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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 188147, 5310]*) (*NotebookOutlinePosition[ 188845, 5334]*) (* CellTagsIndexPosition[ 188801, 5330]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ StyleBox["Lab 1B - Graphing Functions of Two Variables and Quadric \ Surfaces", FontSize->18, FontWeight->"Bold", FontVariations->{"Underline"->True}], "\nMath 2374 - University of Minnesota\nhttp://www.math.umn.edu/math2374\n\ Questions to: rogness@math.umn.edu" }], "Text", CellFrame->True, TextAlignment->Center, FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Introduction", FontSize->16, FontWeight->"Bold"]], "Section"], Cell["\<\ Remember to evaluate the initialization cells in math2374.nb before \ continuing on with this lab.\ \>", "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell[TextData[{ "At the end of Lab 1A you learned how to use ", StyleBox["Mathematica", FontSlant->"Italic"], " to plot the graph of equations such as ", Cell[BoxData[ \(TraditionalForm\`y = \((x - 1)\)\^2\)]], " or ", Cell[BoxData[ \(TraditionalForm\`x\^2 + y\^2 = 1\)]], ". As you hopefully remember, we have to use different commands to plot \ the graphs of these two equations. In the first case, we have ", StyleBox["explicitly", FontSlant->"Italic"], " solved for y as a function of x; there is a single y on the left hand \ side of the equation, and no occurrences of y on the right hand side. In \ cases like this we can use the ", StyleBox["Plot", FontWeight->"Bold"], " function to show a graph of y. In the second case we have an ", StyleBox["implicit", FontSlant->"Italic"], " function of y. We can't solve explicitly for y because we end up with \ \[PlusMinus]", Cell[BoxData[ \(TraditionalForm\`\@\(1 - x\^2\)\)]], "on the right hand side. (This is not a well defined function because for \ a given value of x we can only have one value, not a positive ", StyleBox["and", FontSlant->"Italic"], " a negative value.) We learned how to use the command ", StyleBox["ImplicitPlot", FontWeight->"Bold"], " to handle equations like this.\n\nIn this lab we're going to work with \ the three-dimensional analogs of these commands and, as you might expect, \ we'll have to consider two different cases. The first is when we have a \ function of x and y which is explicitly solved for z, e.g.\n\n", Cell[BoxData[ \(TraditionalForm\`z = x\^2 + \(\(y\^2\)\(.\)\)\)]], "\n\nWe'll also consider implicit functions of z, such as\n\n", Cell[BoxData[ \(TraditionalForm\`x\^2 + y\^2 + z\^\(\(2\)\(\ \)\) = 1. \)]], "\n\n(Note that if you tried to solve this last equation for z, you'd have \ the same problem with a \[PlusMinus] sign.)\n\nThe first case is considerably \ easier, and we'll deal with that one first. One other comment before we move \ on: some of the equations in this lab might look a bit small on your screen. \ If you're having trouble reading them, try the menu option Format : Screen \ Style Environment : Presentation." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Plot3D", FontSize->16]], "Section"], Cell["Suppose we define a function z=f(x,y) of two variables, e.g.", "Text"], Cell[BoxData[ \(f[x_, y_] = x^2\ + \ y^2\)], "Input"], Cell[TextData[{ "What does a graph of this function even ", StyleBox["mean", FontSlant->"Italic"], "? The definition usually given is \"the set of all points of the form ", StyleBox["(x, y, f(x,y))", FontSlant->"Italic"], " such that ", StyleBox["(x,y)", FontSlant->"Italic"], " is in the domain of ", StyleBox["f", FontSlant->"Italic"], ",\" but you may not find this particularly enlightening.\n\nNotice that \ the function f takes two inputs, x and y, and returns a single number, which \ we call z. If we draw the x-y-z coordinate axes in the standard way, the \ z-axis represents height, and this is the key to graphing ", StyleBox["f(x,y)", FontSlant->"Italic"], ". If you choose a point ", StyleBox["(x,y)", FontSlant->"Italic"], " in the xy-plane, then ", StyleBox["z=f(x,y)", FontSlant->"Italic"], " represents the height of the graph at that point. For example, here's \ the graph of a simple function, g(x,y)=1. This means that no matter what \ values you choose for x and y, the function g will always return (\"a height \ of\") one." }], "Text"], Cell[BoxData[{ \(g[x_, y_] = 1\), "\n", \(Plot3D[g[x, y], {x, \(-1\), 1}, {y, \(-1\), 1}]\)}], "Input"], Cell[TextData[{ "As you can see, we're using the command ", StyleBox["Plot3D", FontWeight->"Bold"], " to create this graph. The syntax of ", StyleBox["Plot3D", FontWeight->"Bold"], " is very similar to that of ", StyleBox["Plot", FontWeight->"Bold"], "; you first give it a function of x and y, and then ranges for x and y.\n\n\ Here's a slightly more complicated function. Before you evaluate this cell, \ see if you can predict what the graph will look like." }], "Text"], Cell[BoxData[{ \(g[x_, y_] = x + y\), "\n", \(Plot3D[g[x, y], {x, \(-1\), 1}, {y, \(-1\), 1}]\)}], "Input"], Cell[TextData[{ "This graph does not have a constant height of one, but if you look at the \ definition of g(x,y) you should be able to make some observations:\n\n\ \[FilledVerySmallSquare] if ", Cell[BoxData[ \(TraditionalForm\`y = \(-x\)\)]], ", then g(x,y)=0.\n\[FilledVerySmallSquare] if x and y are both positive, \ then ", Cell[BoxData[ \(TraditionalForm\`g(x, y) \[Succeeds] 0\)]], ".\n\[FilledVerySmallSquare] if x and y are both negative, then ", Cell[BoxData[ \(TraditionalForm\`g(x, y) \[Precedes] 0\)]], ".\n\nLooking back at the picture, are these things true? One thing you \ have to notice is that the z-axis goes from -2 to 2, which means the \"height\ \" is zero halfway up the box, not at the bottom. Another thing worth \ pointing out is that the z-axis is scaled differently than the x- and y-axes. \ If you want to change the scaling, you can use the option ", StyleBox["BoxRatios", FontWeight->"Bold"], ", which you can look up in the Help Browser", ":" }], "Text"], Cell[BoxData[ \(Plot3D[g[x, y], {x, \(-1\), 1}, {y, \(-1\), 1}, BoxRatios \[Rule] {1, 1, 2}]\)], "Input"], Cell[TextData[{ "Let's plot the function ", Cell[BoxData[ \(TraditionalForm\`f(x, y) = x\^2 + y\^2\)]], " we considered previously so that we have something slightly more \ interesting to work with. " }], "Text"], Cell[BoxData[ \(Plot3D[x^2 + y^2, \ {x, \(-2\), 2}, {y, \(-2\), 2}]\)], "Input"], Cell[TextData[{ "Another useful option is ", StyleBox["ViewPoint", FontWeight->"Bold"], ". The effect of this option is to pretend you're floating in space at \ some point (x,y,z) -- which you specify -- looking back at the origin. For \ instance, if we want to look at the previous graph from straight above, we \ can set the viewpoint to be (0,0,10):" }], "Text"], Cell[BoxData[ \(Plot3D[x^2 + y^2, \ {x, \(-2\), 2}, {y, \(-2\), 2}, ViewPoint \[Rule] {0, 0, 10}]\)], "Input"], Cell[TextData[{ "The output isn't terribly interesting. (Why?) You should try graphing \ f(x,y) on your own a few times, choosing different viewpoints each time. You \ should also try using the ", StyleBox["BoxRatios", FontWeight->"Bold"], " option and see if it makes a difference.\n\nWe could spend an entire lab \ having you plot the graphs of all sorts of functions. Some of the most \ interesting involve trigonometric functions like ", StyleBox["Sin", FontWeight->"Bold"], ":" }], "Text"], Cell[BoxData[ \(Plot3D[Sin[x*y], {x, \(-Pi\), Pi}, {y, \(-Pi\), Pi}]\)], "Input"], Cell[TextData[{ "This graph moves up and down so much that ", StyleBox["Mathematica", FontSlant->"Italic"], "'s picture is kind of choppy. We can fix this by forcing ", StyleBox["Mathematica", FontSlant->"Italic"], " to compute the height of the graph (i..e. the z-value) at more points \ before it starts to fill in the gridlines. This is done with the ", StyleBox["PlotPoints", FontWeight->"Bold"], " option:" }], "Text"], Cell[BoxData[ \(Plot3D[Sin[x*y], {x, \(-Pi\), Pi}, {y, \(-Pi\), Pi}, \ PlotPoints \[Rule] 40]\)], "Input"], Cell[TextData[{ "If you don't like the gridlines, you can turn them off with the ", StyleBox["Mesh", FontWeight->"Bold"], " option:" }], "Text"], Cell[BoxData[ \(Plot3D[Sin[x*y], {x, \(-Pi\), Pi}, {y, \(-Pi\), Pi}, PlotPoints \[Rule] 40, Mesh \[Rule] False]\)], "Input"], Cell[TextData[{ "Sometimes when you're changing viewpoints it's easy to lose track of which \ axis is the x-axis, and which is the y-axis. You can label them with the ", StyleBox["AxesLabel", FontWeight->"Bold"], " option:" }], "Text"], Cell[BoxData[ \(Plot3D[Sin[x*y], {x, \(-Pi\), Pi}, {y, \(-Pi\), Pi}, PlotPoints \[Rule] 40, Mesh \[Rule] False, \ AxesLabel \[Rule] {"\", "\", "\"}]\)], "Input"], Cell[TextData[{ "Or, if you don't want to have the axes numbered, you can use the ", StyleBox["Axes", FontWeight->"Bold"], " option. (If you want the box to disappear entirely, try adding ", StyleBox["Boxed\[Rule]False", FontWeight->"Bold"], " to this command.)" }], "Text"], Cell[BoxData[ \(Plot3D[Sin[x*y], {x, \(-Pi\), Pi}, {y, \(-Pi\), Pi}, PlotPoints \[Rule] 40, Mesh \[Rule] False, Axes \[Rule] False]\)], "Input"], Cell[TextData[{ StyleBox["Exercise 1", FontSize->16, FontWeight->"Bold"], "\n\nDownload the \"Addendum to Exercise 1\" for this lab from the course \ website. Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to graph the functions in that document with the pictures below. You \ will make your life easier if you match the x- and y- ranges in the ", StyleBox["Plot3D", FontWeight->"Bold"], " commands to the ranges in the figures. The ", StyleBox["PlotPoints", FontWeight->"Bold"], " and ", StyleBox["BoxRatios", FontWeight->"Bold"], " options might also be useful. Also note that the viewpoints in the \ figures might be different than the default viewpoint in ", StyleBox["Mathematica", FontSlant->"Italic"], ". \n\nIf you name the functions before plotting them, remember to use ", StyleBox["lower-case", FontSlant->"Italic"], " names as discussed last week or you will run into problems.\n\nWhile we'd \ like to you to put some effort into identifying each graph, you will only \ hand in a careful write up one particular match. For example, if you're told \ to write about B(x,y), you should carefully explain why B(x,y) produces the \ graph that it does. Use the definition of the function to explain the shape \ -- why is it high in some areas, low in others? Is it ever equal to zero? \ Is it ever negative, or is it always positive? etc.\n\nYour TA will tell you \ which function has been chosen for you to describe." }], "Text", CellFrame->True, Background->RGBColor[0.996109, 0.500008, 0.500008]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["ContourPlot", FontSize->16]], "Section"], Cell[TextData[{ "Before we work with implicit functions, we're going to introduce one more \ way to examine the height of a surface. If you've ever done any kind of \ hiking outdoors or fishing on a big lake you're probably familiar with \ topographic maps. These maps use so-called \"contour lines\" to represent \ elevation. For instance, on standard United States Geological Survey maps, \ each contour line represents 10 feet of elevation. If you'd like to see an \ example of a topographic map, copy the following link and paste it into a web \ browser to see to a topographic map of Eagle Mountain, the highest mountain \ (well... hill) in Minnesota.\n\n\ http://www.dnr.state.mn.us/maps/tomo.html?mode=recenter&size=3&layer=24k&col=\ 513&row=243\n\nThe contour lines on this map represent elevation above sea \ level. Notice how in some places the lines are very close together, which \ represents a steep slope. In other places the lines are further apart, which \ represents a more gradual slope. Not surprisingly, the steepest slopes seem \ to be very near the summits of Eagle Mountain and Moose Mountain.\n\n", StyleBox["Mathematica", FontSlant->"Italic"], " can draw a topographic map of a surface for us. For example, let's look \ at a map of this function:" }], "Text"], Cell[BoxData[ \(\(f[x_, y_] = x^2\ + \ y^2;\)\)], "Input"], Cell[TextData[{ "To draw the topographic map, or \"contour diagram,\" of the function, we \ use the command ", StyleBox["ContourPlot", FontWeight->"Bold"], ". We have to give the command the function we want to plot, and ranges for \ x and y:" }], "Text"], Cell[BoxData[ \(ContourPlot[x^2 + y^2, {x, \(-2\), 2}, {y, \(-2\), 2}]\)], "Input"], Cell[TextData[{ "Notice that ", StyleBox["Mathematica", FontSlant->"Italic"], " shades the picture according to elevation. The darker regions represent \ the lower points; the lighter shades represent higher points. Scroll back \ and look at the graph of f(x,y) generated by ", StyleBox["Plot3D", FontWeight->"Bold"], " and see if this contour diagram makes sense to you. Notice in particular \ that the contour lines get closer to each other near the edges of the \ diagram. Why is this and what does this mean? (If you're not sure, talk to \ the students next to you and/or your TA before you go on.)" }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["ContourPlot3D", FontSize->16]], "Section"], Cell[TextData[{ "The three-dimensional equivalent of ", StyleBox["ImplicitPlot", FontWeight->"Bold"], ", which we used to graph equations such as ", Cell[BoxData[ \(TraditionalForm\`x\^2 + y\^2 = 1\)]], ", is ", StyleBox["not", FontSlant->"Italic"], " called ", StyleBox["ImplicitPlot3D", FontWeight->"Bold"], " as you might expect. It's actually called ", StyleBox["ContourPlot3D", FontWeight->"Bold"], ", and it's technically the three-dimensional analog of ", StyleBox["ContourPlot", FontWeight->"Bold"], ". (As it turns out, ", StyleBox["ImplicitPlot", FontWeight->"Bold"], " is really a special case of ", StyleBox["ContourPlot", FontWeight->"Bold"], ", but you don't need to understand why. The reason has something to do \ with ", StyleBox["level curves", FontSlant->"Italic"], ", which are defined in chapter 3. The reason ", StyleBox["ContourPlot3D", FontWeight->"Bold"], " is the generalization of ", StyleBox["ImplicitPlot", FontWeight->"Bold"], " has something to do with ", StyleBox["level surfaces", FontSlant->"Italic"], ", which are also defined in chapter 3.)" }], "Text"], Cell[TextData[{ "Suppose we want to graph an implicit function of z like this:\n\n", Cell[BoxData[ \(TraditionalForm\`x\^2 + y\^2 + z\^2 = 1. \)]], "\n\n(Hopefully you recognize that this is the equation for the unit \ sphere.) To graph this function using ", StyleBox["ContourPlot3D", FontWeight->"Bold"], ", you'll have to accept on faith alone that you should move everything to \ the left hand side of the equation, so you have a zero on the right hand \ side:\n\n", Cell[BoxData[ \(TraditionalForm\`x\^2 + y\^2 + z\^2 - 1 = 0\)]], "\n\nThe reasons for doing this aren't really important now, and we can't \ really explain it until we've covered level surfaces later in the course, so \ you'll just have to trust us for now. \n\nNext we pass the expression on the \ ", StyleBox["left hand side", FontSlant->"Italic"], " of this equation to ", StyleBox["ContourPlot3D", FontWeight->"Bold"], ", along with ranges for x, y, ", StyleBox["and", FontSlant->"Italic"], " z!" }], "Text"], Cell[BoxData[ \(ContourPlot3D[ x^2 + y^2 + z^2 - 1, {x, \(-1\), 1}, {y, \(-1\), 1}, {z, \(-1\), 1}]\)], "Input"], Cell[TextData[{ "Notice that we chose the ranges for x, y, and z to go from -1 to 1. These \ ranges have a great influence over the resulting picture. If you make them \ too large, the command will run much faster but the picture will look awful. \ If you make them too small, you won't see your sphere (because you'll \ actually be ", StyleBox["inside", FontSlant->"Italic"], " it). Try changing all of the ranges above to {_,-3,3} and {_,-1/2,1/2} \ to see examples of this.\n\nThe point is this: when you use ", StyleBox["ContourPlot3D", FontWeight->"Bold"], " you should put careful thought into your ranges.\n\nOne other note: you \ can plot any kind of equation with x, y, and z using ", StyleBox["ContourPlot3D", FontWeight->"Bold"], ". It doesn't have to be an implicit function of z like the equation of \ the sphere. For example, to graph this equation from above,\n\nz = ", Cell[BoxData[ \(TraditionalForm\`x\^2 + y\^2\)]], ",\n\nwe first move everything to the left hand side:\n\nz - ", Cell[BoxData[ \(TraditionalForm\`x\^2 + y\^2\)]], " = 0.\n\nThen we use the command:" }], "Text"], Cell[BoxData[ \(ContourPlot3D[ z - x^2 - y^2, {x, \(-1\), 1}, {y, \(-1\), 1}, {z, 0, 2}]\)], "Input"], Cell[TextData[{ "However, if you can use ", StyleBox["Plot3D", FontWeight->"Bold"], ", it generally makes better pictures. Compare the picture from the \ previous command to this one:" }], "Text"], Cell[BoxData[ \(Plot3D[x^2 + y^2, {x, \(-1\), 1}, {y, \(-1\), 1}, BoxRatios \[Rule] {1, 1, 2}]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Interactive 3D Graphics", FontSize->16]], "Section"], Cell[TextData[{ "The commands in this section are defined in math2374.nb, and they depend \ on a certain Java program being located in the correct place. If you're \ trying to use ", StyleBox["Mathematica", FontSlant->"Italic"], " at home, this command may not work. You can look at the course web page \ to find tips about how to make it function correctly." }], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell[TextData[{ "The three-dimensional graphs that ", StyleBox["Mathematica", FontSlant->"Italic"], " can produce are great, but it can be very tedious to adjust the \ ViewPoints if you want to get a different look at your function. In today's \ interactive world, we could argue, we ought to be able to rotate the \ pictures, zoom in and out, and so on. While ", StyleBox["Mathematica", FontSlant->"Italic"], " has never put much effort into these capabilities, we've incorporated a \ ", StyleBox["Mathematica", FontSlant->"Italic"], " add-on called \"LiveGraphics3D\" which will let you rotate things to your \ hearts content.\n\nIt works like this. Suppose you want to look at a certain \ picture, like this one from earlier:" }], "Text"], Cell[BoxData[ \(graph = Plot3D[Sin[x*y], {x, \(-Pi\), Pi}, {y, \(-Pi\), Pi}, PlotPoints \[Rule] 40, \ AxesLabel \[Rule] {"\", "\", "\"}]\)], "Input"], Cell[TextData[{ "Notice that we used a name -- \"graph\" -- for the picture. Now, instead \ of using the ", StyleBox["Show", FontWeight->"Bold"], " command, we'll use a new command called ", StyleBox["ShowLive", FontWeight->"Bold"], ":" }], "Text"], Cell[BoxData[ \(ShowLive[graph]\)], "Input"], Cell[TextData[{ "If all goes well, evaluating the previous cell will make an extra window \ pop up on your monitor. Now you can interact with the picture in this new \ window. Click on it and move your mouse to rotate it. If you hold the shift \ key down, you can click on it and move your mouse up and down to zoom in and \ out. To reset the image, press the \"Home\" key.\n\nYou can basically use \ the ", StyleBox["ShowLive", FontWeight->"Bold"], " command anytime that you would use the ", StyleBox["Show", FontWeight->"Bold"], " command with a 3D picture, although it a few things may behave \ differently. If something goes wrong you'll get a small windows with a \ (possibly incomprehensible) error message. Just close the window and try \ something different.\n\nNote that every time you use the ", StyleBox["ShowLive", FontWeight->"Bold"], " command, you'll create a new little window. Make sure you close them \ when you're done, or you'll end up with a lot of clutter on your virtual \ desktop.\n\nThere's also an interactive version of ", StyleBox["Plot3D", FontWeight->"Bold"], " you can use, called ", StyleBox["Plot3DLive", FontWeight->"Bold"], ", which means you can use LiveGraphics3D to look at a graph without having \ to plot it separately before using ", StyleBox["ShowLive", FontWeight->"Bold"], ":" }], "Text"], Cell[BoxData[ \(Plot3DLive[Cos[x*y], {x, \(-Pi\), Pi}, {y, \(-Pi\), Pi}, PlotPoints \[Rule] 40, \ AxesLabel \[Rule] {"\", "\", "\"}]\)], "Input"], Cell[TextData[{ "Note that using the ", StyleBox["Live", FontWeight->"Bold"], " commands can really slow down some computers, especially if you use a \ high ", StyleBox["PlotPoints", FontWeight->"Bold"], " setting." }], "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell[TextData[{ "The downside to using ", StyleBox["Plot3DLive", FontWeight->"Bold"], " is that the input and output is not sorted next to each other, so you can \ lose track of what command created what picture. Also, if you'd like to \ print out a picture, you'll have to use the regular old ", StyleBox["Plot3D", FontWeight->"Bold"], " command." }], "Text"], Cell[TextData[{ "LiveGraphics3D can do some other really cool things. For example, instead \ of just adjusting your viewpoint and magnification of a surface, we've set up \ some examples where you can interactively ", StyleBox["change", FontSlant->"Italic"], " the picture! You may have already used some of the interactive demos \ posted online. Although it's possible to run these from within ", StyleBox["Mathematica", FontSlant->"Italic"], ", we'll usually ask you to go to a web page instead." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Quadric Surfaces", FontSize->16]], "Section"], Cell[TextData[{ "The graphs of functions of two variables are examples of what we call ", StyleBox["surfaces", FontWeight->"Bold"], ". More generally, a set of points (x,y,z) that satisfy an equation \ relating all three variables is often a surface. A simple example is the \ unit sphere, which you graphed above with ", StyleBox["ContourPlot3D", FontWeight->"Bold"], ". (The sphere is the set of points which satisfy the equation ", Cell[BoxData[ \(TraditionalForm\`x\^2 + y\^2 + z\^2 = 1. \)]], ")\n\nIn this section we're going to analyze some equations which involve \ ", Cell[BoxData[ \(TraditionalForm\`x, \ y, \ z, \ x\^2, \ y\^2, \ and\ z\^2, \)]], " and we'll look at their graphs. The surfaces in question are known as ", StyleBox["quadric", FontSlant->"Italic"], " surfaces, and will provide important examples for the rest of the course. \ There are six different quadric surfaces: the ellipsoid, the elliptic \ paraboloid, the hyperbolic paraboloid, the elliptic cone, and hyperboloids of \ one and two sheets. \n\nIn the first lecture of the semester, you talked \ about 3D graphs and how to analyze ", StyleBox["cross-sections", FontSlant->"Italic"], " of quadric surfaces. This is where you choose a specific value for ", Cell[BoxData[ \(TraditionalForm\`x, \ y, \)]], "or ", Cell[BoxData[ \(TraditionalForm\`z\)]], ". For example, consider the quadric surface given by the equation" }], "Text", TextAlignment->Left], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`z = 4 x\^2 + 9\ y\^2\)]]], "Text", TextAlignment->Center], Cell[TextData[{ "To check the horizontal cross-sections, we'd choose values for ", Cell[BoxData[ \(TraditionalForm\`z\)]], ", such as ", Cell[BoxData[ \(TraditionalForm\`z = 36\)]], ". In this case, \n" }], "Text", TextAlignment->Left], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`36 = 4 x\^2 + 9\ y\^2\)]]], "Text", TextAlignment->Center], Cell["or", "Text"], Cell[BoxData[ \(TraditionalForm\`x\^2\/9 + y\^2\/4 = 1\)], "DisplayFormula", TextAlignment->Center], Cell["\<\ So we see that the cross section in the plane z=36 is an ellipse \ which stretches 3 units in the positive and negative x-direction, and 2 units \ in the positive and negative y-direction.\ \>", "Text", TextAlignment->Left], Cell["\<\ If you don't remember how to graph ellipses and hyperbolas, this \ might be a good time to ask your TA to review those things.\ \>", "Text", CellFrame->True, TextAlignment->Left, Background->GrayLevel[0.833326]], Cell["\<\ Rather than have you read through a long section where we work out \ the cross sections of all the quadric surfaces, we're going to send you to an \ interactive gallery instead. Go to the following page in your web browser. http://www.math.umn.edu/~rogness/quadrics/ The gallery also includes another short review about cross sections, along \ with a picture. Once you've spent some time browsing through the gallery, \ you're ready to continue on with this lab. You can use the gallery as a \ reference throughout the semester.\ \>", "Text", TextAlignment->Left], Cell[CellGroupData[{ Cell["Graphing Quadric Surfaces in Mathematica", "Subsection"], Cell[TextData[{ "You can use the commands you learned earlier in this lab to graph any of \ the quadric surfaces. Some of the equations, such as ", Cell[BoxData[ \(TraditionalForm\`z = x\^2 + y\^2\)]], ", are explicitly solved for z, so you can graph using ", StyleBox["Plot3D", FontWeight->"Bold"], ". Others, like ", Cell[BoxData[ \(TraditionalForm\`z = x\^2 + y\^2\)]], "or ", Cell[BoxData[ \(TraditionalForm\`x\^2 + y\^2 + z\^2 = 1\)]], ", are implicit functions, so you have to use ", StyleBox["ContourPlot3D", FontWeight->"Bold"], ". Below you can see an example of graphing each kind of surface." }], "Text"], Cell[CellGroupData[{ Cell["Elliptic Paraboloid", "Subsubsection"], Cell[TextData[{ "This shows the graph of ", Cell[BoxData[ \(TraditionalForm\`z = x\^2 + y\^2\)]], "." }], "Text"], Cell[BoxData[ \(Plot3D[x^2 + y^2, {x, \(-1\), 1}, {y, \(-1\), 1}, AxesLabel \[Rule] {"\", "\", "\"}]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Hyperbolic Paraboloid (Saddle)", "Subsubsection"], Cell[TextData[{ "This shows the graph of ", Cell[BoxData[ \(TraditionalForm\`z = x\^2 - y\^2\)]], "." }], "Text"], Cell[BoxData[ \(Plot3D[x^2 - y^2, {x, \(-1\), 1}, {y, \(-1\), 1}, AxesLabel \[Rule] {"\", "\", "\"}]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Ellipsoid", "Subsubsection"], Cell[TextData[{ "This displays the graph of ", Cell[BoxData[ \(TraditionalForm\`x\^2\/9 + y\^2\/4 + z\^2\/1 = 1\)]], "." }], "Text"], Cell[BoxData[ \(ContourPlot3D[ x^2\ /\ 9 + \ y^2\ /\ 4\ + \ z^2\ /\ 1\ - 1, {x, \(-3\), 3}, {y, \(-2\), 2}, {z, \(-1\), 1}, Axes \[Rule] True, AxesLabel \[Rule] {"\", "\", "\"}]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Double Cone", "Subsubsection"], Cell[TextData[{ "This shows the graph of ", Cell[BoxData[ \(TraditionalForm\`z\^2 = x\^2 + y\^2\)]], "." }], "Text"], Cell[BoxData[ \(ContourPlot3D[ x^2 + y^2 - z^2, {x, \(-1\), 1}, {y, \(-1\), 1}, {z, \(-1\), 1}, Axes \[Rule] True, AxesLabel \[Rule] {"\", "\", "\"}]\)], "Input"], Cell["\<\ Alternatively, you can solve for z and graph the upper and lower \ parts separately before putting them together.\ \>", "Text"], Cell[BoxData[{ \(upper = Plot3D[Sqrt[x^2 + y^2], {x, \(-1\), 1}, {y, \(-1\), 1}]\), "\[IndentingNewLine]", \(lower = Plot3D[\(-Sqrt[x^2 + y^2]\), {x, \(-1\), 1}, {y, \(-1\), 1}]\), "\[IndentingNewLine]", \(Show[upper, lower, BoxRatios \[Rule] {1, 1, 2}]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Hyperboloid of One Sheet", "Subsubsection"], Cell[TextData[{ "This shows the graph of ", Cell[BoxData[ \(TraditionalForm\`x\^2 + y\^2 - z\^2 = 1\)]], "." }], "Text"], Cell[BoxData[ \(ContourPlot3D[ x^2 + y^2 - z^2 - 1, {x, \(-1.5\), 1.5}, {y, \(-1.5\), 1.5}, {z, \(-1\), 1}, Axes \[Rule] True, AxesLabel \[Rule] {"\", "\", "\"}]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Hyperboloid of Two Sheets", "Subsubsection"], Cell[TextData[{ "This shows the graph of ", Cell[BoxData[ \(TraditionalForm\`\(-x\^2\) - y\^2 + z\^2 = 1\)]], "." }], "Text"], Cell[BoxData[ \(ContourPlot3D[\(-x^2\) - y^2 + z^2 - 1, {x, \(-1\), 1}, {y, \(-1\), 1}, {z, \(-3\), 3}, Axes \[Rule] True, AxesLabel \[Rule] {"\", "\", "\"}]\)], "Input"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Wrapup and Exercises", "Subsection"], Cell[TextData[{ StyleBox["A preview of what's to come...", FontWeight->"Bold"], "\n\nSome people feel that using ", StyleBox["Plot3D", FontWeight->"Bold"], " to graph a paraboloid or cone results in an ugly picture, because the \ surfaces are plotted over a square in the xy-plane.", " (For example, the square where -1\[LessEqual]x\[LessEqual]1 and -1\ \[LessEqual]y\[LessEqual]1.) To get nicer pictures, like those in the book, \ we'd like plot these surfaces over a ", StyleBox["circular", FontSlant->"Italic"], " domain in the xy-plane. We don't have the resources to do that yet, but \ you'll learn how in Lab 2B. If you want a little preview, you can evaluate \ the followingcells. Don't bother trying to understand the commands yet!" }], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell[BoxData[ \(paraboloid = ParametricPlot3D[{r*Cos[t], \ r*Sin[t], r^2}, {r, 0, 2}, {t, 0, 2 Pi}]\)], "Input"], Cell[BoxData[ \(paraboloid = ParametricPlot3D[{r*Cos[t], \ r*Sin[t], r^2 \((Cos[t]^2 - Sin[t]^2)\)}, {r, 0, 1}, {t, 0, 2 Pi}]\)], "Input"], Cell[BoxData[{ \(\(uppercone = ParametricPlot3D[{s\ Cos[t], \ s\ Sin[t], s}, {s, 0, 2}, {t, 0, 2 Pi}, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(lowercone = ParametricPlot3D[{s\ Cos[t], \ s\ Sin[t], \(-s\)}, {s, 0, 2}, {t, 0, 2 Pi}, DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(Show[uppercone, lowercone, DisplayFunction \[Rule] $DisplayFunction]\)}], "Input"], Cell[TextData[{ StyleBox["Exercise 2", FontSize->16, FontWeight->"Bold"], "\n\nFind a function whose contour diagram is as similar as possible to the \ following picture. 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