(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2"; MacintoshStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8, 24, "Times"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, "Times"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6, 14, "Times"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20, 18, "Times"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15, 14, "Times"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12, 12, "Times"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-4, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-4, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R32768, L-4, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-4, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B32768, L-4, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = leftheader, inactive, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, 12, "Times"; fontset = leftfooter, inactive, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; paletteColors = 128; showRuler; currentKernel; ] :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Math 225: Calculus III Assignment 3 :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Name: :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Section: :[font = text; inactive; preserveAspect; startGroup] I affirm that the solutions presented in this assignment are entirely my own work. :[font = subsection; inactive; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450; endGroup] Signature: :[font = subsection; inactive; preserveAspect; startGroup] Instructions :[font = text; inactive; preserveAspect] This assignment contains problems on vector functions, including their derivatives and integrals, from Chapter 12 in Finney & Thomas and Chapter 2 in the Lecture Notes. Remember, % stands for the previous output. For example, ;[s] 3:0,0;179,1;180,0;226,-1; 2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect; startGroup] r[t_] = t i + t^2 j + t^3 k :[font = output; output; inactive; preserveAspect; endGroup] {t, t^2, t^3} ;[o] 2 3 {t, t , t } :[font = input; preserveAspect; startGroup] D[%,t] :[font = output; output; inactive; preserveAspect; endGroup] {1, 2*t, 3*t^2} ;[o] 2 {1, 2 t, 3 t } :[font = input; preserveAspect; startGroup] Integrate[r[t],t] :[font = output; output; inactive; preserveAspect; endGroup] {t^2/2, t^3/3, t^4/4} ;[o] 2 3 4 t t t {--, --, --} 2 3 4 :[font = input; preserveAspect; startGroup] % /. t->4 :[font = output; output; inactive; preserveAspect; endGroup] {8, 64/3, 64} ;[o] 64 {8, --, 64} 3 :[font = text; inactive; preserveAspect] This last line shows how to substitute a value for a variable in an expression: expression /. variable -> value The double symbol /. applies the substitution that follows to the expression on the left. The substitution itself uses another double symbol -> (looks like an arrow). This example also shows that D[] and Integrate[] work correctly with vector functions. You can combine plots with the Show[] command. First save the plots with names and list the names in Show[]: p1 = Plot3D[...] p2 = Plot3D[...] Show[p1,p2] To get a 3D line segment representing a vector from p to q, use v1 = Vector[p,q] and include v1 in a Show[] command. See the notebooks AdvancedCalculusDemo, and TangentsNormals in the Notebook folder or the back of the Lecture Notes. Be sure to type in comments explaining what you are doing. Remember to uncheck Show In/Out Names in the File menu and close this group before printing. ;[s] 37:0,0;84,1;118,0;137,1;139,0;260,1;262,0;315,1;318,0;323,1;334,0;405,1;411,0;475,1;481,0;483,1;542,0;594,1;595,0;599,1;600,0;606,1;622,0;636,1;638,0;644,1;650,0;678,2;698,0;703,2;719,0;776,1;777,0;858,1;875,0;883,1;887,0;931,-1; 3:19,13,9,Times,0,12,0,0,0;16,13,10,Courier,1,12,0,0,0;2,13,9,Times,1,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Initialization :[font = input; initialization; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] *) <= 0. b) Find the particle's maximum speed for t >= 0. ;[s] 13:0,0;48,1;49,0;53,1;104,0;106,2;108,0;147,1;153,0;155,2;157,0;196,1;202,0;204,-1; 3:7,13,9,Times,0,12,0,0,0;4,13,10,Courier,1,12,0,0,0;2,13,9,Times,2,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450] Solution :[font = input; preserveAspect; endGroup] :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450; startGroup] Problem 5 :[font = text; inactive; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450] Let r[t_] = (3/4)t i + (1+t)^(3/2)j + (1-t)^(3/2)k. a) Find the unit tangent vector T[t], the unit normal vector n[t] (remember, the letter N is reserved for the function N[] in Mathematica), and the binormal vector B[t_] = Cross[T[t],n[t]] . b) Plot T[0], n[0] and B[0] along with the plot of r[t] near t == 0. ;[s] 29:0,0;3,1;50,0;52,2;54,0;84,1;88,0;113,1;117,0;140,1;141,0;171,1;174,0;178,2;189,0;216,1;240,0;243,2;245,0;251,1;255,0;257,1;261,0;266,1;270,0;294,1;298,0;304,1;310,0;313,-1; 3:15,13,9,Times,0,12,0,0,0;11,13,10,Courier,1,12,0,0,0;3,13,9,Times,2,12,0,0,0; :[font = subsubsection; inactive; noKeepOnOnePage; preserveAspect; leftNameWrapOffset = 18; rightWrapOffset = 450] Solution :[font = input; noKeepOnOnePage; preserveAspect; leftWrapOffset = 18; leftNameWrapOffset = 18; rightWrapOffset = 450; endGroup] ^*)