(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 97340, 4557] NotebookOptionsPosition[ 93299, 4433] NotebookOutlinePosition[ 93704, 4450] CellTagsIndexPosition[ 93661, 4447] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) 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[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[ FormBox[ RowBox[{"y", "=", SuperscriptBox[ RowBox[{"(", RowBox[{"x", "-", "1"}], ")"}], "2"]}], TraditionalForm]]], " or ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]}], "=", "1"}], TraditionalForm]]], ". 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[ FormBox[ SqrtBox[ RowBox[{"1", "-", SuperscriptBox["x", "2"]}]], TraditionalForm]]], "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["ContourPlot", 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[ FormBox[ RowBox[{"z", "=", RowBox[{ SuperscriptBox["x", "2"], "+", RowBox[{ SuperscriptBox["y", "2"], "."}]}]}], TraditionalForm]]], "\n\nWe'll also consider implicit functions of z, such as\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", SuperscriptBox["z", RowBox[{"2", " "}]]}], "=", "1."}], TraditionalForm]]], "\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 \ Environment : Presentation." }], "Text", CellChangeTimes->{{3.442061943435672*^9, 3.442061963858821*^9}}] }, Open ]], 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[ RowBox[{ RowBox[{"f", "[", RowBox[{"x_", ",", "y_"}], "]"}], "=", RowBox[{ RowBox[{"x", "^", "2"}], " ", "+", " ", RowBox[{"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[{ RowBox[{ RowBox[{"g", "[", RowBox[{"x_", ",", "y_"}], "]"}], "=", "1"}], "\n", RowBox[{"Plot3D", "[", RowBox[{ RowBox[{"g", "[", RowBox[{"x", ",", "y"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "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." }], "Text", CellChangeTimes->{3.409326278583324*^9}], Cell[TextData[{ StyleBox["3D Graph Controls", FontSize->16, FontWeight->"Bold"], "\n\nIn version 6.0, ", StyleBox["Mathematica", FontSlant->"Italic"], " finally implemented interactive controls for its 3D graphics. You can \ click and drag the picture above to rotate it. If you hold the control key \ while clicking on the picture and dragging up/down you will zoom in/out of \ the picture. Holding the shift key while dragging will move the picture \ around the window. (Note that the ", StyleBox["Mathematica", FontSlant->"Italic"], " documentation says to use shift to zoom and control to move the picture, \ so if you have problems try both keys.)" }], "Text", CellFrame->True, CellChangeTimes->{{3.409326341856931*^9, 3.409326472010054*^9}, { 3.409326582866475*^9, 3.40932664045139*^9}}, Background->RGBColor[ 0.6862745098039216, 0.6862745098039216, 0.6862745098039216]], Cell["\<\ Here's a slightly more complicated function. Before you evaluate this cell, \ see if you can predict what the graph will look like.\ \>", "Text", CellChangeTimes->{3.409326278583324*^9}], Cell[BoxData[{ RowBox[{ RowBox[{"g", "[", RowBox[{"x_", ",", "y_"}], "]"}], "=", RowBox[{"x", "+", "y"}]}], "\n", RowBox[{"Plot3D", "[", RowBox[{ RowBox[{"g", "[", RowBox[{"x", ",", "y"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "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[ FormBox[ RowBox[{"y", "=", RowBox[{"-", "x"}]}], TraditionalForm]]], ", then g(x,y)=0.\n\[FilledVerySmallSquare] if x and y are both positive, \ then ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"g", "(", RowBox[{"x", ",", "y"}], ")"}], "\[Succeeds]", "0"}], TraditionalForm]]], ".\n\[FilledVerySmallSquare] if x and y are both negative, then ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"g", "(", RowBox[{"x", ",", "y"}], ")"}], "\[Precedes]", "0"}], TraditionalForm]]], ".\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[ RowBox[{"Plot3D", "[", RowBox[{ RowBox[{"g", "[", RowBox[{"x", ",", "y"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"BoxRatios", "\[Rule]", RowBox[{"{", RowBox[{"1", ",", "1", ",", "2"}], "}"}]}]}], "]"}]], "Input"], Cell[TextData[{ "Let's plot the function ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", RowBox[{"x", ",", "y"}], ")"}], "=", RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]}]}], TraditionalForm]]], " we considered previously so that we have something slightly more \ interesting to work with. " }], "Text"], Cell[BoxData[ RowBox[{"Plot3D", "[", RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"y", "^", "2"}]}], ",", " ", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}]}], "]"}]], "Input"], Cell[TextData[{ "Another useful option is ", StyleBox["ViewPoint", FontWeight->"Bold"], ", which allows you to specify the location of your \"eyes\" as you look at \ the surface. For example, the following command puts you at the point \ (0,0,10) on the ", StyleBox["z", FontSlant->"Italic"], "-axis, looking down at the surface." }], "Text", CellChangeTimes->{{3.409326797230639*^9, 3.409326849372702*^9}}], Cell[BoxData[ RowBox[{"Plot3D", "[", RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"y", "^", "2"}]}], ",", " ", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"ViewPoint", "\[Rule]", RowBox[{"{", RowBox[{"0", ",", "0", ",", "10"}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{3.409326782535254*^9}], Cell[TextData[{ StyleBox["ViewPoint", FontWeight->"Bold"], " isn't so necessary now that you can rotate 3D graphics in ", StyleBox["Mathematica", FontSlant->"Italic"], " -- you can just plot the picture and rotate it as needed. But you might \ find it helpful at times to specify the viewpoint so you don't always have to \ re-rotate the picture. \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", CellChangeTimes->{{3.409326863007665*^9, 3.409326983465628*^9}}], Cell[BoxData[ RowBox[{"Plot3D", "[", RowBox[{ RowBox[{"Sin", "[", RowBox[{"x", "*", "y"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "Pi"}], ",", "Pi"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "Pi"}], ",", "Pi"}], "}"}]}], "]"}]], "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[ RowBox[{"Plot3D", "[", RowBox[{ RowBox[{"Sin", "[", RowBox[{"x", "*", "y"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "Pi"}], ",", "Pi"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "Pi"}], ",", "Pi"}], "}"}], ",", RowBox[{"Mesh", "\[Rule]", "False"}]}], "]"}]], "Input", CellChangeTimes->{3.409327006840989*^9}], 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[ RowBox[{"Plot3D", "[", RowBox[{ RowBox[{"Sin", "[", RowBox[{"x", "*", "y"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "Pi"}], ",", "Pi"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "Pi"}], ",", "Pi"}], "}"}], ",", RowBox[{"Mesh", "\[Rule]", "False"}], ",", " ", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\"", ",", "\"\\""}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{3.409327013642555*^9}], 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[ RowBox[{"Plot3D", "[", RowBox[{ RowBox[{"Sin", "[", RowBox[{"x", "*", "y"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "Pi"}], ",", "Pi"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "Pi"}], ",", "Pi"}], "}"}], ",", RowBox[{"Mesh", "\[Rule]", "False"}], ",", RowBox[{"Axes", "\[Rule]", "False"}]}], "]"}]], "Input", CellChangeTimes->{3.409327024871572*^9}], 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[ RowBox[{ RowBox[{ RowBox[{"f", "[", RowBox[{"x_", ",", "y_"}], "]"}], "=", RowBox[{ RowBox[{"x", "^", "2"}], " ", "+", " ", RowBox[{"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[ RowBox[{"ContourPlot", "[", RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"y", "^", "2"}]}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "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.)\n\nIf you have \ trouble remembering which points are high and which points are lower, move \ your mouse over the contour lines in the picture. ", StyleBox["Mathematica", FontSlant->"Italic"], " will show you the \"elevation\" on each line. If you'd like to have these \ displayed at all times, use the ", StyleBox["ContourLabels", FontWeight->"Bold"], " option:" }], "Text", CellChangeTimes->{{3.409327089681204*^9, 3.409327163036321*^9}}], Cell[BoxData[ RowBox[{"ContourPlot", "[", RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"y", "^", "2"}]}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", RowBox[{"ContourLabels", "\[Rule]", "Automatic"}]}], "]"}]], "Input", CellChangeTimes->{{3.409327168042221*^9, 3.409327200775418*^9}}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["ContourPlot3D", FontSize->16]], "Section"], Cell[TextData[{ "The term ", StyleBox["ContourPlot", FontWeight->"Bold"], " is used in many different contexts in ", StyleBox["Mathematica", FontSlant->"Italic"], ". In the previous section we used it to graph the contour lines of a \ function, ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "x", ")"}], "=", RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]}]}], TraditionalForm]]], ". Last week you used ", StyleBox["ContourPlot", FontWeight->"Bold"], " to graph an implicit function, ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]}], "=", "1"}], TraditionalForm]]], ". There is a 3D analog called ", StyleBox["ContourPlot3D", FontWeight->"Bold"], " which you can use to plot implicit functions with three variables." }], "Text", CellChangeTimes->{{3.409327259304872*^9, 3.409327377156669*^9}}], Cell[TextData[{ "Suppose we want to graph an implicit function of z like this:\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", SuperscriptBox["z", "2"]}], "=", "1."}], TraditionalForm]]], "\n\nWe use the following command, which is very similar to how we graphed \ ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]}], "=", "1"}], TraditionalForm]]], " with ", StyleBox["ContourPlot", FontWeight->"Bold"], " last week; the only differences are the ", StyleBox["3D", FontWeight->"Bold"], " at the end of the command together with a range for ", StyleBox["z", FontSlant->"Italic"], ":" }], "Text", CellChangeTimes->{{3.409327417424265*^9, 3.409327462901242*^9}}], Cell[BoxData[ RowBox[{"ContourPlot3D", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"y", "^", "2"}], "+", RowBox[{"z", "^", "2"}]}], "==", "1"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.409327399609902*^9, 3.409327400063568*^9}, { 3.409327477945408*^9, 3.409327480569737*^9}}], 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[ FormBox[ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]}], TraditionalForm]]], ",\n\nwe first move everything to the left hand side:\n\nz - ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]}], TraditionalForm]]], " = 0.\n\nThen we use the command:" }], "Text"], Cell[BoxData[ RowBox[{"ContourPlot3D", "[", RowBox[{ RowBox[{ RowBox[{"z", "-", RowBox[{"x", "^", "2"}], "-", RowBox[{"y", "^", "2"}]}], "\[Equal]", "0"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"z", ",", "0", ",", "2"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.409327524770698*^9, 3.409327525079494*^9}}], Cell[TextData[{ "This makes slightly different pictures than ", StyleBox["Plot3D", FontWeight->"Bold"], ". You can compare the picture from the previous command to this one, and \ see which you prefer:" }], "Text", CellChangeTimes->{{3.409327553643466*^9, 3.409327581639973*^9}}], Cell[BoxData[ RowBox[{"Plot3D", "[", RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"y", "^", "2"}]}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"BoxRatios", "\[Rule]", RowBox[{"{", RowBox[{"1", ",", "1", ",", "2"}], "}"}]}]}], "]"}]], "Input"] }, 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[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", SuperscriptBox["z", "2"]}], "=", "1."}], TraditionalForm]]], ")\n\nIn this section we're going to analyze some equations which involve ", Cell[BoxData[ FormBox[ RowBox[{"x", ",", " ", "y", ",", " ", "z", ",", " ", SuperscriptBox["x", "2"], ",", " ", SuperscriptBox["y", "2"], ",", " ", RowBox[{"and", " ", SuperscriptBox["z", "2"]}], ","}], TraditionalForm]]], " 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[ FormBox[ RowBox[{"x", ",", " ", "y", ","}], TraditionalForm]]], "or ", Cell[BoxData[ FormBox["z", TraditionalForm]]], ". 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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[ FormBox[ RowBox[{"z", "=", RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]}]}], TraditionalForm]]], ", are explicitly solved for z, so you can graph using ", StyleBox["Plot3D", FontWeight->"Bold"], ". 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.85352 L .0857 .82051 L .07142 .78045 L .05943 .74038 L .05929 .7399 L .04953 .70032 L .04159 .66026 L .03551 .62019 L .03121 .58013 L .02865 .54006 L .02779 .5 L .02865 .45994 L .03121 .41987 L .03551 .37981 L .04159 .33974 L .04953 .29968 L .05929 .2601 L .05943 .25962 L .07142 .21955 L .0857 .17949 L .09936 .14648 L .10252 .13942 L .12217 .09936 L .13942 .06876 L .14518 .05929 L .17223 .01923 L .98077 .01923 L .98077 .98077 L F 0 g .17223 .98077 m .14518 .94071 L .13942 .93124 L .12217 .90064 L .10252 .86058 L .09936 .85352 L .0857 .82051 L .07142 .78045 L .05943 .74038 L .05929 .7399 L .04953 .70032 L .04159 .66026 L .03551 .62019 L .03121 .58013 L .02865 .54006 L .02779 .5 L .02865 .45994 L .03121 .41987 L .03551 .37981 L .04159 .33974 L .04953 .29968 L .05929 .2601 L .05943 .25962 L .07142 .21955 L .0857 .17949 L .09936 .14648 L .10252 .13942 L .12217 .09936 L .13942 .06876 L .14518 .05929 L .17223 .01923 L s .5 g .22733 .98077 m .21955 .97168 L .19535 .94071 L .17949 .91756 L .16886 .90064 L .14658 .86058 L .13942 .8461 L .12778 .82051 L .11194 .78045 L .09936 .74245 L .09874 .74038 L .08788 .70032 L .0792 .66026 L .07258 .62019 L .06791 .58013 L .06514 .54006 L .06422 .5 L .06514 .45994 L .06791 .41987 L .07258 .37981 L .0792 .33974 L .08788 .29968 L .09874 .25962 L .09936 .25755 L .11194 .21955 L .12778 .17949 L .13942 .1539 L .14658 .13942 L .16886 .09936 L .17949 .08244 L .19535 .05929 L .21955 .02832 L .22733 .01923 L .98077 .01923 L .98077 .98077 L F 0 g .22733 .98077 m .21955 .97168 L .19535 .94071 L .17949 .91756 L .16886 .90064 L .14658 .86058 L .13942 .8461 L .12778 .82051 L .11194 .78045 L .09936 .74245 L .09874 .74038 L .08788 .70032 L .0792 .66026 L .07258 .62019 L .06791 .58013 L .06514 .54006 L .06422 .5 L .06514 .45994 L .06791 .41987 L .07258 .37981 L .0792 .33974 L .08788 .29968 L .09874 .25962 L .09936 .25755 L .11194 .21955 L .12778 .17949 L .13942 .1539 L .14658 .13942 L .16886 .09936 L .17949 .08244 L .19535 .05929 L .21955 .02832 L .22733 .01923 L s .6 g .29691 .98077 m .25962 .94502 L .25565 .94071 L .22331 .90064 L .21955 .89537 L .19698 .86058 L .17949 .82889 L .17527 .82051 L .15722 .78045 L .14234 .74038 L .13942 .73154 L .1302 .70032 L .12056 .66026 L .11322 .62019 L .10807 .58013 L .10502 .54006 L .10401 .5 L .10502 .45994 L .10807 .41987 L .11322 .37981 L .12056 .33974 L .1302 .29968 L .13942 .26846 L .14234 .25962 L .15722 .21955 L .17527 .17949 L .17949 .17111 L .19698 .13942 L .21955 .10463 L .22331 .09936 L .25565 .05929 L .25962 .05498 L .29691 .01923 L .98077 .01923 L .98077 .98077 L F 0 g .29691 .98077 m .25962 .94502 L .25565 .94071 L .22331 .90064 L .21955 .89537 L .19698 .86058 L .17949 .82889 L .17527 .82051 L .15722 .78045 L .14234 .74038 L .13942 .73154 L .1302 .70032 L .12056 .66026 L .11322 .62019 L .10807 .58013 L .10502 .54006 L .10401 .5 L .10502 .45994 L .10807 .41987 L .11322 .37981 L .12056 .33974 L .1302 .29968 L .13942 .26846 L .14234 .25962 L .15722 .21955 L .17527 .17949 L .17949 .17111 L .19698 .13942 L .21955 .10463 L .22331 .09936 L .25565 .05929 L .25962 .05498 L .29691 .01923 L s .7 g .40901 .98077 m .37981 .96745 L .33974 .94278 L .33686 .94071 L .29968 .90887 L .29149 .90064 L .25962 .86312 L .2577 .86058 L .23098 .82051 L .21955 .80024 L .20948 .78045 L .19204 .74038 L .17949 .70489 L .17805 .70032 L .16701 .66026 L .15867 .62019 L .15285 .58013 L .1494 .54006 L .14826 .5 L .1494 .45994 L .15285 .41987 L .15867 .37981 L .16701 .33974 L .17805 .29968 L .17949 .29511 L .19204 .25962 L .20948 .21955 L .21955 .19976 L .23098 .17949 L .2577 .13942 L .25962 .13688 L .29149 .09936 L .29968 .09113 L .33686 .05929 L .33974 .05722 L .37981 .03255 L .40901 .01923 L .98077 .01923 L .98077 .98077 L F 0 g .40901 .98077 m .37981 .96745 L .33974 .94278 L .33686 .94071 L .29968 .90887 L .29149 .90064 L .25962 .86312 L .2577 .86058 L .23098 .82051 L .21955 .80024 L .20948 .78045 L .19204 .74038 L .17949 .70489 L .17805 .70032 L .16701 .66026 L .15867 .62019 L .15285 .58013 L .1494 .54006 L .14826 .5 L .1494 .45994 L .15285 .41987 L .15867 .37981 L .16701 .33974 L .17805 .29968 L .17949 .29511 L .19204 .25962 L .20948 .21955 L .21955 .19976 L .23098 .17949 L .2577 .13942 L .25962 .13688 L .29149 .09936 L .29968 .09113 L .33686 .05929 L .33974 .05722 L .37981 .03255 L .40901 .01923 L s .8 g .41987 .08963 m .45994 .07806 L .5 .07427 L .54006 .07806 L .58013 .08963 L .60288 .09936 L .62019 .10968 L .66004 .13942 L .66026 .13963 L .69816 .17949 L .70032 .18222 L .72656 .21955 L .74038 .2437 L .74848 .25962 L .76567 .29968 L .77889 .33974 L .78045 .34513 L .7888 .37981 L .79568 .41987 L .79972 .45994 L .80106 .5 L .79972 .54006 L .79568 .58013 L .7888 .62019 L .78045 .65487 L .77889 .66026 L .76567 .70032 L .74848 .74038 L .74038 .7563 L .72656 .78045 L .70032 .81778 L .69816 .82051 L .66026 .86037 L .66004 .86058 L .62019 .89032 L .60288 .90064 L .58013 .91037 L .54006 .92194 L .5 .92573 L .45994 .92194 L .41987 .91037 L .39712 .90064 L .37981 .89032 L .33996 .86058 L .33974 .86037 L .30184 .82051 L .29968 .81778 L .27344 .78045 L .25962 .7563 L .25152 .74038 L .23433 .70032 L .22111 .66026 L .21955 .65487 L .2112 .62019 L .20432 .58013 L .20028 .54006 L .19894 .5 L .20028 .45994 L .20432 .41987 L .2112 .37981 L .21955 .34513 L .22111 .33974 L .23433 .29968 L .25152 .25962 L .25962 .2437 L .27344 .21955 L .29968 .18222 L .30184 .17949 L .33974 .13963 L .33996 .13942 L .37981 .10968 L .39712 .09936 L F 0 g .41987 .08963 m .45994 .07806 L .5 .07427 L .54006 .07806 L .58013 .08963 L .60288 .09936 L .62019 .10968 L .66004 .13942 L .66026 .13963 L .69816 .17949 L .70032 .18222 L .72656 .21955 L .74038 .2437 L .74848 .25962 L .76567 .29968 L .77889 .33974 L .78045 .34513 L .7888 .37981 L .79568 .41987 L .79972 .45994 L .80106 .5 L .79972 .54006 L .79568 .58013 L .7888 .62019 L .78045 .65487 L .77889 .66026 L .76567 .70032 L .74848 .74038 L .74038 .7563 L .72656 .78045 L .70032 .81778 L .69816 .82051 L .66026 .86037 L .66004 .86058 L .62019 .89032 L .60288 .90064 L .58013 .91037 L .54006 .92194 L .5 .92573 L .45994 .92194 L .41987 .91037 L .39712 .90064 L .37981 .89032 L .33996 .86058 L .33974 .86037 L .30184 .82051 L .29968 .81778 L .27344 .78045 L .25962 .7563 L .25152 .74038 L Mistroke .23433 .70032 L .22111 .66026 L .21955 .65487 L .2112 .62019 L .20432 .58013 L .20028 .54006 L .19894 .5 L .20028 .45994 L .20432 .41987 L .2112 .37981 L .21955 .34513 L .22111 .33974 L .23433 .29968 L .25152 .25962 L .25962 .2437 L .27344 .21955 L .29968 .18222 L .30184 .17949 L .33974 .13963 L .33996 .13942 L .37981 .10968 L .39712 .09936 L .41987 .08963 L Mfstroke .9 g .45994 .16556 m .5 .1608 L .54006 .16556 L .57847 .17949 L .58013 .18031 L .62019 .20645 L .63522 .21955 L .66026 .24761 L .66931 .25962 L .69363 .29968 L .70032 .31334 L .71145 .33974 L .72436 .37981 L .7331 .41987 L .73818 .45994 L .73984 .5 L .73818 .54006 L .7331 .58013 L .72436 .62019 L .71145 .66026 L .70032 .68666 L .69363 .70032 L .66931 .74038 L .66026 .75239 L .63522 .78045 L .62019 .79355 L .58013 .81969 L .57847 .82051 L .54006 .83444 L .5 .8392 L .45994 .83444 L .42153 .82051 L .41987 .81969 L .37981 .79355 L .36478 .78045 L .33974 .75239 L .33069 .74038 L .30637 .70032 L .29968 .68666 L .28855 .66026 L .27564 .62019 L .2669 .58013 L .26182 .54006 L .26016 .5 L .26182 .45994 L .2669 .41987 L .27564 .37981 L .28855 .33974 L .29968 .31334 L .30637 .29968 L .33069 .25962 L .33974 .24761 L .36478 .21955 L .37981 .20645 L .41987 .18031 L .42153 .17949 L F 0 g .45994 .16556 m .5 .1608 L .54006 .16556 L .57847 .17949 L .58013 .18031 L .62019 .20645 L .63522 .21955 L .66026 .24761 L .66931 .25962 L .69363 .29968 L .70032 .31334 L .71145 .33974 L .72436 .37981 L .7331 .41987 L .73818 .45994 L .73984 .5 L .73818 .54006 L .7331 .58013 L .72436 .62019 L .71145 .66026 L .70032 .68666 L .69363 .70032 L .66931 .74038 L .66026 .75239 L .63522 .78045 L .62019 .79355 L .58013 .81969 L .57847 .82051 L .54006 .83444 L .5 .8392 L .45994 .83444 L .42153 .82051 L .41987 .81969 L .37981 .79355 L .36478 .78045 L .33974 .75239 L .33069 .74038 L .30637 .70032 L .29968 .68666 L .28855 .66026 L .27564 .62019 L .2669 .58013 L .26182 .54006 L .26016 .5 L .26182 .45994 L .2669 .41987 L .27564 .37981 L .28855 .33974 L .29968 .31334 L .30637 .29968 L Mistroke .33069 .25962 L .33974 .24761 L .36478 .21955 L .37981 .20645 L .41987 .18031 L .42153 .17949 L .45994 .16556 L Mfstroke 1 g .45994 .28628 m .5 .27888 L .54006 .28628 L .56605 .29968 L .58013 .3101 L .60861 .33974 L .62019 .35835 L .63143 .37981 L .646 .41987 L .6539 .45994 L .65642 .5 L .6539 .54006 L .646 .58013 L .63143 .62019 L .62019 .64165 L .60861 .66026 L .58013 .6899 L .56605 .70032 L .54006 .71372 L .5 .72112 L .45994 .71372 L .43395 .70032 L .41987 .6899 L .39139 .66026 L .37981 .64165 L .36857 .62019 L .354 .58013 L .3461 .54006 L .34358 .5 L .3461 .45994 L .354 .41987 L .36857 .37981 L .37981 .35835 L .39139 .33974 L .41987 .3101 L .43395 .29968 L F 0 g .45994 .28628 m .5 .27888 L .54006 .28628 L .56605 .29968 L .58013 .3101 L .60861 .33974 L .62019 .35835 L .63143 .37981 L .646 .41987 L .6539 .45994 L .65642 .5 L .6539 .54006 L .646 .58013 L .63143 .62019 L .62019 .64165 L .60861 .66026 L .58013 .6899 L .56605 .70032 L .54006 .71372 L .5 .72112 L .45994 .71372 L .43395 .70032 L .41987 .6899 L .39139 .66026 L .37981 .64165 L .36857 .62019 L .354 .58013 L .3461 .54006 L .34358 .5 L .3461 .45994 L .354 .41987 L .36857 .37981 L .37981 .35835 L .39139 .33974 L .41987 .3101 L .43395 .29968 L .45994 .28628 L s .6 g .59099 .98077 m .62019 .96745 L .66026 .94278 L .66314 .94071 L .70032 .90887 L .70851 .90064 L .74038 .86312 L .7423 .86058 L .76902 .82051 L .78045 .80024 L .79052 .78045 L .80796 .74038 L .82051 .70489 L .82195 .70032 L .83299 .66026 L .84133 .62019 L .84715 .58013 L .8506 .54006 L .85174 .5 L .8506 .45994 L .84715 .41987 L .84133 .37981 L .83299 .33974 L .82195 .29968 L .82051 .29511 L .80796 .25962 L .79052 .21955 L .78045 .19976 L .76902 .17949 L .7423 .13942 L .74038 .13688 L .70851 .09936 L .70032 .09113 L .66314 .05929 L .66026 .05722 L .62019 .03255 L .59099 .01923 L .98077 .01923 L .98077 .98077 L F 0 g .59099 .98077 m .62019 .96745 L .66026 .94278 L .66314 .94071 L .70032 .90887 L .70851 .90064 L .74038 .86312 L .7423 .86058 L .76902 .82051 L .78045 .80024 L .79052 .78045 L .80796 .74038 L .82051 .70489 L .82195 .70032 L .83299 .66026 L .84133 .62019 L .84715 .58013 L .8506 .54006 L .85174 .5 L .8506 .45994 L .84715 .41987 L .84133 .37981 L .83299 .33974 L .82195 .29968 L .82051 .29511 L .80796 .25962 L .79052 .21955 L .78045 .19976 L .76902 .17949 L .7423 .13942 L .74038 .13688 L .70851 .09936 L .70032 .09113 L .66314 .05929 L .66026 .05722 L .62019 .03255 L .59099 .01923 L s .5 g .70309 .98077 m .74038 .94502 L .74435 .94071 L .77669 .90064 L .78045 .89537 L .80302 .86058 L .82051 .82889 L .82473 .82051 L .84278 .78045 L .85766 .74038 L .86058 .73154 L .8698 .70032 L .87944 .66026 L .88678 .62019 L .89193 .58013 L .89498 .54006 L .89599 .5 L .89498 .45994 L .89193 .41987 L .88678 .37981 L .87944 .33974 L .8698 .29968 L .86058 .26846 L .85766 .25962 L .84278 .21955 L .82473 .17949 L .82051 .17111 L .80302 .13942 L .78045 .10463 L .77669 .09936 L .74435 .05929 L .74038 .05498 L .70309 .01923 L .98077 .01923 L .98077 .98077 L F 0 g .70309 .98077 m .74038 .94502 L .74435 .94071 L .77669 .90064 L .78045 .89537 L .80302 .86058 L .82051 .82889 L .82473 .82051 L .84278 .78045 L .85766 .74038 L .86058 .73154 L .8698 .70032 L .87944 .66026 L .88678 .62019 L .89193 .58013 L .89498 .54006 L .89599 .5 L .89498 .45994 L .89193 .41987 L .88678 .37981 L .87944 .33974 L .8698 .29968 L .86058 .26846 L .85766 .25962 L .84278 .21955 L .82473 .17949 L .82051 .17111 L .80302 .13942 L .78045 .10463 L .77669 .09936 L .74435 .05929 L .74038 .05498 L .70309 .01923 L s .4 g .77267 .98077 m .78045 .97168 L .80465 .94071 L .82051 .91756 L .83114 .90064 L .85342 .86058 L .86058 .8461 L .87222 .82051 L .88806 .78045 L .90064 .74245 L .90126 .74038 L .91212 .70032 L .9208 .66026 L .92742 .62019 L .93209 .58013 L .93486 .54006 L .93578 .5 L .93486 .45994 L .93209 .41987 L .92742 .37981 L .9208 .33974 L .91212 .29968 L .90126 .25962 L .90064 .25755 L .88806 .21955 L .87222 .17949 L .86058 .1539 L .85342 .13942 L .83114 .09936 L .82051 .08244 L .80465 .05929 L .78045 .02832 L .77267 .01923 L .98077 .01923 L .98077 .98077 L F 0 g .77267 .98077 m .78045 .97168 L .80465 .94071 L .82051 .91756 L .83114 .90064 L .85342 .86058 L .86058 .8461 L .87222 .82051 L .88806 .78045 L .90064 .74245 L .90126 .74038 L .91212 .70032 L .9208 .66026 L .92742 .62019 L .93209 .58013 L .93486 .54006 L .93578 .5 L .93486 .45994 L .93209 .41987 L .92742 .37981 L .9208 .33974 L .91212 .29968 L .90126 .25962 L .90064 .25755 L .88806 .21955 L .87222 .17949 L .86058 .1539 L .85342 .13942 L .83114 .09936 L .82051 .08244 L .80465 .05929 L .78045 .02832 L .77267 .01923 L s .3 g .82777 .98077 m .85482 .94071 L .86058 .93124 L .87783 .90064 L 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"\n\nConsider the surface given by ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "-", RowBox[{"2", SuperscriptBox["z", "2"]}]}], "=", "4"}], TraditionalForm]]], ". Make an accurate picture of this quadric surface by drawing the cross \ sections given by z=0, z=\[PlusMinus]2, x=0, x=\[PlusMinus]2, y=0, y=\ \[PlusMinus]2, and any others which might be helpful. Note that this \ exercise is easier to do by hand than with ", StyleBox["Mathematica", FontSlant->"Italic"], ", although the online gallery of quadric surfaces could be useful.\n\n", StyleBox["Exercise 5", FontSize->16, FontWeight->"Bold"], "\n\nConsider the surface given by ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{ RowBox[{"2", SuperscriptBox["x", "2"]}], "-", SuperscriptBox["y", "2"]}]}], TraditionalForm]]], ". Make an accurate picture of this surface by drawing the cross sections \ given by x=0, x=\[PlusMinus]1, y=0, y=\[PlusMinus]1, z=0, z=\[PlusMinus]1, \ and any others which might be helpful. As with Exercise 4, this is probably \ easier to do by hand, but the online gallery could help.\n\n", StyleBox["Exercise 6", FontSize->16, FontWeight->"Bold"], "\n\nSo far all of our quadric surfaces have been centered at the origin, \ but that's not always the case. You can move them around using the same \ methods you've used in Calculus and PreCalculus -- replacing \"x\" with \ \"(x-h)\" and so on. Find the equation for a double cone which:\n\n(1) has a \ vertex at (1,1,2),\n(2) and intersects the xy-plane (i.e. the plane z=0) in a \ circle of radius 4." }], "Text", CellFrame->True, Background->RGBColor[0.996109, 0.500008, 0.500008]] }, Closed]], Cell[CellGroupData[{ Cell["Credits", "Subsection"], Cell[TextData[{ "This lab was written from scratch in January 2002. Most of the sections \ were written with the future in mind; I know what we're going to do with \ future labs, and therefore know what we'd like students to learn about \ plotting functions in ", StyleBox["Mathematica", FontSlant->"Italic"], ". The commands used in the Quadric Surfaces section were also written from \ scratch. I'm not entirely happy with them, but they do provide some insight \ with the cross sections. Please send me any comments or questions!\n\n\ Update: I rewrote parts of this lab in January 2004. I edited exercises, \ removed references to the unsupported and undocumented RealTime3D package, \ and replaced it with the section on ShowLive. I also revamped the section on \ quadric surfaces. The old version was a regurgitation of the textbook. The \ new version is more interactive and will hopefully be more interesting.\n\n\ (Minor changes made in September 2004; just housekeeping stuff. Bigger \ changes in January 2007, removing the ShowLive section and updating to \ reflect ", StyleBox["Mathematica", FontSlant->"Italic"], " 6.0's 3D graphics capabilities.)\n\nThis lab is copyright 2002, 2004 by \ Jonathan Rogness (rogness@math.umn.edu) and is protected by the Creative \ Commons Attribution-NonCommercial-ShareAlike License. You can find more \ information on this license at \ http://creativecommons.org/licenses/by-nc-sa/1.0/\n\nAlthough it's not \ specifically required by the license, I'd appreciate it if you let me know if \ you use parts of our labs, just so I can keep track of it. 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