ParametricPlot[{t^2+1, t^3+2}, {t, -10, 10},PlotRange->1500]

ParametricPlot[{t^2+1, t^3+2}, {t, -5, 5},PlotRange->10, Axes -> None]

ParametricPlot[{{t^2+1, t^3+2}, {-t^3,2t}}, {t, -5, 5},PlotRange->10]

Show[{ParametricPlot[{t^2+1, t^3+2}, {t, -5, 5},PlotRange->10], ParametricPlot[{Exp[t] ,t^3+2}, {t, -2, 2}, PlotStyle-> {RGBColor[0.5,0.9,0.5]}]}]

ContourPlot[x^2/3+y^2/9==1, {x,-3,3},{y,-3,3}]

ContourPlot[x^3*y+2x*y^5+y^2==1, {x,-10,10},{y,-10,10}](*jfssajkhfkajsh*)

ellisse[x0_,y0_][a_, b_][t_] := {x0+a*Cos[t], y0+b*Sin[t]}

ParametricPlot[ellisse[2,3][2,5][t], {t,0, 2Pi}, PlotRange->8]

Manipulate[ParametricPlot[ellisse[2,3][2,5][t],{t,0.01,k}, PlotRange->8], {k, 0, 10Pi}]

Manipulate[ParametricPlot[ellisse[2,3][k,5][t],{t,0,2Pi}, PlotRange-> 8], {k, 0.1, 6}]

D[x^4*Cos[x], x]

D[x^4*Cos[x], {x,2}]

vt[gamma_][t_] := Simplify[D[gamma[tt],tt]]/.tt-> t

nvt[gamma_][t_] :=Sqrt[Simplify[D[gamma[tt],tt].D[gamma[tt],tt]]]/.tt-> t

lung[a_,b_][gamma_] :=Integrate[nvt[gamma][u],{u, a,b}]

lemniscata[t_]:={Cos[t]/(1+Sin[t]^2), Sin[2t]/(1+Sin[t]^2)}

vt[lemniscata][t]

parabola[t_] := {t,t^2}

ParametricPlot[lemniscata[t], {t, 0, 2Pi}]

lung[0,1][parabola]

retta[t_]:= {t, 3t}

lung[0,1][retta]

nlung[a_,b_][gamma_] :=NIntegrate[nvt[gamma][u],{u, a,b}]

nlung[1,2][lemniscata]

ellisse[x0_,y0_][a_,b_][t_] := {x0+a*Cos[t], y0+b*Sin[t]}

ParametricPlot[ellisse[1,3][2,5][t], {t,0,2Pi}]

ParametricPlot[ellisse[1,3][2,2][t], {t,0,2Pi}, (*sjkskghsd*)
PlotRange->{{-6, 6}, {-6, 6}}] (*sjkskghsd*)

cicloide[a_][h_][t_]:={a*t-h*Sin[t], a-h*Cos[t]}

ParametricPlot[cicloide[1][1][t], {t, 0, 4Pi}]

Manipulate[ParametricPlot[cicloide[1][1][t], {t,0, k}, PlotRange->{{0,6Pi}, {0,2}}, Epilog->{Circle[{k,1},1],Point[cicloide[1][1][k]], Line[{{k,1},cicloide[1][1][k]}]}], {k, 0.0001, 6Pi}]

epicicloide[a_,b_][t_]:={(a+b)*Cos[t]-b*Cos[(a+b)/b *t], (a+b)*Sin[t]-b*Sin[(a+b)/b *t]}

ParametricPlot[epicicloide[2,3][t], {t,0,6Pi}, PlotRange -> {{-9, 9}, {-9, 9}}]

Manipulate[ParametricPlot[epicicloide[2,3][t],{t,0,k}, PlotRange -> {{-9, 9}, {-9, 9}}, Epilog -> {Circle[{0,0}, 2], Circle[{5*Cos[k],5*Sin[k]},3], Point[epicicloide[2,3][k]], Line[{{5*Cos[k],5*Sin[k]}, epicicloide[2,3][k]}]}], {k, 0.0001, 8Pi}]

ParametricPlot[epicicloide[2,Pi][t], {t,0,20Pi}, PlotRange -> {{-9, 9}, {-9, 9}}]

ParametricPlot[epicicloide[2,Sqrt[2]][t], {t,0,20Pi}, PlotRange -> {{-9, 9}, {-9, 9}}]

Manipulate[ParametricPlot[lemniscata[t], {t,0,2Pi},PlotRange->{{-1.2, 1.2}, {-1.2, 1.2}},Epilog -> {{PointSize[0.03], Red, Point[lemniscata[k]]},  Thick, Green, Arrow[{lemniscata[k], lemniscata[k]+vt[lemniscata][k]/4}]}],{k, 0, 2Pi}]

ParametricPlot[lemniscata[t], {t,0,2Pi},PlotRange->{{-1, 1}, {-1, 1}},Epilog -> {Thick, Green, Arrow[{lemniscata[1], lemniscata[2]}]}]

Show[Graphics[Arrow[{lemniscata[1], lemniscata[2]}]]]

vt[lemniscata][k]