]},{x,0,15},PlotStyle->{{RGBColor[0.5,0,0],Thickness[th1]}},PlotRange->{{-4.9,4.9},{-4.9,4.9}},AspectRatio->Automatic,Ticks->{Range[-10,10,1],Range[-10,10,1]}];

figExg=Show[figExga,figExgb,PlotRange->{{-4.9,4.9},{-4.9,4.9}},AspectRatio->Automatic,Ticks->{Range[-10,10,1],Range[-10,10,1]}]

(* Export["figExg.png",figExg] *)

figExpSq= Plot[{x Exp[x^2],Sign[x]((Sqrt[ProductLog[2 (x^2)]])/(Sqrt[2])),x},{x,-15,15},PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]},{RGBColor[0.5,0,0],Thickness[th1]},{RGBColor[0.5,0.5,.5],Thickness[th0]}},PlotRange->{{-4.9,4.9},{-4.9,4.9}},AspectRatio->Automatic,Ticks->{Range[-10,10,1],Range[-10,10,1]}]

(* Export["figExpSq.png",figExpSq] *)

Cycloid as a function

Solve[1-Cos[t]==y,t]

(t-Sin[t])/.{t-> ArcCos[1-y]}

figCyxc=Plot[{x-Sin[x]},{x,-2Pi,5 Pi},PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]}},PlotRange->{{-0.6-Pi,4 Pi+.6},{-0.6- Pi,4 Pi+.6}},Ticks->{Range[-2Pi,5Pi,(Pi/2)],Range[-2Pi,5Pi,(Pi/2)]},AspectRatio->Automatic,GridLines->{Range[-Pi,4Pi,Pi],Range[-Pi,4Pi,Pi]},GridLinesStyle->Directive[GrayLevel[.8],Thickness[0.001]],ImageSize->600]

tblCyu=Table[{t-Sin[t],t},{t,-3Pi,8 Pi,(Pi/(2*256))}];

Cyui=Interpolation[tblCyu]

figCyu=Plot[{x-Sin[x],Cyui[x],x},{x,-2Pi,6 Pi},PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]},{RGBColor[0.5,0,0],Thickness[th1]},{RGBColor[0.5,0.5,0.5],Thickness[th0]}},PlotRange->{{-0.6-Pi,4 Pi+.6},{-0.6-Pi,4 Pi+.6}},Ticks->{Range[-2Pi,5Pi,(Pi/2)],Range[-2Pi,5Pi,(Pi/2)]},AspectRatio->Automatic,GridLines->{Range[-Pi,4Pi,Pi],Range[-Pi,4Pi,Pi]},GridLinesStyle->Directive[GrayLevel[.8],Thickness[0.001]],ImageSize->600]

(* Export["figCyu.png",figCyu]; Export["figCyxc.png",figCyxc]; Export["figCyu.gif",figCyu]; Export["figCyxc.gif",figCyxc] *)

(*  Export["figCyu.png",figCyu] *)

Linear functions

Manipulate[Plot[{m x+b,(1/m)x-(b/m),x},{x,-10,10},Exclusions->None,PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]},{RGBColor[0.5,0,0],Thickness[th1]},{RGBColor[0.5,0.5,0.5],Thickness[th0]}},PlotRange->{{-3.1,3.1},{-3.1,3.1}},AspectRatio->Automatic],{m,-1,3},{b,-5,5}]

LinBijection1p=Plot[Evaluate[{m x+b,(1/m)x-(b/m),x}/.{m->1,b-> 1/2}],{x,-10,10},Exclusions->None,PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]},{RGBColor[0.5,0,0],Thickness[th1]},{RGBColor[0.5,0.5,0.5],Thickness[th0]}},PlotRange->{{-3.1,3.1},{-3.1,3.1}},AspectRatio->Automatic,ImageSize->800,TicksStyle->Directive[FontSize->18]]

LinBijectionpp=Plot[Evaluate[{m x+b,(1/m)x-(b/m),x}/.{m->2,b-> 1}],{x,-10,10},Exclusions->None,PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]},{RGBColor[0.5,0,0],Thickness[th1]},{RGBColor[0.5,0.5,0.5],Thickness[th0]}},PlotRange->{{-3.1,3.1},{-3.1,3.1}},AspectRatio->Automatic,ImageSize->800,TicksStyle->Directive[FontSize->18]]

LinBijectionpn=Plot[Evaluate[{m x+b,(1/m)x-(b/m),x}/.{m->3,b->-1/2}],{x,-10,10},Exclusions->None,PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]},{RGBColor[0.5,0,0],Thickness[th1]},{RGBColor[0.5,0.5,0.5],Thickness[th0]}},PlotRange->{{-3.1,3.1},{-3.1,3.1}},AspectRatio->Automatic,ImageSize->800,TicksStyle->Directive[FontSize->18]]

LinBijectionnp=Plot[Evaluate[{m x+b,(1/m)x-(b/m),x}/.{m->-1/2,b->1}],{x,-10,10},Exclusions->None,PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]},{RGBColor[0.5,0,0],Thickness[th1]},{RGBColor[0.5,0.5,0.5],Thickness[th0]}},PlotRange->{{-3.1,3.1},{-3.1,3.1}},AspectRatio->Automatic,ImageSize->800,TicksStyle->Directive[FontSize->18]]

LinBijectionnn=Plot[Evaluate[{m x+b,(1/m)x-(b/m),x}/.{m->-2,b-> -1/2}],{x,-10,10},Exclusions->None,PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]},{RGBColor[0.5,0,0],Thickness[th1]},{RGBColor[0.5,0.5,0.5],Thickness[th0]}},PlotRange->{{-3.1,3.1},{-3.1,3.1}},AspectRatio->Automatic,ImageSize->800,TicksStyle->Directive[FontSize->18]]

LinBijectionnb=Plot[Evaluate[{m x+b,(1/m)x-(b/m),x}/.{m->.02,b-> -1}],{x,-10,10},Exclusions->None,PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]},{RGBColor[0.5,0,0],Thickness[th1]},{RGBColor[0.5,0.5,0.5],Thickness[th0]}},PlotRange->{{-3.1,3.1},{-3.1,3.1}},AspectRatio->Automatic,ImageSize->800,TicksStyle->Directive[FontSize->18]]

(* Export["LinBijection1p.png",LinBijection1p];Export["LinBijectionpp.png",LinBijectionpp];Export["LinBijectionpn.png",LinBijectionpn];Export["LinBijectionnp.png",LinBijectionnp];Export["LinBijectionnn.png",LinBijectionnn];Export["LinBijectionnb.png",LinBijectionnb] *)

Steps

figSign=Plot[Sign[x],{x,-10,10},PlotStyle->{{RGBColor[0,0,0.5],Thickness[th1]}},Epilog->{PointSize[ps1],RGBColor[0,0,0.5],Point[{0,0}],Point[{0,-1}],Point[{0,1}],White,PointSize[ps0],Point[{0,-1}],Point[{0,1}]},PlotRange->{{-5,5},{-2,2}},Ticks->{Union[Range[-10,10,1]], Range[-2,2 ,1]},AspectRatio->Automatic,ImageSize->500]

NotebookDirectory[]

(* Export["figSign.png",figSign] *)

figUS=Plot[UnitStep[x],{x,-10,10},PlotStyle->{{RGBColor[0,0,0],Thickness[th1]}},Epilog->{PointSize[ps1],Black,Point[{0,0}],Point[{0,1}],White,PointSize[ps0],Point[{0,0}]},PlotRange->{{-5,5},{-2,2}},Ticks->{Union[Range[-10,10,1]], Range[-2,2 ,1]},AspectRatio->Automatic]

(* Export["figUS.png",figUS] *)

figFloor=Plot[Floor[x],{x,-10,10},PlotStyle->{{RGBColor[0,0,0],Thickness[th1]}},Epilog->{PointSize[ps1],Black,Table[Point[{n,n}],{n,-10,10}],Table[Point[{n,n-1}],{n,-10,10}],White,PointSize[ps0],Table[Point[{n,n-1}],{n,-10,10}]},PlotRange->{{-4.5,4.5},{-4.5,4.5}},Ticks->{Union[Range[-10,10,1]], Range[-10,10 ,1]},AspectRatio->Automatic]

(* Export["figFloor.png",figFloor] *)

figCeiling=Plot[Ceiling[x],{x,-10,10},PlotStyle->{{RGBColor[0,0,0],Thickness[th1]}},Epilog->{PointSize[ps1],Black,Table[Point[{n,n}],{n,-10,10}],Table[Point[{n,n+1}],{n,-10,10}],White,PointSize[ps0],Table[Point[{n,n+1}],{n,-10,10}]},PlotRange->{{-4.5,4.5},{-4.5,4.5}},Ticks->{Union[Range[-10,10,1]], Range[-10,10 ,1]},AspectRatio->Automatic]

(* Export["figCeiling.png",figCeiling] *)

Circle - function?

UnitCircle=ParametricPlot[{Cos[s],Sin[s]},{s,0,2 Pi},PlotStyle->{{Thickness[0.007]}},PlotRange->{{-3,3},{-2,2}},ImageSize->800,TicksStyle->Directive[FontSize->18]]

(* Export["UnitCircle.png",UnitCircle] *)

Considet the set F={(x,y) ∈ℝ×ℝ | (x^2)+(y^2)=1}  which is a subset of the Cartesian product ℝ×ℝ. Answer the following questions: (a) Is the subset F a function from ℝ to ℝ?  (b) Find the largest subsets X⊆ℝ and Y⊆ℝ  such that  the subset  f = {(x,y) ∈F| x∈X, y∈Y}   is a function from X to Y.  Find all such sets  X and Y. (c)

Hyperbolic functions

HypCos=Plot[{Cosh[x],(1/2)Exp[-x],(1/2)Exp[x]},{x,-10,10},Exclusions->None,PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]},{RGBColor[1,0.5,0.5],Thickness[th0]},{RGBColor[0.5,1,0.5],Thickness[th0]}},PlotRange->{{-3.1,3.1},{-0.1,8.1}},AspectRatio->Automatic,ImageSize->800,TicksStyle->Directive[FontSize->20]]

HypSin=Plot[{Sinh[x],-(1/2)Exp[-x],(1/2)Exp[x]},{x,-10,10},Exclusions->None,PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]},{RGBColor[1,0.5,0.5],Thickness[th0]},{RGBColor[0.5,1,0.5],Thickness[th0]}},PlotRange->{{-3.1,3.1},{-4.1,4.1}},AspectRatio->Automatic,ImageSize->800,TicksStyle->Directive[FontSize->20]]

HypTan=Plot[{Tanh[x],-1,1},{x,-10,10},Exclusions->None,PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]},{RGBColor[1,0.5,0.5],Thickness[th0]},{RGBColor[0.5,1,0.5],Thickness[th0]}},PlotRange->{{-3.1,3.1},{-1.1,1.1}},AspectRatio->Automatic,ImageSize->800,Ticks->{Automatic,{-1,1}},TicksStyle->Directive[FontSize->18]]

(* Export["HypCos.png",HypCos];Export["HypSin.png",HypSin];Export["HypTan.png",HypTan]; *)

Hyperbola=ParametricPlot[{{-Cosh[t],Sinh[t]},{Cosh[t],Sinh[t]}},{t,-10,10},PlotRange->{{-3.1,3.1},{-3.1,3.1}},PlotStyle->{{Blue,Thickness[0.007]}},ImageSize->800,TicksStyle->Directive[FontSize->18]]

(* Export["Hyperbola.png",Hyperbola];*)

HypTanInv=Plot[{x,Tanh[x],ArcTanh[x]},{x,-6,6},Exclusions->None,PlotStyle->{{RGBColor[0.5,0.5,0.5],Thickness[th0]},{RGBColor[0,0,.5],Thickness[th1-0.002]},{RGBColor[0.5,0,0],Thickness[th1-0.002]}},Epilog->{
{Thickness[0.001],Dashing[{0.003,0.002}],RGBColor[0,.5,.5],Line[{{1,-10},{1,10}}]},{Thickness[0.001],Dashing[{0.003,0.002}],RGBColor[0,0.5,0.5],Line[{{-1,-10},{-1,10}}]},{Thickness[0.001],Dashing[{0.003,0.002}],RGBColor[.5,.5,0],Line[{{-10,1},{10,1}}]},{Thickness[0.001],Dashing[{0.003,0.002}],RGBColor[.5,.5,0],Line[{{-10,-1},{10,-1}}]}

},PlotRange->{{-4.1,4.1},{-4.1,4.1}},AspectRatio->Automatic,ImageSize->600,Ticks->{Range[-10,10,1],Range[-10,10,1]},TicksStyle->Directive[FontSize->14]]

(* Export["HypTanInv.png",HypTanInv];*)

Circle

Solve[(z^3)-2 (z^2)+2 z -2==0,z,Reals]

Solve[(z^3)-2 (z^2)+2 z -2==0,z]

N[(1/3) (2-(2/(((17+3 (Sqrt[33])))^(1/3)))+(((17+3 (Sqrt[33])))^(1/3)))]

TeXForm[(1/3) (2-(2/(((17+3 (Sqrt[33])))^(1/3)))+(((17+3 (Sqrt[33])))^(1/3)))]

x0s=Log[Root[-2+2 #1-2 ((#1)^2)+((#1)^3)&,1]]

N[x0s,12]

N[FullSimplify[x0s+Exp[2 x0s]-Exp[x0s]],12]

Solve[(-2+2 ((#1)^2)+((#1)^3))&[z]==0,z]

N[%]

TeXForm[(1/3) (-2+(((19-3 (Sqrt[33])))^(1/3))+(((19+3 (Sqrt[33])))^(1/3)))]

(t+Exp[2 t]-((Exp[2t])/(1+(1/(Sqrt[1+Exp[2t]])))))/.FindRoot[((Exp[t])/(1+(1/(Sqrt[1+Exp[2t]]))))==1,{t,.5}]

figExpUC=Plot[{Exp[x],-Exp[-x0s](x-x0s)+Exp[x0s],Exp[x0s](x-x0s)+Exp[x0s]},{x,-4,10},PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]},{RGBColor[0,.5,.5],Thickness[th0]},{RGBColor[0,.5,0],Thickness[th0]}},Prolog->{{GrayLevel[.7],Table[{Circle[{t+Exp[2 t]-((Exp[2t])/(1+(1/(Sqrt[1+Exp[2t]])))),((Exp[t])/(1+(1/(Sqrt[1+Exp[2t]]))))},((Exp[t])/(1+(1/(Sqrt[1+Exp[2t]]))))]},{t,-3,3,.04}]},{Thickness[0.001],Dashing[{0.005,.01}],Line[{{x0s,0},{x0s,Exp[x0s]}}],Line[{{x0s+Exp[2x0s]-Exp[x0s],1},{x0s+Exp[2x0s]-Exp[x0s],0}}]}},
Epilog->{{RGBColor[0.5,0,0.5],Thickness[th1],Circle[{x0s+Exp[2 x0s]-Exp[x0s],1},1]},{Red,PointSize[0.015],Point[{x0s,Exp[x0s]}]},{RGBColor[0.5,0,0.5],PointSize[0.015],Point[{x0s+Exp[2 x0s]-Exp[x0s],1}]},{Text[Style["A",Italic,FontSize->20],{x0s,Exp[x0s]}+{-.03,.12}]},{PointSize[0.012],Point[{x0s,0}],Point[{x0s+Exp[2x0s]-Exp[x0s],0}],Point[{x0s+Exp[2x0s],0}]},
{Text[Style["B",Italic,FontSize->20],{x0s,0}+{-.03,-.12}]},
{Text[Style["C",Italic,FontSize->20],{x0s+Exp[2 x0s],0}+{-.03,-.12}]},
{Text[Style["D",Italic,FontSize->20],{x0s+Exp[2 x0s]-Exp[x0s],1}+{.03,.12}]},
{Text[Style["E",Italic,FontSize->20],{x0s+Exp[2 x0s]-Exp[x0s],0}+{.0,-.12}]}},PlotRange->{{-2,3},{-.5,4}},AspectRatio->Automatic,ImageSize->700]

(* Export["figExpUC.png",figExpUC] *)

Plot[z^3-2z^2+2 z -2,{z,-1,3}]

Unit circle in the graph of the square

Manipulate[Plot[{(x^2)},{x,-3,3},PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]}},Prolog->{{GrayLevel[.5],Thickness[th1],Circle[{0,Abs[t]+5/4},1]}},

PlotRange->{{-2.5,2.5},{-0.02,6}},AspectRatio->Automatic],
{t,-6,6,.05}]

aniUCSquare=Table[Plot[{(x^2)},{x,-3,3},PlotStyle->{{RGBColor[0,0,.5],Thickness[th1]}},Prolog->{{GrayLevel[.5],Thickness[th1],Circle[{0,Abs[t]+5/4},1]}},

PlotRange->{{-2.5,2.5},{-0.02,6}},AspectRatio->Automatic,ImageSize->600],{t,-6,6,.05}];

Length[aniUCSquare]

Show[aniUCSquare[[34]]]

(* Export["aniUCSquare.gif",aniUCSquare,"GIF","AnimationRepetitions"->1,"ImageSize"->600,"DisplayDurations"->(((20/(Length[aniUCSquare])))&/@Range[Length[aniUCSquare]])]  *)

figCirclesinSquare =Plot[{(x^2)},{x,-3,3},PlotStyle->{{RGBColor[0,0,.5],Thickness[0.007]}},Prolog->{{GrayLevel[.7],Table[{Circle[{0, ((t^2)+1/2)},((Sqrt[(t^2)+1/4]))]},{t,0,4,.04}]}},
Epilog->{{Thickness[0.005],Red,Circle[{0,(1/2)+(((((Sqrt[3])/2)))^2)},1],PointSize[0.012],Point[{0,(1/2)+(((((Sqrt[3])/2)))^2)}]}},

 PlotRange->{{-2.75,2.75},{-0.02,7}},AspectRatio->Automatic,ImageSize->800]

(* Export["figCirclesinSquare.png",figCirclesinSquare] *)

Many tangens

64 16

figSinTLs1=Plot[{Sin[x]},{x,-10,10},PlotStyle->{{RGBColor[0,0,0.5],Thickness[0.004]}}];

figSinTLs2=Plot[Evaluate[Table[Cos[x0](x-x0)+Sin[x0],{x0,-8 Pi,8 Pi,Pi/64}]],{x,-10,10},PlotStyle->{{GrayLevel[0.6],Thickness[0.001]}}];
figSinTLs=Show[figSinTLs2,figSinTLs1,PlotRange->{{-3Pi,3Pi},{-3 Pi,3Pi}},Ticks->{Range[-7/2 Pi,7/2 Pi,(Pi/2)],Range[-10 Pi,10 Pi,Pi/2]},AspectRatio->Automatic,ImageSize->800,TicksStyle->Directive[Black,14]]

(* Export["figSinTLs.png",figSinTLs] *)

(* figSinTLs1=Plot[{Sin[x]},{x,-10,10},PlotStyle->{{RGBColor[0,0,0.5],Thickness[0.006]}}];


aniSinTLs=Table[Show[Plot[Evaluate[Table[Cos[x0](x-x0)+Sin[x0],{x0,-8 Pi,Xend,Pi/64}]],{x,-10,10},PlotStyle->{{GrayLevel[0.6],Thickness[0.001]}}],figSinTLs1,PlotRange->{{-3Pi,3Pi},{-3 Pi,3Pi}},Ticks->{Range[-7/2 Pi,7/2 Pi,(Pi/2)],Range[-10 Pi,10 Pi,Pi/2]},AspectRatio->Automatic,ImageSize->800,TicksStyle->Directive[Black,14]],{Xend,-8 Pi+(Pi/64),8 Pi,(Pi/16)}]; *)

(* Length[aniSinTLs] *)

(* ListAnimate[aniSinTLs,AnimationRate->8
] *)

(* Export["aniSinTLs.gif",aniSinTLs,"GIF","AnimationRepetitions"->1,"ImageSize"->800,"DisplayDurations"->(((60/(Length[aniSinTLs])))&/@Range[Length[aniSinTLs]])] *)

Composition

(((1/(2(8-2 a)))(a 4(x-(1/2)) - (((4 (x-(1/2))))^3)))+(1/2))

(* gg[x_]:=(((1/(2(8-2 a)))(a 4(x-(1/2)) - (((4 (x-(1/2))))^3)))+(1/2))/.{a->2}*)

ff[x_]:=-2.2(x-.15)((x-.15)-1)+.45;

gg[x_]:=.65+1.2(x+.1)((x+.1)-1);

Solve[gg'[x]==0,x]

Solve[ff[x]==0.4,x]

Solve[ff'[x]==0,x]

tt=0.12776703213290647`;figCompMin=Show[Graphics[{
{Thickness[0.0001],Dashing[{.01,.005}],RGBColor[0.5,0.5,.5],Line[{{tt,0},{tt,Max[ff[tt],gg[ff[tt]]]}}]},{Thickness[0.0025],Dashing[{.01,.005}],RGBColor[0,0,.5],Arrowheads[0.025],Arrow[{{tt,ff[tt]},{ff[tt],ff[tt]}}]},{Thickness[0.0001],Dashing[{.01,.005}],RGBColor[0.5,0.5,.5],Line[{{ff[tt],0},{ff[tt],Max[ff[tt],gg[ff[tt]]]}}]},{Thickness[0.0025],Dashing[{.01,.005}],RGBColor[0,.4,0],Arrowheads[0.025],Arrow[{{ff[tt],gg[ff[tt]]},{tt,gg[ff[tt]]}}]},
{RGBColor[.5,.5,.5],Thickness[0.002],Line[{{-1,-1},{2,2}}]}
}],
Plot[{ff[x],gg[x] ,gg[ff[x]]},{x,0,1.3},PlotStyle->{{RGBColor[0,0,.5],Thickness[0.005]},{RGBColor[0,.4,0],Thickness[0.005]},{RGBColor[0.5,0,0],Thickness[0.001]}}],
Graphics[{
{RGBColor[.5,0,0],PointSize[0.02],Point[{tt,gg[ff[tt]]}]},
{RGBColor[0,0,.5],PointSize[0.02],Point[{tt,ff[tt]}]},
{RGBColor[0,0,.5],PointSize[0.01],Point[{ff[tt],ff[tt]}]},
{RGBColor[0,0.5,0],PointSize[0.02],Point[{ff[tt],gg[ff[tt]]}]},
{Text[Style["x",Italic,FontSize-> 14],{tt,-.03}],Text[Style["f(x)",Italic,FontSize-> 14],{ff[tt],-.03}],Text[Style["y=x",Italic,FontSize-> 14],{1.05,1.02}],Text[Style["f",Italic,FontSize-> 14,FontColor->RGBColor[0,0,.5]],{.4,.9}],Text[Style["g",Italic,FontSize-> 14,FontColor->RGBColor[0,0.4,0]],{.7,.43}],Text[Style["g∘f",Italic,FontSize-> 14,FontColor->RGBColor[0.5,0,0]],{tt/2,gg[ff[tt/2]]+.05Sign[tt-.48]}]}
}],Plot[gg[ff[x]],{x,0,tt},PlotStyle->{{RGBColor[.5,0,0],Thickness[0.005]}}],
PlotRange->{{-0.05,1.073},{-0.05,1.07}},Axes->True,Frame->False,Ticks->None,AxesOrigin->{0,0},AspectRatio->Automatic,ImageSize->600]

tt=0.65;figCompMax=Show[Graphics[{
{Thickness[0.0001],Dashing[{.01,.005}],RGBColor[0.5,0.5,.5],Line[{{tt,0},{tt,Max[ff[tt],gg[ff[tt]]]}}]},{Thickness[0.0025],Dashing[{.01,.005}],RGBColor[0,0,.5],Arrowheads[0.025],Arrow[{{tt,ff[tt]},{ff[tt],ff[tt]}}]},{Thickness[0.0001],Dashing[{.01,.005}],RGBColor[0.5,0.5,.5],Line[{{ff[tt],0},{ff[tt],Max[ff[tt],gg[ff[tt]]]}}]},{Thickness[0.0025],Dashing[{.01,.005}],RGBColor[0,.4,0],Arrowheads[0.025],Arrow[{{ff[tt],gg[ff[tt]]},{tt,gg[ff[tt]]}}]},
{RGBColor[.5,.5,.5],Thickness[0.002],Line[{{-1,-1},{2,2}}]}
}],
Plot[{ff[x],gg[x] ,gg[ff[x]]},{x,0,1.3},PlotStyle->{{RGBColor[0,0,.5],Thickness[0.005]},{RGBColor[0,.4,0],Thickness[0.005]},{RGBColor[0.5,0,0],Thickness[0.001]}}],
Graphics[{
{RGBColor[.5,0,0],PointSize[0.02],Point[{tt,gg[ff[tt]]}]},
{RGBColor[0,0,.5],PointSize[0.02],Point[{tt,ff[tt]}]},
{RGBColor[0,0,.5],PointSize[0.01],Point[{ff[tt],ff[tt]}]},
{RGBColor[0,0.5,0],PointSize[0.02],Point[{ff[tt],gg[ff[tt]]}]},
{Text[Style["x",Italic,FontSize-> 14],{tt,-.03}],Text[Style["f(x)",Italic,FontSize-> 14],{ff[tt],-.03}],Text[Style["y=x",Italic,FontSize-> 14],{1.05,1.02}],Text[Style["f",Italic,FontSize-> 14,FontColor->RGBColor[0,0,.5]],{.4,.9}],Text[Style["g",Italic,FontSize-> 14,FontColor->RGBColor[0,0.4,0]],{.7,.43}],Text[Style["g∘f",Italic,FontSize-> 14,FontColor->RGBColor[0.5,0,0]],{tt/2,gg[ff[tt/2]]+.05Sign[tt-.48]}]}
}],Plot[gg[ff[x]],{x,0,tt},PlotStyle->{{RGBColor[.5,0,0],Thickness[0.005]}}],
PlotRange->{{-0.05,1.073},{-0.05,1.07}},Axes->True,Frame->False,Ticks->None,AxesOrigin->{0,0},AspectRatio->Automatic,ImageSize->600]

tt=0.8;figCompGenb=Show[Graphics[{
{Thickness[0.0001],Dashing[{.01,.005}],RGBColor[0.5,0.5,.5],Line[{{tt,0},{tt,Max[ff[tt],gg[ff[tt]]]}}]},{Thickness[0.0025],Dashing[{.01,.005}],RGBColor[0,0,.5],Arrowheads[0.025],Arrow[{{tt,ff[tt]},{ff[tt],ff[tt]}}]},{Thickness[0.0001],Dashing[{.01,.005}],RGBColor[0.5,0.5,.5],Line[{{ff[tt],0},{ff[tt],Max[ff[tt],gg[ff[tt]]]}}]},{Thickness[0.0025],Dashing[{.01,.005}],RGBColor[0,.4,0],Arrowheads[0.025],Arrow[{{ff[tt],gg[ff[tt]]},{tt,gg[ff[tt]]}}]},
{RGBColor[.5,.5,.5],Thickness[0.002],Line[{{-1,-1},{2,2}}]}
}],
Plot[{ff[x],gg[x] ,gg[ff[x]]},{x,0,1.3},PlotStyle->{{RGBColor[0,0,.5],Thickness[0.005]},{RGBColor[0,.4,0],Thickness[0.005]},{RGBColor[0.5,0,0],Thickness[0.001]}}],
Graphics[{
{RGBColor[.5,0,0],PointSize[0.02],Point[{tt,gg[ff[tt]]}]},
{RGBColor[0,0,.5],PointSize[0.02],Point[{tt,ff[tt]}]},
{RGBColor[0,0,.5],PointSize[0.01],Point[{ff[tt],ff[tt]}]},
{RGBColor[0,0.5,0],PointSize[0.02],Point[{ff[tt],gg[ff[tt]]}]},
{Text[Style["x",Italic,FontSize-> 14],{tt,-.03}],Text[Style["f(x)",Italic,FontSize-> 14],{ff[tt],-.03}],Text[Style["y=x",Italic,FontSize-> 14],{1.05,1.02}],Text[Style["f",Italic,FontSize-> 14,FontColor->RGBColor[0,0,.5]],{.4,.9}],Text[Style["g",Italic,FontSize-> 14,FontColor->RGBColor[0,0.4,0]],{.7,.43}],Text[Style["g∘f",Italic,FontSize-> 14,FontColor->RGBColor[0.5,0,0]],{tt/2,gg[ff[tt/2]]+.05Sign[tt-.48]}]}
}],Plot[gg[ff[x]],{x,0,tt},PlotStyle->{{RGBColor[.5,0,0],Thickness[0.005]}}],
PlotRange->{{-0.05,1.073},{-0.05,1.07}},Axes->True,Frame->False,Ticks->None,AxesOrigin->{0,0},AspectRatio->Automatic,ImageSize->600]

tt=0.3;figCompGena=Show[Graphics[{
{Thickness[0.0001],Dashing[{.01,.005}],RGBColor[0.5,0.5,.5],Line[{{tt,0},{tt,Max[ff[tt],gg[ff[tt]]]}}]},{Thickness[0.0025],Dashing[{.01,.005}],RGBColor[0,0,.5],Arrowheads[0.025],Arrow[{{tt,ff[tt]},{ff[tt],ff[tt]}}]},{Thickness[0.0001],Dashing[{.01,.005}],RGBColor[0.5,0.5,.5],Line[{{ff[tt],0},{ff[tt],Max[ff[tt],gg[ff[tt]]]}}]},{Thickness[0.0025],Dashing[{.01,.005}],RGBColor[0,.4,0],Arrowheads[0.025],Arrow[{{ff[tt],gg[ff[tt]]},{tt,gg[ff[tt]]}}]},
{RGBColor[.5,.5,.5],Thickness[0.002],Line[{{-1,-1},{2,2}}]}
}],
Plot[{ff[x],gg[x] ,gg[ff[x]]},{x,0,1.3},PlotStyle->{{RGBColor[0,0,.5],Thickness[0.005]},{RGBColor[0,.4,0],Thickness[0.005]},{RGBColor[0.5,0,0],Thickness[0.001]}}],
Graphics[{
{RGBColor[.5,0,0],PointSize[0.02],Point[{tt,gg[ff[tt]]}]},
{RGBColor[0,0,.5],PointSize[0.02],Point[{tt,ff[tt]}]},
{RGBColor[0,0,.5],PointSize[0.01],Point[{ff[tt],ff[tt]}]},
{RGBColor[0,0.5,0],PointSize[0.02],Point[{ff[tt],gg[ff[tt]]}]},
{Text[Style["x",Italic,FontSize-> 14],{tt,-.03}],Text[Style["f(x)",Italic,FontSize-> 14],{ff[tt],-.03}],Text[Style["y=x",Italic,FontSize-> 14],{1.05,1.02}],Text[Style["f",Italic,FontSize-> 14,FontColor->RGBColor[0,0,.5]],{.4,.9}],Text[Style["g",Italic,FontSize-> 14,FontColor->RGBColor[0,0.4,0]],{.7,.43}],Text[Style["g∘f",Italic,FontSize-> 14,FontColor->RGBColor[0.5,0,0]],{tt/2,gg[ff[tt/2]]+.05Sign[tt-.48]}]}
}],Plot[gg[ff[x]],{x,0,tt},PlotStyle->{{RGBColor[.5,0,0],Thickness[0.005]}}],
PlotRange->{{-0.05,1.073},{-0.05,1.07}},Axes->True,Frame->False,Ticks->None,AxesOrigin->{0,0},AspectRatio->Automatic,ImageSize->600]

(* Export["figCompMin.gif",figCompMin,"GIF",{"ImageSize"->600}] *)

(* Export["figCompMax.gif",figCompMax,"GIF",{"ImageSize"->600}] *)

(* Export["figCompGenb.gif",figCompGenb,"GIF",{"ImageSize"->600}] *)

(* Export["figCompGena.gif",figCompGena,"GIF",{"ImageSize"->600}] *)

tt=.85;Clear[tt];Manipulate[ Show[Plot[gg[ff[x]],{x,0,tt},PlotStyle->{{RGBColor[.5,0,0],Thickness[0.005]}}],Plot[{ff[x],gg[x] ,gg[ff[x]]},{x,0,1.3},PlotStyle->{{RGBColor[0,0,.5],Thickness[0.005]},{RGBColor[0,.5,0],Thickness[0.005]},{RGBColor[0.5,0,0],Thickness[0.002]}}],
Graphics[{
{RGBColor[.5,0,0],PointSize[0.02],Point[{tt,gg[ff[tt]]}]},
{RGBColor[0,0,.5],PointSize[0.02],Point[{tt,ff[tt]}]},
{RGBColor[0,0,.5],PointSize[0.01],Point[{ff[tt],ff[tt]}]},
{RGBColor[0,0.5,0],PointSize[0.02],Point[{ff[tt],gg[ff[tt]]}]},
{Thickness[0.0001],Dashing[{.01,.005}],RGBColor[0.5,0.5,.5],Line[{{tt,0},{tt,Max[ff[tt],gg[ff[tt]]]}}]},{Thickness[0.0025],Dashing[{.01,.005}],RGBColor[0,0,.5],Arrowheads[0.025],Arrow[{{tt,ff[tt]},{ff[tt],ff[tt]}}]},{Thickness[0.0001],Dashing[{.01,.005}],RGBColor[0.5,0.5,.5],Line[{{ff[tt],0},{ff[tt],Max[ff[tt],gg[ff[tt]]]}}]},{Thickness[0.0025],Dashing[{.01,.005}],RGBColor[0,.5,0],Arrowheads[0.025],Arrow[{{ff[tt],gg[ff[tt]]},{tt,gg[ff[tt]]}}]},
{RGBColor[.5,.5,.5],Thickness[0.002],Line[{{-1,-1},{2,2}}]},
{Text[Style["x",Italic],{tt,-.03}],Text[Style["f(x)",Italic],{ff[tt],-.03}],Text[Style["y=x",Italic],{1.05,1.02}],Text[Style["f",Italic,FontSize-> 14,FontColor->RGBColor[0,0,.5]],{.4,.9}],Text[Style["g",Italic,FontSize-> 14,FontColor->RGBColor[0,0.5,0]],{.7,.43}],Text[Style["g∘f",Italic,FontSize-> 14,FontColor->RGBColor[0.5,0,0]],{tt/2,gg[ff[tt/2]]+.05Sign[tt-.48]}]}
}],
PlotRange->{{-0.05,1.07},{-0.05,1.07}},Axes->True,Frame->False,Ticks->None,AxesOrigin->{0,0},AspectRatio->Automatic],{{tt,0.3},0.001,1.07}
]

CompMovie=Table[Show[Graphics[{
{Thickness[0.0001],Dashing[{.01,.005}],RGBColor[0.5,0.5,.5],Line[{{tt,0},{tt,Max[ff[tt],gg[ff[tt]]]}}]},{Thickness[0.0025],Dashing[{.01,.005}],RGBColor[0,0,.5],Arrowheads[0.025],Arrow[{{tt,ff[tt]},{ff[tt],ff[tt]}}]},{Thickness[0.0001],Dashing[{.01,.005}],RGBColor[0.5,0.5,.5],Line[{{ff[tt],0},{ff[tt],Max[ff[tt],gg[ff[tt]]]}}]},{Thickness[0.0025],Dashing[{.01,.005}],RGBColor[0,.4,0],Arrowheads[0.025],Arrow[{{ff[tt],gg[ff[tt]]},{tt,gg[ff[tt]]}}]},
{RGBColor[.5,.5,.5],Thickness[0.002],Line[{{-1,-1},{2,2}}]}
}],
Plot[{ff[x],gg[x] },{x,0,1.3},PlotStyle->{{RGBColor[0,0,.5],Thickness[0.005]},{RGBColor[0,.4,0],Thickness[0.005]}}],
Graphics[{
{RGBColor[.5,0,0],PointSize[0.02],Point[{tt,gg[ff[tt]]}]},
{RGBColor[0,0,.5],PointSize[0.02],Point[{tt,ff[tt]}]},
{RGBColor[0,0,.5],PointSize[0.01],Point[{ff[tt],ff[tt]}]},
{RGBColor[0,0.5,0],PointSize[0.02],Point[{ff[tt],gg[ff[tt]]}]},
{Text[Style["x",Italic,FontSize-> 14],{tt,-.03}],Text[Style["f(x)",Italic,FontSize-> 14],{ff[tt],-.03}],Text[Style["y=x",Italic,FontSize-> 14],{1.05,1.02}],Text[Style["f",Italic,FontSize-> 14,FontColor->RGBColor[0,0,.5]],{.4,.9}],Text[Style["g",Italic,FontSize-> 14,FontColor->RGBColor[0,0.4,0]],{.7,.43}],Text[Style["g∘f",Italic,FontSize-> 14,FontColor->RGBColor[0.5,0,0]],{tt/2,gg[ff[tt/2]]+.05Sign[tt-.48]}]}
}],Plot[gg[ff[x]],{x,0,tt},PlotStyle->{{RGBColor[.5,0,0],Thickness[0.005]}}],
PlotRange->{{-0.05,1.073},{-0.05,1.07}},Axes->True,Frame->False,Ticks->None,AxesOrigin->{0,0},AspectRatio->Automatic,ImageSize->600],{tt,0.001,1.07,0.005}
];

CompMovie[[80]]

ListAnimate[CompMovie,AnimationRate->2]

CompMovie[[1]]

Last[CompMovie]

(* Export["CompMovieS1.gif",CompMovie[[1]],"GIF",{"ImageSize"->600}] *)

(* Export["CompMovieSl.gif",Last[CompMovie],"GIF",{"ImageSize"->600}] *)

Length[CompMovie]

(* Export["CompMovieAniS.gif",CompMovie,"GIF","AnimationRepetitions"->1,"ImageSize"->600,"DisplayDurations"->(((60/(Length[CompMovie])))&/@Range[Length[CompMovie]])]  *)

Limit figures

xx=.75;figCosLimUC=Graphics[
{{Thickness[0.004],RGBColor[.5,.5,.5],Circle[]},
{Thickness[0.004],RGBColor[0.5,0.5,0.5],Line[{{Cos[xx],Sin[xx]},{0,0}}]},
{Thickness[0.007],RGBColor[0,0,0.7],Line[{{Cos[xx],0},{0,0}}]},
{PointSize[0.017],Point[{0,0}],Point[{Cos[xx],0}],Point[{1,0}]},
{Thickness[0.005],Line[{{Cos[xx],0},{1,0},{Cos[xx],Sin[xx]},{Cos[xx],0}}]},
{RGBColor[0,.5,.5],Thickness[0.007],Circle[{0,0},1,{0,xx}]},{PointSize[0.017],RGBColor[0,.5,.5],Point[{Cos[xx],Sin[xx]}]},
{Text[Style["A",Italic,FontSize-> 16,FontColor->Black],{Cos[xx],Sin[xx]}+{.04,.04}],Text[Style["B",Italic,FontSize-> 16,FontColor->Black],{Cos[xx],0}-{0,.05}],Text[Style["C",Italic,FontSize-> 16,FontColor->Black],{1,0}+{.05,
.03}],
Text[Style["0",Italic,FontSize-> 16,FontColor->Black],{0,0}-{0.035,.035}],
Text[Style[1,FontSize-> 16,FontColor->Black],{1,0}+{0.02,-.06}],
Text[Style[1,FontSize-> 16,FontColor->Black],{0,1}+{-0.03,.0}],
Text[Style["x",Italic,FontSize-> 18,FontColor->RGBColor[0,.5,.5]],{Cos[xx/2],Sin[xx/2]}+{.05,.05}],Text[Style["x",Italic,FontSize-> 16,FontColor->RGBColor[0,0,0.7]],{Cos[xx]/2,0}+{0.075,-.05}],Text[Style["cos",FontSize-> 16,FontColor->RGBColor[0,0,0.7]],{Cos[xx]/2,0}+{0.,-.052}]}},
PlotRange->{{-0.25,1.1},{-0.25,1.1}},Axes->True,Ticks->None,ImageSize->325]

(* Export["figCosLimUC.gif",figCosLimUC,"GIF",{"ImageSize"->325}] *)

xx=.75;figSincLimUC=Graphics[
{
{RGBColor[1,1,.4],Polygon[Union[{{0,0}},Table[{Cos[t],Sin[t]},{t,0,xx,xx/100}],{{0,0}}]]},
{Thickness[0.004],RGBColor[.5,.5,.5],Circle[]},
{Thickness[0.004],RGBColor[0.5,0.5,0.5],Line[{{Cos[xx],Sin[xx]},{0,0}}]}, 

{Thickness[0.002],RGBColor[0.5,0.5,0.5],Dashing[{0.0075,0.005}],Line[{{Cos[xx],Sin[xx]},{0,Sin[xx]}}]},

{Thickness[0.007],RGBColor[0,0,0.7],Line[{{Cos[xx],0},{0,0}}]},
{Thickness[0.007],RGBColor[0,0.7,0],Line[{{Cos[xx],Sin[xx]},{Cos[xx],0}}],Line[{{0,Sin[xx]},{0,0}}]},

{PointSize[0.017],Point[{0,0}],Point[{Cos[xx],0}],Point[{1,0}],Point[{1,Tan[xx]}]},

{Thickness[0.005],Line[{{Cos[xx],0},{1,0},{Cos[xx],Sin[xx]},{0,0}}],Line[{{1,0},{1,Tan[xx]},{Cos[xx],Sin[xx]}}]},

{RGBColor[0,.5,.5],Thickness[0.007],Circle[{0,0},1,{0,xx}]},{PointSize[0.017],RGBColor[0,.5,.5],Point[{Cos[xx],Sin[xx]}]},
{Text[Style["A",Italic,FontSize-> 16,FontColor->Black],{Cos[xx],Sin[xx]}+{.0,.05}],Text[Style["B",Italic,FontSize-> 16,FontColor->Black],{Cos[xx],0}-{0,.05}],Text[Style["C",Italic,FontSize-> 16,FontColor->Black],{1,0}+{.05,
.03}],
Text[Style["D",Italic,FontSize-> 16,FontColor->Black],{1,Tan[xx]}+{-.02,.04}],
Text[Style["0",Italic,FontSize-> 16,FontColor->Black],{0,0}-{0.035,.035}],
Text[Style[1,FontSize-> 16,FontColor->Black],{1,0}+{0.02,-.06}],
Text[Style[1,FontSize-> 16,FontColor->Black],{0,1}+{-0.03,.0}],
Text[Style["x",Italic,FontSize-> 18,FontColor->RGBColor[0,.5,.5]],{Cos[xx/2],Sin[xx/2]}+{.02,.06}],
Text[Style["x",Italic,FontSize-> 16,FontColor->RGBColor[0,0,0.7]],{Cos[xx]/2,0}+{0.075,-.05}],Text[Style["cos",FontSize-> 16,FontColor->RGBColor[0,0,0.7]],{Cos[xx]/2,0}+{0.,-.052}],Text[Style["sin",FontSize-> 16,FontColor->RGBColor[0,0.7,0]],{0,Sin[xx]/2}+{-0.05,0},Automatic,{0,1}],Text[Style["x",Italic,FontSize-> 16,FontColor->RGBColor[0,0.7,0]],{0,Sin[xx]/2}+{-0.05,0.07},Automatic,{0,1}]},Text[Style["tan",FontSize-> 16,FontColor->RGBColor[0,0,0]],{1,Tan[xx]/2}+{0.05,0},Automatic,{0,1}],Text[Style["x",Italic,FontSize-> 16,FontColor->RGBColor[0,0,0]],{1,Tan[xx]/2}+{0.05,0.07},Automatic,{0,1}]},
PlotRange->{{-0.25,1.1},{-0.25,1.1}},Axes->True,Ticks->None,ImageSize->325]

(* Export["figSincLimUC.gif",figSincLimUC,"GIF",{"ImageSize"->325}] *)

xx=.75;figCosLimUC=Graphics[
{
{RGBColor[1,1,.4],Polygon[Union[{{0,0}},Table[{Cos[t],Sin[t]},{t,0,xx,xx/100}],{{0,0}}]]},
{Thickness[0.004],RGBColor[.5,.5,.5],Circle[]},
{Thickness[0.004],RGBColor[0.5,0.5,0.5],Line[{{Cos[xx],Sin[xx]},{0,0}}]}, 

{Thickness[0.002],RGBColor[0.5,0.5,0.5],Dashing[{0.0075,0.005}],Line[{{Cos[xx],Sin[xx]},{0,Sin[xx]}}]},

{Thickness[0.007],RGBColor[0,0,0.7],Line[{{Cos[xx],0},{0,0}}]},
{Thickness[0.007],RGBColor[0,0.7,0],Line[{{Cos[xx],Sin[xx]},{Cos[xx],0}}],Line[{{0,Sin[xx]},{0,0}}]},

{PointSize[0.017],Point[{0,0}],Point[{Cos[xx],0}],Point[{1,0}]},

{Thickness[0.005],Line[{{Cos[xx],0},{1,0},{Cos[xx],Sin[xx]},{0,0}}]},

{RGBColor[0,.5,.5],Thickness[0.007],Circle[{0,0},1,{0,xx}]},{PointSize[0.017],RGBColor[0,.5,.5],Point[{Cos[xx],Sin[xx]}]},
{Text[Style["A",Italic,FontSize-> 16,FontColor->Black],{Cos[xx],Sin[xx]}+{.0,.05}],Text[Style["B",Italic,FontSize-> 16,FontColor->Black],{Cos[xx],0}-{0,.05}],Text[Style["C",Italic,FontSize-> 16,FontColor->Black],{1,0}+{.05,
.03}],
Text[Style["0",Italic,FontSize-> 16,FontColor->Black],{0,0}-{0.035,.035}],
Text[Style[1,FontSize-> 16,FontColor->Black],{1,0}+{0.02,-.06}],
Text[Style[1,FontSize-> 16,FontColor->Black],{0,1}+{-0.03,.0}],
Text[Style["x",Italic,FontSize-> 18,FontColor->RGBColor[0,.5,.5]],{Cos[xx/2],Sin[xx/2]}+{.02,.06}],
Text[Style["x",Italic,FontSize-> 16,FontColor->RGBColor[0,0,0.7]],{Cos[xx]/2,0}+{0.075,-.05}],Text[Style["cos",FontSize-> 16,FontColor->RGBColor[0,0,0.7]],{Cos[xx]/2,0}+{0.,-.052}],Text[Style["sin",FontSize-> 16,FontColor->RGBColor[0,0.7,0]],{0,Sin[xx]/2}+{-0.05,0},Automatic,{0,1}],Text[Style["x",Italic,FontSize-> 16,FontColor->RGBColor[0,0.7,0]],{0,Sin[xx]/2}+{-0.05,0.07},Automatic,{0,1}]}},
PlotRange->{{-0.25,1.1},{-0.25,1.1}},Axes->True,Ticks->None,ImageSize->325]

(* Export["figCosLimUC.gif",figCosLimUC,"GIF",{"ImageSize"->325}] *)

Folium of Decartes

FoliumOfDecartes=Show[{Plot[-x-2,{x,-5,5},PlotStyle->{{Thickness[0.005],GrayLevel[0.75]}}],Graphics[{{GrayLevel[0.7],Dashing[{0.005,0.005}],Line[{{0,2 (2^(1/3))},{2 (2^(2/3)),2 (2^(1/3))}}],Line[{{2 (2^(2/3)),2 (2^(1/3))},{2 (2^(2/3)),0}}],
Line[{{2 (2^(1/3)),2 (2^(2/3))},{0,2 (2^(2/3))}}],
Line[{{2 (2^(1/3)),2 (2^(2/3))},{2 (2^(1/3)),0}}],Line[{{-14,0},{14,0}}],Line[{{0,-14},{0,14}}]}}],ContourPlot[(x^3)+(y^3)==6 x y,{x,-6 ,4},{y,-6,4},ContourStyle->{{Thickness[0.007],RGBColor[0,0,.5]}},PlotPoints-> 200,ClippingStyle->Automatic],
ContourPlot[(x^3)+(y^3)==6 x y,{x,-6 ,0},{y,0,4},ContourStyle->{{Thickness[0.0025],RGBColor[1,1,0]}},PlotPoints-> 200,ClippingStyle->Automatic],
ContourPlot[(x^3)+(y^3)==6 x y,{x,0 ,4},{y,-6,0},ContourStyle->{{Thickness[0.0025],RGBColor[1,1,0]}},PlotPoints-> 200,ClippingStyle->Automatic],Graphics[{PointSize[0.012],Red,Point[{0,0}],Point[{3,3}],Point[{2 (2^(1/3)),2 (4^(1/3))}],Point[{2 (2^(2/3)),2 (2^(1/3))}]}]},Axes->True,Ticks->{Range[-9,9,1],Range[-9,9,1]},Frame->False,PlotRange->{{-4.5,3.5},{-4.5,3.5}},FrameTicks->{None,None,None,None},AspectRatio->Automatic,ImageSize->600,PlotRangeClipping->True]

Export["FoliumOfDecartes.png",FoliumOfDecartes]

Manipulate[Show[{Graphics[{{GrayLevel[0.7],Dashing[{0.005,0.005}],Line[{{-14,2 (2^(1/3))},{14,2 (2^(1/3))}}],Line[{{2 (2^(2/3)),-14},{2 (2^(2/3)),14}}],
Line[{{-14,2 (2^(2/3))},{14,2 (2^(2/3))}}],
Line[{{2 (2^(1/3)),-14},{2 (2^(1/3)),14}}],Line[{{-14,0},{14,0}}],Line[{{0,-14},{0,14}}]}}],ContourPlot[(x^3)+(y^3)==6 x y,{x,-6 ,4},{y,-6,4},ContourStyle->{{Thickness[0.007],RGBColor[0,0,.5]}},PlotPoints-> 200,ClippingStyle->Automatic],Graphics[{{PointSize[0.03],Blue,Point[{((6 t)/(1+(t^3))),((6 (t^2))/(1+(t^3)))}]},{Text[t,{0,3}]}}]},Frame->True,PlotRange->{{-4.5,3.5},{-4.5,3.5}},FrameTicks->{Range[-9,9],Range[-9,9],None,None},AspectRatio->Automatic,ImageSize->400,PlotRangeClipping->True],{t,-20,20}]

ParametricPlot[{((6 t)/(1+(t^3))),((6 (t^2))/(1+(t^3)))},{t,-20,20},Exclusions->{-1},PlotRange->{{-4.5,3.5},{-4.5,3.5}},FrameTicks->{Range[-9,9],Range[-9,9],None,None},AspectRatio->Automatic,ImageSize->600,PlotRangeClipping->True]

Osculating circle pictures

Clear[SecLine,PoSlLineF,PoSlLine,f,a,b,x]; 

SecLine[f_,a_,b_,x_]:=If[a==b,f'[a](x-a)+f[a],((f[b]-f[a])/(b-a))(x-a)+f[a]];
PoSlLineF[{a_,b_},sl_,x_]:=sl*(x-a)+b ;
PoSlLine[{a_,b_},sl_,ll_]:=Line[{{a,b}-(ll/(Sqrt[1+(sl^2)])){1,sl},{a,b}+(ll/(Sqrt[1+(sl^2)])){1,sl}}];

ff[x_]=((x)^3-.999 x+1.3)/.{x->.65 (x-.2)-.2}; aa=1.8; x0=1.9; 

Plot[Evaluate[{SecLine[ff[#]&,aa,x0,x],ff[x],ff'[aa](x-aa)+ff[aa]}],{x,-2,5.5},PlotRange->{{-1,4},{-.5,3.5}},
PlotStyle->{{Thickness[th0+.002],RGBColor[0,0.5,0]},{Thickness[th1],RGBColor[0,0,.5]},{Thickness[th1],RGBColor[0,0.5,0.5]}},Epilog->{
{PointSize[0.017],RGBColor[0,0.5,0],Point[{x0,ff[x0]}]},
{PointSize[0.017],RGBColor[0,0,.5],Point[{aa,ff[aa]}],PointSize[0.009],RGBColor[1,1,1],Point[{aa,ff[aa]}]}
},AspectRatio->Automatic]

AllLines=Plot[Evaluate[{ff[x],ff'[aa](x-aa)+ff[aa]}],{x,-2,5.5},PlotRange->{{-1,4},{-.5,3.5}},
PlotStyle->{{Thickness[th1],RGBColor[0,0,.5]},{Thickness[th1],RGBColor[0,0.5,0.5]}},
Prolog->{{Table[{Hue[((Mod[t+.48,Pi])/Pi)],PoSlLine[{aa,ff[aa]},((Sin[t])/(Cos[t])),12]},{t,0,Pi,(Pi/63)}]}},
Epilog->{
{PointSize[0.017],RGBColor[0,0,.5],Point[{aa,ff[aa]}],PointSize[0.009],RGBColor[1,1,1],Point[{aa,ff[aa]}]}
},AspectRatio->Automatic,ImageSize->600]

(* Export["AllLines.png",AllLines] *)

Module[{x0},Manipulate[
Plot[Evaluate[{SecLine[ff[#]&,aa,x0,x],ff[x],ff'[aa](x-aa)+ff[aa]}],{x,-2,5.5},PlotRange->{{-1,4},{-.5,3.5}},
PlotStyle->{{Thickness[th0+.002],RGBColor[0,1,0]},{Thickness[th1],RGBColor[0,0,.5]},{Thickness[th1],RGBColor[0,0.5,0.5]}},Epilog->{
{PointSize[0.017],RGBColor[0,0.5,0],Point[{x0,ff[x0]}]},
{PointSize[0.017],RGBColor[0,0,.5],Point[{aa,ff[aa]}],PointSize[0.009],RGBColor[1,1,1],Point[{aa,ff[aa]}]}
},AspectRatio->Automatic]
,{{x0,1,"x0"},0,3}]
]

TanLineAniR=Table[
Plot[Evaluate[{SecLine[ff[#]&,aa,x0,x],ff[x],ff'[aa](x-aa)+ff[aa]}],{x,-2,5.5},PlotRange->{{-1,4},{-.5,3.5}},
PlotStyle->{{Thickness[th0+.002],RGBColor[0,1,0]},{Thickness[th1],RGBColor[0,0,.5]},{Thickness[th1],RGBColor[0,0.5,0.5]}},Epilog->{
{PointSize[0.017],RGBColor[0,0.5,0],Point[{x0,ff[x0]}]},
{PointSize[0.017],RGBColor[0,0,.5],Point[{aa,ff[aa]}],PointSize[0.009],RGBColor[1,1,1],Point[{aa,ff[aa]}]}
},AspectRatio->Automatic,ImageSize->600]
,{x0,aa+1,aa,-(1/120)}]; TanLineAniR =Join[TanLineAniR,{Last[TanLineAniR],Last[TanLineAniR],Last[TanLineAniR],Last[TanLineAniR],Last[TanLineAniR]}];

ListAnimate[TanLineAniR]

(* Export["TanLineAniRS1.gif",First[TanLineAniR],"GIF",{"ImageSize"->600}] *)

(* Export["TanLineAniRSl.gif",Last[TanLineAniR],"GIF",{"ImageSize"->600}] *)

(* Export["TanLineAniR.gif",TanLineAniR,"GIF","AnimationRepetitions"->0,"ImageSize"->600,"DisplayDurations"->(((60/(Length[TanLineAniR])))&/@Range[Length[TanLineAniR]])]  *)

Clear[OC,f,a]; OC[f_,a_]:=If[f''[a]==0,Line[{{a,f[a]}-10{1,f'[a]},{a,f[a]}+10{1,f'[a]}}],Circle[{a,f[a]}+(((1+(((f'[a]))^2)))/(f''[a])){-f'[a],1},((((1+(((f'[a]))^2)))^(3/2))/(Abs[f''[a]]))]];

Clear[OCCe,f,a];OCCe[f_,a_]:=If[f''[a]==0,{100,100},{a,f[a]}+(((1+(((f'[a]))^2)))/(f''[a])){-f'[a],1}];

Clear[SecC,f,a,b];
 

SecC[f_,a_,b_]:=If[a==b, OC[f,a],Circle[{a,f[a]}+{((f'[a])/2)*(((((b-a))^2)+(((f[b]-f[a]))^2))/(-(f[b]-f[a])+f'[a](b-a))),(1/2)*(((((b-a))^2)+(((f[b]-f[a]))^2))/((f[b]-f[a])-f'[a](b-a)))},(1/2)*(((((b-a))^2)+(((f[b]-f[a]))^2))/(Abs[-(f[b]-f[a])+f'[a](b-a)]))(Sqrt[1+(((f'[a]))^2)])]];

SecCCe[f_,a_,b_]:=If[a==b, OCCe[f,a],{a,f[a]}+{((f'[a])/2)*(((((b-a))^2)+(((f[b]-f[a]))^2))/(-(f[b]-f[a])+f'[a](b-a))),(1/2)*(((((b-a))^2)+(((f[b]-f[a]))^2))/((f[b]-f[a])-f'[a](b-a)))}];

x0=aa+.5;
Plot[Evaluate[{-((ff'[aa])^(-1))(x-aa)+ff[aa],ff[x],ff'[aa](x-aa)+ff[aa]}],{x,-2,5.5},PlotRange->{{-1,4},{-.5,3.5}},
PlotStyle->{{Thickness[th0],RGBColor[0,1,0]},{Thickness[th1],RGBColor[0,0,.5]},{Thickness[th1-0.003],RGBColor[0,0.5,0.5]}},AspectRatio->Automatic,
Prolog->{{RGBColor[0,1,0],SecC[ff[#]&,aa,x0]},{RGBColor[.7,0.7,.7],Dashing[{0.01,0.003}],Line[{{aa,ff[aa]},{x0,ff[x0]}}],If[x0==aa,PoSlLine[{aa,ff[aa]},-(1/(ff'[aa])),12],PoSlLine[{((x0+aa)/2),((ff[x0]+ff[aa])/2)},((x0-aa)/(ff[aa]-ff[x0])),12]]}},Epilog->{{PointSize[0.017],RGBColor[0,1,0],Point[SecCCe[ff[#]&,aa,x0]]},{PointSize[0.017],RGBColor[0,0.5,0],Point[{x0,ff[x0]}]},
{PointSize[0.017],RGBColor[0,0,.5],Point[{aa,ff[aa]}],PointSize[0.009],RGBColor[1,1,1],Point[{aa,ff[aa]}]},{RGBColor[0,.5,.5],OC[ff[#]&,aa]},{PointSize[0.017],RGBColor[0,0.5,0.5],Point[OCCe[ff[#]&,aa]]}}]

aa

aa=(18/10)

AllCircles=Plot[Evaluate[{-((ff'[aa])^(-1))(x-aa)+ff[aa],ff[x],ff'[aa](x-aa)+ff[aa]}],{x,-2,5.5},PlotRange->{{-1,4},{-.5,3.5}},
PlotStyle->{{Thickness[th0],RGBColor[0,1,0]},{Thickness[th1],RGBColor[0,0,.5]},{Thickness[th1-0.003],RGBColor[0,0.5,0.5]}},AspectRatio->Automatic,
Prolog->{Table[{Hue[Exp[2t]],Circle[{aa,ff[aa]}+((Exp[2t])/(Sqrt[1+(((ff'[aa]))^2)])){-ff'[aa],1},Exp[2t]]},{t,-1.3,2.5,(1/20)}],Table[{Hue[Exp[2t]],Circle[{aa,ff[aa]}-((Exp[2t])/(Sqrt[1+(((ff'[aa]))^2)])){-ff'[aa],1},Exp[2t]]},{t,-1.3,2.5,(1/20)}]},Epilog->{
{Thickness[th1],RGBColor[0,.5,.5],OC[ff[#]&,aa]},{PointSize[0.017],RGBColor[0,0,.5],Point[{aa,ff[aa]}],PointSize[0.009],RGBColor[1,1,1],Point[{aa,ff[aa]}]}},ImageSize->600]

(* Export["AllCircles.png",AllCircles] *)

Module[{x0},
Manipulate[
Plot[Evaluate[{-((ff'[aa])^(-1))(x-aa)+ff[aa],ff[x],ff'[aa](x-aa)+ff[aa]}],{x,-2,5.5},PlotRange->{{-1,4},{-.5,3.5}},
PlotStyle->{{Thickness[th0],RGBColor[0,1,0]},{Thickness[th1],RGBColor[0,0,.5]},{Thickness[th1-0.003],RGBColor[0,0.5,0.5]}},AspectRatio->Automatic,
Prolog->{{Thickness[th0+.002],RGBColor[0,1,0],SecC[ff[#]&,aa,x0]},{RGBColor[.5,0.5,.5],Dashing[{0.01,0.005}],Line[{{aa,ff[aa]},{x0,ff[x0]}}],If[x0==aa,PoSlLine[{aa,ff[aa]},-(1/(ff'[aa])),12],PoSlLine[{((x0+aa)/2),((ff[x0]+ff[aa])/2)},((x0-aa)/(ff[aa]-ff[x0])),12]]}},Epilog->{{PointSize[0.017],RGBColor[0,1,0],Point[SecCCe[ff[#]&,aa,x0]]},{PointSize[0.017],RGBColor[0,0.5,0],Point[{x0,ff[x0]}]},
{PointSize[0.017],RGBColor[0,0,.5],Point[{aa,ff[aa]}],PointSize[0.009],RGBColor[1,1,1],Point[{aa,ff[aa]}]},{RGBColor[0,.5,.5],Thickness[th1-0.003],OC[ff[#]&,aa]},{PointSize[0.017],RGBColor[0,0.5,0.5],Point[OCCe[ff[#]&,aa]]}}]

,{{x0,1,"x0"},0,4}]
]

OscCircleAniR=Table[
Plot[Evaluate[{-((ff'[aa])^(-1))(x-aa)+ff[aa],ff[x],ff'[aa](x-aa)+ff[aa]}],{x,-2,5.5},PlotRange->{{-1,4},{-.5,3.5}},
PlotStyle->{{Thickness[th0],RGBColor[0,1,0]},{Thickness[th1],RGBColor[0,0,.5]},{Thickness[th1-0.003],RGBColor[0,0.5,0.5]}},AspectRatio->Automatic,
Prolog->{{Thickness[th0+.002],RGBColor[0,1,0],SecC[ff[#]&,aa,x0]},{RGBColor[.5,0.5,.5],Dashing[{0.01,0.005}],Line[{{aa,ff[aa]},{x0,ff[x0]}}],If[x0==aa,PoSlLine[{aa,ff[aa]},-(1/(ff'[aa])),12],PoSlLine[{((x0+aa)/2),((ff[x0]+ff[aa])/2)},((x0-aa)/(ff[aa]-ff[x0])),12]]}},Epilog->{{PointSize[0.017],RGBColor[0,1,0],Point[SecCCe[ff[#]&,aa,x0]]},{PointSize[0.017],RGBColor[0,0.5,0],Point[{x0,ff[x0]}]},
{RGBColor[0,.5,.5],Thickness[th1-0.003],OC[ff[#]&,aa]},{PointSize[0.017],RGBColor[0,0.5,0.5],Point[OCCe[ff[#]&,aa]]},
{PointSize[0.017],RGBColor[0,0,.5],Point[{aa,ff[aa]}],PointSize[0.009],RGBColor[1,1,1],Point[{aa,ff[aa]}]}},ImageSize->600]

,{x0,aa+1,aa,-(1/120)}];

OscCircleAniR =Join[OscCircleAniR,{Last[OscCircleAniR],Last[OscCircleAniR],Last[OscCircleAniR],Last[OscCircleAniR],Last[OscCircleAniR]}];

Last[OscCircleAniR]

First[OscCircleAniR]

(* Export["OscCircleAniRS1.gif",First[OscCircleAniR],"GIF",{"ImageSize"->600}] *)

(* Export["OscCircleAniRSl.gif",Last[OscCircleAniR],"GIF",{"ImageSize"->600}] *)

(* Export["OscCircleAniR.gif",OscCircleAniR,"GIF","AnimationRepetitions"->0,"ImageSize"->600,"DisplayDurations"->(((60/(Length[OscCircleAniR])))&/@Range[Length[OscCircleAniR]])] *)

OscCircleAniL=Table[
Plot[Evaluate[{-((ff'[aa])^(-1))(x-aa)+ff[aa],ff[x],ff'[aa](x-aa)+ff[aa]}],{x,-2,5.5},PlotRange->{{-1,4},{-.5,3.5}},
PlotStyle->{{Thickness[th0],RGBColor[0,1,0]},{Thickness[th1],RGBColor[0,0,.5]},{Thickness[th1-0.003],RGBColor[0,0.5,0.5]}},AspectRatio->Automatic,
Prolog->{{Thickness[th0+.002],RGBColor[0,1,0],SecC[ff[#]&,aa,x0]},{RGBColor[.5,0.5,.5],Dashing[{0.01,0.005}],Line[{{aa,ff[aa]},{x0,ff[x0]}}],If[x0==aa,PoSlLine[{aa,ff[aa]},-(1/(ff'[aa])),12],PoSlLine[{((x0+aa)/2),((ff[x0]+ff[aa])/2)},((x0-aa)/(ff[aa]-ff[x0])),12]]}},Epilog->{{PointSize[0.017],RGBColor[0,1,0],Point[SecCCe[ff[#]&,aa,x0]]},{PointSize[0.017],RGBColor[0,0.5,0],Point[{x0,ff[x0]}]},
{RGBColor[0,.5,.5],Thickness[th1-0.003],OC[ff[#]&,aa]},{PointSize[0.017],RGBColor[0,0.5,0.5],Point[OCCe[ff[#]&,aa]]},
{PointSize[0.017],RGBColor[0,0,.5],Point[{aa,ff[aa]}],PointSize[0.009],RGBColor[1,1,1],Point[{aa,ff[aa]}]}},ImageSize->600]

,{x0,aa-(210/120),aa,(1/120)}];

OscCircleAniL =Join[OscCircleAniL,{Last[OscCircleAniL],Last[OscCircleAniL],Last[OscCircleAniL],Last[OscCircleAniL],Last[OscCircleAniL]}];

Last[OscCircleAniL]

First[OscCircleAniL]

(* Export["OscCircleAniLS1.gif",First[OscCircleAniL],"GIF",{"ImageSize"->600}] *)

(* Export["OscCircleAniLSl.gif",Last[OscCircleAniL],"GIF",{"ImageSize"->600}] *)

(* Export["OscCircleAniL.gif",OscCircleAniL,"GIF","AnimationRepetitions"->0,"ImageSize"->600,"DisplayDurations"->(((60/(Length[OscCircleAniL])))&/@Range[Length[OscCircleAniL]])] *)

Many osculating circles

See the notebook Osculating_circles.nb

Quadratic and Cubic trichotomies as plane colorings

Quadratic012=Graphics[{
{Yellow,Polygon[Append[Prepend[{#,(((#/2))^2)}&/@Range[-14,14,.1],{0,(((-14/2))^2)}],{0,(((-14/2))^2)}]]},

{RGBColor[(1-.45){0,.5,.5}+0.45{1,1,1}],Polygon[Join[{{14,(((14/2))^2)},{14,-11},{-14,-11},{-14,(((-14/2))^2)}},{#,(((#/2))^2)}&/@Range[-14,14,.1]]]},

{Thickness[0.007],RGBColor[0,0,.7],Line[{#,(((#/2))^2)}&/@Range[-14,14,.1]]},
{Text[Style[α,FontSize-> 16],{9.3,0.3}],Text[Style[β,FontSize-> 16],{0.3,9.3}]}
},
PlotRange->{{-9.9,9.9},{-9.9,9.9}},Frame->True,Axes->True,AxesStyle->Directive[GrayLevel[0.7]],FrameTicks->{Range[-10,9,2],Range[-10,9,2],Range[-10,9,2],Range[-10,9,2]},PlotRangeClipping->True,ImageSize->500]

(* Export["Quadratic012.png",Quadratic012] *)

Cubic123=Graphics[{
{Yellow,Polygon[Join[{{#,-((2 (((-#))^(3/2)))/(3 (Sqrt[3])))}&[-14],{12,-((2 (((14))^(3/2)))/(3 (Sqrt[3])))},{12,((2 (((14))^(3/2)))/(3 (Sqrt[3])))},{#,((2 (((-#))^(3/2)))/(3 (Sqrt[3])))}&[-14]},{#,((2 (((-#))^(3/2)))/(3 (Sqrt[3])))}&/@Range[-14,0,.1],{#,-((2 (((-#))^(3/2)))/(3 (Sqrt[3])))}&/@Range[0,-14,-.1]]]},

{RGBColor[(1-.45){0,.5,.5}+0.45{1,1,1}],Polygon[Join[{{#,-((2 (((-#))^(3/2)))/(3 (Sqrt[3])))}&[-14],{-12,-((2 (((14))^(3/2)))/(3 (Sqrt[3])))},{-12,((2 (((14))^(3/2)))/(3 (Sqrt[3])))},{#,((2 (((-#))^(3/2)))/(3 (Sqrt[3])))}&[-14]},{#,((2 (((-#))^(3/2)))/(3 (Sqrt[3])))}&/@Range[-14,0,.1],{#,-((2 (((-#))^(3/2)))/(3 (Sqrt[3])))}&/@Range[0,-14,-.1]]]},


{Thickness[0.007],RGBColor[0,0,.7],Line[Join[{#,((2 (((-#))^(3/2)))/(3 (Sqrt[3])))}&/@Range[-14,0,.1],{#,-((2 (((-#))^(3/2)))/(3 (Sqrt[3])))}&/@Range[0,-14,-.1]]]},
{Yellow,PointSize[0.012],Point[{0,0}]},

{Text[Style[α,FontSize-> 16],{9.3,0.3}],Text[Style[β,FontSize-> 16],{0.3,9.3}]}
},
PlotRange->{{-9.9,9.9},{-9.9,9.9}},Frame->True,Axes->True,AxesStyle->Directive[GrayLevel[0.7]],FrameTicks->{Range[-10,9,2],Range[-10,9,2],Range[-10,9,2],Range[-10,9,2]},PlotRangeClipping->True,ImageSize->500]

(* Export["Cubic123.png",Cubic123] *)

Yellow

Orange

Cubic123YOB=Graphics[{
{RGBColor[.94,.94,0],Polygon[Join[{{#,-((2 (((-#))^(3/2)))/(3 (Sqrt[3])))}&[-14],{12,-((2 (((14))^(3/2)))/(3 (Sqrt[3])))},{12,((2 (((14))^(3/2)))/(3 (Sqrt[3])))},{#,((2 (((-#))^(3/2)))/(3 (Sqrt[3])))}&[-14]},{#,((2 (((-#))^(3/2)))/(3 (Sqrt[3])))}&/@Range[-14,0,.1],{#,-((2 (((-#))^(3/2)))/(3 (Sqrt[3])))}&/@Range[0,-14,-.1]]]},

{RGBColor[0,0,.8],Polygon[Join[{{#,-((2 (((-#))^(3/2)))/(3 (Sqrt[3])))}&[-14],{-12,-((2 (((14))^(3/2)))/(3 (Sqrt[3])))},{-12,((2 (((14))^(3/2)))/(3 (Sqrt[3])))},{#,((2 (((-#))^(3/2)))/(3 (Sqrt[3])))}&[-14]},{#,((2 (((-#))^(3/2)))/(3 (Sqrt[3])))}&/@Range[-14,0,.1],{#,-((2 (((-#))^(3/2)))/(3 (Sqrt[3])))}&/@Range[0,-14,-.1]]]},


{Thickness[0.007],RGBColor[.9{1,0.5`,0}],Line[Join[{#,((2 (((-#))^(3/2)))/(3 (Sqrt[3])))}&/@Range[-14,0,.1],{#,-((2 (((-#))^(3/2)))/(3 (Sqrt[3])))}&/@Range[0,-14,-.1]]]},
{Yellow,PointSize[0.012],Point[{0,0}]},

{Text[Style[α,FontSize-> 16],{9.3,0.3}],Text[Style[β,FontSize-> 16],{0.3,9.3}]}
},
PlotRange->{{-9.9,9.9},{-9.9,9.9}},Frame->True,Axes->True,AxesStyle->Directive[GrayLevel[0.7]],FrameTicks->{Range[-10,9,2],Range[-10,9,2],Range[-10,9,2],Range[-10,9,2]},PlotRangeClipping->True,ImageSize->500]

(* Export["Cubic123YOB.png",Cubic123YOB] *)

Manipulate[Plot[(x^3)-Abs[a] x+Sign[a]((2Abs[a](Sqrt[Abs[a]]))/(3 (Sqrt[3]))),{x,-4,4},PlotRange->{-6,6}],{a,-4,4}]

Manipulate[Plot[(x^3)-3 (a^2) x+2 (a^3),{x,-4,4},PlotRange->{-9,9}],{a,-2,2}]

(Manipulate[Plot[((x^3)+α x+β),{x,-3,3},PlotRange->{-6,6},Epilog->{{PointSize[0.02],Point[{(((Abs[α])^(1/2))/( (Sqrt[3]))),β-((2 ((Abs[α])^(3/2)))/(3 (Sqrt[3])))}],Point[{-(((Abs[α])^(1/2))/( (Sqrt[3]))),β+((2 ((Abs[α])^(3/2)))/(3 (Sqrt[3])))}]}}],{β,-((2 ((Abs[α])^(3/2)))/(3 (Sqrt[3]))),((2 ((Abs[α])^(3/2)))/(3 (Sqrt[3])))}])/.{α->-3}

FullSimplify[Log[{a^x,x^β}],And[a>1,x>0,β>0]]

FullSimplify[{x Log[a],β Log[x]}/.{x->Exp[z]},z>0]

{- ((Log[a])/β),-z (E^(-z)) }

(* two solutions *)

{-(1/E),- ((Log[a])/β),0}

{0,((Log[a])/β),(1/E)}

{0,E Log[a],β}

{0, Log[a],(β/E)}

{0, a,Exp[(β/E)]}

(* solutions *)

{z, -ProductLog[- ((Log[a])/β)],-ProductLog[-1,- ((Log[a])/β)]}

{x,Exp[ -ProductLog[- ((Log[a])/β)]],Exp[-ProductLog[-1,- ((Log[a])/β)]]}

FullSimplify[{x,Exp[ -ProductLog[- ((Log[a])/β)]],Exp[-ProductLog[-1,- ((Log[a])/β)]]},And[a>1,β>0]]

Solve[a^x==x^β,x]

(* unique solution *)

FullSimplify[Solve[a^x==x^(E Log[a]),x],a>1]

a^E==E^(E Log[a])

ExpPow012=Graphics[{
{RGBColor[(1-.45){0,.5,.5}+0.45{1,1,1}],Polygon[Join[{{14,E Log[14]},{14,10},{1,10},{1,0}},{#,E Log[#]}&/@Range[1,14,.1]]]},

{Yellow,Polygon[Join[{{14,E Log[14]},{14,0},{1,0}},{#,E Log[#]}&/@Range[1,14,.1]]]},

{Thickness[0.004],RGBColor[(1-.35){0,.5,.5}+0.35{1,1,1}],Line[{#,#}&/@Range[1,14,.1]]},


{Thickness[0.007],RGBColor[0,0,.7],Line[{#,E Log[#]}&/@Range[1,14,.1]]},

{PointSize[0.012],RGBColor[{0,0,.6}],Point[{E,E}]},

{Text[Style[a,FontSize-> 16],{9.3,0.3}],Text[Style[β,FontSize-> 16],{1.3,8.3}]}
},
PlotRange->{{1,9.9},{0,8.9}},Frame->True,FrameTicks->{Range[1,9,1],Range[0,9,1],Range[1,9,1],Range[0,9,1]},PlotRangeClipping->True,AspectRatio->Automatic,ImageSize->500]

(* Export["ExpPow012.png",ExpPow012] *)

Quadratic approximation: an intuitive image

Clear[f];ffd[x_]:=(1.6(z^3)-3.1z+0.9)/.{z->1.6x-1.5};aa=0.105;xx=0.25;st=0.005;
QuadraticApp=Plot[
{ffd[x],ffd[aa]+ffd'[aa](x-aa)},{x,-1,15} ,
PlotStyle->{{RGBColor[0,0,0.5],Thickness[0.006]},{RGBColor[0,0.5,0],Thickness[0.005]}},Prolog->{{Thickness[0.001],GrayLevel[0.5],Dashing[{0.006,0.006}],Line[{{aa,ffd[aa]},{0,ffd[aa]}}],Line[{{xx,ffd[xx]},{0,ffd[xx]}}]},{RGBColor[0.95,0.95,0],Polygon[Join[{{xx,ffd[xx]},{xx,0},{aa,0}},{#,ffd[#]}&/@Range[aa,xx,((xx-aa)/100)]]]},{Black,Thickness[0.004],Line[{{aa,0},{aa,ffd[aa]-st},{xx,ffd[aa]-st},{xx,0},{aa,0}}]}},Epilog->{{RGBColor[0.85,0,0],Thickness[0.0035],Line[{{aa,ffd[aa]+st},{xx,ffd[aa]+st},{xx,ffd[aa]+ffd'[aa](xx-aa)},{aa,ffd[aa]}}]},{PointSize[0.01],RGBColor[0.85,0,0],Point[{aa,ffd[aa]}],Point[{xx,ffd[aa]}],,Point[{xx,Evaluate[ffd[aa]+ffd'[aa](xx-aa)]}]},{Text[A,{aa,ffd[aa]}+{-0.01,0.06},BaseStyle->{FontSize->16}],Text[B,{xx,ffd[aa]}+{0.015,0.04},BaseStyle->{FontSize->16}],Text["C",{xx,Evaluate[ffd[aa]+ffd'[aa](xx-aa)]}+{-0.013,0.06},BaseStyle->{FontSlant->Italic,FontSize->16}],Text["slope",{xx+0.03,Evaluate[ffd[aa]+ffd'[aa](xx+0.05-aa)]}+{0.05,0},BaseStyle->{FontSlant->Plain,FontSize->16,FontColor-> RGBColor[0,0.5,0]}],Text[f''[a],{xx+0.03,Evaluate[ffd[aa]+ffd'[aa](xx+0.05-aa)]}+{0.11,0},BaseStyle->{FontSlant->Plain,FontSize->16,FontColor-> Black}]}},
AxesOrigin->{0,0},Ticks->{{{aa,a},{xx,x}},{{ffd[aa],f'[a]},{ffd[xx],f'[x]}}},PlotRange->{{-0.1,0.5},{-.2,3.2}},AspectRatio->1.13,BaseStyle->{FontSize->16},ImageSize->400]

The navy blue graph is the graph of the function f'. The quantity f(x)-f(a) is represented by the yellow area under the graph of f'.  The quantity f'(a)(x-a) is represented by the area enclosed by the black rectangle.  Thus the difference

f(x) - f(a) - f' (a) (x - a)

is well approximated by the area of the red triangle ABC.  The height of the red triangle, the lenght of the line segment BC, is f'(a)(x-a) and the base of this triangle, the length of the line segment AB,  is  x-a. Thus the area of this triangle is

(1/2)f''(a)(((x-a))^2).

This strongly indicates that

(lim_(x->a)) ((f(x)-f(a)-f'(a)(x-a))/(((x-a))^2))= (1/2)f''(a)

However, the easiest proof of the last limit (assuming f is differentiable in a neighbourhood of a) seems to be using Cauchy’s mean value theorem.

The smallest osculating circle in Exp

aso=(1/2) (-Log[2]);

ExpMinOC=Plot[{-Exp[-aso](x-aso)+Exp[aso],Exp[aso](x-aso)+Exp[aso],Exp[x]},{x,-9,4},
PlotRange->{{-6,3},{-1,7}},PlotStyle->{{RGBColor[0.5,1,.5],Thickness[0.003]},{RGBColor[0,0.5,.5],Thickness[0.003]},{RGBColor[0,0,.5],Thickness[0.006]}},Epilog->{{RGBColor[0,.5,.5],Thickness[0.003],Circle[{(1/2) (-3-Log[2]),2 (Sqrt[2])},((3 (Sqrt[3]))/2)]},{RGBColor[0,.5,.5],PointSize[0.015],Point[{(1/2) (-3-Log[2]),2 (Sqrt[2])}],Point[{(1/2) (-Log[2]),(1/(Sqrt[2]))}]}
},AspectRatio->Automatic,Ticks->{Range[-7,7],Range[-7,7]},ImageSize->600]

(* Export["ExpMinOC.png",ExpMinOC] *)

Parametric Equations

Cycloid

Manipulate[Graphics[{
{Line[{{-2,0},{19,0}}]},
{GrayLevel[0.7],Thickness[0.005],Circle[{t,1},1],Line[{{t,1},{t,1}-{Sin[t],Cos[t]}}]},
{RGBColor[0,0,.5],Thickness[0.007],Line[Table[{s,1}-{Sin[s],Cos[s]},{s,-4,t,(Pi/64)}]]},
{RGBColor[0,0,.5],PointSize[0.02],Point[{t,1}-{Sin[t],Cos[t]}]}

},
Axes->{True,False},Ticks->{Range[0,3Pi,Pi],{}},
PlotRange->{{-1.5,3Pi+1},{-.2,2.2}},AspectRatio->Automatic
],{t,-4,4 Pi}]

CycloidAni=Table[Graphics[{
{Line[{{-2,0},{19,0}}]},
{GrayLevel[0.7],Thickness[0.005],Circle[{t,1},1],Line[{{t,1},{t,1}-{Sin[t],Cos[t]}}]},
{RGBColor[0,0,.5],Thickness[0.007],Line[Table[{s,1}-{Sin[s],Cos[s]},{s,-4,t,(Pi/64)}]]},
{RGBColor[0,0,.5],PointSize[0.02],Point[{t,1}-{Sin[t],Cos[t]}]}

},
Axes->{True,False},Ticks->{Range[0,3Pi,Pi],{}},
PlotRange->{{-1.5,3Pi+1},{-.2,2.2}},AspectRatio->Automatic,ImageSize->600
],{t,-Pi,4 Pi,(Pi/64)}];

ListAnimate[CycloidAni]

CycloidAniS1=CycloidAni[[87]]

(* Export["CycloidAniS1.gif",CycloidAni[[87]],"GIF",{"ImageSize"->600}]*)

(* Export["CycloidAni.gif",CycloidAni,"GIF","AnimationRepetitions"->0,"ImageSize"->600,"DisplayDurations"->(((30/(Length[CycloidAni])))&/@Range[Length[CycloidAni]])]  *)

CycloidPic=Graphics[{
{RGBColor[0,0,.5],Thickness[0.004],Line[Table[{s,1}-{Sin[s],Cos[s]},{s,-4,7 Pi,(Pi/64)}]]}

},
Axes->{True,True},Ticks->{Range[-Pi,9Pi,Pi],{0,1,2}},
PlotRange->{{-1.5,4Pi+2},{-.2,2.5}},AspectRatio->Automatic,ImageSize->500
]

(* Export["CycloidPic.gif",CycloidPic,"GIF",{"ImageSize"->600}] *)

{s,1}-{Sin[s],Cos[s]}

D[{s-Sin[s],1-Cos[s]},s]

CycloidNorms=Graphics[{
{Thickness[0.001],GrayLevel[0.7],Table[Line[{{t0-Sin[t0]+(-100)(-Sin[t0]),1-Cos[t0]+(-100)(1-Cos[t0])},{t0-Sin[t0]+(100)(-Sin[t0]),1-Cos[t0]+(100)(1-Cos[t0])}}],{t0,-4 Pi,6 Pi,(Pi/48)}]},
{RGBColor[0,0,.5],Thickness[0.007],Line[Table[{s,1}-{Sin[s],Cos[s]},{s,-4Pi,5 Pi,(Pi/64)}]]}
},
Axes->{True,False},Ticks->{Range[-3 Pi,7Pi,Pi],{}},
PlotRange->{{-Pi-.2,5Pi},{-1Pi,1 Pi}},AspectRatio->Automatic,ImageSize->600
]

(* Export["CycloidNorms.gif",CycloidNorms,"GIF",{"ImageSize"->600}] *)

Cardioid

Manipulate[Graphics[{
{GrayLevel[0.3],Thickness[0.005],Circle[{0,0},1]},
{Point[{0,0}]},
{GrayLevel[0.7],Thickness[0.005],Circle[{2Cos[t],2 Sin[t]},1],Line[{{2Cos[t],2 Sin[t]},{2Cos[t],2 Sin[t]}-{Cos[2t],Sin[2 t]}}]},
{RGBColor[0,0,.5],Thickness[0.007],Line[Table[{2Cos[s],2 Sin[s]}-{Cos[2s],Sin[2s]},{s,0,t,(Pi/64)}]]},
{RGBColor[0,0,.5],PointSize[0.02],Point[{2Cos[t],2 Sin[t]}-{Cos[2t],Sin[2 t]}]}

},
Axes-> False,
PlotRange->{{-3.25,3.25},{-3.25,3.25}},AspectRatio->Automatic
],{t,0,2 Pi}]

CardioidAni=Table[Graphics[{
{GrayLevel[0.3],Thickness[0.005],Circle[{0,0},1]},
{Point[{0,0}]},
{GrayLevel[0.7],Thickness[0.005],Circle[{2Cos[t],2 Sin[t]},1],Line[{{2Cos[t],2 Sin[t]},{2Cos[t],2 Sin[t]}-{Cos[2t],Sin[2 t]}}]},
{RGBColor[0,0,.5],Thickness[0.007],Line[Table[{2Cos[s],2 Sin[s]}-{Cos[2s],Sin[2s]},{s,0,t,(Pi/64)}]]},
{RGBColor[0,0,.5],PointSize[0.02],Point[{2Cos[t],2 Sin[t]}-{Cos[2t],Sin[2 t]}]}

},
Axes-> False,
PlotRange->{{-3.25,3.25},{-3.25,3.25}},AspectRatio->Automatic,ImageSize->600
],{t,0,2 Pi,(Pi/64)}];

CardioidAni[[14]]

(* Export["CardioidAniS1.gif",CardioidAni[[14]],"GIF",{"ImageSize"->600}] *)

(* Export["CardioidAni.gif",CardioidAni,"GIF","AnimationRepetitions"->0,"ImageSize"->600,"DisplayDurations"->(((30/(Length[CardioidAni])))&/@Range[Length[CardioidAni]])]  *)

CardioidTans=ParametricPlot[{2 Cos[t]- Cos[2 t],2 Sin[t]-Sin[2 t]},{t,0,2 Pi},PlotStyle->{{RGBColor[0,0,.5],Thickness[0.004]}},PlotRange->{{-5,5},{-5,5}},Prolog-> {{Thickness[0.001],GrayLevel[0.7],Table[Line[{{2 Cos[t0]-Cos[2 t0]+(-200) (-2 Sin[t0]+2 Sin[2 t0]),(-200)(2 Cos[t0]-2 Cos[2 t0])+2 Sin[t0]-Sin[2 t0]},{2 Cos[t0]-Cos[2 t0]+200 (-2 Sin[t0]+2 Sin[2 t0]),200(2 Cos[t0]-2 Cos[2 t0])+2 Sin[t0]-Sin[2 t0]}}],{t0,(Pi/128),2 Pi-(Pi/128),(Pi/128)}]}},ImageSize->600,Ticks->{Range[-5,5],Range[-5,5]}]

(* Export["CardioidTans.gif",CardioidTans,"GIF",{"ImageSize"->600}] *)

CircleTans=ParametricPlot[{Cos[t],Sin[t]},{t,0,2 Pi},PlotStyle->{{RGBColor[0,0,.5],Thickness[0.004]}},PlotRange->{{-5,5},{-5,5}},Axes->False,ImageSize->600,Prolog-> {{Thickness[0.001],GrayLevel[0.7],Table[Line[{{Cos[t0]+(-200) (- Sin[t0]),Sin[t0]+(-200)(Cos[t0])},{Cos[t0]+(-200) ( Sin[t0]),Sin[t0]+(200)(Cos[t0])}}],{t0,0,2 Pi-(Pi/128),(Pi/128)}]}}]

(* Export["CircleTans.gif",CircleTans,"GIF",{"ImageSize"->600}] *)

Manipulate[ParametricPlot[{Cos[t],Sin[t]},{t,0,2 Pi},PlotStyle->{{RGBColor[0,0,.5],Thickness[0.004]}},PlotRange->{{-5,5},{-5,5}},Axes->False,ImageSize->600,Prolog-> {{Thickness[0.001],GrayLevel[0.7],Table[Line[{{Cos[t0]+(-200) (- Sin[t0]),Sin[t0]+(-200)(Cos[t0])},{Cos[t0]+(-200) ( Sin[t0]),Sin[t0]+(200)(Cos[t0])}}],{t0,0,s0,(Pi/128)}]}}],{s0,0,2 Pi-(Pi/128)}]

CardioidNorms=ParametricPlot[{2 Cos[t]- Cos[2 t],2 Sin[t]-Sin[2 t]},{t,0,2 Pi},PlotStyle->{{RGBColor[0,0,.5],Thickness[0.004]}},PlotRange->{{-5,5},{-5,5}},Prolog-> {{Thickness[0.001],GrayLevel[0.7],Table[Line[{{2 Cos[t0]-Cos[2 t0]+(-200) (-(2 Cos[t0]-2 Cos[2 t0])),(-200)(-2 Sin[t0]+2 Sin[2 t0])+2 Sin[t0]-Sin[2 t0]},{2 Cos[t0]-Cos[2 t0]+200 (-(2 Cos[t0]-2 Cos[2 t0])),200(-2 Sin[t0]+2 Sin[2 t0])+2 Sin[t0]-Sin[2 t0]}}],{t0,(Pi/256),2 Pi-(Pi/256),(Pi/128)}]}},ImageSize->600,Ticks->{Range[-5,5],Range[-5,5]}]

(* Export["CardioidNorms.gif",CardioidNorms,"GIF",{"ImageSize"->600}] *)

Nephroid

Manipulate[Graphics[{
{GrayLevel[0.3],Thickness[0.005],Circle[{0,0},1]},
{Point[{0,0}]},
{GrayLevel[0.7],Thickness[0.005],Circle[{(1+(1/2)) Cos[t],(1+(1/2))  Sin[t]},(1/2)],Line[{{(1+(1/2))Cos[t],(1+(1/2)) Sin[t]},{(1+(1/2))Cos[t],(1+(1/2)) Sin[t]}-(1/2){Cos[t+(t/(1/2))],Sin[t+(t/(1/2))]}}]},
{RGBColor[0,0,.5],Thickness[0.007],Line[Table[{(1+(1/2))Cos[s],(1+(1/2)) Sin[s]}-(1/2){Cos[s+(s/(1/2))],Sin[s+(s/(1/2))]},{s,0,t,(Pi/64)}]]},
{RGBColor[0,0,.5],PointSize[0.02],Point[{(1+(1/2))Cos[t],(1+(1/2)) Sin[t]}-(1/2){Cos[t+(t/(1/2))],Sin[t+(t/(1/2))]}]}

},
Axes-> False,
PlotRange->{{-2.2,2.2},{-2.5,2.5}},AspectRatio->Automatic
],{t,0,2 Pi}]

NephroidAni=Table[Graphics[{
{GrayLevel[0.3],Thickness[0.005],Circle[{0,0},1]},
{Point[{0,0}]},
{GrayLevel[0.7],Thickness[0.005],Circle[{(1+(1/2)) Cos[t],(1+(1/2))  Sin[t]},(1/2)],Line[{{(1+(1/2))Cos[t],(1+(1/2)) Sin[t]},{(1+(1/2))Cos[t],(1+(1/2)) Sin[t]}-(1/2){Cos[t+(t/(1/2))],Sin[t+(t/(1/2))]}}]},
{RGBColor[0,0,.5],Thickness[0.007],Line[Table[{(1+(1/2))Cos[s],(1+(1/2)) Sin[s]}-(1/2){Cos[s+(s/(1/2))],Sin[s+(s/(1/2))]},{s,0,t,(Pi/64)}]]},
{RGBColor[0,0,.5],PointSize[0.02],Point[{(1+(1/2))Cos[t],(1+(1/2)) Sin[t]}-(1/2){Cos[t+(t/(1/2))],Sin[t+(t/(1/2))]}]}

},
Axes-> False,
PlotRange->{{-2.2,2.2},{-2.5,2.5}},AspectRatio->Automatic,ImageSize->600
],{t,0,2 Pi,(Pi/64)}];

NephroidAni[[11]]

(* Export["NephroidAniS1.gif",NephroidAni[[11]],"GIF",{"ImageSize"->600}] *)

(* Export["NephroidAni.gif",NephroidAni,"GIF","AnimationRepetitions"->0,"ImageSize"->600,"DisplayDurations"->(((30/(Length[NephroidAni])))&/@Range[Length[NephroidAni]])]  *)

One third

Manipulate[Graphics[{
{GrayLevel[0.3],Thickness[0.005],Circle[{0,0},1]},
{Point[{0,0}]},
{GrayLevel[0.7],Thickness[0.005],Circle[{(1+(1/3)) Cos[t],(1+(1/3))  Sin[t]},(1/3)],Line[{{(1+(1/3))Cos[t],(1+(1/3)) Sin[t]},{(1+(1/3))Cos[t],(1+(1/3)) Sin[t]}-(1/3){Cos[t+(t/(1/3))],Sin[t+(t/(1/3))]}}]},
{RGBColor[0,0,.5],Thickness[0.007],Line[Table[{(1+(1/3))Cos[s],(1+(1/3)) Sin[s]}-(1/3){Cos[s+(s/(1/3))],Sin[s+(s/(1/3))]},{s,0,t,(Pi/64)}]]},
{RGBColor[0,0,.5],PointSize[0.02],Point[{(1+(1/3))Cos[t],(1+(1/3)) Sin[t]}-(1/3){Cos[t+(t/(1/3))],Sin[t+(t/(1/3))]}]}

},
Axes-> False,
PlotRange->{{-2.,2.},{-1.8,1.8}},AspectRatio->Automatic
],{t,0,2 Pi}]

OneThirdAni=Table[Graphics[{
{GrayLevel[0.3],Thickness[0.005],Circle[{0,0},1]},
{Point[{0,0}]},
{GrayLevel[0.7],Thickness[0.005],Circle[{(1+(1/3)) Cos[t],(1+(1/3))  Sin[t]},(1/3)],Line[{{(1+(1/3))Cos[t],(1+(1/3)) Sin[t]},{(1+(1/3))Cos[t],(1+(1/3)) Sin[t]}-(1/3){Cos[t+(t/(1/3))],Sin[t+(t/(1/3))]}}]},
{RGBColor[0,0,.5],Thickness[0.007],Line[Table[{(1+(1/3))Cos[s],(1+(1/3)) Sin[s]}-(1/3){Cos[s+(s/(1/3))],Sin[s+(s/(1/3))]},{s,0,t,(Pi/64)}]]},
{RGBColor[0,0,.5],PointSize[0.02],Point[{(1+(1/3))Cos[t],(1+(1/3)) Sin[t]}-(1/3){Cos[t+(t/(1/3))],Sin[t+(t/(1/3))]}]}

},
Axes-> False,
PlotRange->{{-2.,2.},{-1.8,1.8}},AspectRatio->Automatic,ImageSize-> 600
],{t,0,2 Pi,(Pi/64)}];

OneThirdAni[[9]]

(* Export["OneThirdAniS1.gif",OneThirdAni[[11]],"GIF",{"ImageSize"->600}] *)

(* Export["OneThirdAni.gif",OneThirdAni,"GIF","AnimationRepetitions"->0,"ImageSize"->600,"DisplayDurations"->(((30/(Length[OneThirdAni])))&/@Range[Length[OneThirdAni]])]  *)

One third-inside

Manipulate[Graphics[{
{GrayLevel[0.3],Thickness[0.005],Circle[{0,0},1]},
{Point[{0,0}]},
{GrayLevel[0.7],Thickness[0.005],Circle[{(1-(1/3)) Cos[t],(1-(1/3))  Sin[t]},(1/3)],Line[{{(1-(1/3))Cos[t],(1-(1/3)) Sin[t]},{(1-(1/3))Cos[t],(1-(1/3)) Sin[t]}+(1/3){Cos[t-(t/(1/3))],Sin[t-(t/(1/3))]}}]},
{RGBColor[0,0,.5],Thickness[0.007],Line[Table[{(1-(1/3))Cos[s],(1-(1/3)) Sin[s]}+(1/3){Cos[s-(s/(1/3))],Sin[s-(s/(1/3))]},{s,0,t,(Pi/64)}]]},
{RGBColor[0,0,.5],PointSize[0.02],Point[{(1-(1/3))Cos[t],(1-(1/3)) Sin[t]}+(1/3){Cos[t-(t/(1/3))],Sin[t-(t/(1/3))]}]}

},
Axes-> False,
PlotRange->{{-1.3,1.3},{-1.3,1.3}},AspectRatio->Automatic
],{t,0,2 Pi}]

{(1-(1/3))Cos[t],(1-(1/3)) Sin[t]}+(1/3){Cos[t-(t/(1/3))],Sin[t-(t/(1/3))]}

OneThirdIAni=Table[Graphics[{
{GrayLevel[0.3],Thickness[0.005],Circle[{0,0},1]},
{Point[{0,0}]},
{GrayLevel[0.7],Thickness[0.005],Circle[{(1-(1/3)) Cos[t],(1-(1/3))  Sin[t]},(1/3)],Line[{{(1-(1/3))Cos[t],(1-(1/3)) Sin[t]},{(1-(1/3))Cos[t],(1-(1/3)) Sin[t]}+(1/3){Cos[t-(t/(1/3))],Sin[t-(t/(1/3))]}}]},
{RGBColor[0,0,.5],Thickness[0.007],Line[Table[{(1-(1/3))Cos[s],(1-(1/3)) Sin[s]}+(1/3){Cos[s-(s/(1/3))],Sin[s-(s/(1/3))]},{s,0,t,(Pi/64)}]]},
{RGBColor[0,0,.5],PointSize[0.02],Point[{(1-(1/3))Cos[t],(1-(1/3)) Sin[t]}+(1/3){Cos[t-(t/(1/3))],Sin[t-(t/(1/3))]}]}

},
Axes-> False,
PlotRange->{{-1.3,1.3},{-1.3,1.3}},AspectRatio->Automatic,ImageSize->600
],{t,0,2 Pi,(Pi/64)}];

OneThirdIAni[[9]]

(* Export["OneThirdIAniS1.gif",OneThirdIAni[[11]],"GIF",{"ImageSize"->600}] *)

(* Export["OneThirdIAni.gif",OneThirdIAni,"GIF","AnimationRepetitions"->0,"ImageSize"->600,"DisplayDurations"->(((30/(Length[OneThirdIAni])))&/@Range[Length[OneThirdIAni]])] *)

One fifth-inside

Manipulate[Graphics[{
{GrayLevel[0.3],Thickness[0.005],Circle[{0,0},1]},
{Point[{0,0}]},
{GrayLevel[0.7],Thickness[0.005],Circle[{(1-(1/5)) Cos[t],(1-(1/5))  Sin[t]},(1/5)],Line[{{(1-(1/5))Cos[t],(1-(1/5)) Sin[t]},{(1-(1/5))Cos[t],(1-(1/5)) Sin[t]}+(1/5){Cos[t-(t/(1/5))],Sin[t-(t/(1/5))]}}]},
{RGBColor[0,0,.5],Thickness[0.007],Line[Table[{(1-(1/5))Cos[s],(1-(1/5)) Sin[s]}+(1/5){Cos[s-(s/(1/5))],Sin[s-(s/(1/5))]},{s,0,t,(Pi/64)}]]},
{RGBColor[0,0,.5],PointSize[0.02],Point[{(1-(1/5))Cos[t],(1-(1/5)) Sin[t]}+(1/5){Cos[t-(t/(1/5))],Sin[t-(t/(1/5))]}]}

},
Axes-> False,
PlotRange->{{-1.3,1.3},{-1.3,1.3}},AspectRatio->Automatic
],{t,0,2 Pi}]

{(1-(1/5))Cos[t],(1-(1/5)) Sin[t]}+(1/5){Cos[t-(t/(1/5))],Sin[t-(t/(1/5))]}

OneFifthIAni=Table[Graphics[{
{GrayLevel[0.3],Thickness[0.005],Circle[{0,0},1]},
{Point[{0,0}]},
{GrayLevel[0.7],Thickness[0.005],Circle[{(1-(1/5)) Cos[t],(1-(1/5))  Sin[t]},(1/5)],Line[{{(1-(1/5))Cos[t],(1-(1/5)) Sin[t]},{(1-(1/5))Cos[t],(1-(1/5)) Sin[t]}+(1/5){Cos[t-(t/(1/5))],Sin[t-(t/(1/5))]}}]},
{RGBColor[0,0,.5],Thickness[0.007],Line[Table[{(1-(1/5))Cos[s],(1-(1/5)) Sin[s]}+(1/5){Cos[s-(s/(1/5))],Sin[s-(s/(1/5))]},{s,0,t,(Pi/64)}]]},
{RGBColor[0,0,.5],PointSize[0.02],Point[{(1-(1/5))Cos[t],(1-(1/5)) Sin[t]}+(1/5){Cos[t-(t/(1/5))],Sin[t-(t/(1/5))]}]}

},
Axes-> False,
PlotRange->{{-1.3,1.3},{-1.3,1.3}},AspectRatio->Automatic,ImageSize->600
],{t,0,2 Pi,(Pi/64)}];

OneFifthIAni[[9]]

(* Export["OneFifthIAniS1.gif",OneFifthIAni[[11]],"GIF",{"ImageSize"->600}] *)

(* Export["OneFifthIAni.gif",OneFifthIAni,"GIF","AnimationRepetitions"->0,"ImageSize"->600,"DisplayDurations"->(((30/(Length[OneThirdIAni])))&/@Range[Length[OneThirdIAni]])] *)

Cylinders

Show[RegionPlot3D[And[y^2+z^2>1,z^2+x^2<1],{x,-1,1},{y,-1,1},{z,-1,1},PlotPoints->{200,200,200},Mesh->None,PlotStyle->{Opacity[.7]}],
RegionPlot3D[And[y^2+z^2<1,z^2+x^2>1],{x,-1,1},{y,-1,1},{z,-1,1},PlotPoints->{200,200,200},Mesh->None,PlotStyle->{Opacity[.7]}],
Ticks->{Range[-2,2,1],Range[-2,2,1],Range[-2,2,1]},PlotRange->{{-1,1},{-1,1},{-1,1}},ImageSize->500
]

ThreeCylC11=RegionPlot3D[And[x^2+y^2>1,y^2+z^2<1,z^2+x^2<1],{x,-1,1},{y,-1,1},{z,-1,1},PlotPoints->{300,300,300},Mesh->None,PlotStyle->{Opacity[1]},Ticks->{Range[-2,2,1],Range[-2,2,1],Range[-2,2,1]},ImageSize->500]

ThreeCylC21=RegionPlot3D[And[x^2+y^2>1,y^2+z^2>1,z^2+x^2<1],{x,-1,1},{y,-1,1},{z,-1,1},PlotPoints->{200,200,200},Mesh->None,PlotStyle->{Opacity[.7]},Ticks->{Range[-2,2,1],Range[-2,2,1],Range[-2,2,1]},ImageSize->500]

(* Export["ThreeCylC21.png",ThreeCylC21];Export["ThreeCylC11.png",ThreeCylC11];
Export["ThreeCylC21.gif",ThreeCylC21];Export["ThreeCylC11.gif",ThreeCylC11]; *)