(*********************************************************************** 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). 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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[ 958249, 43584]*) (*NotebookOutlinePosition[ 959304, 43620]*) (* CellTagsIndexPosition[ 959260, 43616]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{Cell[TextData[{ StyleBox["RATES OF CHANGE", Evaluatable->False, AspectRatioFixed->True, FontSize->11, FontWeight->"Plain", FontColor->GrayLevel[0]], StyleBox["\nSecant and Tangent Lines", Evaluatable->False, AspectRatioFixed->True, FontSize->12, FontColor->GrayLevel[0.333333]] }], "Title", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["<True, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[{ StyleBox["Rates of Change", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[1, 0, 0]], StyleBox["", Evaluatable->False, AspectRatioFixed->True] }], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "The VELOCITY of an object moving in a straight line is something\nwe \ should all be able to calculate, especially when we relate this\nto a car \ traveling on the interstate.", Evaluatable->False, AspectRatioFixed->True, FontColor->GrayLevel[0.333333]], StyleBox["", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Let d(t) be the distance in miles that a car travels due east in time t (in \ hours after noon). 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", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[1, 0, 0]], StyleBox["Below is the graphical interpretation.", Evaluatable->False, AspectRatioFixed->True, FontSize->10, FontColor->GrayLevel[0.499992]], StyleBox["", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[1, 0, 0]] }], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData[StyleBox["Instantaneous Rate of Change", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[1, 0, 0]]], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "What happens if we calculate the average rates of change of the function \ f(x)=x\:2039 over the intervals from x=1 to x=1+h for various values of h as \ h is getting smaller and smaller, that is h is \napproaching zero?", Evaluatable->False, AspectRatioFixed->True], StyleBox[ " (By doing these calculations, we are zeroing in on the behavior of the \ curve for a fixed value of x.)", Evaluatable->False, AspectRatioFixed->True, FontSize->10, FontColor->GrayLevel[0.499992]] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox[ "First let us start with x=1 and h=.05 as h is decreasing by .1 on each \ interval. ", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[1, 0, 0]]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Use ", Evaluatable->False, AspectRatioFixed->True, FontSize->10, FontColor->GrayLevel[0.499992]], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSize->10, FontSlant->"Italic", FontColor->GrayLevel[0.499992]], StyleBox[ " to make the calculation. Click on the input cell and hit enter.", Evaluatable->False, AspectRatioFixed->True, FontSize->10, FontColor->GrayLevel[0.499992]] }], "SmallText", Evaluatable->False, AspectRatioFixed->True, FontSize->10, FontColor->GrayLevel[0.499992]], Cell[CellGroupData[{Cell[TextData[ "\nClear[f,x,h]\nf[x_]:=x^3\nx=1;\n\ TableForm[Table[{h,{{\"from\",x,\"to\",x+h}},(f[x+h]-f[x])/h},\n\ {h,.5,.01,-.1}],\nTableSpacing->{0,10},\nTableHeadings->{{},{\"h\",\"Interval\ \",\n\"Average Rate of Change\"}}]"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ TableForm[{{0.5, {{\"from\", 1, \"to\", 1.5}}, 4.75}, {0.4, {{\"from\", 1, \"to\", 1.4}}, 4.36}, {0.3, {{\"from\", 1, \"to\", 1.3}}, 3.989999999999999999}, {0.2, {{\"from\", 1, \"to\", 1.2}}, 3.64}, {0.1, {{\"from\", 1, \"to\", 1.1}}, 3.31}}, TableSpacing -> {0, 10}, TableHeadings -> {{}, {\"h\", \"Interval\", \"Average Rate of Change\"}}]\ \>", "\<\ h Interval Average Rate of Change 0.5 from 1 to 1.5 4.75 0.4 from 1 to 1.4 4.36 0.3 from 1 to 1.3 3.99 0.2 from 1 to 1.2 3.64 0.1 from 1 to 1.1 3.31\ \>"], "Output", Evaluatable->False, LineSpacing->{0.6, 0}, AspectRatioFixed->True]}, Open]], Cell[TextData[StyleBox[ "Now let's see the results if we let x=1 and h=.1 and let h decrease by .002 \ on each interval.", Evaluatable->False, AspectRatioFixed->True, FontColor->RGBColor[1, 0, 0]]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "\nClear[f,x,h]\nf[x_]:=x^3\nx=1;\n\ TableForm[Table[{h,{{\"from\",x,\"to\",x+h}},(f[x+h]-f[x])/h},\n\ {h,.01,.001,-.002}],\nTableSpacing->{0,10},\n\ TableHeadings->{{},{\"h\",\"Interval\",\n\"Average Rate of Change\"}}]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ TableForm[{{0.01, {{\"from\", 1, \"to\", 1.01}}, 3.030100000000000003}, {0.008, {{\"from\", 1, \"to\", 1.008}}, 3.024063999999999987}, {0.006, {{\"from\", 1, \"to\", 1.006}}, 3.018036000000000001}, {0.003999999999999999999, {{\"from\", 1, \"to\", 1.004}}, 3.012015999999999983}, {0.002, {{\"from\", 1, \"to\", 1.002}}, 3.006004000000000055}}, TableSpacing -> {0, 10}, TableHeadings -> {{}, {\"h\", \"Interval\", \"Average Rate of Change\"}}]\ \>", "\<\ h Interval Average Rate of Change 0.01 from 1 to 1.01 3.0301 0.008 from 1 to 1.008 3.02406 0.006 from 1 to 1.006 3.01804 0.004 from 1 to 1.004 3.01202 0.002 from 1 to 1.002 3.006\ \>"], "Output", Evaluatable->False, LineSpacing->{0.6, 0}, AspectRatioFixed->True]}, Open]], Cell[TextData[{ StyleBox["Use ", Evaluatable->False, AspectRatioFixed->True, FontSize->10, FontColor->GrayLevel[0.499992]], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSize->10, FontSlant->"Italic", FontColor->GrayLevel[0.499992]], StyleBox[ " to make the calculation. 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