(*********************************************************************** 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[ 8737, 307]*) (*NotebookOutlinePosition[ 9791, 343]*) (* CellTagsIndexPosition[ 9747, 339]*) (*WindowFrame->Normal*) Notebook[{ Cell["", "Input", AspectRatioFixed->False], Cell[TextData["Name: Jo Anne Kenyon"], "Section", Evaluatable->False, AspectRatioFixed->False], Cell[TextData["Title: Derivative of the Tangent"], "Section", Evaluatable->False, AspectRatioFixed->False], Cell[TextData[ "Description: Using the definition of derivative and \ngraphing to find the \ derivative of the tangent \n"], "Section", Evaluatable->False, AspectRatioFixed->False], Cell["", "Title", Evaluatable->False, AspectRatioFixed->False], Cell["", "Input", AspectRatioFixed->False], Cell[TextData[{ StyleBox["Derivative of Tangent", AspectRatioFixed->False, FontFamily->"New York", FontSize->24, FontColor->RGBColor[0, 1, 1]], StyleBox["\n\n\n", AspectRatioFixed->False] }], "Input", AspectRatioFixed->False], Cell[TextData[ "In the previous lesson, we saw that if f[x]=sin[x], then f'[x]=cos[x]. Now, \ let's find f'[x] when f[x]=tan[x].\nRemember, f'[x] represents instantaneous \ growth. We need to examine the limiting case of average growth rates as h \ closes in on 0. "], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[TextData["\n\n\n\n\n"], "Input", AspectRatioFixed->False], Cell[TextData["Clear[x,f]\nf[x_]=Tan[x]"], "Input", AspectRatioFixed->False], Cell["", "Input", AspectRatioFixed->False], Cell["", "Input", AspectRatioFixed->False], Cell[TextData[ "Remember, the average growth rate on the interval [x,x+h] in units on the \ y-axis per unit on the x-axis is:"], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[TextData["(f[x+h]-f[x])/h"], "Input", AspectRatioFixed->False], Cell["", "Input", AspectRatioFixed->False], Cell[CellGroupData[{Cell[TextData["(f[x+h] - f[x])/h as h->0"], "Section", Evaluatable->False, AspectRatioFixed->False], Cell[TextData[ "As with sin[x], the limiting case is hard to see from what we have above and \ it is going to be difficult to spot f'[x] by looking at this. Let's look at \ some plots of (f[x+h] - f[x])/h as h approaches 0.\n"], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[TextData["Let's start with h=.1. Here is the plot."], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[TextData[ "plot1=Plot[(f[x+.1]-f[x])/.1,{x,0,2Pi},PlotStyle->{RGBColor[1,0,0]}];"], "Input", AspectRatioFixed->False], Cell[TextData[ "Now, let h get closer to 0. Here is a graph when h=.01. "], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[TextData[ "plot2=Plot[(f[x+.01]-f[x])/.01,{x,0,2Pi},PlotStyle->{RGBColor[0,1,0]}]"], "Input", AspectRatioFixed->False], Cell[TextData["One last plot. Let h=001."], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[TextData[ "plot3=Plot[(f[x+.001]-f[x])/.001,{x,0,2Pi},PlotStyle->{RGBColor[0,0,1]}]"], "Input", AspectRatioFixed->False]}, Open]], Cell[CellGroupData[{Cell[TextData["A Possible Choice for f'[x]"], "Section", Evaluatable->False, AspectRatioFixed->False], Cell[CellGroupData[{Cell[TextData[ "Do these curves look like the curves of another trigonometric function? "], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[TextData[{ StyleBox["\n", AspectRatioFixed->False], StyleBox[ "That's correct, it does remind us of the graph of the secant except for \ the fact that these curves are all above the x-axis. Here is the graph of \ the secant.", AspectRatioFixed->False, FontWeight->"Plain"] }], "Input", AspectRatioFixed->False]}, Open]], Cell[TextData["Plot[Sec[x],{x,0,2Pi}]"], "Input", AspectRatioFixed->False], Cell[TextData[ "What might we do to the secant in order to stay above the x-axis?"], "Text", Evaluatable->False, AspectRatioFixed->False], Cell["", "Text", Evaluatable->False, AspectRatioFixed->False], Cell[TextData["\nLet's look at the graph of secant squared."], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[TextData["secantsqplot=Plot[(Sec[x])^2,{x,0,2Pi}]"], "Input", AspectRatioFixed->False]}, Open]], Cell[CellGroupData[{Cell[TextData["Verification of our Choice"], "Section", Evaluatable->False, AspectRatioFixed->False], Cell[TextData[ "Let's check this out by superimposing a plot of secant squared on each of \ the three average rate of growth graphs of the tangent."], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[TextData[ "Here is the graph of (f[x+.1] - f[x])/.1 and secant squared on the same \ axis:"], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[CellGroupData[{Cell[TextData["Show[plot1,secantsqplot]"], "Input", AspectRatioFixed->False], Cell[OutputFormData[ "\<\ The Unformatted text for this cell was not generated. Use options in the Actions Settings dialog box to control when Unformatted text is generated.\ \>", "\<\ -Graphics-\ \>"], "Output", Evaluatable->False, AspectRatioFixed->False]}, Open]], Cell[TextData["\n"], "Input", AspectRatioFixed->False], Cell[TextData["These graphs are pretty close. \n"], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[TextData[ "Here is a plot of (f[x+.01]-f[x])/.01 and secant squared:"], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[CellGroupData[{Cell[TextData["Show[plot2,secantsqplot]"], "Input", AspectRatioFixed->False], Cell[OutputFormData[ "\<\ The Unformatted text for this cell was not generated. Use options in the Actions Settings dialog box to control when Unformatted text is generated.\ \>", "\<\ -Graphics-\ \>"], "Output", Evaluatable->False, AspectRatioFixed->False], Cell[TextData[ "The two curves are almost concurrent. Last, but not least, here is the \ graph of (f[x+.001] - f[x])/.001 and secant squared on the same axes:"], "Text", Evaluatable->False, AspectRatioFixed->False], Cell[CellGroupData[{Cell[TextData["Show[plot3,secantsqplot]"], "Input", AspectRatioFixed->False], Cell[OutputFormData[ "\<\ The Unformatted text for this cell was not generated. Use options in the Actions Settings dialog box to control when Unformatted text is generated.\ \>", "\<\ -Graphics-\ \>"], "Output", Evaluatable->False, AspectRatioFixed->False]}, Open]]}, Open]], Cell["", "Input", AspectRatioFixed->False], Cell[TextData[ "The evidence tells us that the limiting case of the average growth rates\n \ (f[x+h]-f[x])/h = (tan[x+h] - tan[x])/h\nas h closes in on 0 is\n \ f'[x]=(sec[x])\[Currency].\n \n \nLet's check this \ numerically."], "Text", Evaluatable->False, AspectRatioFixed->False], Cell["", "Text", Evaluatable->False, AspectRatioFixed->False], Cell[TextData["Clear [f,x]\nf[x_]=Tan[x]\n\nf'[x]"], "Input", AspectRatioFixed->False], Cell[TextData[ "\n\nHurray!! We now know the derivative of tangent is secant squared.\n"], "Text", Evaluatable->False, AspectRatioFixed->False], Cell["", "Input", AspectRatioFixed->False], Cell["", "Input", AspectRatioFixed->False], Cell["", "Input", AspectRatioFixed->False]}, Open]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 640}, {0, 460}}, WindowToolbars->{}, CellGrouping->Manual, WindowSize->{520, 365}, WindowMargins->{{4, Automatic}, {30, Automatic}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, MacintoshSystemPageSetup->"\<\ AVU/IFiQKFD000000V:^/09R]g0000000OVaH097bCP0AP1Y06`0I@1^0642HZj` 0V:gT0000001nK500TO9>000000000000000009R[[0000000000000000000000 00000000000000000000000000000000\>" ] (*********************************************************************** Cached data follows. 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