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Notebook[{

Cell[CellGroupData[{
Cell["\<\
11 - How High the Moon  
or 
How Low Can You Go?\
\>", "Title"],

Cell["Maxima and Minima in Three Dimensions", "Subtitle"],

Cell[CellGroupData[{

Cell["\<\
William D. Emerson\t
Metropolitan State College of Denver\
\>", "Section"],

Cell[CellGroupData[{

Cell["First and Second Order Partial Derivatives", "Subsection"],

Cell[TextData[{
  "Given a function we can compute the partial derivatives using ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  ".  Here's an example."
}], "Text"],

Cell[BoxData[{
    \(Clear[f, x, y]\), "\[IndentingNewLine]", 
    \(f[x_, y_]\  = \ x^3\ \  + \ Sqrt[y] + \ x^2\ y^2\)}], "Input"],

Cell["\<\
We can compute the partial derivative of f with respect to x \
:\
\>", "Text"],

Cell[BoxData[
    \(D[f[x, y], x]\)], "Input"],

Cell["\<\
Similarly, we can compute the partial derivative with respect to y:\
\
\>", "Text"],

Cell[BoxData[
    \(D[f[x, y], y]\)], "Input"],

Cell["\<\
The syntax for second partial derivatives is similar.  Here is the \
second partial with respect to x twice and the second partial with respect to \
y twice.\
\>", "Text"],

Cell[BoxData[
    \(D[f[x, y], x, x]\)], "Input"],

Cell[BoxData[
    \(D[f[x, y], y, y]\)], "Input"],

Cell["\<\
Finally, here is the mixed second partial derivative, first with \
respect to x and then with respect to y.\
\>", "Text"],

Cell[BoxData[
    \(D[f[x, y], x, y]\)], "Input"],

Cell["\<\
If we want the mixed second partial, first with respect to y and \
then with respect to x we should get the same answer.\
\>", "Text"],

Cell[BoxData[
    \(D[f[x, y], y, x]\)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Tangent Planes to Surfaces", "Subsection"],

Cell[TextData[{
  "Given a surface ",
  Cell[BoxData[
      \(TraditionalForm\`z\  = \ f[x, y]\)]],
  " and a point ",
  Cell[BoxData[
      \(TraditionalForm\`\((a, b, f[a, b])\)\)]],
  " the equation of the tangent plane at this point is ",
  Cell[BoxData[
      \(TraditionalForm\`z\  = \ 
        f[a, b]\  + \ f\_x[a, b] \((x - a)\)\  + \ 
          f\_y[a, b] \(\((y - b)\)\(.\)\)\)]],
  "  Here is an example."
}], "Text"],

Cell[BoxData[{
    \(Clear[f, x, y, surface]\), "\[IndentingNewLine]", 
    \(f[x_, y_]\  = \ 16\  - \ x^2\  - \ y^2\), "\[IndentingNewLine]", 
    \(surface\  = \ 
      Plot3D[f[x, y], {x, \(-3\), 3}, {y, \(-3\), 3}]\)}], "Input"],

Cell["\<\
We will find the tangent plane at the point (2, -2, f[2,-1]).  \
First we need to find the partial derivatives.\
\>", "Text"],

Cell[BoxData[{
    \(Clear[fx, fy]\), "\[IndentingNewLine]", 
    \(fx[x_, y_]\  = \ D[f[x, y], x]\), "\[IndentingNewLine]", 
    \(fy[x_, y_]\  = \ D[f[x, y], y]\)}], "Input"],

Cell["Now we can define and plot the tangent plane.", "Text"],

Cell[BoxData[{
    \(Clear[tangent, tanPlane]\), "\[IndentingNewLine]", 
    \(tangent[x_, y_]\  = \ 
      f[2, \(-2\)]\  + \ fx[2, \(-2\)] \((x - 2)\)\  + \ 
        fy[2, \(-2\)] \((y + 2)\)\), "\[IndentingNewLine]", 
    \(tanPlane\  = \ 
      Plot3D[tangent[x, y], {x, \(-3\), 3}, {y, \(-3\), 3}]\)}], "Input"],

Cell["Let's look at point the surface and the plane together.", "Text"],

Cell[BoxData[
    \(Show[surface, tanPlane]\)], "Input"],

Cell["Here's a view from a different angle.", "Text"],

Cell[BoxData[
    \(Show[surface, tanPlane, 
      ViewPoint \[Rule] {\(-3\), \(-3\), 0}]\)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
Max/Min and Saddle Points and their Horizontal Tangent \
Planes.\
\>", "Subsection"],

Cell[CellGroupData[{

Cell["First Example", "Subsubsection"],

Cell["\<\
Let's look at an example of a function that clearly has a \
maximum.\
\>", "Text"],

Cell[BoxData[{
    \(Clear[f, x, y, surface]\), "\[IndentingNewLine]", 
    \(f[x_, y_]\  = \ 9\  - \ x^2\  - y^2\), "\[IndentingNewLine]", 
    \(surface\  = \ 
      Plot3D[f[x, y], {x, \(-2\), 2}, {y, \(-2\), 2}]\)}], "Input"],

Cell["\<\
This function has a maximum at (0,0).  Let's find the tangent plane \
at that point.\
\>", "Text"],

Cell[BoxData[{
    \(Clear[fx, fy, tangent, tanPlane]\), "\[IndentingNewLine]", 
    \(Clear[tangent, tanPlane]\), "\[IndentingNewLine]", 
    \(fx[x_, y_]\  = \ D[f[x, y], x]\), "\[IndentingNewLine]", 
    \(fy[x_, y_]\  = \ D[f[x, y], y]\), "\n", 
    \(tangent[x_, y_]\  = \ 
      f[0, 0]\  + \ fx[0, 0] \((x - 0)\)\  + \ fy[0, 0] \((y - 0)\)\), "\n", 
    \(tanPlane\  = \ 
      Plot3D[tangent[x, y], {x, \(-2\), 2}, {y, \(-2\), 
          2}]\[IndentingNewLine]\)}], "Input"],

Cell["\<\
Now we can look at the suface and the tangent plane at the same \
time.\
\>", "Text"],

Cell[BoxData[
    \(Show[surface, tanPlane]\)], "Input"],

Cell["\<\
Notice that since the tangent plane is horizontal its equation is \
just a constant.\
\>", "Text"],

Cell[BoxData[
    \(tangent[x, y]\)], "Input"],

Cell[TextData[{
  "This means that the partial derivatives must be 0, otherwise the plane \
would not be a constant.  That means that the gradient of ",
  Cell[BoxData[
      \(TraditionalForm\`f\)]],
  " must be the zero vector.  Let's check"
}], "Text"],

Cell[BoxData[
    \(gradf[x_, y_]\  = \ {fx[x, y], fy[x, y]}\)], "Input"],

Cell[BoxData[
    \(gradf[0, 0]\)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Second Example", "Subsubsection"],

Cell["We can also look at a saddle point.  Here is an example.", "Text"],

Cell[BoxData[{
    \(Clear[f, x, y, surface]\), "\[IndentingNewLine]", 
    \(f[x_, y_]\  = \ x^2 - y^2\), "\[IndentingNewLine]", 
    \(surface\  = \ 
      Plot3D[f[x, y], {x, \(-2\), 2}, {y, \(-2\), 2}]\)}], "Input"],

Cell["Let's find and plot the tangent plane at (0,0).", "Text"],

Cell[BoxData[{
    \(Clear[fx, fy, tangent, tanPlane]\), "\[IndentingNewLine]", 
    \(Clear[tangent, tanPlane]\), "\[IndentingNewLine]", 
    \(fx[x_, y_]\  = \ D[f[x, y], x]\), "\[IndentingNewLine]", 
    \(fy[x_, y_]\  = \ D[f[x, y], y]\), "\n", 
    \(tangent[x_, y_]\  = \ 
      f[0, 0]\  + \ fx[0, 0] \((x - 0)\)\  + \ fy[0, 0] \((y - 0)\)\), "\n", 
    \(tanPlane\  = \ 
      Plot3D[tangent[x, y], {x, \(-2\), 2}, {y, \(-2\), 2}]\)}], "Input"],

Cell["Let's look at both the surface and the plane.", "Text"],

Cell[BoxData[
    \(Show[surface, tanPlane]\)], "Input"],

Cell[TextData[{
  "In this case we also have a horizontal tangent plane given by ",
  Cell[BoxData[
      \(TraditionalForm\`z\  = \ k\)]],
  " (here ",
  Cell[BoxData[
      \(TraditionalForm\`k\  = \ 0\)]],
  ").  Again, this means that the partial derivatives must be 0 and that \
means that the gradient of ",
  Cell[BoxData[
      \(TraditionalForm\`f\)]],
  " mst be the zero vector. Let's check."
}], "Text"],

Cell[BoxData[
    \(tangent[x, y]\)], "Input"],

Cell[BoxData[
    \(gradf[x_, y_]\  = \ {fx[x, y], fy[x, y]}\)], "Input"],

Cell[BoxData[
    \(gradf[0, 0]\)], "Input"],

Cell["\<\
We could continue by looking at a surface with a minimum, \
but....well, you get the point by now.  Surfaces that have maximums, minimums \
or saddle points have horizontal (level) tangent planes.  A point, \
{a,b,f[a,g]}, with a horizontal tangent plane corresponds to a point,{a,b}, \
where the gradient of the function is {0,0}.\
\>", "Text"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Finding Critical Points and Testing with the Discriminant", "Subsection"],

Cell["Let's start with an example.", "Text"],

Cell[BoxData[{
    \(Clear[f, x, y]\), "\[IndentingNewLine]", 
    \(f[x_, y_]\  = \((4  x\  - \ x^2)\) \((4  y - y^2)\)\)}], "Input"],

Cell["Here's a plot of this function.", "Text"],

Cell[BoxData[
    \(Plot3D[f[x, y], {x, \(-5\), 5}, {y, \(-5\), 5}]\)], "Input"],

Cell["\<\
To find the critical points we find where the gradient is \
{0,0}.\
\>", "Text"],

Cell[BoxData[
    \(gradf[x_, y_]\  = {D[f[x, y], x], D[f[x, y], y]}\)], "Input"],

Cell[BoxData[
    \(Solve[gradf[x, y] \[Equal] {0, 0}, {x, y}]\)], "Input"],

Cell["\<\
To test the critical points for maxima, minima, and saddle points \
we need to define the discriminant and that requires that we have a few first \
and second order partial derivatives.\
\>", "Text"],

Cell[BoxData[{
    \(fx[x_, y_]\ \  = \ D[f[x, y], x]\), "\[IndentingNewLine]", 
    \(fy[x_, y_]\ \  = \ D[f[x, y], y]\), "\[IndentingNewLine]", 
    \(fxx[x_, y_]\ \  = \ D[f[x, y], x, x]\), "\[IndentingNewLine]", 
    \(fyy[x_, y_]\ \  = \ D[f[x, y], y, y]\), "\[IndentingNewLine]", 
    \(fxy[x_, y_]\ \  = \ D[f[x, y], x, y]\), "\[IndentingNewLine]", 
    \(discriminant[x_, y_]\  = \ 
      fxx[x, y] fyy[x, y] - \((fxy[x, y])\)^2\)}], "Input"],

Cell["Now we can test our critical points.", "Text"],

Cell[BoxData[
    \(discriminant[0, 0]\)], "Input"],

Cell["This is a saddle point.", "Text"],

Cell[BoxData[
    \(discriminant[0, 4]\)], "Input"],

Cell["This is a saddle point.", "Text"],

Cell[BoxData[
    \(discriminant[4, 0]\)], "Input"],

Cell["This is a saddle point.", "Text"],

Cell[BoxData[
    \(discriminant[4, 4]\)], "Input"],

Cell["This is a saddle point.", "Text"],

Cell[BoxData[
    \(discriminant[2, 2]\)], "Input"],

Cell["\<\
Ah! We have finally located a maximum or minimum.  We'll use the \
second partial with respect to x to determine which.\
\>", "Text"],

Cell[BoxData[
    \(fxx[2, 2]\)], "Input"],

Cell["\<\
Since the second partial with respect to x is negative the surface \
is concave down in the x direction and this indicates that the critical \
points is a maximum.\
\>", "Text"],

Cell["\<\
Another way to determine whether we have a maximum, minimum or \
saddle point is to graph the function in a small neighborhood of the point.   \
 First, we look at the point (4,4).\
\>", "Text"],

Cell[BoxData[
    \(Plot3D[f[x, y], {x, 3.9, 4.1}, {y, 3.9, 4.1}]\)], "Input"],

Cell["\<\
Yep, that's a saddle point.  Now we look closely at the point at \
(2,2).\
\>", "Text"],

Cell[BoxData[
    \(Plot3D[f[x, y], {x, 1.9, 2.1}, {y, 1.9, 2.1}]\)], "Input"],

Cell["Clearly, we have a maximum at (2,2).", "Text"],

Cell[TextData[{
  "Sometimes, we are able to find a portion of the surface which clearly \
depicts the behavior.  Check out the following.  We are able to avoid some of \
the very large values of ",
  Cell[BoxData[
      \(TraditionalForm\`f\)]],
  " for negative values of ",
  Cell[BoxData[
      \(TraditionalForm\`x\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`y\)]],
  ".  As a result we can see more detail of the surface and the maximum and \
saddle points become much more visible."
}], "Text"],

Cell[BoxData[
    \(Plot3D[f[x, y], {x, \(-1\), 5}, {y, \(-1\), 5}]\)], "Input"],

Cell["\<\
Another way to examine our surface for critical points is to look \
at a contour map.  Here is one for our function.\
\>", "Text"],

Cell[BoxData[
    \(ContourPlot[f[x, y], {x, \(-1\), 5}, {y, \(-1\), 5}, 
      Contours \[Rule] 30]\)], "Input"],

Cell["\<\
We can see that we have saddle points at (0,0), (4,0),(0,4) and \
(4,4) and a maximum at (0,0).  The lighter shading at (0,0) shows that we \
have a maximum there and not a minimum.  The darker shading indicates a lower \
position on the surface.\
\>", "Text"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Problems for Notebook 11 - MaxMin Problems", "Section"],

Cell["\<\
Name:
Name:
Name:\
\>", "Text"],

Cell[TextData[{
  "You may want to cut and paste the following definitions to save yourself \
some typing.  Of course, you will need to modify the definition of the \
function ",
  Cell[BoxData[
      \(TraditionalForm\`f\)]],
  " depending on which problem you are working."
}], "Text"],

Cell[BoxData[{
    \(Clear[f, x, y, fx, fy, fxx, fyy, fxy, gradf, 
      discriminant]\), "\[IndentingNewLine]", 
    \(f[x_, y_]\  = \ 
      x^4\  + y^4\  - \ 2  x\ y\  + \ 3\), "\[IndentingNewLine]", 
    \(fx[x_, y_]\ \  = \ D[f[x, y], x]\), "\n", 
    \(fy[x_, y_]\ \  = \ D[f[x, y], y]\), "\n", 
    \(fxx[x_, y_]\ \  = \ D[f[x, y], x, x]\), "\n", 
    \(fyy[x_, y_]\ \  = \ D[f[x, y], y, y]\), "\n", 
    \(fxy[x_, y_]\ \  = \ D[f[x, y], x, y]\), "\[IndentingNewLine]", 
    \(gradf[x_, y_]\  = \ {fx[x, y], fy[x, y]}\), "\n", 
    \(discriminant[x_, y_]\  = \ 
      fxx[x, y] fyy[x, y] - \((fxy[x, y])\)^2\)}], "Input"],

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Cell["Problem 1", "Subsubsection"],

Cell[TextData[{
  "For the function ",
  Cell[BoxData[
      \(TraditionalForm\`f[x, y]\  = \ 
        x\^4\  + \ y\^4\  - \ 2  xy\  + \ 3\)]],
  " find the critical points and use the discriminant to test them for max, \
min., and saddle points.  (You can cut and paste the definitions from above \
here.)"
}], "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Problem 2", "Subsubsection"],

Cell[TextData[{
  "For the function ",
  Cell[BoxData[
      \(TraditionalForm\`f[x, y]\  = \ 
        3  x\^3\  + \ xy\^2\  + \ 4  x\^2\  + \ y\^2\)]],
  " find the critical points and use the discriminant to test them for max, \
min and saddle points."
}], "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Problem 3", "Subsubsection"],

Cell[TextData[{
  "For the function ",
  Cell[BoxData[
      \(TraditionalForm\`f[x, y]\  = \ Cos[x\  + \ y]\)]],
  " use a contour plot to find the critical points in the region ",
  Cell[BoxData[
      \(TraditionalForm\`R\  = \ \([\(-4\), 
            4]\)\[Times]\(\([\(-4\), 4]\)\(.\)\)\)]],
  "  Try to determine the nature of the critical point from the contour plot. \
 Verify your results by \"zooming in\" on the surface near the point."
}], "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Problem 4", "Subsubsection"],

Cell[TextData[{
  "For the function ",
  Cell[BoxData[
      \(TraditionalForm\`f[x . y]\  = \ 
        Sin[x]\  + \ Sin[y]\  + \ Sin[x + y]\)]],
  " use a contour plot to find the critical points in the region ",
  Cell[BoxData[
      \(TraditionalForm\`R\  = \ \([0, 2  \[Pi]]\)\[Times]\([0, 
            2  \[Pi]]\)\)]],
  ".  Try to determine the nature of the critical point from the contour \
plot.  Verify your results by \"zooming in\" on the surface near the point."
}], "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Problem 5", "Subsubsection"],

Cell[TextData[{
  "Show that the function ",
  Cell[BoxData[
      \(TraditionalForm\`f[x, 
          y]\  = \ \((y\^2 - 1)\)\^\(\(2\)\(\ \)\) + \ \((xy\^2 - y - 1)\)\^2\
\)]],
  " has two critical points which are both minimums.  Plot this function so \
that this phenomena can be clearly seen."
}], "Text"]
}, Open  ]]
}, Open  ]]
}, Open  ]]
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*)




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End of Mathematica Notebook file.
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