(*^

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:[font = special1; inactive; preserveAspect; ]
                                                Partial Derivatives

The cell below looks at the same function as the movie in this module -- the function

                                                                f(x, y) = x^2 - y^2

It produces three graphs showing progressively smaller and smaller neighborhoods of the point
(0, 0, 0)  on the surface under progressively greater and greater magnification.  Evaluate the next cell now and then notice that as the magnification increases the surface looks more and more like a flat plane.
;[s]
8:0,1;67,0;220,1;239,0;335,1;345,0;417,2;443,0;547,-1;
3:4,13,9,Times,0,12,0,0,0;3,13,9,Times,1,12,0,0,0;1,13,9,Times,1,12,65535,0,0;
:[font = input; preserveAspect; ]
M := 1

Plot3D[x^2 - y^2, {x, -M, M}, {y, -M, M},
       PlotRange -> {-M, M},
       AspectRatio -> Automatic]
       
M := 0.25

Plot3D[x^2 - y^2, {x, -M, M}, {y, -M, M},
       PlotRange -> {-M, M},
       AspectRatio -> Automatic]
       
M := 0.0625

Plot3D[x^2 - y^2, {x, -M, M}, {y, -M, M},
       PlotRange -> {-M, M},
       AspectRatio -> Automatic]
:[font = special1; inactive; preserveAspect; ]
The next cell looks at another point  (-0.5, 0.5, 0)  on the same surface.  Evaluate it now.  Notice that once again as the magnification increases the surface looks more and more like a flat plane.
;[s]
5:0,0;38,2;52,0;76,1;91,0;199,-1;
3:3,13,9,Times,0,12,0,0,0;1,13,9,Times,1,12,65535,0,0;1,13,9,Times,1,12,0,0,0;
:[font = input; preserveAspect; ]
M := 1

Plot3D[x^2 - y^2, 
       {x, -M + 0.5, M + 0.5}, {y, -M - 0.5, M - 0.5},
       PlotRange -> {-3 M, 3 M},
       AspectRatio -> Automatic]
       
M := 0.25

Plot3D[x^2 - y^2, 
      {x, -M + 0.5, M + 0.5}, {y, -M - 0.5, M - 0.5},
       PlotRange -> {-3 M, 3 M},
       AspectRatio -> Automatic]
       
M := 0.0625

Plot3D[x^2 - y^2, 
       {x, -M + 0.5, M + 0.5}, {y, -M - 0.5, M - 0.5},
       PlotRange -> {-3 M, 3 M},
       AspectRatio -> Automatic]
:[font = special1; inactive; preserveAspect; ]
The next cell illustrates how easy it is to use Mathematica  to look at the neighborhood of a point on a graph under higher and higher magnification.  In this case we look at the point  (0, 0)  on the graph of the function  f(x) = |x|.  Evaluate the next cell now.
;[s]
9:0,0;48,1;59,0;185,2;192,0;224,2;233,0;237,3;263,0;265,-1;
4:5,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0;2,13,9,Times,1,12,0,0,0;1,13,9,Times,1,12,65535,0,0;
:[font = input; preserveAspect; ]
Mag[M_] :=
   Plot[Abs[x], {x, -M, M}, 
        PlotRange -> {-M, M},
        AspectRatio -> Automatic]
     
Mag[1]
Mag[0.1]
Mag[0.01]
:[font = special1; inactive; preserveAspect; ]
The cell below illustrates how we can compare functions like those in this module.  Evaluate it now.
;[s]
3:0,0;84,1;99,0;101,-1;
2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,1,12,65535,0,0;
:[font = input; preserveAspect; ]
Clear[m, b]

f[x_] := x^2

x0 := 0.5

m = D[f[x],x] /. x -> x0
b = f[x0] - m x0
T[x_] := m x + b

Plot[{f[x], T[x]},{x, -1, 1},
     PlotRange -> {-1, 1},
     PlotStyle -> {{RGBColor[0, 0, 0]},
                   {RGBColor[1, 0, 0]}}]
:[font = special1; inactive; preserveAspect; ]
The cell below is also shown in your browser window.  This cell illustrates how Mathematica  can be used to draw the tangent plane to a surface at a point.  Evaluate it now.
;[s]
5:0,0;80,1;91,0;157,2;172,0;174,-1;
3:3,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0;1,13,9,Times,1,12,65535,0,0;
:[font = input; preserveAspect; ]
Clear[f, fx, fy, a, b, c]

f[x_, y_] := x^2 + y^2

x0 = -0.5;
y0 = -0.5;

fx[x_, y_] = D[f[x, y], x];
fy[x_, y_] = D[f[x, y], y];

a = fx[x0, y0];
b = fy[x0, y0];
c = f[x0, y0] - a x0 - b y0;

T[x_, y_] := a x + b y + c

Show[Plot3D[f[x, y], {x, -1, 1}, {y, -1, 1},
              PlotRange -> {-4, 2}],
             Plot3D[T[x, y], {x, -1, 1}, {y, -1, 1},
              PlotRange -> {-4, 2}]]
^*)