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Enter the following command to define the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["ListPlotVectorField", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" command we will use in this lab.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["<True], Cell[TextData["Part 1: A Simple Geometric Example"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "We begin by finding the minimum and maximum values of a function on the \ unit circle. That is, we wish to find the extreme values of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["(", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[") given that (", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[") is a point on the circle where ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["2", Evaluatable->False, AspectRatioFixed->True, FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" + ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["2", Evaluatable->False, AspectRatioFixed->True, FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" = 1. Consider the function ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["(", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[") = ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["2", Evaluatable->False, AspectRatioFixed->True, FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[". Recall that minimum values of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["(", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ ") occur at critical points of the function (Lab 5 and yesterday's \ lecture); describe the set of all the critical points of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" (in the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["xy", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["-plane) below.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Describe the intersection of the set of critical points of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" with the unit circle ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["2", Evaluatable->False, AspectRatioFixed->True, FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" + ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["2", Evaluatable->False, AspectRatioFixed->True, FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[ " = 1. Explain why this intersection must contain all points where ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["(", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[") attains its minimum on the circle.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["\n\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Finally, before turning to ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", where does the function ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["(", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[") = ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["2", Evaluatable->False, AspectRatioFixed->True, FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[ " attain its maximum on the unit circle? Explain (in one sentence) how you \ know.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Here is the problem: Checking for critical points found some of the \ extreme values, but not all of them, we need to find all the extreme values \ of a function ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" on a curve ", Evaluatable->False, AspectRatioFixed->True], StyleBox["C", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ ". The next commands plot the unit circle as a parametric curve. 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Then enter the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["ListPlotVectorField", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" command to get a plot of the gradient vectors of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" around the unit circle. ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["gradf[x_,y_] := {your gradient here}"], "Input", AspectRatioFixed->True], Cell[TextData[ "fgradplot = ListPlotVectorField[\n \ Table[{{x[t],y[t]},gradf[x[t],y[t]]},\n {t,0,2Pi,Pi/12}],\n \ HeadScaling->Relative]"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "The command below plots the unit circle and the gradient vectors together, \ but without the coordinate axes. Print a copy of the plot and draw in the \ points on the circle where ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " attains its maximum value. On the bottom out the printout: Explain the \ relationship between the unit circle and the gradient vectors at these \ points. What would the rate of change of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " be as we leave one of these points in a direction tangential to the \ circle? (Think about the directional derivative.) Explain why the statement \ \"This is a critical point of the function ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" restricted to the circle.\" is reasonable.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Show[fgradplot, unitcirc]"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "We have drawn the unit circle parametrically, but it is also a level curve \ of a function. There are many functions that have the unit circle as a level \ curve, but one of the simplest is\n", Evaluatable->False, AspectRatioFixed->True], StyleBox["g", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["(", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[") = (", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["2", Evaluatable->False, AspectRatioFixed->True, FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" + ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["2", Evaluatable->False, AspectRatioFixed->True, FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[")/2. Compute the gradient of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["g", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ ", type it into the first command below, and enter the command. Enter the \ next command to get a plot of the gradient vectors of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["r", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " around the unit circle. The last command plots the unit circle and \ gradient vectors together. What basic relationship do the level curves of a \ function and the gradient vectors of that function have? Does your plot \ demonstrate this relationship?", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["\n\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["gradg[x_,y_] := {your gradient here}"], "Input", AspectRatioFixed->True], Cell[TextData[ "ggradplot = ListPlotVectorField[\n \ Table[{{x[t],y[t]},gradr[x[t],y[t]]},\n {t,0,2Pi,Pi/12}],\n \ HeadScaling->Relative]"], "Input", AspectRatioFixed->True], Cell[TextData["Show[ggradplot,unitcirc]"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "From your answers to to the questions above, it should be clear that the \ maximum values of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" on the unit circle are attained when ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Del]", Evaluatable->False, AspectRatioFixed->True], StyleBox["f ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Del]", Evaluatable->False, AspectRatioFixed->True], StyleBox["g", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " point in the same direction (or in opposite directions). Use a ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Show", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " command to create a plot of the unit circle with gradient vectors of both \ functions. Print a copy and circle the points where the gradients ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Del]", Evaluatable->False, AspectRatioFixed->True], StyleBox["f ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Del]", Evaluatable->False, AspectRatioFixed->True], StyleBox["g", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " point in the same direction. Are these the same points you circled on \ your first printout?", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Part 2: The Multiplier in the Method"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Two vectors point in the same direction when one is a nonzero scalar \ multiple of the other. That is, ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Del]", Evaluatable->False, AspectRatioFixed->True], StyleBox["f ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Del]", Evaluatable->False, AspectRatioFixed->True], StyleBox["g", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" point in the same direction when there is a nonzero number ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Lambda]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" so that ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Del]", Evaluatable->False, AspectRatioFixed->True], StyleBox["f ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Lambda]\[Del]", Evaluatable->False, AspectRatioFixed->True], StyleBox["g", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[". The number ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Lambda]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" is called the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Lagrange multiplier", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" (", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Lambda]", Evaluatable->False, AspectRatioFixed->True], StyleBox[ " is the Greek letter \"l\"). Below explain (in a sentence or two) why, if \ we are hunting for places where ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" takes on its extreme values, we should look at places where ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Del]", Evaluatable->False, AspectRatioFixed->True], StyleBox["f ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = 0", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Del]", Evaluatable->False, AspectRatioFixed->True], StyleBox["g", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" as well. That is, ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Lambda]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" = 0 still gives a place where ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" might take on an extreme value.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[" \n \n "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Let's try to tackle a problem where we don't already know the answer. We \ will find the extreme values of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["F ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["(", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[") = ", Evaluatable->False, AspectRatioFixed->True], StyleBox["xy", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" on the circle where ", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["2", Evaluatable->False, AspectRatioFixed->True, FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" + ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["2", Evaluatable->False, AspectRatioFixed->True, FontSize->9, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" = 2. This circle is also a level curve of the function ", Evaluatable->False, AspectRatioFixed->True], StyleBox["g", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " introduced in Part 1, so we can use it again here. Our goal is to find \ all the points where ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Del]", Evaluatable->False, AspectRatioFixed->True], StyleBox["F ", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Lambda]\[Del]", Evaluatable->False, AspectRatioFixed->True], StyleBox["g", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" for some number ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Lambda]", Evaluatable->False, AspectRatioFixed->True], StyleBox[". This is really shorthand for two equations: ", Evaluatable->False, AspectRatioFixed->True], StyleBox["F", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSize->9, FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" = ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Lambda]", Evaluatable->False, AspectRatioFixed->True], StyleBox["g", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSize->9, FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["F", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSize->9, FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" = ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Lambda]", Evaluatable->False, AspectRatioFixed->True], StyleBox["g", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSize->9, FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[". We begin by defining ", Evaluatable->False, AspectRatioFixed->True], StyleBox["F", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" and its partial derivatives. Enter the commands below.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["F[x_,y_] := x y"], "Input", AspectRatioFixed->True], Cell[TextData["Fx[x_,y_] = D[F[x,y],x]"], "Input", AspectRatioFixed->True], Cell[TextData["Fy[x_,y_] = D[F[x,y],y]"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Using ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Del]", Evaluatable->False, AspectRatioFixed->True], StyleBox["g", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["(", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[") = <", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ "> (you computed this in Part 1), this gives us the first two equations in \ the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Solve", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " command below. The third equation makes sure we stay on the correct \ level curve of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["g", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[". Enter this command to request that ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" find the solutions", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Solve[{Fx[x,y] == L x, Fy[x,y] == L y,\n x^2 + y^2 == 2},{x,y,L}]"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Solve this system of three equations by hand in the space below; make sure \ that ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " has found all possible solutions, and has not given any \"extraneous \ solutions\" (incorrect answers).", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["\n\n\n\n\n\n\n\n\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "In order to get a feeling for which points will give the maximum value \ and which will give the minimum, let's examine a contour plot of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["F", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " superimposed on the circle. The following three commands will generate \ this plot.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "lcF = ContourPlot[F[x,y],{x,-2,2},{y,-2,2},\n Contours->20]"], "Input", AspectRatioFixed->True], Cell[TextData[ "newcirc = ParametricPlot[{Sqrt[2]Cos[t],Sqrt[2]Sin[t]},\n \ {t, 0, 2Pi},AspectRatio->Automatic]"], "Input", AspectRatioFixed->True], Cell[TextData["Show[lcF, newcirc]"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Print a copy of this plot and circle the points that correspond to \ solutions of the multiplier problem. On the bottom of the sheet, explain \ why the level curves of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["F", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " are tangent to the circle at these points. 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