(*^ ::[ 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, e8, 24, "B Univers 65 Bold"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, "Garamond"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, e6, 14, "I Garamond LightItalic"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, a20, 18, "B Univers 65 Bold"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, a15, 14, "AGaramond Semibold"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12, 12, "Garamond"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "AGaramond"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "AGaramond"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-5, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L-5, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B65535, L-5, 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, "Garamond"; fontset = leftheader, inactive, L2, 12, "Garamond"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, 12, "Garamond"; fontset = leftfooter, inactive, L2, 12, "Garamond"; 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"; showRuler; currentKernel; ] :[font = text; inactive; preserveAspect; leftWrapOffset = 17; leftNameWrapOffset = 1] Mathematics 161 Laboratory 5 October 1, 1992 Name: _____________________________ Lab Partner: ___________________________ Consulted with: ____________________________________________________________ :[font = smalltext; inactive; preserveAspect; right] © Lafayette College, 1994 :[font = title; inactive; preserveAspect] Properties of Derivatives :[font = text; inactive; preserveAspect] In this laboratory you will use Mathematica's symbolic differentiation and graphing facilities to begin exploring some properties of derivatives. ;[s] 3:0,0;33,1;44,0;147,-1; 2:2,14,10,AGaramond,0,12,0,0,0;1,14,10,AGaramond,2,12,0,0,0; :[font = section; inactive; preserveAspect] Part 0: Differentiation by Rules :[font = subsection; inactive; preserveAspect] Preliminaries :[font = text; inactive; preserveAspect] As in last week's lab, we will use some special commands contained in the "DerivativeGraphics" package; recall, we input the package as follows: :[font = input; preserveAspect] < 1, while c(x) is decreasing for Ð1 < x < 1. ;[s] 32:0,0;35,1;36,0;37,1;38,0;42,1;43,2;44,0;48,1;49,0;133,1;134,0;154,1;155,0;179,1;180,0;225,1;226,0;293,1;294,0;295,1;296,0;316,1;317,0;331,1;332,0;344,1;345,0;346,1;347,0;372,1;373,0;379,-1; 3:16,14,10,AGaramond,0,12,0,0,0;15,14,10,AGaramond,2,12,0,0,0;1,11,7,AGaramond,32,9,0,0,0; :[font = text; inactive; preserveAspect] (b) Use Plot[{c[x],c'[x]}, {x,-3,3}, PlotStyle->{Red, Blue}] to plot the graphs of c and c' on the same set of axes. Use the plot to answer the following questions: ;[s] 7:0,0;8,1;60,0;84,2;85,0;90,2;91,0;167,-1; 3:4,14,10,AGaramond,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0;2,14,10,AGaramond,2,12,0,0,0; :[font = text; inactive; preserveAspect] i) What is the sign of c' in the intervals where c(x) is increasing? ;[s] 7:0,0;28,1;29,0;54,1;55,0;56,1;57,0;77,-1; 2:4,14,10,AGaramond,0,12,0,0,0;3,14,10,AGaramond,2,12,0,0,0; :[font = text; inactive; preserveAspect] ii) What is the sign of c' in the interval where c(x) is decreasing? ;[s] 7:0,0;29,1;30,0;54,1;55,0;56,1;57,0;77,-1; 2:4,14,10,AGaramond,0,12,0,0,0;3,14,10,AGaramond,2,12,0,0,0; :[font = text; inactive; preserveAspect] (c) Modify the Plot command from part (b) for the function g(x) = x2 Ð 2x Ð 3 and its derivative. Use the interval [Ð2,4] for your plot. ;[s] 12:0,0;15,3;19,0;59,1;60,0;61,1;62,0;66,1;67,2;68,0;72,1;73,0;138,-1; 4:6,14,10,AGaramond,0,12,0,0,0;4,14,10,AGaramond,2,12,0,0,0;1,12,9,AGaramond,32,10,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = text; inactive; preserveAspect] i) On what interval(s) is g increasing? What is the sign of g' there? ;[s] 5:0,0;32,1;33,0;66,1;67,0;79,-1; 2:3,14,10,AGaramond,0,12,0,0,0;2,14,10,AGaramond,2,12,0,0,0; :[font = text; inactive; preserveAspect] ii) On what interval(s) is g decreasing? What is the sign of g' there? ;[s] 5:0,0;33,1;34,0;67,1;68,0;80,-1; 2:3,14,10,AGaramond,0,12,0,0,0;2,14,10,AGaramond,2,12,0,0,0; :[font = text; inactive; preserveAspect] Based on your investigations in this part, make a conjecture on the relationship between graph of any function f and the sign of its derivative. Support your conjecture with reasoning beyond the "empirical evidence" you collected above. ;[s] 3:0,0;111,1;112,0;250,-1; 2:2,14,10,AGaramond,0,12,0,0,0;1,14,10,AGaramond,2,12,0,0,0; :[font = section; inactive; preserveAspect] Part 3: Tangent Lines, Secant Lines, & Velocity :[font = text; inactive; preserveAspect] In this part of the lab, h is the function h(t) = t/(t+1). Define it in Mathematica now. ;[s] 13:0,0;26,1;27,0;44,1;45,0;46,1;47,0;51,1;52,0;54,1;55,0;74,1;85,0;91,-1; 2:7,14,10,AGaramond,0,12,0,0,0;6,14,10,AGaramond,2,12,0,0,0; :[font = text; inactive; preserveAspect] (a) The command SecantLine[h, 0, 5, {x, -1/2, 6}] will generate a plot of the graph of h and the secant line for the graph passing through (0, h(0)) and (5, h(5)). Assign this plot the name plot1 by entering plot1 = %. (Mathematica will respond with -Graphics-, indicating it has assigned a graphic image as the "value" of plot1.) ;[s] 19:0,0;16,1;49,0;87,2;88,0;143,2;144,0;157,2;158,0;191,1;196,0;209,1;218,0;222,2;233,0;252,3;262,0;325,1;330,0;333,-1; 4:10,14,10,AGaramond,0,12,0,0,0;4,13,10,Courier,1,12,0,0,0;4,14,10,AGaramond,2,12,0,0,0;1,13,10,Courier,0,12,0,0,0; :[font = text; inactive; preserveAspect] (b) Use Mathematica to determine the exact slope of the secant line in this plot. (Do not try to read coordinates off the plot; use the two points on the line whose coordinates you know.) The slope is: :[font = text; inactive; preserveAspect] (c) Then find a point c in the interval 0 < t < 5 where the tangent line to the graph of h is parallel to that secant line. (Use Mathematica's Solve command for this computation. Recall, this command has the form Solve[expr1 == expr2, var] where you are trying to solve the equation expr1 = expr2 for the variable var.) A satisfactory c is: ;[s] 27:0,0;22,1;23,0;44,1;45,0;89,1;90,0;130,1;141,0;144,2;149,0;216,2;222,3;227,2;231,3;236,2;238,3;241,2;242,0;288,3;293,4;296,3;301,0;320,3;323,0;342,1;343,0;349,-1; 5:10,14,10,AGaramond,0,12,0,0,0;5,14,10,AGaramond,2,12,0,0,0;5,13,10,Courier,1,12,0,0,0;6,13,10,Courier,2,12,0,0,0;1,13,10,Courier,0,12,0,0,0; :[font = text; inactive; preserveAspect] (d) The command TangentLine[h, c, {x, -1/2, 6}] will generate a plot of the graph of h and the tangent line at the point c. (You must substitute the value for c you have found in order for the command to work.) Assign this plot the name plot2. To see the secant line and the tangent line on the same plot, enter Show[plot1, plot2]. Print out this plot and, based on the plot, answer the following question: do the two lines have the same slope? Explain how you can tell. ;[s] 15:0,0;16,1;31,2;32,1;47,0;85,2;86,0;121,2;122,0;160,2;161,0;239,1;244,0;315,1;333,0;482,-1; 3:7,14,10,AGaramond,0,12,0,0,0;4,13,10,Courier,1,12,0,0,0;4,14,10,AGaramond,2,12,0,0,0; :[font = text; inactive; preserveAspect] (e) Suppose now that the function h(t) is the position function of an object moving along a coordinate line. Interpret the equality of slopes of the two lines in terms of the velocity of that object. Your response should require only one or two sentences. ;[s] 5:0,0;34,1;35,0;36,1;37,0;258,-1; 2:3,14,10,AGaramond,0,12,0,0,0;2,14,10,AGaramond,2,12,0,0,0; ^*)