Some basic commands:

Plot[ x Sin[x],{x,10,10}]  Graph function of one variable 
Plot3D[ Sin[x y],{x,2,2},{y,2,2}]  Graph function of two variables 
ParametricPlot[ {Cos[3 t],Sin[5 t]} ,{t,0,2Pi}]  Plot planar curve 
ParametricPlot3D[{Cos[t],Sin[t],t} ,{t,0,4Pi},AspectRatio>1]  Plot space curve 
ParametricPlot3D[{Cos[t] Sin[s],Sin[t] Sin[s],Cos[s]},{t,0,2Pi},{s,0,Pi}]  Parametric Surface 
SphericalPlot3D[(2+Sin[2 t] Sin[3 s]),{t,0,Pi},{s,0,2 Pi}]  Spherical Plot 
RevolutionPlot3D[{2 + Cos[t], t}, {t,0,2 Pi}]  Revolution Plot 
ContourPlot[Sin[x y],{x,2,2},{y,2,2} ]  Contour lines (traces) 
ContourPlot3D[x^2+2y^2z^2,{x,2,2},{y,2,2},{z,2,2}]  Implicit surface 
VectorPlot[{xy,x+y},{x,3,3},{y,3,3}]  Vectorfield plot 
VectorPlot3D[{xy,x+y,z},{x,3,3},{y,3,3},{z,0,1}]  Vectorfield plot 3D 
Integrate[x Sin[x], x]  Integrate symbolically 
Integrate[x y^2z,{x,0,2},{y,0,x},{z,0,y}]  3D Integral 
NIntegrate[Exp[x^2],{x,0,10}]  Integrate numerically 
D[ Cos^5[x],x ]  Differentiate symbolically 
Series[Exp[x],{x,0,3} ]  Taylor series 
DSolve[ x''[t]==x[t],x,t ]  Solution to ODE 
DSolve[{D[u[x,t],t]==D[u[x,t],x],u[x,0]==Sin[x]},u[x,t],{x,t}]  Solution to PDE 
Classify extrema:
ClassifyCriticalPoints[f_,{x_,y_}]:=Module[{X,P,H,g,d,S}, X={x,y};
P=Sort[Solve[Thread[D[f,#] & /@ X==0],X]]; H=Outer[D[f,#1,#2]&,X,X];g=H[[1,1]];d=Det[H];
S[d_,g_]:=If[d<0,"saddle",If[g>0,"minimum","maximum"]];
TableForm[{x,y,d,g,S[d,g],f} /.P,TableHeadings>{None,{x,y,"D","f_xx","Type","f"}}]]
ClassifyCriticalPoints[4 x y  x^3 y  x y^3,{x,y}]
Solve a Lagrange problem:
F[x_,y_]:=2x^2+4 x y; G[x_,y_]:=x^2 y;
Solve[{D[F[x,y],x]==L*D[G[x,y],x],D[F[x,y],y]==L*D[G[x,y],y],G[x,y]==1},{x,y,L}]
Check that a function solves a PDE:
f[t_,x_]:=(x/t)*Sqrt[1/t]*Exp[x^2/(4 t)]/(1+ Sqrt[1/t] Exp[x^2/(4 t)]);
D[f[t,x],t]+f[t,x]*D[f[t,x],x]D[f[t,x],{x,2}]
Simplify[%] Chop[%]
