Initialization Cells
Close (don't delete) this Cell grouping before Printing

Needs["Graphics`"]
Needs["Calculus`"]
Needs["DiscreteMath`"]

LabelLine[InLine_Line,  InLabel_] := Module[
{Data},
Data  =  
              InLine[[1]];
Graphics[ {
InLine, 
Text[
StringForm["\!\(
\*FrameBox[\(\(\\ \)\(``\)\(\\ \)\),
BoxMargins->{{0.2, 0.2}, {0.4, 0.4}}]\)", InLabel], 
Data[[ 
                        Floor[Length[Data]/2]]],
Background -> White]
}
]
]

LabeledContourPlot[
f_, 
{x_, 
          xMin_, xMax_}, 
{y_, yMin_, 
          yMax_}, 
ContourValues_List, 
options___
] :=  
      Module[
{i, 
          TempLines}, 
Show[
Table[
TempLines  = 
              Cases[
Graphics[
ContourPlot[f, {x, xMin, xMax}, {y, yMin, 
                      yMax}, 
Contours -> {ContourValues[[
                          i]]}, 
DisplayFunction -> 
                      Identity, 
ContourShading -> 
                      False]], 
_Line,  ∞]; 
Map[
              LabelLine[#, 
                  ContourValues[[i]]] &, 
              TempLines], 
{i,  
              Length[ContourValues]}], 
DisplayFunction -> $DisplayFunction, 
options
]
]

This input cell automatically loads two "Add-On" packages.  It is called an initialization cell.  Look in the help or under the Cell menu for further information.

({{Name,                                }, {Class, 3160}, {Exam, One}, {Section #,  }, {Instructor, Struthers}})

({{Question, Score, Total}, {1 ,  ,  10}, {2,  ,  10}, {3 ,  ,  10}, { 4,  ,  10}, { 5,  ,  10}, { 6,  ,  10}, { 7,  ,  10}, { 8,  ,  10}, { 9,  ,  10}, { 10,  , 10}, { Total,  , 100 }})

Identify the graphs below.         (({{f(x, y), "Graph"}, {-(Sqrt[(x^2) + (y^2)]), "_____"}, {cos(x), "_____"}, {-(x^2) - (y^2), "_____"}, {-(e^(-(x^2) - (y^2))) x y, "_____"}, {((sin((x^2) + (y^2)))/((x^2) + (y^2))), "_____"}, {(1/((x^2) + (y^2))), "_____"}}))

SetOptions[
          Plot3D,  
DisplayFunction -> 
            Identity, 
Shading -> 
            False, 
Ticks -> 
            False, 
ClipFill -> 
            None
];

f1[x_, y_] = -x*y*
          Exp[-(x^2 + y^2)];
f2[x_, y_] = 
        Cos[x];
f3[x_, 
          y_] = (x^2 + y^2)^(-1);
f4[
          x_, y_] = 
        Sin[(x^2 + y^2)]/(x^2 + y^2);
f5[
          x_, y_] = -(x^2 + y^2)^(1/
                2);
f6[x_, 
          y_] = -(x^2 + y^2);
Pic1  = 
        Plot3D[f1[x, y], {x, -2, 2}, {y, -2, 
            2}, 
PlotLabel -> "A"];
Pic2  = 
        Plot3D[f2[x, y], {x, -2 Pi, 2  Pi}, {y, -2 Pi, 
            2  Pi}, 
PlotLabel -> "B"];
\(Pic3\  = Plot3D[
          f3[x, y], {x, \(-1\), 1}, {y, \(-1\), 
            1}, 
PlotLabel -> "C"];\)
Pic4 = 
        Plot3D[f4[x, y], {x, -Pi, Pi}, {y, -Pi, 
            Pi}, 
PlotLabel -> "D"];
Pic5 = 
        Plot3D[f5[x, y], {x, -1, 1}, {y, -1, 
            1}, 
PlotLabel -> "E"];
Pic6 = 
          Plot3D[f6[x, y], {x, -1, 1}, {y, -1, 
              1}, 
PlotLabel -> "F"];

Show[
GraphicsArray[
{{Pic1, Pic2, 
              Pic3}, {Pic4, Pic5, 
              Pic6}}, 
GraphicsSpacing -> 
            0], 
Frame -> 
          True];
({{f[x, y], "Graph"}, {f5[x, y], "_____"}, {f2[x, y], "_____"}, {f6[x, y], "_____"}, {f1[x, y], "_____"}, {f4[x, y], "_____"}, {f3[x, y], "_____"}})

Match the contour and 3D plots.  
({{3  DPlot, A, B, C, D, E, F}, {ContourPlot,      ,      ,      ,      ,      ,      }})
Is dark high or low?

SetOptions[
        Plot3D,  
DisplayFunction -> 
          Identity, 
Shading -> 
          False, 
Ticks -> 
          False, 
ClipFill -> 
          None
];

f1[x_, y_] = -x*y*
          Exp[Abs[x] + Abs[y]];
f2[x_, y_] = Sin[0.1 x y];
f3[x_, y_] = (x^2 + y^2);
f4[x_, y_] = Sin[0.5 (x^2 + y^2)];
f5[x_, y_] = -x y ;
f6[x_, y_] = -(x^2 - y^2);
Pic1  = 
        Plot3D[f1[x, y], {x, -2, 2}, {y, -2, 
            2}, 
PlotLabel -> "A"];
Pic2  = 
        Plot3D[f2[x, y], {x, -2 Pi, 2  Pi}, {y, -2 Pi, 
            2  Pi}, 
PlotLabel -> "B"];
Pic3  = 
        Plot3D[f3[x, y], {x, -1, 1}, {y, -1, 
            1}, 
PlotLabel -> "C"];
Pic4 = 
        Plot3D[f4[x, y], {x, -Pi, Pi}, {y, -Pi, 
            Pi}, 
PlotLabel -> "D"];
Pic5 = 
        Plot3D[f5[x, y], {x, -1, 1}, {y, -1, 
            1}, 
PlotLabel -> "E"];
\(\(Pic6 = 
        Plot3D[f6[x, y], {x, \(-1\), 1}, {y, \(-1\), 
            1}, 
PlotLabel -> "F"];\)

\)
\(\({CPic1, \ CPic2, \ CPic3, CPic4, CPic5, CPic6}\  = \ 
        MapThread[\ 
Show[
              ContourGraphics[#1], 
PlotLabel -> #2, 
FrameTicks -> 
                None] &, 
{
Permute[{Pic1, \ Pic2, \ Pic2, Pic4, 
                Pic5, Pic6}, 
              RandomPermutation[
                6]], 
{"1", "2", "3", "4", "5", "6"}
}
];\)
\)
Show[
GraphicsArray[
{{Pic1, 
              Pic2, Pic3}, {Pic4, Pic5, 
              Pic6}}, 
GraphicsSpacing -> 
            0], 
Frame -> 
          True];
Show[
GraphicsArray[
{{CPic1, 
              CPic2, CPic3}, {CPic4, CPic5, 
              CPic6}}, 
GraphicsSpacing -> {0, 
              0.1}], 
Frame -> True];

Let A = (2, -1, -1), B = (3, 1, 0),and C = (1, 2, 1).
      (a) Find an equation for the plane that passes through these three points.
      (b) Find the area of the triangle determined by these three points.

Given 
	 (Overscript[a, →]) = 
        2  (Overscript[i, →]) - 3  (Overscript[j, →]) + 4  (Overscript[k, →]) and (Overscript[b, →]) = 
        1  (Overscript[i, →]) + 2  (Overscript[j, →]) + 3  (Overscript[k, →])  
write (Overscript[a, ⇀])  as a sum of two vectors (Subscript[(Overscript[a, →]), par]) and (Subscript[(Overscript[a, →]), perp ]) with
	 (Subscript[(Overscript[a, →]), par]) parallel to (Overscript[b, →])
and
	(Subscript[(Overscript[a, →]), perp]) perpendicular to (Overscript[b, ⇀]).

For f(x, y) = x (tan^(-1))(x  + y) compute (Subscript[f, x])

Let f(x, y) = 15  x  +  13 (x^3) + 1  x (y^2)

a) Compute enough values (to at least 3 decimal places) in the following table to estimate 
	(Subscript[f, x])(1, 2) and (Subscript[f, y])(1, 2).
	
	({{\(y\x\), x = 0.95, x = 1.00, x = 1.05}, {y = 1.99,  ,  ,  }, {y = 2.00,  ,  ,  }, {y = 2.01,  ,  ,  }})
	
Show your estimates for the two partial derivatives to 3 decimal places.

b) Estimate (Subscript[f, x])(1, 2) and (Subscript[f, y])(1, 2) using the contour diagram below.  Show where all the computations come from.

c) Compute exact values for (Subscript[f, x])(1, 2) and (Subscript[f, y])(1, 2) .

f[x_, y_] = 
        15  x  +  13 (x^3) + 1  x (y^2);
Hmm  =  
        LabeledContourPlot[
f[x, y], {x, 0, 2}, {y, 0, 
            3}, 
{30, 32, 
            34}, 
Frame -> 
            True, 
Epilog -> {PointSize[0.05], 
              Point[{1, 2}]}
];

Complete the table describing the signs of the following partial derivatives at the point P on the contour plot below.

(∂_(x, y))f[x, y]

f[x_, y_] = -(x^3)  + 3  x y ;
Hmm  =  
        LabeledContourPlot[
f[x, y], {x, 0, 2}, {y, 0, 
            3}, 
{2, 4, 6,  
            8}, 
Frame -> 
            True, 
Epilog -> {PointSize[0.05], 
              Point[{1, 2}]}
];

Hmm = 
      Cases[
Graphics[
ContourPlot[ 
            f[x, y], {x, 0, 2}, {y, 0, 
              3}, 
Contours -> {2}
]
], 
_Line, 
∞]

Show[
        LabelLine[Hmm[[1]], "AA"]] // 
      FullForm

A) Find the second order  Taylor polynomial centered at (0,0) for the function 
	f (x, y) = (Sqrt[x + 2  y + 1])

B) Compute the values of this quadratic approximation, the linear tangent plane approximation and the function at the point (-0.01, 0.1).

C) Compare the accuracy of the linear and quadratic approximations.

Describe the level surfaces of the function 
	f(x, y, z) = (x^2) - (y^2) + (z^2)
in words.

Find the local linearization of the function 
	f(x, y) = ln(x - 3  y) 
at (7, 2) and use it to approximate f(6.9,  2.06).

For the function 
	f(x, y) = 2  x (y^2) + y
a) Compute ∇f.
b) Compute the maximum rate of change in (1, -1)?
c) Find  the directional derivative (Overscript[(Subscript[f, u]), ⇀])(-5, 1) where  (Overscript[u, ⇀]) = 
        8  (Overscript[i, ⇀]) + 6  (Overscript[j, ⇀]).

Estimate the equation for the tangent plane from the contour diagram below.

Estimate the equation for the tangent plane from the contour diagram below.

Find the equation of the linear function
	z = c+mx+ny 
whose graph intersects the xz-plane in the line
	z = 3  x + 4 
and intersects the yz-plane in the line 
	z = y + 4.

A boat heads due the east at 25 km/hr(relative to the water) while a current flows southwest at 10km/hr.

A) Write down the velocity vector (relative to the ground) of the boat.
B) Write down the speed of the boat relative to the ground.
C) What angle (in radians) does the velocity vector make with the vector pointing due east?