$PlotTheme="Default"

Plot[{1/x^2,
Piecewise[{{1/x^2,1≤x≤2}},_]}
,{x,.5,3},Filling->{2->Axis},
PlotRange->{0,1.5}]

Plot[{1/x^2,
Piecewise[{{1/x^2,1≤x≤5}},_]}
,{x,.5,3},Filling->{2->Axis},
PlotRange->{0,1.5}]

Table[
Plot[{1/x^2,
Piecewise[{{1/x^2,1≤x≤n}},_]}
,{x,.5,8},Filling->{2->Axis},
PlotRange->{0,1}],
{n,3,8}]

ContourPlot[{x==2},{x,0,5},{y,0,5}]

Plot[{1/x,
Piecewise[{{1/x,1≤x≤8}},_]}
,{x,.5,3},Filling->{2->Axis},
PlotRange->{0,1.5}]

f[x_]:=Sin[x]
Table[
Plot[{f[x],
Piecewise[{{f[x],-n Pi/2≤x≤Pi}},_]}
,{x,-3Pi,Pi},Filling->{2->Axis},
PlotRange->{-1,1}],
{n,1,6}]

f[x_]:=1/(1+x^2)
Plot[f[x],{x,-3,3},Filling->Axis,PlotRange->{0,1}]

f[x_]:=2x E^(-x^2)
Plot[f[x],{x,-3,3},Filling->Axis,PlotRange->{-1,1}]

f[x_]:=1/Sqrt[x]
Plot[{f[x],
Piecewise[{{f[x],-1≤x≤1}},_]}
,{x,0,2},Filling->{2->Axis},
PlotRange->{0,5}]
Table[
Plot[{f[x],
Piecewise[{{f[x],1/n≤x≤1}},_]}
,{x,0,2},Filling->{2->Axis},
PlotRange->{0,5}],
{n,2,7}]

f[x_]:=Tan[x]
Plot[{f[x],
Piecewise[{{f[x],0≤x≤Pi/2}},_]}
,{x,-Pi/2,Pi},Filling->{2->Axis},
PlotRange->{-10,10}]
Table[
Plot[{f[x],
Piecewise[{{f[x],0≤x≤Pi/2-1/(2n)}},_]}
,{x,-Pi/2,Pi},Filling->{2->Axis},
PlotRange->{-10,10}],
{n,1,6}]

f[x_]:=Tan[x]
Plot[{f[x],
Piecewise[{{f[x],0≤x≤Pi}},_]}
,{x,-Pi/2,3Pi/2},Filling->{2->Axis},
PlotRange->{-10,10}]

f[x_]:=1/CubeRoot[x-1]
Plot[{f[x],
Piecewise[{{f[x],0≤x≤2}},_]}
,{x,-1,3},Filling->{2->Axis},
PlotRange->{-5,5}]

f[x_]:=1/(x-6)^2
Plot[{f[x],
Piecewise[{{f[x],0≤x≤10}},_]}
,{x,0,10},Filling->{2->Axis},
PlotRange->{0,5}]

f[x_]:=E^(-x^2)
Plot[{f[x],
Piecewise[{{f[x],0≤x≤10}},_]}
,{x,-3,3},Filling->{2->Axis},
PlotRange->{0,1}]
Plot[E^(-x^2),{x,0,1},Filling->Axis,PlotRange->{0,1}]

Plot[{E^(-x^2),
Piecewise[{{ 2x E^(-x^2),x≥1}},_],
2x E^(-x^2)}
,{x,0,3},Filling->{2->Axis,1->Axis},PlotRange->{0,1}]

Integrate[E^(-x^2),{x,0,Infinity}]

n=4;a=1;b=3;d = (b-a)/n;
f[x_]:=1/x
points=Table[{x,f[x]},{x,a,b,d}];
approxPlot=ListPlot[points,Joined->True,PlotStyle->{Thick,Red}];
fnPlot=Plot[{f[x]},{x,a,b},PlotStyle->Thickness[.01],PlotRange->{f[a],f[b]}];
Show[fnPlot,approxPlot]
fparam[x_]:={x,f[x]}
curve=ParametricPlot[fparam[x],{x,a,b},PlotStyle->{Black,Thickness[.02]}];
lines=Graphics[Join[{Thickness[.01],Red},Table[Arrow[{fparam[k],fparam[k]+fparam'[k]*d}],{k,a,b,d}]]];
Show[fnPlot,lines]

ParametricPlot3D[{s,2Cos[t],2Sin[t]},{s,0,1},{t,0,2Pi}]
ParametricPlot3D[{s,(1+2s)Cos[t],(1+2s)Sin[t]},{s,0,1},{t,0,2Pi}]

d=.5;
Show[
Table[
ParametricPlot3D[{x,Sqrt[1-x^2]Cos[t],Sqrt[1-x^2]Sin[t]},{x,k,k+d},{t,0,2 Pi},PlotRange->{-1,1},BoundaryStyle->Directive[Thickness[.01],Black]],
{k,-1,1-d,d}]]
Show[
Table[
ParametricPlot3D[{x,(Sqrt[1-k^2](k+d-x)/d + Sqrt[1-(k+d)^2](x-k)/d )Cos[t],(Sqrt[1-k^2](k+d-x)/d + Sqrt[1-(k+d)^2](x-k)/d )Sin[t]},
{x,k,k+d},{t,0,2 Pi},PlotRange->{-1,1},BoundaryStyle->Directive[Thickness[.01],Black]],
{k,-1,1-d,d}]]

d=.5;a=0;b=2;f[x_]:=CubeRoot[3x];
r=MaxValue[{f[x],a≤x≤b},x];
Show[
Table[
ParametricPlot3D[{x,f[x]Cos[t],f[x]Sin[t]},{x,k,k+d},{t,0,2 Pi},PlotRange->{{a,b},{-r,r},{-r,r}},BoundaryStyle->Directive[Thickness[.01],Black]],
{k,a,b-d,d}]]
Show[
Table[
ParametricPlot3D[{x,(f[k](k+d-x)/d + f[k+d](x-k)/d )Cos[t],(f[k](k+d-x)/d + f[k+d](x-k)/d )Sin[t]},
{x,k,k+d},{t,0,2 Pi},PlotRange->{{a,b},{-r,r},{-r,r}},
BoundaryStyle->Directive[Thickness[.01],Black]],
{k,a,b-d,d}]]

d=1;a=1;b=25;f[x_]:=1/x;
r=MaxValue[{f[x],a≤x≤b},x];
Show[
Table[
ParametricPlot3D[{x,f[x]Cos[t],f[x]Sin[t]},{x,k,k+d},{t,0,2 Pi},PlotRange->{{a,b},{-r,r},{-r,r}},BoundaryStyle->Directive[Thickness[.01],Black]],
{k,a,b-d,d}]]
Show[
Table[
ParametricPlot3D[{x,(f[k](k+d-x)/d + f[k+d](x-k)/d )Cos[t],(f[k](k+d-x)/d + f[k+d](x-k)/d )Sin[t]},
{x,k,k+d},{t,0,2 Pi},PlotRange->{{a,b},{-r,r},{-r,r}},
BoundaryStyle->Directive[Thickness[.01],Black]],
{k,a,b-d,d}]]

Plot[BesselJ[0,x],{x,-15,15}]

Plot[-Sin[t] + 3 Cos[t],{t,-2 Pi,2 Pi}]

DSolve[p'[t]==100 - p[t],p[t],t]

Plot[{10 + 5E^(-10t),10},{t,0,1},PlotRange->{0,15},PlotStyle->{Black,Dashed}]

Plot[Cos[t],{t,-2Pi,2Pi}]

Plot[2Sin[t]+2Cos[t],{t,-2Pi,2Pi}]

Plot[Sqrt[x^2+4],{x,-5,5}]

ContourPlot[y^2+y^3/3 == 3x^2/2 +Sin[x] +18,{x,-5,5},{y,0,10}]

Plot[CubeRoot[3 Log[Abs[1+x]]],{x,-5,5}]

Plot[(t^(3/2)/3 + Sqrt[2])^2,{t,-5,5}]

Plot[{20-10E^(-t/100),20},{t,0,600},PlotStyle->{Black,Dashed},PlotRange->{5,20}]

Plot[{.5/(.5 E^(-t) + .5)},{t,-10,10}]

P[t_]:= 3M/( (M-3) E^(-M k t) +3)

Solve[{P[15]==4,P[60]==7.8},{k,M},Reals]

{{k->0.0018395563970610522`,M->13.909839362953823`}}

Solve[{P[t]==12,k == .00183956,M==13.9098},t]

k=  .00183956;M=13.9098;
Plot[{P[t],13.9098},{t,0,140},PlotStyle->{Black,Dashed}]

Plot[{P[t],13.9098},{t,-260,140},PlotStyle->{Black,Dashed}]