(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 4.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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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[ 96552, 1924]*) (*NotebookOutlinePosition[ 97241, 1948]*) (* CellTagsIndexPosition[ 97197, 1944]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ Cell[BoxData[{ StyleBox[\(MATH\ 257\ Calculus\ III\t\t\t\tWeek\ of\ June\ 2, \ 2003\), FontSize->14], "\[IndentingNewLine]", RowBox[{"\t\t\t\t\t\t\t\t\t\t", \(The\ LAST\ \(Lab!\)\)}], "\n", StyleBox[\(Lab\ 10 : \ Max/Min\ on\ Surfaces\), "Title"], "\n", StyleBox[\(Name\ 1\), "Section"], "\n", StyleBox[\(Name\ 2\), "Section"], "\n", StyleBox[\(\(Section\)\(:\)\), "Section"]}], "Input"], "\n" }], "Text", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], Cell["\<\ At some point during this lab, make sure you fill out a teacher \ evaluation form for me! Yeah.\ \>", "Text", FontSize->16], Cell[CellGroupData[{ Cell[TextData[StyleBox[" Partial derivatives on Mathematica", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]]], "Subtitle"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " has a couple of ways to do partial derivatives. One way is to use the \ \"partial\" symbol ", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\_\[Placeholder]\ \[Placeholder]\)]], " on the Basic Inputs Pallette. ", StyleBox["Example:", FontWeight->"Bold"], " to find ", Cell[BoxData[ \(TraditionalForm\`\(\[PartialD]\ f\/\[PartialD]\ x\) \((\(x\^2\) y\ + \ Sin[x*y])\)\)]], " we can do the following:" }], "Text", FontSize->14], Cell[BoxData[ \(\[PartialD]\_x\ \((\(x\^2\) y\ + \ Sin[x*y])\)\)], "Input"], Cell[TextData[{ StyleBox["Note:", FontWeight->"Bold"], " you NEED the parentheses. Another way to do this same thing would be to \ use the ", StyleBox["D[{", "Input"], StyleBox["function", FontColor->RGBColor[0, 0, 1]], StyleBox["},x]", "Input"], " command:" }], "Text", FontSize->14], Cell[BoxData[ \(\(\(D[\(x\^2\) y + Sin[x*y], x]\)\(\[IndentingNewLine]\) \)\)], "Input"], Cell[TextData[{ StyleBox["Question 1:", FontWeight->"Bold"], " Find the two partial derivatives (with respect to x and with respect to \ y) of the following functions:\n(a) f(x,y)=", Cell[BoxData[ FormBox[ StyleBox[\(x\/\[ExponentialE]\^y\), FontSize->24], TraditionalForm]], FontSize->18], "\n(b) f(x,y)=", Cell[BoxData[ \(TraditionalForm\`y\^3\/\(x\^3 + 1\)\)], FontSize->24] }], "Text", FontSize->14, FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ \(\[IndentingNewLine]\)], "Input"], Cell[TextData[{ "One of the big concepts in Chapter 11 in the book is trying to find \ maximum and minimum points on a 3D surface. 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This means that the partial \ derivatives in the \"x direction\" and the \"y direction\" should be zero. \ We can use this to find the max point:" }], "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[\[IndentingNewLine]{\[PartialD]\_x\ \((\(-y\^2\) + .2 x*y - x\^2 + x + 10)\) \[Equal] 0, \[IndentingNewLine]\[PartialD]\_y\ \((\(-y\^2\) + .2 x*y - x\^2 + x + 10)\) \[Equal] 0}, \ {x, y}]\)], "Input"], Cell[BoxData[ \({{x \[Rule] 0.5050505050505051`, y \[Rule] 0.05050505050505051`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(-y\^2\) + .2 x*y - x\^2 + x + 10 /. %\)], "Input"], Cell[BoxData[ \({10.252525252525253`}\)], "Output"] }, Open ]], Cell[TextData[{ "So it looks like this function has a max at approximately {0.505051, \ 0.0505051, 10.2525}.\n\n", StyleBox["Be warned", FontWeight->"Bold"], ", though! Just because a surface has ", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\ f\/\[PartialD]\ x = 0\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\ f\/\[PartialD]\ y = 0\)]], " at a point doesn't mean it has a max or a min at that point. It could \ also be a ", StyleBox["saddle point", FontWeight->"Bold"], "." }], "Text", FontSize->14], Cell[TextData[{ StyleBox["Question 2:", FontWeight->"Bold"], " Find the coordinates of all local Max/Min points of the surface f(x,y)=", Cell[BoxData[ \(x\^4\)], FontColor->RGBColor[1, 0, 0]], " + ", Cell[BoxData[ \(y\^3\)], FontColor->RGBColor[1, 0, 0]], " - 3 y. Be sure to ", StyleBox["state", FontWeight->"Bold"], " if they are maximum or minimum points. ", "Double-check yourself by making a plot of the surface. (It might help to \ use the setting ", StyleBox["AspectRatio->1", "Input"], " command in your plot.)" }], "Text", FontSize->14, FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ \(\[IndentingNewLine]\)], "Input"], Cell[TextData[{ StyleBox["Question 3:", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox[" Do the same thing (find all local max/min points, state which is \ which, and plot the surface) for the function f(x,y)=", FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ \(x\^2 - x\^4 - y\^4\)], FontColor->RGBColor[1, 0, 0]], StyleBox[". (Again, ", FontColor->RGBColor[1, 0, 0]], StyleBox["AspectRatio->1", "Input", FontColor->RGBColor[1, 0, 0]], StyleBox[" might help.)", FontColor->RGBColor[1, 0, 0]] }], "Text", FontSize->14], Cell[BoxData[ \(\[IndentingNewLine]\)], "Input"], Cell[TextData[{ StyleBox["Question 4:", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox[" Do the same thing (find all local max/min points, state which is \ which, and plot the surface) for the function f(x,y)=", FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ StyleBox[\(\(15 \( x\^2\) \(y\^2\) \[ExponentialE]\^\(\(-x\^2\) - \ y\^2\)\)\/\(x\^2 + y\^2\)\), FontSize->24]], FontColor->RGBColor[1, 0, 0]], StyleBox[". (Here you might want to play with the ", FontColor->RGBColor[1, 0, 0]], StyleBox["PlotPoints", "Input", FontColor->RGBColor[1, 0, 0]], StyleBox[" and ", FontColor->RGBColor[1, 0, 0]], StyleBox["PlotRange", "Input", FontColor->RGBColor[1, 0, 0]], StyleBox[" settings.)", FontColor->RGBColor[1, 0, 0]] }], "Text", FontSize->14], Cell[BoxData[ \(\[IndentingNewLine]\)], "Input"], Cell[TextData[{ StyleBox["Now that you're done, ", FontSize->18], "\n\t(1) \t", StyleBox["Clean up your work", FontWeight->"Bold"], " by deleting everything that's not needed. \n\t\tThere should only be the \ title, your names, section number, and the questions and your answers\n\t\t\ (with any needed explanations, graphs, etc.).\n\t(2) \tSave this to your \ disk.\n\t(3)\tGo back to Blackboard and upload your final lab report to the \ ", StyleBox["Digital Drop Box", FontWeight->"Bold"], ", located\n\t\tin the ", StyleBox["User Tools", FontWeight->"Bold"], " area. Remember that you have to first ", StyleBox["ADD", FontWeight->"Bold"], " your file to your digital drop box and \n\t\tthen ", StyleBox["SEND it to me!", FontWeight->"Bold"] }], "Text", CellFrame->{{0, 0}, {0, 2}}, FontSize->12] }, Open ]] }, FrontEndVersion->"4.2 for Macintosh", ScreenRectangle->{{0, 754}, {0, 578}}, AutoGeneratedPackage->None, WindowToolbars->"EditBar", WindowSize->{671, 545}, WindowMargins->{{1, Automatic}, {Automatic, 0}} ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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