> 
  A Surface and its Tangent Plane


We need plots
> with(plots):
Look at the function f(x,y)=x^2 + y^2
> f:=x^2 + y^2;

                                   2    2
                             f := x  + y

Set S to be the surface
> S := plot3d(f,x=0..2,y=0..2,color='red'):
To get the equation of the tangent plane, we need to find
the partial derivatives of f
> dfdx:=diff(f,x);  dfdy:=diff(f,y);

                             dfdx := 2 x


                             dfdy := 2 y

Let's look at the tangent plane at (x,y)=(1,1):
The equation of the tangent plane is

z = f(1,1)  + dfdx(1,1) * (x-1) + dfdx(1,1) * (y-1)

> z := subs(x=1,y=1,f)+subs(x=1,y=1,dfdx)*(x-1)+subs(x=1,y=1,dfdy)*(y-1);

                         z := -2 + 2 x + 2 y

Call the tangent plane P
> P := plot3d(z,x=0..2,y=0..2,color='yellow'):
Show the surface and its tangent plane on the same graph
> display3d({S,P});

>