Example 1

Clear[f]
f[x_,y_] = 18-(x^2+y^2);
Plot3D[f[x,y], {x,-3,3},{y,-3,3}]

ContourPlot[f[x,y], {x,-3,3},{y,-3,3}, Contours->7]

Plot3D[f[x,y],{x,-3,3},{y,-2,2},MeshFunctions->{#3&},Mesh->5]

Plot3D[f[x,y],{x,-3,3},{y,-2,2},MeshFunctions->{#3&},Mesh->5,AxesLabel->{"x","y","z"}]

Example 2

Clear[f]
f[x_,y_] = y^3 Log[3 x^2 - 9];
Plot3D[f[x,y], {x,-3,3},{y,-3,3}]

Plot3D[f[x,y],{x,-3,3},{y,-2,2},MeshFunctions->{#3&},Mesh->5,AxesLabel->{"x","y","z"}]

Example 3

(2 x^2 + y^2)/(x^2 + y^2)

Clear[f]
f[x_,y_] = ((2 (x^2)+(y^2))/((x^2)+(y^2)));
Plot3D[f[x,y], {x,-3,3},{y,-3,3},AxesLabel->{"x","y","z"}]

ContourPlot[f[x,y], {x,-3,3},{y,-3,3}, Contours->7]

Plot3D[f[x,y],{x,-3,3},{y,-2,2},MeshFunctions->{#3&},Mesh->5]

Plot3D[f[x,y],{x,-3,3},{y,-2,2},MeshFunctions->{#3&},Mesh->5,AxesLabel->{"x","y","z"}]

Example 4

Clear[f]
f[x_,y_] =Sin[x+y]/(x+y);
Plot3D[f[x,y], {x,-8π,8π},{y,-8π,8π},AxesLabel->{"x","y","z"}]

ContourPlot[f[x,y],{x,-8π,8π},{y,-8π,8π}, Contours->7]

Plot3D[f[x,y],{x,-8π,8π},{y,-8π,8π},MeshFunctions->{#3&},Mesh->5]

Plot3D[f[x,y],{x,-3,3},{y,-2,2},MeshFunctions->{#3&},Mesh->5,AxesLabel->{"x","y","z"}]

Limit Exercises

Limit[Sin[x]/x,x-> 0]

Limit[Sin[x+y]/(x+y),{x-> 0,y->0}]

TableForm[
Table[{x,y,Sin[x+y]/(x+y)},
{x,-.1,.1,.02},{y,-.1,.1,.02}]
]

Derivative Play

D[f[x,y],x]

D[  Sin[x+y]/(x+y),y]

Together[((Cos[x+y])/(x+y))-((Sin[x+y])/(((x+y))^2))]

D[ y^3 Log[3 x^2 - 9],x]

D[ y^3 Log[3 x^2 - 9],y]

the gradient vector

{D[ y^3 Log[3 x^2 - 9],x],D[ y^3 Log[3 x^2 - 9],y]}