(*^ ::[ 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, L3, e8, 24, "Calculus"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L2, e6, 18, "Calculus"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L2, e6, 14, "Calculus"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, L2, a20, 14, "Calculus"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, L2, a15, 12, "Calculus"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, L2, a12, 12, "Calculus"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L2, 12, "Calculus"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, B65535, L2, 12, "Calculus"; fontset = input, noPageBreakInGroup, preserveAspect, groupLikeInput, M42, N23, bold, L2, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, preserveAspect, groupLikeOutput, M42, N23, L2, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L2, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L2, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L2, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L2, 12, "Courier"; fontset = name, inactive, nowordwrap, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, B65535, L2, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7, L2, 10, "Times"; fontset = leftheader, inactive, L2, 10, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, L2, 12, "Times"; fontset = leftfooter, inactive, center, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L2, 10, "Geneva"; fontset = clipboard, inactive, noKeepOnOnePage, preserveAspect, M7, L2, 12, "New York"; fontset = completions, inactive, nowordwrap, noKeepOnOnePage, preserveAspect, M7, L2, 12, "New York"; fontset = special1, inactive, nowordwrap, noKeepOnOnePage, preserveAspect, M7, L2, 12, "New York"; fontset = special2, inactive, nowordwrap, noKeepOnOnePage, preserveAspect, center, M7, L2, 12, "New York"; fontset = special3, inactive, nowordwrap, noKeepOnOnePage, preserveAspect, right, M7, L2, 12, "New York"; fontset = special4, inactive, noKeepOnOnePage, preserveAspect, M7, bold, superscript, R21845, G21845, B21845, L2, 12, "C&M"; fontset = special5, inactive, noKeepOnOnePage, preserveAspect, M7, bold, superscript, R19708, G31200, B40959, L2, 12, "C&M"; paletteColors = 128; currentKernel; ] :[font = title; inactive; Cclosed; noPageBreak; dontPreserveAspect; startGroup] Mini-Project for Calculus and Mathematica Workshop ;[s] 3:0,2;31,1;43,2;52,-1; 3:0,47,32,Calculus,1,24,0,0,0;1,47,32,Calculus,3,24,65535,0,0;2,47,32,Calculus,1,24,65535,0,0; :[font = subsection; inactive; noPageBreak; dontPreserveAspect] Name: Shirley Barrette ;[s] 1:0,0;25,-1; 1:1,27,19,Calculus,1,12,65535,0,0; :[font = subsection; inactive; noPageBreak; dontPreserveAspect] Title: The Distance Function ;[s] 1:0,0;32,-1; 1:1,27,19,Calculus,1,12,65535,0,0; :[font = subsection; inactive; noPageBreak; dontPreserveAspect; endGroup] Description: This lesson introduces the Distance Function (d = rt) to 8æî graders. It also introduces the function for free falling objects (d=16tÛ) and an acceleration function v = rt + at. Solving systems of equations is shown by graphing. ;[s] 1:0,0;245,-1; 1:1,27,19,Calculus,1,12,65535,0,0; :[font = section; inactive; Cclosed; dontPreserveAspect; startGroup] IT KEEPS ON TICKING-Introduction ;[s] 1:0,0;34,-1; 1:1,31,22,Calculus,1,14,0,0,65535; :[font = subtitle; inactive; dontPreserveAspect] D = RT ;[s] 1:0,0;7,-1; 1:1,37,26,Calculus,1,18,65535,0,0; :[font = text; inactive; dontPreserveAspect; center] Contrary to popular belief, D = RT is not an abbreviation for ;[s] 3:0,2;28,1;36,2;64,-1; 3:0,27,19,Calculus,0,12,0,0,65535;1,27,19,Calculus,1,12,65535,0,0;2,27,19,Calculus,1,12,0,0,65535; :[font = text; inactive; dontPreserveAspect; center] Disaster equals a Raving Tiger. ;[s] 1:0,0;32,-1; 1:1,27,19,Calculus,1,12,65535,0,0; :[font = text; inactive; dontPreserveAspect; center] Au contraire, mon ami. Distance = (Rate) x (Time) IS THE ;[s] 3:0,0;26,1;52,0;61,-1; 2:2,27,19,Calculus,1,12,0,0,65535;1,27,19,Calculus,1,12,65535,0,0; :[font = subsubtitle; inactive; dontPreserveAspect] WONDERFUL DISTANCE FUNCTION. ;[s] 1:0,0;32,-1; 1:1,31,22,Calculus,1,14,65535,0,0; :[font = text; inactive; dontPreserveAspect; center] WHAT does this WONDERFUL function do? HOW does it WORK? ;[s] 1:0,0;56,-1; 1:1,27,19,Calculus,1,12,0,0,65535; :[font = text; inactive; dontPreserveAspect; center] Let's use Mathematica and find out. ;[s] 3:0,0;11,1;22,0;37,-1; 2:2,27,19,Calculus,1,12,65535,0,0;1,27,19,Calculus,3,12,65535,0,0; :[font = text; inactive; dontPreserveAspect] :[font = text; inactive; dontPreserveAspect] The function D = RT tells us that the DISTANCE an object travels VARIES DIRECTLY as the product of the constant RATE at which it travels and the TIME it travels. Therefore, DISTANCE is a function of RATE and TIME. ;[s] 3:0,0;13,1;19,0;216,-1; 2:2,27,19,Calculus,1,12,0,0,65535;1,27,19,Calculus,1,12,65535,0,0; :[font = text; inactive; dontPreserveAspect; center] WHOOPEE, YOU SAY? ;[s] 1:0,0;18,-1; 1:1,27,19,Calculus,1,12,65535,0,0; :[font = text; inactive; dontPreserveAspect; center; endGroup] Well, try this on for size. ;[s] 1:0,0;29,-1; 1:1,27,19,Calculus,1,12,0,0,65535; :[font = section; inactive; Cclosed; dontPreserveAspect; startGroup] SOME FUN WITH D = RT ;[s] 1:0,0;22,-1; 1:1,31,22,Calculus,1,14,65535,0,0; :[font = text; inactive; dontPreserveAspect] A car travels at a steady rate of 55 miles per hour. How far will the car travel in 4 hours? To solve, evaluate the function for r = 55 and t = 4. ;[s] 1:0,0;150,-1; 1:1,27,19,Calculus,1,12,0,0,65535; :[font = input; dontPreserveAspect] f[t] = 55 * 4 :[font = text; inactive; dontPreserveAspect] The car will travel 220 miles in 4 hours. How far will the car travel in 90 minutes? Change 90 minutes to hours, since the rate is given in miles per hour, and evaluate the the function. ;[s] 1:0,0;189,-1; 1:1,27,19,Calculus,1,12,0,0,65535; :[font = input; dontPreserveAspect] f[h] = N[90/60] :[font = input; dontPreserveAspect] Clear f[t] f[t] = 55 f[h] :[font = text; inactive; dontPreserveAspect; endGroup] The car will travel 82.5 miles in 90 minutes. ;[s] 1:0,0;47,-1; 1:1,27,19,Calculus,1,12,65535,0,0; :[font = section; inactive; Cclosed; dontPreserveAspect; startGroup] EXTENSIONS ;[s] 1:0,0;11,-1; 1:1,31,22,Calculus,1,14,65535,0,0; :[font = text; inactive; dontPreserveAspect] The function for distance D in feet traveled by a free-falling object in T seconds is: ;[s] 1:0,0;88,-1; 1:1,27,19,Calculus,1,12,0,0,65535; :[font = text; inactive; dontPreserveAspect; center] D = 16TÛ ;[s] 1:0,0;10,-1; 1:1,27,19,Calculus,1,12,65535,0,0; :[font = subsection; inactive; dontPreserveAspect; startGroup] How far will an object fall in 2 seconds? in 4 seconds? in 8 seconds? A GOOD TIME FOR A GRAPH! ;[s] 2:0,0;73,1;96,-1; 2:1,27,19,Calculus,1,12,0,0,65535;1,27,19,Calculus,1,12,65535,0,0; :[font = input; dontPreserveAspect] Clear[d,r,t] f[t_] = 16 t^2 Plot[f[t],{t,0,10}, AxesLabel->{"TIME","DISTANCE"}, PlotStyle->{RGBColor[0,0,1]}, PlotRange->All, Ticks->{Range[0,10,1],Range[64,1500,96]}]; :[font = input; dontPreserveAspect] Clear[d,r,t] :[font = input; dontPreserveAspect] f[t] = 16 * t^2 :[font = text; inactive; dontPreserveAspect] Now, substitute for T in the function. ;[s] 1:0,0;40,-1; 1:1,27,19,Calculus,1,12,0,0,65535; :[font = input; dontPreserveAspect; endGroup] f[T] ;[s] 1:0,0;6,-1; 1:1,14,10,Courier,1,12,65535,0,0; :[font = subsection; inactive; dontPreserveAspect; startGroup] 2. Does the object fall the same distance each second? Explain your answer. ;[s] 1:0,0;78,-1; 1:1,27,19,Calculus,1,12,0,0,65535; :[font = text; inactive; dontPreserveAspect; endGroup] No, the 1êæ second the object falls 16 feet; the 2öë second it falls 64 feet. ;[s] 1:0,0;80,-1; 1:1,27,19,Calculus,0,12,65535,0,0; :[font = text; inactive; dontPreserveAspect] Well, enough of this. Let's do some GOOD STUFF! ;[s] 3:0,0;38,1;48,0;50,-1; 2:2,27,19,Calculus,1,12,0,0,65535;1,27,19,Calculus,1,12,65535,0,0; :[font = subsubsection; inactive; dontPreserveAspect] You know how to graph a linear equation, so now let's graph TWO of them in order to solve problems. ;[s] 1:0,0;100,-1; 1:1,27,19,Calculus,1,12,0,0,65535; :[font = text; inactive; dontPreserveAspect] Here is a problem that can be solved by using a graph. ;[s] 1:0,0;56,-1; 1:1,27,19,Calculus,1,12,0,0,65535; :[font = text; inactive; dontPreserveAspect] At noon, Kelly begins a brisk walk at 4 miles per hour. At 1 P.M., Melissa leaves and runs after her at 6 miles per hour. At what time will Melissa catch up with Kelly? Make a graph showing the distances ;[s] 1:0,0;209,-1; 1:1,27,19,Calculus,1,12,0,0,65535; :[font = input; dontPreserveAspect; endGroup] Clear[f,r,t] plot1 = Plot[4 t,{t,0,10},PlotRange -> {0,60}, PlotStyle->{RGBColor[0,0,1]},DisplayFunction->Identity]; plot2 = Plot[6(t-1),{t,0,10},PlotRange->{0,60}, PlotStyle->{RGBColor[0,1,0]},DisplayFunction->Identity]; Show[{plot1,plot2}, AxesLabel->{"TIME","DISTANCE"}, Ticks->{Range[0,10,1],Range[5,60,5]}, DisplayFunction->$DisplayFunction]; :[font = section; inactive; Cclosed; dontPreserveAspect; startGroup] THINK AND TRY ;[s] 1:0,0;14,-1; 1:1,31,22,Calculus,1,14,65535,0,0; :[font = text; inactive; dontPreserveAspect] 1. A train leaves a station at 3 p.m. traveling at 40 mph. Another train leaves at 5:00 p.m. from the same station on a parallel track. The second train travels at 60 mph. When will the second train overtake the first train? ;[s] 1:0,0;231,-1; 1:1,27,19,Calculus,1,12,0,0,65535; :[font = input; dontPreserveAspect] Clear[f,r,t] plot1 = Plot[40(t),{t,0,10},PlotRange -> {0,600}, PlotStyle->{RGBColor[0,0,1]},DisplayFunction->Identity]; plot2 = Plot[60(t-2),{t,0,10},PlotRange->{0,600}, PlotStyle->{RGBColor[0,1,0]},DisplayFunction->Identity]; Show[{plot1,plot2}, AxesLabel->{"TIME","DISTANCE"}, Ticks->{Range[0,10,1],Range[5,600,50]}, DisplayFunction->$DisplayFunction]; :[font = text; inactive; dontPreserveAspect] WA-HA-TIZZY! The second train overtakes the first train at 9:00 pm. The function D = RT, can be solved for R or T. Divide both sides of the function by R AND THE RESULT IS: ;[s] 2:0,1;12,0;175,-1; 2:1,27,19,Calculus,1,12,0,0,65535;1,27,19,Calculus,1,12,65535,0,0; :[font = text; inactive; dontPreserveAspect; center] D/R = T {Distance/Rate = Time} or T = D/R {Time = Distance/Rate} ;[s] 1:0,0;66,-1; 1:1,27,19,Calculus,1,12,65535,0,0; :[font = subsection; inactive; dontPreserveAspect] 1. A satellite travels at 12,500 miles per hour. How far will it travel in a week? ;[s] 1:0,0;85,-1; 1:1,27,19,Calculus,1,12,0,0,65535; :[font = input; dontPreserveAspect] f[r] = 12500 * 7 * 24 :[font = subsection; inactive; dontPreserveAspect; startGroup] 2. A missle travels at 1,725 miles per hour. Find the distance traveled in 90 seconds. ;[s] 1:0,0;90,-1; 1:1,27,19,Calculus,1,12,0,0,65535; :[font = input; dontPreserveAspect; endGroup] f[t] = N[1725 * 90/60] :[font = subsection; inactive; dontPreserveAspect; startGroup] 3. Light travels at 186,000 miles per second. Find the distance traveled in 1 minute. ;[s] 1:0,0;88,-1; 1:1,27,19,Calculus,1,12,0,0,65535; :[font = input; dontPreserveAspect; endGroup] f[t] = 186000 * 60 :[font = subsection; inactive; dontPreserveAspect; center] 4. The formula for the distance traveled in meters by a freely falling object on the moon in t seconds is: d = 0.8tÛ ;[s] 4:0,0;94,1;95,0;108,2;118,-1; 3:2,27,19,Calculus,1,12,0,0,65535;1,27,19,Calculus,3,12,0,0,65535;1,27,19,Calculus,1,12,65535,0,0; :[font = text; inactive; dontPreserveAspect] The planet Earth's gravity is 6 times stronger than that of the moon. In the same period of time, objects will fall 6 times farther on the Earth than theyu will on the moon. How far in meters will an object on the Earth fall in 5 5 5 seconds?, 12 seconds? ;[s] 1:0,0;260,-1; 1:1,27,19,Calculus,1,12,0,0,65535; :[font = input; dontPreserveAspect] Clear[t,f] f[t] = 6 .8 5^2 :[font = input; dontPreserveAspect] Clear[t,f] f[t] = 6 .8 12^2 :[font = input; dontPreserveAspect] Plot[6 .8(t)^2,{t,0,12},PlotRange -> {0,600}, PlotStyle->{RGBColor[0,0,1]}, AxesLabel->{"TIME","DISTANCE"}, Ticks->{Range[0,12,1],Range[5,600,50]}, DisplayFunction->$DisplayFunction]; :[font = text; inactive; dontPreserveAspect] I LOVE IT! ;[s] 1:0,0;11,-1; 1:1,27,19,Calculus,1,12,65535,0,0; :[font = subsection; inactive; dontPreserveAspect] 5. A car traveling 30 feet per second accelerates by increasing it speed 8 feet per second each second. Find its new speed in 6 seconds. Use the formula v = rt + at. ;[s] 3:0,0;156,1;167,0;169,-1; 2:2,27,19,Calculus,1,12,0,0,65535;1,27,19,Calculus,1,12,65535,0,0; :[font = input; dontPreserveAspect] Clear[v,r,t,a] f[v] = t(r + a) f[v] = 6(30 + 8) :[font = text; inactive; dontPreserveAspect] This car's new speed is 228 ft/s. ;[s] 1:0,0;35,-1; 1:1,27,19,Calculus,1,12,65535,0,0; :[font = text; inactive; dontPreserveAspect; center] :[font = text; inactive; dontPreserveAspect; center] CHECK THIS OUT, DUDES! ;[s] 1:0,1;23,-1; 2:0,27,19,Calculus,0,12,0,0,0;1,27,19,Calculus,1,12,65535,0,0; :[font = text; inactive; dontPreserveAspect] I can't help it. I have to BOGGLE your BRAIN. I heard the following conversation the other day. ;[s] 1:0,1;100,-1; 2:0,27,19,Calculus,0,12,0,0,0;1,27,19,Calculus,1,12,65535,0,0; :[font = text; inactive; dontPreserveAspect; center] Alicia: How old are your 3 brothers? Kelly: The product of their ages is 36. The sum of their ages equals my age. Alicia: I know your age, but I need more facts to your brothers' ages. Kelly: The oldest boy is not a twin. Alicia: Now I can tell you their ages! FIND THE BOYS' AGES. ;[s] 9:0,1;39,2;123,0;125,2;131,1;219,2;256,1;298,2;319,1;388,-1; 3:1,27,19,Calculus,3,12,65535,0,0;4,27,19,Calculus,0,12,0,0,65535;4,27,19,Calculus,0,12,65535,0,0; :[font = text; inactive; dontPreserveAspect; center; endGroup] CHECK MARQUEE FOR COMING ATTRACTIONS! ;[s] 1:0,1;38,-1; 2:0,27,19,Calculus,0,12,0,0,0;1,27,19,Calculus,1,12,0,0,65535; ^*)