(*^ ::[ 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 = "X Window System Mathematica Notebook Front End Version 2.2"; X11StandardFontEncoding; fontset = title, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8, 24, fontName, "times"; fontset = subtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, fontName, "times"; fontset = subsubtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6, 14, fontName, "times"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20, 18, fontName, "times"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15, 14, fontName, "times"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12, 12, fontName, "times"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, fontName, "times"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, 12, fontName, "courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, fontName, "courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, 12, fontName, "courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, 12, fontName, "courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, 12, fontName, "courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, 12, fontName, "courier"; fontset = name, inactive, noPageBreakInGroup, nohscroll, preserveAspect, M7, italic, B65535, 10, fontName, "times"; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, 12, fontName, "times"; fontset = leftheader, 12, fontName, "times"; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, italic, 12, fontName, "times"; fontset = leftfooter, 12, fontName, "times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "courier"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times";paletteColors = 128; currentKernel; ] :[font = title; inactive; preserveAspect] Tangent Line Animation :[font = section; inactive; preserveAspect] The following is an animation which shows a "zoom in" on the graph of f(x)=3x^3 - 16x - 20 at the point x=3. The darker curve is the graph of the function f(x) while the lighter line is the secant line from (3,f(3)) to the point on the curve which is above the rightmost x value in the plot region. :[font = section; inactive; preserveAspect] To see the animation, do the following: If you are running Mathematica: double click on the picture below. The speed and other parameters in the animation are controlled by buttons which will appear at the bottom of the window. If you are running MathReader, select the cell bracket to the right of the picture below, and then select the "animate selected graphics" option under the Graphics menu. 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%%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -2.5 1 0.249443 0.0045826 [ [(2.75)] .25 0 0 2 Msboxa [(3)] .5 0 0 2 Msboxa [(3.25)] .75 0 0 2 Msboxa [(13)] -0.0125 .30902 1 0 Msboxa [(Slope of secant = 79.25)] .5 .61803 0 -2 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .25 0 m .25 .00625 L s P [(2.75)] .25 0 0 2 Mshowa p .002 w .5 0 m .5 .00625 L s P [(3)] .5 0 0 2 Mshowa p .002 w .75 0 m .75 .00625 L s P [(3.25)] .75 0 0 2 Mshowa p .002 w 0 0 m 1 0 L s P p .002 w 0 .30902 m .00625 .30902 L s P [(13)] -0.0125 .30902 1 0 Mshowa p .002 w 0 0 m 0 .61803 L s P P p p .002 w .25 .61178 m .25 .61803 L s P p .002 w .5 .61178 m .5 .61803 L s P p .002 w .75 .61178 m .75 .61803 L s P p .002 w 0 .61803 m 1 .61803 L s P [(Slope of secant = 79.25)] .5 .61803 0 -2 Mshowa p .002 w .99375 .30902 m 1 .30902 L s P p .002 w 1 0 m 1 .61803 L s P P p P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .01 w 0 .1893 m .04167 .19716 L .08333 .20539 L .125 .21399 L .16667 .22297 L .20833 .23232 L .25 .24207 L .29167 .25221 L .33333 .26275 L .375 .27369 L .41667 .28505 L .45833 .29682 L .5 .30902 L .54167 .32164 L .58333 .33471 L .625 .34821 L .66667 .36216 L .70833 .37657 L .75 .39143 L .79167 .40676 L .83333 .42256 L .875 .43884 L .91667 .4556 L .95833 .47286 L 1 .4906 L s P P p [ .01 .01 ] 0 setdash .0005 w .45039 .27112 m .45039 .34691 L .54961 .34691 L .54961 .27112 L .45039 .27112 L s P p p .02 w .5 .30902 Mdot 1 .4906 Mdot P P p .0005 w .5 .30902 m 1 .4906 L s P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174.25; animationSpeed = 3.75] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -3.27976 1.25992 0.233959 0.00577372 [ [(2.80157)] .25 0 0 2 Msboxa [(3)] .5 0 0 2 Msboxa [(3.19843)] .75 0 0 2 Msboxa [(13)] -0.0125 .30902 1 0 Msboxa [(Slope of secant = 76.1874)] .5 .61803 0 -2 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .25 0 m .25 .00625 L s P [(2.80157)] .25 0 0 2 Mshowa p .002 w .5 0 m .5 .00625 L s P [(3)] .5 0 0 2 Mshowa p .002 w .75 0 m .75 .00625 L s P [(3.19843)] .75 0 0 2 Mshowa p .002 w 0 0 m 1 0 L s P p .002 w 0 .30902 m .00625 .30902 L s P [(13)] -0.0125 .30902 1 0 Mshowa p .002 w 0 0 m 0 .61803 L s P P p p .002 w .25 .61178 m .25 .61803 L s P p .002 w .5 .61178 m .5 .61803 L s P p .002 w .75 .61178 m .75 .61803 L s P p .002 w 0 .61803 m 1 .61803 L s P [(Slope of secant = 76.1874)] .5 .61803 0 -2 Mshowa p .002 w .99375 .30902 m 1 .30902 L s P p .002 w 1 0 m 1 .61803 L s P P p P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .01 w 0 .18355 m .04167 .19229 L .08333 .20133 L .125 .21067 L .16667 .22032 L .20833 .23028 L .25 .24055 L .29167 .25114 L .33333 .26206 L .375 .2733 L .41667 .28487 L .45833 .29678 L .5 .30902 L .54167 .3216 L .58333 .33453 L .625 .3478 L .66667 .36143 L .70833 .37541 L .75 .38976 L .79167 .40446 L .83333 .41954 L .875 .43498 L .91667 .45081 L .95833 .467 L 1 .48359 L s P P p [ .01 .01 ] 0 setdash .0005 w .4375 .26127 m .4375 .35676 L .5625 .35676 L .5625 .26127 L .4375 .26127 L s P p p .02 w .5 .30902 Mdot 1 .48359 Mdot P P p .0005 w .5 .30902 m 1 .48359 L s P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174.25; animationSpeed = 3.75] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -4.2622 1.5874 0.214449 0.00727443 [ [(2.84251)] .25 0 0 2 Msboxa [(3)] .5 0 0 2 Msboxa [(3.15749)] .75 0 0 2 Msboxa [(13)] -0.0125 .30902 1 0 Msboxa [(Slope of secant = 73.8021)] .5 .61803 0 -2 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .25 0 m .25 .00625 L s P [(2.84251)] .25 0 0 2 Mshowa p .002 w .5 0 m .5 .00625 L s P [(3)] .5 0 0 2 Mshowa p .002 w .75 0 m .75 .00625 L s P [(3.15749)] .75 0 0 2 Mshowa p .002 w 0 0 m 1 0 L s P p .002 w 0 .30902 m .00625 .30902 L s P [(13)] -0.0125 .30902 1 0 Mshowa p .002 w 0 0 m 0 .61803 L s P P p p .002 w .25 .61178 m .25 .61803 L s P p .002 w .5 .61178 m .5 .61803 L s P p .002 w .75 .61178 m .75 .61803 L s P p .002 w 0 .61803 m 1 .61803 L s P [(Slope of secant = 73.8021)] .5 .61803 0 -2 Mshowa p .002 w .99375 .30902 m 1 .30902 L s P p .002 w 1 0 m 1 .61803 L s P P p P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .01 w 0 .17889 m .04167 .18834 L .08333 .19804 L .125 .20799 L .16667 .21819 L .20833 .22863 L .25 .23934 L .29167 .25029 L .33333 .26151 L .375 .27299 L .41667 .28473 L .45833 .29674 L .5 .30902 L .54167 .32156 L .58333 .33438 L .625 .34748 L .66667 .36085 L .70833 .37451 L .75 .38844 L .79167 .40266 L .83333 .41717 L .875 .43197 L .91667 .44706 L .95833 .46244 L 1 .47812 L s P P p [ .01 .01 ] 0 setdash .0005 w .42125 .24886 m .42125 .36917 L .57875 .36917 L .57875 .24886 L .42125 .24886 L s P p p .02 w .5 .30902 Mdot 1 .47812 Mdot P P p .0005 w .5 .30902 m 1 .47812 L s P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174.25; animationSpeed = 3.75] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -5.5 2 0.189869 0.00916521 [ [(2.875)] .25 0 0 2 Msboxa [(3)] .5 0 0 2 Msboxa [(3.125)] .75 0 0 2 Msboxa [(13)] -0.0125 .30902 1 0 Msboxa [(Slope of secant = 71.9375)] .5 .61803 0 -2 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .25 0 m .25 .00625 L s P [(2.875)] .25 0 0 2 Mshowa p .002 w .5 0 m .5 .00625 L s P [(3)] .5 0 0 2 Mshowa p .002 w .75 0 m .75 .00625 L s P [(3.125)] .75 0 0 2 Mshowa p .002 w 0 0 m 1 0 L s P p .002 w 0 .30902 m .00625 .30902 L s P [(13)] -0.0125 .30902 1 0 Mshowa p .002 w 0 0 m 0 .61803 L s P P p p .002 w .25 .61178 m .25 .61803 L s P p .002 w .5 .61178 m .5 .61803 L s P p .002 w .75 .61178 m .75 .61803 L s P p .002 w 0 .61803 m 1 .61803 L s P [(Slope of secant = 71.9375)] .5 .61803 0 -2 Mshowa p .002 w .99375 .30902 m 1 .30902 L s P p .002 w 1 0 m 1 .61803 L s P P p P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .01 w 0 .17512 m .04167 .18516 L .08333 .1954 L .125 .20583 L .16667 .21647 L .20833 .22732 L .25 .23836 L .29167 .24961 L .33333 .26107 L .375 .27274 L .41667 .28462 L .45833 .29671 L .5 .30902 L .54167 .32154 L .58333 .33427 L .625 .34722 L .66667 .3604 L .70833 .37379 L .75 .3874 L .79167 .40124 L .83333 .41531 L .875 .4296 L .91667 .44412 L .95833 .45887 L 1 .47385 L s P P p [ .01 .01 ] 0 setdash .0005 w .40079 .23323 m .40079 .38481 L .59921 .38481 L .59921 .23323 L .40079 .23323 L s P p p .02 w .5 .30902 Mdot 1 .47385 Mdot P P p .0005 w .5 .30902 m 1 .47385 L s P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174.25; animationSpeed = 3.75] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -7.05953 2.51984 0.1589 0.0115474 [ [(2.90079)] .25 0 0 2 Msboxa [(3)] .5 0 0 2 Msboxa [(3.09921)] .75 0 0 2 Msboxa [(13)] -0.0125 .30902 1 0 Msboxa [(Slope of secant = 70.4756)] .5 .61803 0 -2 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .25 0 m .25 .00625 L s P [(2.90079)] .25 0 0 2 Mshowa p .002 w .5 0 m .5 .00625 L s P [(3)] .5 0 0 2 Mshowa p .002 w .75 0 m .75 .00625 L s P [(3.09921)] .75 0 0 2 Mshowa p .002 w 0 0 m 1 0 L s P p .002 w 0 .30902 m .00625 .30902 L s P [(13)] -0.0125 .30902 1 0 Mshowa p .002 w 0 0 m 0 .61803 L s P P p p .002 w .25 .61178 m .25 .61803 L s P p .002 w .5 .61178 m .5 .61803 L s P p .002 w .75 .61178 m .75 .61803 L s P p .002 w 0 .61803 m 1 .61803 L s P [(Slope of secant = 70.4756)] .5 .61803 0 -2 Mshowa p .002 w .99375 .30902 m 1 .30902 L s P p .002 w 1 0 m 1 .61803 L s P P p P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .01 w 0 .17209 m .04167 .1826 L .08333 .19327 L .125 .20411 L .16667 .2151 L .20833 .22626 L .25 .23758 L .29167 .24907 L .33333 .26073 L .375 .27255 L .41667 .28453 L .45833 .29669 L .5 .30902 L .54167 .32151 L .58333 .33418 L .625 .34702 L .66667 .36004 L .70833 .37322 L .75 .38659 L .79167 .40013 L .83333 .41384 L .875 .42774 L .91667 .44181 L .95833 .45606 L 1 .4705 L s P P p [ .01 .01 ] 0 setdash .0005 w .375 .21353 m .375 .40451 L .625 .40451 L .625 .21353 L .375 .21353 L s P p p .02 w .5 .30902 Mdot 1 .4705 Mdot P P p .0005 w .5 .30902 m 1 .4705 L s P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174.25; animationSpeed = 3.75] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -9.02441 3.1748 0.119882 0.0145489 [ [(2.92125)] .25 0 0 2 Msboxa [(3)] .5 0 0 2 Msboxa [(3.07875)] .75 0 0 2 Msboxa [(13)] -0.0125 .30902 1 0 Msboxa [(Slope of secant = 69.3266)] .5 .61803 0 -2 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .25 0 m .25 .00625 L s P [(2.92125)] .25 0 0 2 Mshowa p .002 w .5 0 m .5 .00625 L s P [(3)] .5 0 0 2 Mshowa p .002 w .75 0 m .75 .00625 L s P [(3.07875)] .75 0 0 2 Mshowa p .002 w 0 0 m 1 0 L s P p .002 w 0 .30902 m .00625 .30902 L s P [(13)] -0.0125 .30902 1 0 Mshowa p .002 w 0 0 m 0 .61803 L s P P p p .002 w .25 .61178 m .25 .61803 L s P p .002 w .5 .61178 m .5 .61803 L s P p .002 w .75 .61178 m .75 .61803 L s P p .002 w 0 .61803 m 1 .61803 L s P [(Slope of secant = 69.3266)] .5 .61803 0 -2 Mshowa p .002 w .99375 .30902 m 1 .30902 L s P p .002 w 1 0 m 1 .61803 L s P P p P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .01 w 0 .16966 m .04167 .18055 L .08333 .19157 L .125 .20272 L .16667 .21401 L .20833 .22542 L .25 .23696 L .29167 .24864 L .33333 .26045 L .375 .27239 L .41667 .28446 L .45833 .29667 L .5 .30902 L .54167 .3215 L .58333 .33411 L .625 .34686 L .66667 .35975 L .70833 .37278 L .75 .38594 L .79167 .39924 L .83333 .41269 L .875 .42627 L .91667 .43999 L .95833 .45386 L 1 .46787 L s P P p [ .01 .01 ] 0 setdash .0005 w .34251 .18871 m .34251 .42933 L .65749 .42933 L .65749 .18871 L .34251 .18871 L s P p p .02 w .5 .30902 Mdot 1 .46787 Mdot P P p .0005 w .5 .30902 m 1 .46787 L s P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174.25; endGroup; animationSpeed = 3.75] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -11.5 4 0.0707217 0.0183304 [ [(2.9375)] .25 0 0 2 Msboxa [(3)] .5 0 0 2 Msboxa [(3.0625)] .75 0 0 2 Msboxa [(13)] -0.0125 .30902 1 0 Msboxa [(Slope of secant = 68.4219)] .5 .61803 0 -2 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .25 0 m .25 .00625 L s P [(2.9375)] .25 0 0 2 Mshowa p .002 w .5 0 m .5 .00625 L s P [(3)] .5 0 0 2 Mshowa p .002 w .75 0 m .75 .00625 L s P [(3.0625)] .75 0 0 2 Mshowa p .002 w 0 0 m 1 0 L s P p .002 w 0 .30902 m .00625 .30902 L s P [(13)] -0.0125 .30902 1 0 Mshowa p .002 w 0 0 m 0 .61803 L s P P p p .002 w .25 .61178 m .25 .61803 L s P p .002 w .5 .61178 m .5 .61803 L s P p .002 w .75 .61178 m .75 .61803 L s P p .002 w 0 .61803 m 1 .61803 L s P [(Slope of secant = 68.4219)] .5 .61803 0 -2 Mshowa p .002 w .99375 .30902 m 1 .30902 L s P p .002 w 1 0 m 1 .61803 L s P P p P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .01 w 0 .16771 m .04167 .17891 L .08333 .19021 L .125 .20162 L .16667 .21313 L .20833 .22475 L .25 .23647 L .29167 .2483 L .33333 .26023 L .375 .27226 L .41667 .28441 L .45833 .29666 L .5 .30902 L .54167 .32148 L .58333 .33405 L .625 .34674 L .66667 .35953 L .70833 .37242 L .75 .38543 L .79167 .39855 L .83333 .41178 L .875 .42511 L .91667 .43856 L .95833 .45212 L 1 .46579 L s P P p [ .01 .01 ] 0 setdash .0005 w .30157 .15743 m .30157 .4606 L .69843 .4606 L .69843 .15743 L .30157 .15743 L s P p p .02 w .5 .30902 Mdot 1 .46579 Mdot P P p .0005 w .5 .30902 m 1 .46579 L s P % End of Graphics MathPictureEnd ^*)