(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 8970, 348]*) (*NotebookOutlinePosition[ 9606, 371]*) (* CellTagsIndexPosition[ 9562, 367]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "Multivariable Calculus and ", StyleBox["Mathematica", FontSlant->"Italic"], ".", StyleBox[" ", FontSlant->"Italic"], "Exam 1." }], "Title"], Cell["\<\ Copyright: Kevin Coombes, Paul Green, and Ronald Lipsman, \ 1998.\ \>", "Subsubtitle"], Cell[CellGroupData[{ Cell["Type Your Name Here: ", "Subtitle"], Cell[CellGroupData[{ Cell["\<\ Immediately save a copy of this exam. When following these \ instructions, replace XX by your id number:\ \>", "Section"], Cell["Choose \"File -> Save As...\" from the menu.", "Subsection"], Cell["\<\ Change to the directory XX, which is a subdirectory of the Exam \ directory. (For example, if your id is 21, then you would change to directory \ 21.)\ \>", "Subsection"], Cell["\<\ Edit the name of the file to \"exam1-XX.nb\". (If your id is 21, \ call it exam1-21.nb.)\ \>", "Subsection"], Cell["\<\ Press the \"Save\" button. You may save the exam as many times as \ you like in this directory.\ \>", "Subsection"] }, Open ]], Cell[CellGroupData[{ Cell["General Instructions", "Section"], Cell["\<\ All questions must be answered in the Notebook. Before beginning, \ look over the entire exam. There are three questions, each containing several \ parts. Do not spend too much time on any one part.\ \>", "Subsection"], Cell["\<\ When answering the questions, make all your comments in text cells. \ Any input and output you create should appear in the corresponding type of \ cell. Please delete any extraneous and incorrect cells. Anything that appears \ will be graded! \ \>", "Subsection"] }, Open ]], Cell[CellGroupData[{ Cell["Question 1", "Section"], Cell[TextData[{ "Input the coordinates of the points ", Cell[BoxData[ \(TraditionalForm\`A\_1\)]], ", ", Cell[BoxData[ \(TraditionalForm\`A\_2\)]], ", ", Cell[BoxData[ \(TraditionalForm\`B\_1\)]], ", ", Cell[BoxData[ \(TraditionalForm\`B\_2\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`B\_3\)]], " by evaluating the following input cell:" }], "Text"], Cell[BoxData[ \(A1\ = \ {1, \ 0, \ 1}; \nA2\ = \ {2, \ 3, \ \(-5\)}; \n B1\ = \ {1, \ 1, \ 2}; \nB2\ = \ {3, \ 2, \ 1}; \n B3\ = \ {1, \ 0, \ 3}; \)], "Input"], Cell["\<\ You should also evaluate the following cell, which defines a new \ command for computing the lengths of vectors:\ \>", "Text"], Cell[BoxData[ \(norm[v_]\ := \ Sqrt[v.v]\)], "Input"], Cell[TextData[{ "(a) Find the length of the line segment ", Cell[BoxData[ \(TraditionalForm\`A\_1\ A\_2\)]], "." }], "Subsection"], Cell[TextData[{ "(b) Find the area of triangle ", Cell[BoxData[ \(TraditionalForm\`B\_1\ B\_2\ B\_3\)]], "." }], "Subsection"], Cell[TextData[{ "(c) Find an equation for the plane through ", Cell[BoxData[ \(TraditionalForm\`B\_1\)]], ", ", Cell[BoxData[ \(TraditionalForm\`B\_2\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`B\_3\)]], "." }], "Subsection"], Cell[TextData[{ "(d) Find parametric equations for the line through ", Cell[BoxData[ \(TraditionalForm\`A\_1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`A\_2\)]], "." }], "Subsection"], Cell[TextData[{ "(e)", Cell[BoxData[ \(\ \)]], "Find the coordinates of the point where the line through ", Cell[BoxData[ \(TraditionalForm\`A\_1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`A\_2\)]], " meets the plane through ", Cell[BoxData[ \(TraditionalForm\`B\_1\)]], ", ", Cell[BoxData[ \(TraditionalForm\`B\_2\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`B\_3\)]], "." }], "Subsection"] }, Open ]], Cell[CellGroupData[{ Cell["Question 2", "Section"], Cell[CellGroupData[{ Cell["\<\ (a) Consider the curve given parametrically in the following input \ cell:\ \>", "Subsection"], Cell[BoxData[ \(curve[t_] := \ {Cos[t], Sqrt[Sin[t]]}\)], "Input"] }, Open ]], Cell[TextData[{ "Verify that ", Cell[BoxData[ \(TraditionalForm\`curve[t]\)]], ", for ", Cell[BoxData[ \(TraditionalForm\`0\ \[LessEqual] \ t\ \ \[LessEqual] \ \[Pi]\)]], ", is a parametrization of the quartic curve\n\t", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`y\^4\)]], " = 1, ", Cell[BoxData[ \(TraditionalForm\`y\ \[GreaterEqual] \ 0\)]], "." }], "Subsection"], Cell[CellGroupData[{ Cell[TextData[{ "(b) Next, consider the parabola ", Cell[BoxData[ \(TraditionalForm\`\((2 y - \ 1)\)\^\(\ 2\)\)]], "= ", Cell[BoxData[ \(TraditionalForm\`x\)]], ". Parametrize this parabola; that is, fill in coordinates in the next \ input cell." }], "Subsection"], Cell[BoxData[ \(parabola[t_] := {\ \ \ \ , \ \ \ }\)], "Input"] }, Open ]], Cell[TextData[{ "(c) Plot the two curves, ", Cell[BoxData[ \(TraditionalForm\`curve[t]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`parabola[t]\)]], ", on the same graph. You should see more than one point of intersection." }], "Subsection"], Cell["\<\ (d) Use FindRoot to determine the coordinates of all points of \ intersection on the two curves.\ \>", "Subsection"] }, Open ]], Cell[CellGroupData[{ Cell["Question 3", "Section"], Cell[TextData[{ "Consider the curve given parametrically on the interval ", Cell[BoxData[ \(TraditionalForm\`\(-5\)\ \[LessEqual] \ t\ \[LessEqual] \ 5\)]], " by the equations in the following input cell:" }], "Text"], Cell[BoxData[ \(newcurve\ = \ {ArcSinh[t], \ \(-1\) + Sqrt[t^2 + 1], \ 3 Sqrt[1 + t^2] + 2 ArcSinh[t]}\)], "Input"], Cell["\<\ (a) Find the velocity, speed, and acceleration of the curve. (Simplify your \ answers.)\ \>", "Subsection"], Cell["(b) Find a numerical estimate of the arclength of the curve.", "Subsection"], Cell["\<\ (c) Graph the curve. Based only on the graph, decide if the curve lies \ entirely in a single plane.\ \>", "Subsection"], Cell[TextData[{ "(c) Recall the following formulas for the Frenet frame and the geometric \ invariants of a curve:\n\t", Cell[BoxData[ \(TraditionalForm\`T\ = \ \(v\ /\)\ || v || \)]], ",\n\t", Cell[BoxData[ \(TraditionalForm\`B\ = \ v\[Times]\(a\ /\)\ || v\[Times]a || \)]], ",\n\t", Cell[BoxData[ \(TraditionalForm\`N\ = \ B\[Times]T\)]], ",\n\t", Cell[BoxData[ \(TraditionalForm \`\[Kappa]\ = \ \( || v\[Times]a\) || \ \(/\ \( || v\( || \^3\)\)\)\)]], ",\n\t", Cell[BoxData[ FormBox[ RowBox[{"\[Tau]", " ", "=", " ", RowBox[{ RowBox[{ RowBox[{\((v\[Times]a)\), ".", RowBox[{"a", StyleBox["'", FontFamily->"Courier"]}]}], "/"}], " ", "||", \(v\[Times]a\), \( || \^2\)}]}], TraditionalForm]]], ".\nCompute as many of these quantities as you need in order to verify your \ answer to part (b). Explain your answer." }], "Subsection"], Cell[TextData[{ "(d) Compute the curvature of the curve. Is the curvature increasing, \ decreasing, or neither when ", Cell[BoxData[ \(TraditionalForm\`t = 0\)]], "?" }], "Subsection"] }, Open ]] }, Open ]] }, Open ]] }, FrontEndVersion->"X 3.0", ScreenRectangle->{{0, 1024}, {0, 768}}, WindowSize->{688, 490}, WindowMargins->{{114, Automatic}, {82, Automatic}} ] (*********************************************************************** 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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