(*^ ::[ 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, cellOutline, groupLikeTitle, center, M18, O486, R65535, e8, 24, "Calculus"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M18, O486, bold, R21845, G21845, B21845, e6, 12, "Calculus"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M18, O486, R21845, G21845, B21845, e6, 12, "Calculus"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M18, O486, bold, R21845, G21845, B21845, a10, 12, "Calculus"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M18, O486, bold, R21845, G21845, B21845, a10, 12, "Calculus"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, O486, bold, R21845, G21845, B21845, a10, 12, "Calculus"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Calculus"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, B65535, 12, "Calculus"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M36, N23, O486, bold, L-5, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M36, N23, O486, L-5, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N23, O486, R65535, L-5, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N23, O486, L-5, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N23, O486, B65535, L-5, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M18, O486, l19, o2, w378, h214, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = leftheader, inactive, M18, O486, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M18, O486, 12, "Times"; fontset = leftfooter, inactive, M18, O486, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, whiteBox, M18, O486, bold, R21845, G21845, B21845, 12, "Calculus"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M18, O486, R21845, G21845, B21845, 12, "Calculus"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M18, O486, 10, "Courier"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M18, O486, 10, "Courier"; automaticGrouping; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] RATES OF CHANGE Secant and Tangent Lines ;[s] 2:0,2;15,1;41,-1; 3:0,21,15,Calculus,0,11,65535,0,0;1,26,19,Calculus,1,12,21845,21845,21845;1,21,15,Calculus,0,11,0,0,0; :[font = input; initialization; preserveAspect] *) <1/GoldenRatio, PlotRange->{-1,200}, Ticks->{{1,2,3,4},{0,100,165}}, AxesLabel->{"Time","Distance"}, DisplayFunction->Identity]; points=Graphics[{Blue,PointSize[.025],Point[{4,165}], Blue,PointSize[.025],Point[{1,0}], Blue,PointSize[.025],Point[{3,100}]}]; taw=Graphics[{Text["d(1)=0 miles",{1,10}], Text["d(3)=100 miles",{3,90}], Text["d(4)=165 miles",{4,155}]}]; Show[trip,points,taw,DisplayFunction-> $DisplayFunction]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 346; pictureHeight = 214; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0515488 0.184928 0.0030748 0.0030748 [ [(1)] .23648 .00307 0 2 Msboxa [(2)] .42141 .00307 0 2 Msboxa [(3)] .60633 .00307 0 2 Msboxa [(4)] .79126 .00307 0 2 Msboxa [(Time)] 1.025 .00307 -1 0 Msboxa [(100)] .03905 .31055 1 0 Msboxa [(165)] .03905 .51042 1 0 Msboxa [(Distance)] .05155 .61803 0 -4 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 .23648 .00307 m .23648 .00932 L s P [(1)] .23648 .00307 0 2 Mshowa p .002 w .42141 .00307 m .42141 .00932 L s P [(2)] .42141 .00307 0 2 Mshowa p .002 w .60633 .00307 m .60633 .00932 L s P [(3)] .60633 .00307 0 2 Mshowa p .002 w .79126 .00307 m .79126 .00932 L s P [(4)] .79126 .00307 0 2 Mshowa [(Time)] 1.025 .00307 -1 0 Mshowa p .002 w 0 .00307 m 1 .00307 L s P p .002 w .05155 .31055 m .0578 .31055 L s P [(100)] .03905 .31055 1 0 Mshowa p .002 w .05155 .51042 m .0578 .51042 L s P [(165)] .03905 .51042 1 0 Mshowa [(Distance)] .05155 .61803 0 -4 Mshowa p .002 w .05155 0 m .05155 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p p .004 w .02381 .00307 m .06349 .00307 L .10317 .00307 L .14286 .00307 L .18254 .00307 L .22222 .00307 L .2619 .00307 L .30159 .00307 L .34127 .00307 L .38095 .00307 L .42063 .00307 L .46032 .00307 L .5 .00307 L .53968 .00307 L .57937 .00307 L .61905 .00307 L .65873 .00307 L .69841 .00307 L .7381 .00307 L .77778 .00307 L .81746 .00307 L .85714 .00307 L .89683 .00307 L .93651 .00307 L .97619 .00307 L s P P p 0 0 1 r .025 w .79126 .51042 Mdot .23648 .00307 Mdot .60633 .31055 Mdot P p [(d\(1\)=0 miles)] .23648 .03382 0 0 Mshowa [(d\(3\)=100 miles)] .60633 .27981 0 0 Mshowa [(d\(4\)=165 miles)] .79126 .47967 0 0 Mshowa P P % End of Graphics MathPictureEnd :[font = input; preserveAspect; startGroup] trip=Plot[0,{x,-.15,5}, AspectRatio->1, PlotRange->{-1,200}, Ticks->{{1,2,3,4},{0,100,165}}, AxesLabel->{"Time","Distance"}, DisplayFunction->Identity]; points=Graphics[{Blue,PointSize[.025],Point[{4,165}], Blue,PointSize[.025],Point[{1,0}], Blue,PointSize[.025],Point[{3,100}]}]; taw=Graphics[{Text["d(1)=0 miles",{1,10}], Text["d(3)=100 miles",{3,90}], Text["d(4)=165 miles",{4,155}]}]; route=Graphics[{RGBColor[1,0,0],Line[{{1,0},{3,100}}]}]; Show[trip,points,taw,route, DisplayFunction-> $DisplayFunction]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 214; pictureHeight = 214; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0515488 0.184928 0.00497512 0.00497512 [ [(1)] .23648 .00498 0 2 Msboxa [(2)] .42141 .00498 0 2 Msboxa [(3)] .60633 .00498 0 2 Msboxa [(4)] .79126 .00498 0 2 Msboxa [(Time)] 1.025 .00498 -1 0 Msboxa [(100)] .03905 .50249 1 0 Msboxa [(165)] .03905 .82587 1 0 Msboxa [(Distance)] .05155 1 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .23648 .00498 m .23648 .01123 L s P [(1)] .23648 .00498 0 2 Mshowa p .002 w .42141 .00498 m .42141 .01123 L s P [(2)] .42141 .00498 0 2 Mshowa p .002 w .60633 .00498 m .60633 .01123 L s P [(3)] .60633 .00498 0 2 Mshowa p .002 w .79126 .00498 m .79126 .01123 L s P [(4)] .79126 .00498 0 2 Mshowa [(Time)] 1.025 .00498 -1 0 Mshowa p .002 w 0 .00498 m 1 .00498 L s P p .002 w .05155 .50249 m .0578 .50249 L s P [(100)] .03905 .50249 1 0 Mshowa p .002 w .05155 .82587 m .0578 .82587 L s P [(165)] .03905 .82587 1 0 Mshowa [(Distance)] .05155 1 0 -4 Mshowa p .002 w .05155 0 m .05155 1 L s P P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p p p .004 w .02381 .00498 m .06349 .00498 L .10317 .00498 L .14286 .00498 L .18254 .00498 L .22222 .00498 L .2619 .00498 L .30159 .00498 L .34127 .00498 L .38095 .00498 L .42063 .00498 L .46032 .00498 L .5 .00498 L .53968 .00498 L .57937 .00498 L .61905 .00498 L .65873 .00498 L .69841 .00498 L .7381 .00498 L .77778 .00498 L .81746 .00498 L .85714 .00498 L .89683 .00498 L .93651 .00498 L .97619 .00498 L s P P p 0 0 1 r .025 w .79126 .82587 Mdot .23648 .00498 Mdot .60633 .50249 Mdot P p [(d\(1\)=0 miles)] .23648 .05473 0 0 Mshowa [(d\(3\)=100 miles)] .60633 .45274 0 0 Mshowa [(d\(4\)=165 miles)] .79126 .77612 0 0 Mshowa P p 1 0 0 r .004 w .23648 .00498 m .60633 .50249 L s P P % End of Graphics MathPictureEnd :[font = input; preserveAspect] How would we calculate the car's velocity-or average rate of change from 1:00p.m. to 3:00p.m. ? ;[s] 3:0,1;33,0;42,1;97,-1; 2:1,25,19,Calculus,0,12,65535,0,0;2,25,19,Calculus,0,12,0,0,65535; :[font = text; inactive; preserveAspect] We need to calculate d(b)-d(a) / b-a . For this problem we have [(d(3)-d(1)] / (3-1) = (100-0) / 2 = 100/2 = 50 . The average velocity or rate of change between these two points is 50 mph. ;[s] 3:0,0;127,1;135,0;193,-1; 2:2,26,19,Calculus,0,12,0,0,65535;1,26,19,Calculus,0,12,65535,0,0; :[font = input; preserveAspect; startGroup] trip=Plot[0,{x,-.15,5}, AspectRatio->1, PlotRange->{-1,200}, Ticks->{{1,2,3,4},{0,100,165}}, AxesLabel->{"Time","Distance"}, DisplayFunction->Identity]; points=Graphics[{Blue,PointSize[.025],Point[{4,165}], Blue,PointSize[.025],Point[{1,0}], Blue,PointSize[.025],Point[{3,100}]}]; taw=Graphics[{Text["d(1)=0 miles",{1,10}], Text["d(3)=100 miles",{3,90}], Text["d(4)=165 miles",{4,155}]}]; route=Graphics[{RGBColor[1,0,0],Line[{{3,100},{4,165}}]}]; Show[trip,points,taw,route, DisplayFunction-> $DisplayFunction]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 214; pictureHeight = 214; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0515488 0.184928 0.00497512 0.00497512 [ [(1)] .23648 .00498 0 2 Msboxa [(2)] .42141 .00498 0 2 Msboxa [(3)] .60633 .00498 0 2 Msboxa [(4)] .79126 .00498 0 2 Msboxa [(Time)] 1.025 .00498 -1 0 Msboxa [(100)] .03905 .50249 1 0 Msboxa [(165)] .03905 .82587 1 0 Msboxa [(Distance)] .05155 1 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 1.001 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .23648 .00498 m .23648 .01123 L s P [(1)] .23648 .00498 0 2 Mshowa p .002 w .42141 .00498 m .42141 .01123 L s P [(2)] .42141 .00498 0 2 Mshowa p .002 w .60633 .00498 m .60633 .01123 L s P [(3)] .60633 .00498 0 2 Mshowa p .002 w .79126 .00498 m .79126 .01123 L s P [(4)] .79126 .00498 0 2 Mshowa [(Time)] 1.025 .00498 -1 0 Mshowa p .002 w 0 .00498 m 1 .00498 L s P p .002 w .05155 .50249 m .0578 .50249 L s P [(100)] .03905 .50249 1 0 Mshowa p .002 w .05155 .82587 m .0578 .82587 L s P [(165)] .03905 .82587 1 0 Mshowa [(Distance)] .05155 1 0 -4 Mshowa p .002 w .05155 0 m .05155 1 L s P P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p p p .004 w .02381 .00498 m .06349 .00498 L .10317 .00498 L .14286 .00498 L .18254 .00498 L .22222 .00498 L .2619 .00498 L .30159 .00498 L .34127 .00498 L .38095 .00498 L .42063 .00498 L .46032 .00498 L .5 .00498 L .53968 .00498 L .57937 .00498 L .61905 .00498 L .65873 .00498 L .69841 .00498 L .7381 .00498 L .77778 .00498 L .81746 .00498 L .85714 .00498 L .89683 .00498 L .93651 .00498 L .97619 .00498 L s P P p 0 0 1 r .025 w .79126 .82587 Mdot .23648 .00498 Mdot .60633 .50249 Mdot P p [(d\(1\)=0 miles)] .23648 .05473 0 0 Mshowa [(d\(3\)=100 miles)] .60633 .45274 0 0 Mshowa [(d\(4\)=165 miles)] .79126 .77612 0 0 Mshowa P p 1 0 0 r .004 w .60633 .50249 m .79126 .82587 L s P P % End of Graphics MathPictureEnd :[font = text; inactive; preserveAspect] Similarly, the average rate of change can be calculated between 3:00 and 4:00pm. We have [d(4)-d(3) ] / (4-3) = (165-100) / 1 = 65mph. ;[s] 1:0,1;163,-1; 2:0,26,19,Calculus,0,12,0,0,0;1,26,19,Calculus,0,12,0,0,65535; :[font = smalltext; inactive; preserveAspect] This definition generalizes to the average rate of change for an arbitrary function. :[font = text; inactive; preserveAspect] THE AVERAGE RATE OF CHANGE of the functiony=f(x) over the intervalfrom x=a to x=b, with a1/GoldenRatio, Ticks->{{-5,-4,-3,-2,-1,0,1,2,3,4,5}, {-16,-12,-8,-4,0,4,8,12,16}}, PlotRange->{-18,18}, AxesLabel->{"x","y"}, DisplayFunction->Identity]; pts=Graphics[{Blue,PointSize[.025],Point[{-1,-4}], Blue,PointSize[.025],Point[{4,11}], Red,PointSize[.025],Point[{1,2}], Red,PointSize[.025],Point[{2,5}]}]; rt=Graphics[{Green,Line[{{-1,-4},{4,11}}],SkyBlue, Line[{{1,2},{2,5}}]}]; Show[tgra,pts,rt, DisplayFunction->$DisplayFunction]; :[font = message; inactive; preserveAspect] General::spell1: Possible spelling error: new symbol name "tawa" is similar to existing symbol "taw". :[font = output; output; inactive; preserveAspect] -1 + 3*x ;[o] -1 + 3 x :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 346; pictureHeight = 214; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0952381 0.309017 0.0171676 [ [(-5)] .02381 .30902 0 2 Msboxa [(-4)] .11905 .30902 0 2 Msboxa [(-3)] .21429 .30902 0 2 Msboxa [(-2)] .30952 .30902 0 2 Msboxa [(-1)] .40476 .30902 0 2 Msboxa [(1)] .59524 .30902 0 2 Msboxa [(2)] .69048 .30902 0 2 Msboxa [(3)] .78571 .30902 0 2 Msboxa [(4)] .88095 .30902 0 2 Msboxa [(5)] .97619 .30902 0 2 Msboxa [(x)] 1.025 .30902 -1 0 Msboxa [(-16)] .4875 .03434 1 0 Msboxa [(-12)] .4875 .10301 1 0 Msboxa [(-8)] .4875 .17168 1 0 Msboxa [(-4)] .4875 .24035 1 0 Msboxa [(4)] .4875 .37769 1 0 Msboxa [(8)] .4875 .44636 1 0 Msboxa [(12)] .4875 .51503 1 0 Msboxa [(16)] .4875 .5837 1 0 Msboxa [(y)] .5 .61803 0 -4 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 .02381 .30902 m .02381 .31527 L s P [(-5)] .02381 .30902 0 2 Mshowa p .002 w .11905 .30902 m .11905 .31527 L s P [(-4)] .11905 .30902 0 2 Mshowa p .002 w .21429 .30902 m .21429 .31527 L s P [(-3)] .21429 .30902 0 2 Mshowa p .002 w .30952 .30902 m .30952 .31527 L s P [(-2)] .30952 .30902 0 2 Mshowa p .002 w .40476 .30902 m .40476 .31527 L s P [(-1)] .40476 .30902 0 2 Mshowa p .002 w .59524 .30902 m .59524 .31527 L s P [(1)] .59524 .30902 0 2 Mshowa p .002 w .69048 .30902 m .69048 .31527 L s P [(2)] .69048 .30902 0 2 Mshowa p .002 w .78571 .30902 m .78571 .31527 L s P [(3)] .78571 .30902 0 2 Mshowa p .002 w .88095 .30902 m .88095 .31527 L s P [(4)] .88095 .30902 0 2 Mshowa p .002 w .97619 .30902 m .97619 .31527 L s P [(5)] .97619 .30902 0 2 Mshowa [(x)] 1.025 .30902 -1 0 Mshowa p .002 w 0 .30902 m 1 .30902 L s P p .002 w .5 .03434 m .50625 .03434 L s P [(-16)] .4875 .03434 1 0 Mshowa p .002 w .5 .10301 m .50625 .10301 L s P [(-12)] .4875 .10301 1 0 Mshowa p .002 w .5 .17168 m .50625 .17168 L s P [(-8)] .4875 .17168 1 0 Mshowa p .002 w .5 .24035 m .50625 .24035 L s P [(-4)] .4875 .24035 1 0 Mshowa p .002 w .5 .37769 m .50625 .37769 L s P [(4)] .4875 .37769 1 0 Mshowa p .002 w .5 .44636 m .50625 .44636 L s P [(8)] .4875 .44636 1 0 Mshowa p .002 w .5 .51503 m .50625 .51503 L s P [(12)] .4875 .51503 1 0 Mshowa p .002 w .5 .5837 m .50625 .5837 L s P [(16)] .4875 .5837 1 0 Mshowa [(y)] .5 .61803 0 -4 Mshowa p .002 w .5 0 m .5 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p p .004 w .02381 .03434 m .06349 .05579 L .10317 .07725 L .14286 .09871 L .18254 .12017 L .22222 .14163 L .2619 .16309 L .30159 .18455 L .34127 .20601 L .38095 .22747 L .42063 .24893 L .46032 .27039 L .5 .29185 L .53968 .31331 L .57937 .33477 L .61905 .35623 L .65873 .37769 L .69841 .39915 L .7381 .42061 L .77778 .44207 L .81746 .46353 L .85714 .48499 L .89683 .50644 L .93651 .5279 L .97619 .54936 L s P P p 0 0 1 r .025 w .40476 .24035 Mdot .88095 .49786 Mdot 1 0 0 r .59524 .34335 Mdot .69048 .39486 Mdot P p 0 1 0 r .004 w .40476 .24035 m .88095 .49786 L s .529 .808 .922 r .59524 .34335 m .69048 .39486 L s P P % End of Graphics MathPictureEnd :[font = input; preserveAspect] :[font = text; inactive; preserveAspect] We see both calculations yield an average rate of change of 3, which happens to be the slope of the line. f(x)=3x+1 ;[s] 3:0,0;113,1;114,0;118,-1; 2:2,26,19,Calculus,0,12,0,0,65535;1,26,19,Calculus,0,12,65535,0,0; :[font = input; preserveAspect] :[font = text; inactive; preserveAspect] Let us consider the function f(x)=xÜ (which is a nonlinear function). Compute the average rate of change of f(x) over the interval a.) from x=1 to x=3 and b.) from x=1 to x=1.5 :[font = smalltext; inactive; preserveAspect; fontColorRed = 32767; fontColorGreen = 32767; fontColorBlue = 32767; fontSize = 10] Use Mathematica to make the calculation. Click on the input cell and hit enter. ;[s] 3:0,0;4,1;15,0;81,-1; 2:2,19,13,Calculus,0,10,32767,32767,32767;1,19,13,Calculus,2,10,32767,32767,32767; :[font = input; preserveAspect; startGroup] Clear[f,x] f[x_]:=x^3 (f[3]-f[1])/(3-1) (f[1.5]-f[1])/(1.5-1) :[font = output; output; inactive; preserveAspect] 13 ;[o] 13 :[font = output; output; inactive; preserveAspect; endGroup] 4.75 ;[o] 4.75 :[font = input; preserveAspect; startGroup] Clear[f,x,tgra,pts,tawa,rt] f[x_]:=x^3 tgra=Plot[f[x],{x,-4,4}, AspectRatio->1/GoldenRatio, PlotRange->{-25,35}, AxesLabel->{"x","y"}, DisplayFunction->Identity]; pts=Graphics[{Blue,PointSize[.025],Point[{1,f[1]}], Blue,PointSize[.025],Point[{3,f[3]}], Red,PointSize[.025],Point[{1.5,f[1.5]}]}]; rt=Graphics[{Green,Thickness[.01],Line[{{1,f[1]},{3,f[3]}}],SkyBlue, Thickness[.01], Line[{{1,f[1]},{1.5,f[1.5]}}]}]; Show[tgra,pts,rt, DisplayFunction->$DisplayFunction]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 346; pictureHeight = 214; endGroup] %! 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Below is the graphical interpretation. ;[s] 3:0,1;85,2;123,1;124,-1; 3:0,26,19,Calculus,0,12,0,65535,0;2,26,19,Calculus,0,12,65535,0,0;1,19,13,Calculus,0,10,32767,32767,32767; :[font = section; inactive; preserveAspect; startGroup] Instantaneous Rate of Change ;[s] 1:0,1;29,-1; 2:0,26,19,Calculus,1,12,21845,21845,21845;1,26,19,Calculus,1,12,65535,0,0; :[font = text; inactive; preserveAspect] What happens if we calculate the average rates of change of the function f(x)=xÜ 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 approaching zero? (By doing these calculations, we are zeroing in on the behavior of the curve for a fixed value of x.) ;[s] 2:0,0;210,1;314,-1; 2:1,26,19,Calculus,0,12,0,0,0;1,19,13,Calculus,0,10,32767,32767,32767; :[font = text; inactive; preserveAspect] First let us start with x=1 and h=.05 as h is decreasing by .1 on each interval. ;[s] 1:0,0;83,-1; 1:1,26,19,Calculus,0,12,65535,0,0; :[font = smalltext; inactive; preserveAspect; fontColorRed = 32767; fontColorGreen = 32767; fontColorBlue = 32767; fontSize = 10] Use Mathematica to make the calculation. Click on the input cell and hit enter. ;[s] 3:0,0;4,1;15,0;81,-1; 2:2,19,13,Calculus,0,10,32767,32767,32767;1,19,13,Calculus,2,10,32767,32767,32767; :[font = input; preserveAspect; startGroup] Clear[f,x,h] f[x_]:=x^3 x=1; TableForm[Table[{h,{{"from",x,"to",x+h}},(f[x+h]-f[x])/h}, {h,.5,.01,-.1}], TableSpacing->{0,10}, TableHeadings->{{},{"h","Interval", "Average Rate of Change"}}] :[font = output; output; inactive; preserveAspect; fontLeading = 0; endGroup] 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"}}] ;[o] 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 :[font = text; inactive; preserveAspect] Now let's see the results if we let x=1 and h=.1 and let h decrease by .002 on each interval. ;[s] 1:0,0;95,-1; 1:1,26,19,Calculus,0,12,65535,0,0; :[font = input; preserveAspect; startGroup] Clear[f,x,h] f[x_]:=x^3 x=1; TableForm[Table[{h,{{"from",x,"to",x+h}},(f[x+h]-f[x])/h}, {h,.01,.001,-.002}], TableSpacing->{0,10}, TableHeadings->{{},{"h","Interval", "Average Rate of Change"}}] :[font = output; output; inactive; preserveAspect; fontLeading = 0; endGroup] 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"}}] ;[o] 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 :[font = smalltext; inactive; preserveAspect; fontColorRed = 32767; fontColorGreen = 32767; fontColorBlue = 32767; fontSize = 10] Use Mathematica to make the calculation. Click on the input cell and hit enter. ;[s] 3:0,0;4,1;15,0;81,-1; 2:2,19,13,Calculus,0,10,32767,32767,32767;1,19,13,Calculus,2,10,32767,32767,32767; :[font = text; inactive; preserveAspect] We see that the Average rate of change seems to be approaching 3 as the interval is getting closer and closer to 1 from the right for the function f(x)=xÜ. What happens if we approach 1 from the left? ;[s] 2:0,0;156,1;201,-1; 2:1,26,19,Calculus,0,12,65535,0,0;1,26,19,Calculus,0,12,0,0,65535; :[font = input; preserveAspect; startGroup] Clear[f,x,h] f[x_]:=x^3 x=1; TableForm[Table[{h,{{"from",x+h,"to",x}},(f[x+h]-f[x])/h}, {h,-.01,-.001,.002}], TableSpacing->{0,10}, TableHeadings->{{},{"h","Interval", "Average Rate of Change"}}] :[font = output; output; inactive; preserveAspect; fontLeading = 0; endGroup] TableForm[{{-0.01, {{"from", 0.99, "to", 1}}, 2.970099999999999997}, {-0.008, {{"from", 0.992, "to", 1}}, 2.976064000000000006}, {-0.006, {{"from", 0.994, "to", 1}}, 2.982036000000000011}, {-0.003999999999999999999, {{"from", 0.996, "to", 1}}, 2.988015999999999982}, {-0.002, {{"from", 0.9980000000000000001, "to", 1}}, 2.99400399999999998}}, TableSpacing -> {0, 10}, TableHeadings -> {{}, {"h", "Interval", "Average Rate of Change"}}] ;[o] h Interval Average Rate of Change -0.01 from 0.99 to 1 2.9701 -0.008 from 0.992 to 1 2.97606 -0.006 from 0.994 to 1 2.98204 -0.004 from 0.996 to 1 2.98802 -0.002 from 0.998 to 1 2.994 :[font = text; inactive; preserveAspect] We see that the average rate of change is approaching 3 from the left as well. ;[s] 1:0,0;79,-1; 1:1,26,19,Calculus,0,12,0,0,65535; :[font = text; inactive; preserveAspect; endGroup] This number that the average rates of change is approaching is said to be the INSTANTANEOUS RATE OF CHANGE for the function y=f(x) at a specific value of x. ;[s] 3:0,0;78,1;107,0;157,-1; 2:2,26,19,Calculus,0,12,0,0,0;1,26,19,Calculus,0,12,65535,0,0; :[font = section; inactive; preserveAspect; startGroup] What does this mean geometrically? ;[s] 1:0,0;35,-1; 1:1,26,19,Calculus,1,12,0,0,65535; :[font = text; inactive; preserveAspect] Thus far we have used algebra to define the instantaneous rate of change in terms of the average rate of change. Let's look at the geometric interpretation. In general a line that passes through two point on a curve is called a secant line. ;[s] 2:0,0;229,1;242,-1; 2:1,26,19,Calculus,0,12,0,0,0;1,26,19,Calculus,0,12,65535,0,0; :[font = input; preserveAspect; startGroup] Clear[f,a,h,m,M,g] f[x_]:=x^2 a:=2 m:=-8 M:=13 l:=-18 L:=18 h:=1 g[x_]:=f[a] + (f[a+h]-f[a])/h(x - a) func:=Plot[f[x],{x,m,M},PlotStyle->{Thickness[.015]}, PlotRange->{l,L},DisplayFunction->Identity]; secant:=Plot[g[x],{x,m,M},PlotStyle->{{Thickness[.01], RGBColor[1,0,0]}},PlotRange->{l,L}, DisplayFunction->Identity]; pts:=Graphics[{PointSize[.02],RGBColor[0,1,0], Point[{a,f[a]}],Point[{a+h,f[a+h]}]}]; sectex=Graphics[{Text["a",{2,-4}], Text["b",{3,-4}], Text["f(a)",{2,8}], Text["f(b)",{3,12}], Text["f(b)-f(a)",{6,8}], Text["_________",{6,7.3}], Text["b-a",{6,5.5}], Text["=",{8,7.1}], Text["Change in y",{10.2,8}], Text["__________",{10.2,7.3}], Text["Change in x",{10.2,5.5}]}]; slo= Graphics[{Red,Text["SLOPE",{8,2}]}]; Show[func,secant,pts,sectex,slo,Ticks->{{-8,-6,-4,-2,0,2,4,6,8}, {-16,-12,-8,-4,0,4,8,12,16}}, DisplayFunction->$DisplayFunction]; ;[s] 5:0,0;524,1;525,0;621,1;622,0;863,-1; 2:3,12,10,Courier,1,12,0,0,0;2,12,10,Courier,0,12,65535,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 346; pictureHeight = 214; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.386621 0.0453515 0.309017 0.0171676 [ [(-8)] .02381 .30902 0 2 Msboxa [(-6)] .11451 .30902 0 2 Msboxa [(-4)] .20522 .30902 0 2 Msboxa [(-2)] .29592 .30902 0 2 Msboxa [(2)] .47732 .30902 0 2 Msboxa [(4)] .56803 .30902 0 2 Msboxa [(6)] .65873 .30902 0 2 Msboxa [(8)] .74943 .30902 0 2 Msboxa [(-16)] .37412 .03434 1 0 Msboxa [(-12)] .37412 .10301 1 0 Msboxa [(-8)] .37412 .17168 1 0 Msboxa [(-4)] .37412 .24035 1 0 Msboxa [(4)] .37412 .37769 1 0 Msboxa 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.84921 .40344 0 0 Mshowa P p 1 0 0 r [(SLOPE)] .74943 .34335 0 0 Mshowa P P % End of Graphics MathPictureEnd :[font = text; inactive; preserveAspect] Using the definition of slope we can see that the average rate of change is equal to the slope of the secant line. ;[s] 1:0,0;116,-1; 1:1,26,19,Calculus,0,12,0,0,65535; :[font = text; inactive; preserveAspect] What happens if we look at the secant lines through ( x, f[x] ) and (x+h, f[x+h] ) as h gets smaller and smaller? ;[s] 3:0,0;52,1;84,0;115,-1; 2:2,26,19,Calculus,0,12,0,0,0;1,26,19,Calculus,0,12,65535,0,0; :[font = text; inactive; preserveAspect] Here we see what happens at x=1 as h approaches zero from the right. ;[s] 1:0,0;69,-1; 1:1,26,19,Calculus,0,12,65535,0,0; :[font = input; preserveAspect; startGroup] Clear[f,a,h,m,M,g] f[x_]:=x^3 a:=1 m:=-10 M:=10 l:=-10 L:=10 h:=.5 g[x_]:=f[a] + (f[a+h]-f[a])/h(x - a) func:=Plot[f[x],{x,m,M},PlotStyle->{Thickness[.015]}, PlotRange->{l,L},DisplayFunction->Identity]; secant:=Plot[g[x],{x,m,M},PlotStyle->{{Thickness[.01], RGBColor[1,0,0]}},PlotRange->{l,L}, DisplayFunction->Identity]; pts:=Graphics[{PointSize[.02],RGBColor[0,1,0], Point[{a,f[a]}],Point[{a+h,f[a+h]}] }]; slo:=Graphics[{Blue, Text[StringForm["SLOPE = ``",(f[a+h]-f[a])/h], {-6,5}], Red, Text[StringForm["h = ``",h], {-6,6.5}] }]; Do[Show[func,secant,pts,slo, DisplayFunction->$DisplayFunction],{h,1,.01,-.05}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; Cclosed; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 346; pictureHeight = 214; startGroup; animationSpeed = 30] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.047619 0.309017 0.0309017 [ [(-10)] .02381 .30902 0 2 Msboxa [(-5)] .2619 .30902 0 2 Msboxa [(5)] .7381 .30902 0 2 Msboxa [(10)] .97619 .30902 0 2 Msboxa [(-10)] .4875 0 1 0 Msboxa [(-7.5)] .4875 .07725 1 0 Msboxa [(-5)] .4875 .15451 1 0 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.3122 L .5248 .31338 L .52976 .31656 L .53472 .321 L .53968 .3269 L .54464 .33448 L .5496 .34394 L .55952 .36937 L .56944 .40486 L .57937 .45208 L .58929 .51271 L Mistroke .59921 .58844 L Mfstroke .60209 .61803 m .59921 .58844 L s s s s s s s s s s s s s P P p 1 0 0 r p .01 w s s s s s s s s s s s s .47279 0 m .5 .12361 L s .5 .12361 m .53968 .30387 L .57937 .48413 L s .60884 .61803 m .57937 .48413 L s s s s s s s s s s s P P p 0 1 0 r .02 w .54762 .33992 Mdot .59524 .55623 Mdot P p 0 0 1 r [(SLOPE = 7)] .21429 .46353 0 0 Mshowa 1 0 0 r [(h = 1)] .21429 .50988 0 0 Mshowa P P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 346; pictureHeight = 214; animationSpeed = 30] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.047619 0.309017 0.0309017 [ [(-10)] .02381 .30902 0 2 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.0309 L s P p .001 w .5 .04635 m .50375 .04635 L s P p .001 w .5 .0618 m .50375 .0618 L s P p .001 w .5 .09271 m .50375 .09271 L s P p .001 w .5 .10816 m .50375 .10816 L s P p .001 w .5 .12361 m .50375 .12361 L s P p .001 w .5 .13906 m .50375 .13906 L s P p .001 w .5 .16996 m .50375 .16996 L s P p .001 w .5 .18541 m .50375 .18541 L s P p .001 w .5 .20086 m .50375 .20086 L s P p .001 w .5 .21631 m .50375 .21631 L s P p .001 w .5 .24721 m .50375 .24721 L s P p .001 w .5 .26266 m .50375 .26266 L s P p .001 w .5 .27812 m .50375 .27812 L s P p .001 w .5 .29357 m .50375 .29357 L s P p .001 w .5 .32447 m .50375 .32447 L s P p .001 w .5 .33992 m .50375 .33992 L s P p .001 w .5 .35537 m .50375 .35537 L s P p .001 w .5 .37082 m .50375 .37082 L s P p .001 w .5 .40172 m .50375 .40172 L s P p .001 w .5 .41717 m .50375 .41717 L s P p .001 w .5 .43262 m .50375 .43262 L s P p .001 w .5 .44807 m .50375 .44807 L s P p .001 w .5 .47898 m .50375 .47898 L s P p .001 w .5 .49443 m .50375 .49443 L s P p .001 w .5 .50988 m .50375 .50988 L s P p .001 w .5 .52533 m .50375 .52533 L s P p .001 w .5 .55623 m .50375 .55623 L s P p .001 w .5 .57168 m .50375 .57168 L s P p .001 w .5 .58713 m .50375 .58713 L s P p .001 w .5 .60258 m .50375 .60258 L s P p .002 w .5 0 m .5 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p p .015 w s s s s s s s s s s s s s .39791 0 m .40079 .0296 L s .40079 .0296 m .41071 .10532 L .42063 .16595 L .43056 .21318 L .44048 .24866 L .44544 .26253 L .4504 .27409 L .45536 .28355 L .46032 .29113 L .46528 .29704 L .47024 .30147 L .47272 .30321 L .4752 .30465 L .47768 .30583 L .48016 .30678 L .48264 .30752 L .48512 .30807 L .48636 .30829 L .4876 .30847 L .48884 .30862 L .49008 .30874 L .49132 .30883 L .49256 .3089 L .4938 .30895 L .49504 .30898 L .5 .30902 L .50124 .30902 L .50248 .30902 L .50372 .30903 L .50496 .30905 L .5062 .30909 L .50744 .30913 L .50868 .3092 L .50992 .3093 L .5124 .30956 L .51364 .30974 L .51488 .30996 L .51736 .31051 L .51984 .31125 L .52232 .3122 L .5248 .31338 L .52976 .31656 L .53472 .321 L .53968 .3269 L .54464 .33448 L .5496 .34394 L .55952 .36937 L .56944 .40486 L .57937 .45208 L .58929 .51271 L Mistroke .59921 .58844 L Mfstroke .60209 .61803 m .59921 .58844 L s s s s s s s s s s s s s P P p 1 0 0 r p .01 w s s s s s s s s s s .38937 0 m .42063 .06716 L s .42063 .06716 m .46032 .1524 L .5 .23763 L .53968 .32287 L .57937 .40811 L .61905 .49335 L .65873 .57858 L s .6771 .61803 m .65873 .57858 L s s s s s s s s s P P p 0 1 0 r .02 w .54762 .33992 Mdot .55238 .35015 Mdot P p 0 0 1 r [(SLOPE = 3.31)] .21429 .46353 0 0 Mshowa 1 0 0 r [(h = 0.1)] .21429 .50988 0 0 Mshowa P P % End of Graphics MathPictureEnd :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 346; pictureHeight = 214; endGroup; endGroup; animationSpeed = 30] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.047619 0.309017 0.0309017 [ [(-10)] .02381 .30902 0 2 Msboxa [(-5)] .2619 .30902 0 2 Msboxa [(5)] .7381 .30902 0 2 Msboxa [(10)] .97619 .30902 0 2 Msboxa [(-10)] .4875 0 1 0 Msboxa [(-7.5)] .4875 .07725 1 0 Msboxa [(-5)] .4875 .15451 1 0 Msboxa [(-2.5)] .4875 .23176 1 0 Msboxa [(2.5)] .4875 .38627 1 0 Msboxa [(5)] .4875 .46353 1 0 Msboxa [(7.5)] .4875 .54078 1 0 Msboxa [(10)] .4875 .61803 1 0 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 .02381 .30902 m .02381 .31527 L s P [(-10)] .02381 .30902 0 2 Mshowa p .002 w .2619 .30902 m .2619 .31527 L s P [(-5)] .2619 .30902 0 2 Mshowa p .002 w .7381 .30902 m .7381 .31527 L s P [(5)] .7381 .30902 0 2 Mshowa p .002 w .97619 .30902 m .97619 .31527 L s P [(10)] .97619 .30902 0 2 Mshowa p .001 w .07143 .30902 m .07143 .31277 L s P p .001 w .11905 .30902 m .11905 .31277 L s P p .001 w .16667 .30902 m .16667 .31277 L s P p .001 w .21429 .30902 m .21429 .31277 L s P p .001 w .30952 .30902 m .30952 .31277 L s P p .001 w .35714 .30902 m .35714 .31277 L s P p .001 w .40476 .30902 m .40476 .31277 L s P p .001 w .45238 .30902 m .45238 .31277 L s P p .001 w .54762 .30902 m .54762 .31277 L s P p .001 w .59524 .30902 m .59524 .31277 L s P p .001 w .64286 .30902 m .64286 .31277 L s P p .001 w .69048 .30902 m .69048 .31277 L s P p .001 w .78571 .30902 m .78571 .31277 L s P p .001 w .83333 .30902 m .83333 .31277 L s P p .001 w .88095 .30902 m .88095 .31277 L s P p .001 w .92857 .30902 m .92857 .31277 L s P p .002 w 0 .30902 m 1 .30902 L s P p .002 w .5 0 m .50625 0 L s P [(-10)] .4875 0 1 0 Mshowa p .002 w .5 .07725 m .50625 .07725 L s P [(-7.5)] .4875 .07725 1 0 Mshowa p .002 w .5 .15451 m .50625 .15451 L s P [(-5)] .4875 .15451 1 0 Mshowa p .002 w .5 .23176 m .50625 .23176 L s P [(-2.5)] .4875 .23176 1 0 Mshowa p .002 w .5 .38627 m .50625 .38627 L s P [(2.5)] .4875 .38627 1 0 Mshowa p .002 w .5 .46353 m .50625 .46353 L s P [(5)] .4875 .46353 1 0 Mshowa p .002 w .5 .54078 m .50625 .54078 L s P [(7.5)] .4875 .54078 1 0 Mshowa p .002 w .5 .61803 m .50625 .61803 L s P [(10)] .4875 .61803 1 0 Mshowa p .001 w .5 .01545 m .50375 .01545 L s P p .001 w .5 .0309 m .50375 .0309 L s P p .001 w .5 .04635 m .50375 .04635 L s P p .001 w .5 .0618 m .50375 .0618 L s P p .001 w .5 .09271 m .50375 .09271 L s P p .001 w .5 .10816 m .50375 .10816 L s P p .001 w .5 .12361 m .50375 .12361 L s P p .001 w .5 .13906 m .50375 .13906 L s P p .001 w .5 .16996 m .50375 .16996 L s P p .001 w .5 .18541 m .50375 .18541 L s P p .001 w .5 .20086 m .50375 .20086 L s P p .001 w .5 .21631 m .50375 .21631 L s P p .001 w .5 .24721 m .50375 .24721 L s P p .001 w .5 .26266 m .50375 .26266 L s P p .001 w .5 .27812 m .50375 .27812 L s P p .001 w .5 .29357 m .50375 .29357 L s P p .001 w .5 .32447 m .50375 .32447 L s P p .001 w .5 .33992 m .50375 .33992 L s P p .001 w .5 .35537 m .50375 .35537 L s P p .001 w .5 .37082 m .50375 .37082 L s P p .001 w .5 .40172 m .50375 .40172 L s P p .001 w .5 .41717 m .50375 .41717 L s P p .001 w .5 .43262 m .50375 .43262 L s P p .001 w .5 .44807 m .50375 .44807 L s P p .001 w .5 .47898 m .50375 .47898 L s P p .001 w .5 .49443 m .50375 .49443 L s P p .001 w .5 .50988 m .50375 .50988 L s P p .001 w .5 .52533 m .50375 .52533 L s P p .001 w .5 .55623 m .50375 .55623 L s P p .001 w .5 .57168 m .50375 .57168 L s P p .001 w .5 .58713 m .50375 .58713 L s P p .001 w .5 .60258 m .50375 .60258 L s P p .002 w .5 0 m .5 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p p .015 w s s s s s s s s s s s s s .39791 0 m .40079 .0296 L s .40079 .0296 m .41071 .10532 L .42063 .16595 L .43056 .21318 L .44048 .24866 L .44544 .26253 L .4504 .27409 L .45536 .28355 L .46032 .29113 L .46528 .29704 L .47024 .30147 L .47272 .30321 L .4752 .30465 L .47768 .30583 L .48016 .30678 L .48264 .30752 L .48512 .30807 L .48636 .30829 L .4876 .30847 L .48884 .30862 L .49008 .30874 L .49132 .30883 L .49256 .3089 L .4938 .30895 L .49504 .30898 L .5 .30902 L .50124 .30902 L .50248 .30902 L .50372 .30903 L .50496 .30905 L .5062 .30909 L .50744 .30913 L .50868 .3092 L .50992 .3093 L .5124 .30956 L .51364 .30974 L .51488 .30996 L .51736 .31051 L .51984 .31125 L .52232 .3122 L .5248 .31338 L .52976 .31656 L .53472 .321 L .53968 .3269 L .54464 .33448 L .5496 .34394 L .55952 .36937 L .56944 .40486 L .57937 .45208 L .58929 .51271 L Mistroke .59921 .58844 L Mfstroke .60209 .61803 m .59921 .58844 L s s s s s s s s s s s s s P P p 1 0 0 r p .01 w s s s s s s s s s s .38146 0 m .42063 .08014 L s .42063 .08014 m .46032 .16132 L .5 .2425 L .53968 .32368 L .57937 .40486 L .61905 .48605 L .65873 .56723 L s .68357 .61803 m .65873 .56723 L s s s s s s s s s P P p 0 1 0 r .02 w .54762 .33992 Mdot .55 .34479 Mdot P p 0 0 1 r [(SLOPE = 3.1525)] .21429 .46353 0 0 Mshowa 1 0 0 r [(h = 0.05)] .21429 .50988 0 0 Mshowa P P % End of Graphics MathPictureEnd :[font = text; inactive; preserveAspect; center; fontColorRed = 32767; fontColorGreen = 32767; fontColorBlue = 32767] Click on the fishhook arrow above. Then press command y. Press the number 2 to slow the animation. ;[s] 1:0,1;100,-1; 2:0,26,19,Calculus,0,12,32767,32767,32767;1,13,10,Courier,0,12,32767,32767,32767; :[font = text; inactive; preserveAspect] And here we see what happens at x=1 as h approaches zero from the left. ;[s] 1:0,0;72,-1; 1:1,26,19,Calculus,0,12,65535,0,0; :[font = input; preserveAspect; startGroup] Clear[f,a,h,m,M,g] f[x_]:=x^3 a:=1 m:=-10 M:=10 l:=-10 L:=10 h:=.5 g[x_]:=f[a] + (f[a+h]-f[a])/h(x - a) func:=Plot[f[x],{x,m,M},PlotStyle->{Thickness[.015]}, PlotRange->{l,L},DisplayFunction->Identity]; secant:=Plot[g[x],{x,m,M},PlotStyle->{{Thickness[.01], RGBColor[1,0,0]}},PlotRange->{l,L}, DisplayFunction->Identity]; pts:=Graphics[{PointSize[.02],RGBColor[0,1,0], Point[{a,f[a]}],Point[{a+h,f[a+h]}]}]; slo:=Graphics[{Blue, Text[StringForm["SLOPE = ``",(f[a+h]-f[a])/h], {-6,5}], Red, Text[StringForm["h = ``",h], {-6,6.5}] }]; Do[Show[func,secant,pts,slo, DisplayFunction->$DisplayFunction],{h,-1,-.01,.05}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; Cclosed; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 346; pictureHeight = 214; startGroup; animationSpeed = 30] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.047619 0.309017 0.0309017 [ [(-10)] .02381 .30902 0 2 Msboxa [(-5)] .2619 .30902 0 2 Msboxa [(5)] .7381 .30902 0 2 Msboxa [(10)] .97619 .30902 0 2 Msboxa [(-10)] .4875 0 1 0 Msboxa [(-7.5)] .4875 .07725 1 0 Msboxa [(-5)] .4875 .15451 1 0 Msboxa [(-2.5)] .4875 .23176 1 0 Msboxa [(2.5)] .4875 .38627 1 0 Msboxa [(5)] .4875 .46353 1 0 Msboxa [(7.5)] .4875 .54078 1 0 Msboxa [(10)] .4875 .61803 1 0 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 .02381 .30902 m .02381 .31527 L s P [(-10)] .02381 .30902 0 2 Mshowa p .002 w .2619 .30902 m .2619 .31527 L s P [(-5)] .2619 .30902 0 2 Mshowa p .002 w .7381 .30902 m .7381 .31527 L s P [(5)] .7381 .30902 0 2 Mshowa p .002 w .97619 .30902 m .97619 .31527 L s P [(10)] .97619 .30902 0 2 Mshowa p .001 w .07143 .30902 m .07143 .31277 L s P p .001 w .11905 .30902 m .11905 .31277 L s P p .001 w .16667 .30902 m .16667 .31277 L s P p .001 w .21429 .30902 m .21429 .31277 L s P p .001 w .30952 .30902 m .30952 .31277 L s P p .001 w .35714 .30902 m .35714 .31277 L s P p .001 w .40476 .30902 m .40476 .31277 L s P p .001 w .45238 .30902 m .45238 .31277 L s P p .001 w .54762 .30902 m .54762 .31277 L s P p .001 w .59524 .30902 m .59524 .31277 L s P p .001 w .64286 .30902 m .64286 .31277 L s P p .001 w .69048 .30902 m .69048 .31277 L s P p .001 w .78571 .30902 m .78571 .31277 L s P p .001 w .83333 .30902 m .83333 .31277 L s P p .001 w .88095 .30902 m .88095 .31277 L s P p .001 w .92857 .30902 m .92857 .31277 L s P p .002 w 0 .30902 m 1 .30902 L s P p .002 w .5 0 m .50625 0 L s P [(-10)] .4875 0 1 0 Mshowa p .002 w .5 .07725 m .50625 .07725 L 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p .001 w .5 .26266 m .50375 .26266 L s P p .001 w .5 .27812 m .50375 .27812 L s P p .001 w .5 .29357 m .50375 .29357 L s P p .001 w .5 .32447 m .50375 .32447 L s P p .001 w .5 .33992 m .50375 .33992 L s P p .001 w .5 .35537 m .50375 .35537 L s P p .001 w .5 .37082 m .50375 .37082 L s P p .001 w .5 .40172 m .50375 .40172 L s P p .001 w .5 .41717 m .50375 .41717 L s P p .001 w .5 .43262 m .50375 .43262 L s P p .001 w .5 .44807 m .50375 .44807 L s P p .001 w .5 .47898 m .50375 .47898 L s P p .001 w .5 .49443 m .50375 .49443 L s P p .001 w .5 .50988 m .50375 .50988 L s P p .001 w .5 .52533 m .50375 .52533 L s P p .001 w .5 .55623 m .50375 .55623 L s P p .001 w .5 .57168 m .50375 .57168 L s P p .001 w .5 .58713 m .50375 .58713 L s P p .001 w .5 .60258 m .50375 .60258 L s P p .002 w .5 0 m .5 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p p .015 w s s s s s s s s s s s s s .39791 0 m .40079 .0296 L s .40079 .0296 m .41071 .10532 L .42063 .16595 L .43056 .21318 L .44048 .24866 L .44544 .26253 L .4504 .27409 L .45536 .28355 L .46032 .29113 L .46528 .29704 L .47024 .30147 L .47272 .30321 L .4752 .30465 L .47768 .30583 L .48016 .30678 L .48264 .30752 L .48512 .30807 L .48636 .30829 L .4876 .30847 L .48884 .30862 L .49008 .30874 L .49132 .30883 L .49256 .3089 L .4938 .30895 L .49504 .30898 L .5 .30902 L .50124 .30902 L .50248 .30902 L .50372 .30903 L .50496 .30905 L .5062 .30909 L .50744 .30913 L .50868 .3092 L .50992 .3093 L .5124 .30956 L .51364 .30974 L .51488 .30996 L .51736 .31051 L .51984 .31125 L .52232 .3122 L .5248 .31338 L .52976 .31656 L .53472 .321 L .53968 .3269 L .54464 .33448 L .5496 .34394 L .55952 .36937 L .56944 .40486 L .57937 .45208 L .58929 .51271 L Mistroke .59921 .58844 L Mfstroke .60209 .61803 m .59921 .58844 L s s s s s s s s s s s s s P P p 1 0 0 r p .01 w .02381 0 m .06349 .02575 L .10317 .0515 L .14286 .07725 L .18254 .10301 L .22222 .12876 L .2619 .15451 L .30159 .18026 L .34127 .20601 L .38095 .23176 L .42063 .25751 L 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Then press command y. Press the number 2 to slow the animation. ;[s] 1:0,1;100,-1; 2:0,26,19,Calculus,0,12,32767,32767,32767;1,13,10,Courier,0,12,32767,32767,32767; :[font = text; inactive; preserveAspect] We see that the secant lines are approaching the tangent line to the curve at x=a as h is approaching zero. We can conclude that the instantaneous rate of change is the slope of the tangent line to the curve at a specific value of x. ;[s] 5:0,0;134,1;162,0;170,1;195,0;236,-1; 2:3,26,19,Calculus,0,12,0,0,65535;2,26,19,Calculus,0,12,65535,0,0; :[font = text; inactive; preserveAspect] The tangent line to a curve is the line that best approximates the behavior of the curve at that specific point. :[font = input; preserveAspect; startGroup] Clear[a,f,g,x] a:=-5 f[x_]:= (x-3)(x+4)(x+2)(x-1) g[x_]:=f'[a](x-a)+f[a] slo:=Graphics[{Blue,Text["Tangent line",{6.2,50}], Red,Text [StringForm["y = ``",Simplify[g[x]]] ,{6.2,40}] }]; pic:=Plot[{f[x],g[x]},{x,-6,8},PlotRange->{-50,75}, PlotStyle->{{Thickness[.01]},{RGBColor[1,0,0],Thickness[.0013]}}, AxesLabel->{"x","f[x]"},DisplayFunction->Identity]; Do[Show[pic,slo,DisplayFunction->$DisplayFunction],{a,-4,4,.5}]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; Cclosed; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 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