(*^ ::[ 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. 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inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; showRuler; currentKernel; ] :[font = text; inactive; preserveAspect; leftWrapOffset = 17; leftNameWrapOffset = 1] Mathematics 162 Laboratory 10 Week of April 5, 1993 Name: _____________________________ Lab Partner: ___________________________ Consulted with: ____________________________________________________________ :[font = smalltext; inactive; preserveAspect; right] © Lafayette College, 1994. :[font = title; inactive; preserveAspect] Power Series Representations of Functions :[font = text; inactive; preserveAspect] We have seen that if a power series :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 154; pictureWidth = 75; pictureHeight = 36] %! %%Creator: Mathematica MathPictureStart % Start of picture % Scaling calculations 0 1 0 1 [ [ 0.000000 0.000000 0 0 ] [ 1.000000 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FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFF0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFF0 pop grestore %% End of Graphics MathPictureEnd %% End of picture :[font = text; inactive; preserveAspect] In this laboratory, we will explore the use of power series to define functions, giving special attention to the interval of convergence of the series. :[font = section; inactive; preserveAspect] Part 1: A Geometric Series :[font = text; inactive; preserveAspect] From our study of geometric series we know that :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 134; pictureWidth = 115; pictureHeight = 36] %! %%Creator: Mathematica MathPictureStart % Start of picture % Scaling calculations 0 1 0 1 [ [ 0.000000 0.000000 0 0 ] [ 1.000000 0.313044 0 0 ] ] MathScale % Start of Graphics 0 setgray 0 setlinewidth gsave 0.000000 0.000000 translate 1.000000 0.313044 scale 1 string 115 36 1 [115 0 0 36 0 0] { currentfile 1 index readhexstring pop } false 3 colorimage E0003FFFFFF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFF80 FC71F8007FC7E3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFF80 FF8E3FFFFFC7E3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFF80 FC01F8007FC7E3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFC7E3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFF81FFFFFFFFFFFFFFFFFFFFFFFE00FFFFFFFFFFF81C0FFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FFFFFFFFFFFFE38FFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FFFFFFFFFFFFFC7FFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FFFE0003FFFFFF8FFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FFFFFFFFFFFFFC71FFFFF 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E07FC7FFFFC7FFFFFFFFFFF8FF80 FFFE07FFFE3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFF80 FFF0000001FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0FFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFC0E07FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFC71C7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFC0E07FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFF80 pop grestore %% End of Graphics MathPictureEnd %% End of picture :[font = text; inactive; preserveAspect] We will examine the convergence of this series representation of the function 1/(1-x) by graphing some partial sums of the series. The fifth partial sum of our series is :[font = input; preserveAspect] f5[x_] = 1 + Sum[x^n, {n,1,4}] :[font = text; inactive; preserveAspect] (Note: We use 1 + Sum[x^n, {n,1,4}] rather than Sum[x^n, {n,0,4}] to define f5(x) because Mathematica does not recognize our convention that x0 = 1 when x = 0.) Use ;[s] 17:0,0;14,1;35,0;48,1;65,0;76,3;77,2;78,0;79,3;80,0;90,3;101,0;141,3;142,4;143,0;153,3;154,0;166,-1; 5:8,16,12,Chicago,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0;1,18,12,Chicago,64,9,0,0,0;5,16,12,Chicago,2,12,0,0,0;1,18,12,Chicago,32,9,0,0,0; :[font = input; preserveAspect] plot5 = Plot[f5[x],{x,-1.1,1.1}, PlotStyle->Red,PlotRange->{-1,5}] :[font = text; inactive; preserveAspect] to graph this partial sum. (We use the PlotRange option to prevent the scale from becoming so large that we cannot detect small variations in the values of our function. Feel free to experiment with other ranges for the plot to see what we are cutting out.) ;[s] 3:0,0;40,1;49,0;260,-1; 2:2,16,12,Chicago,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = text; inactive; preserveAspect] Now modify these two commands to define functions f10(x), f15(x), and f20(x) as the 10th, 15th, and 20th partial sums of the series, respectively, and make plots plot10, plot15, and plot20 of those functions over the interval [Ð1.1, 1.1]. Use different colors for each of the plots. (Some good choices are Blue, Green, Magenta, and Cyan.) Then use ;[s] 30:0,0;50,1;51,2;53,0;54,1;55,0;58,1;59,2;61,0;62,1;63,0;70,1;71,2;73,0;74,1;75,0;162,3;168,0;170,3;176,0;182,3;188,0;308,3;312,0;314,3;319,0;321,3;328,0;334,3;338,0;351,-1; 4:14,16,12,Chicago,0,12,0,0,0;6,16,12,Chicago,2,12,0,0,0;3,18,12,Chicago,64,9,0,0,0;7,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] Show[plot5, plot10, plot15, plot20] :[font = text; inactive; preserveAspect] to display the four plots simultaneously; print this one out. By examining your last plot, decide for what values of x you would be willing to use f20(x) to estimate f (x). Explain your conclusion by referring to your plot. ;[s] 12:0,0;118,1;119,0;148,1;149,2;151,0;152,1;153,0;167,1;168,0;170,1;171,0;226,-1; 3:6,16,12,Chicago,0,12,0,0,0;5,16,12,Chicago,2,12,0,0,0;1,18,12,Chicago,64,9,0,0,0; :[font = text; inactive; preserveAspect] :[font = text; inactive; preserveAspect] Use :[font = input; preserveAspect] plot = Plot[1/(1-x), {x,-1,1}] :[font = text; inactive; preserveAspect] to plot the graph of f (x) = 1/(1 Ð x) and then use ;[s] 7:0,0;21,1;22,0;24,1;25,0;36,1;37,0;52,-1; 2:4,16,12,Chicago,0,12,0,0,0;3,16,12,Chicago,2,12,0,0,0; :[font = input; preserveAspect] Show[plot20, plot] :[font = text; inactive; preserveAspect] to display the graphs of f20(x) and f (x) together. Does this plot support the conclusion you drew above? ;[s] 10:0,0;25,1;26,2;28,0;29,1;30,0;36,1;37,0;39,1;40,0;107,-1; 3:5,16,12,Chicago,0,12,0,0,0;4,16,12,Chicago,2,12,0,0,0;1,18,12,Chicago,64,9,0,0,0; :[font = text; inactive; preserveAspect] :[font = text; inactive; preserveAspect] Reexamine your first plot and notice the behaviors of f5(x), f10(x), f15(x) and f20(x) near x = Ð1. Explain why these functions behave as they do there. (You may want to look again at the defining equations of these functions.) ;[s] 23:0,0;54,1;55,2;56,0;57,1;58,0;61,1;62,2;64,0;65,1;66,0;69,1;70,2;72,0;73,1;74,0;80,1;81,2;83,0;84,1;85,0;92,1;93,0;230,-1; 3:10,16,12,Chicago,0,12,0,0,0;9,16,12,Chicago,2,12,0,0,0;4,18,12,Chicago,64,9,0,0,0; :[font = text; inactive; preserveAspect] :[font = section; inactive; preserveAspect] Part 2: Estimating an Interval of Convergence :[font = text; inactive; preserveAspect] We will examine the function :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 153; pictureWidth = 76; pictureHeight = 36] %! %%Creator: Mathematica MathPictureStart % Start of picture % Scaling calculations 0 1 0 1 [ [ 0.000000 0.000000 0 0 ] [ 1.000000 0.473684 0 0 ] ] MathScale % Start of Graphics 0 setgray 0 setlinewidth gsave 0.000000 0.000000 translate 1.000000 0.473684 scale 1 string 76 36 1 [76 0 0 36 0 0] { currentfile 1 index readhexstring pop } false 3 colorimage 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FFFFFFFFFFFFFFFFFF1F8FFFFFFFFFFFFFFF03FFFFFC7FFFFFFFFFFFFFFF FFFFFFFFFFC7FFFFFFFFFFFFFF8FF8FC7FFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFE38000FFFFFFFFFFFFE3FFFFE3FFFFFFFFFFFFFFFF FFFFFFFFFFF8FFFFFFFFFFFFFFF007E3FFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF0 pop grestore %% End of Graphics MathPictureEnd %% End of picture :[font = text; inactive; preserveAspect] Our objective is to determine the interval of convergence of this series. Since we do not have a closed-form representation for the coefficients an, it is not easy to see how we can use the ratio test to determine the interval of convergence. In addition, unlike the case in Part 1, we do not know an alternative formula for the function f (x). Therefore, we will try to guess the interval of convergence of the series by examining some partial sums as we did at first in Part 1. First we will tell Mathematica how to compute the coefficients of the series. ;[s] 10:0,0;146,1;147,2;148,0;340,1;341,0;343,1;344,0;503,1;514,0;562,-1; 3:5,16,12,Chicago,0,12,0,0,0;4,16,12,Chicago,2,12,0,0,0;1,18,12,Chicago,66,9,0,0,0; :[font = input; preserveAspect] Clear[a]; a[0] = 2; a[1] = 1/2; a[n_] := a[n] = ((5 - 4n)a[n-1] - (n - 2)a[n-2])/(4n) :[font = text; inactive; preserveAspect] Now we can compute, say, the 5th partial sum of the series using :[font = input; preserveAspect] f5[x_] = 2 + Sum[a[n] x^n, {n,1,4}] :[font = text; inactive; preserveAspect] Compute some other partial sums of the series representation for f (x) and plot those sums over some interval. By experimenting with the choice of interval, try to determine for what values of x the series will converge. Justify your answer by referring to appropriate graphs. (Write this up on a separate sheet and attach printouts of the graphs.) ;[s] 7:0,0;65,1;66,0;68,1;69,0;194,1;195,0;352,-1; 2:4,16,12,Chicago,0,12,0,0,0;3,16,12,Chicago,2,12,0,0,0; :[font = text; inactive; preserveAspect] HINT: You know that the (open) interval of convergence is of the form (Ðr, r). (Include in your solution an explanation of how you know this.) You might begin by examining the partial sums on the interval (Ð3, 3). ;[s] 5:0,0;73,1;74,0;76,1;77,0;217,-1; 2:3,16,12,Chicago,0,12,0,0,0;2,16,12,Chicago,2,12,0,0,0; :[font = section; inactive; preserveAspect] Part 3: Taylor Series :[font = text; inactive; preserveAspect] In last week's lab we saw many functions can be well approximated by polynomials. In that lab, we approximated a function f (x) by a polynomial p(x) of degree k chosen so ;[s] 11:0,0;123,1;125,0;126,1;127,0;145,1;146,0;147,1;148,0;160,1;161,0;172,-1; 2:6,16,12,Chicago,0,12,0,0,0;5,16,12,Chicago,2,12,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 25; pictureWidth = 334; pictureHeight = 15] %! %%Creator: Mathematica MathPictureStart % Start of picture % Scaling calculations 0 1 0 1 [ [ 0.000000 0.000000 0 0 ] [ 1.000000 0.044910 0 0 ] ] MathScale % Start of Graphics 0 setgray 0 setlinewidth gsave 0.000000 0.000000 translate 1.000000 0.044910 scale 1 string 334 15 1 [334 0 0 15 0 0] { currentfile 1 index readhexstring pop } false 3 colorimage FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFC0 FFFFC7FFFFF81F8FFFFFFFFFFFF007FFF1FFFFFE07E3FE3FFFFFFFFFFFFF 1FFFFFE07E3FFFFFFFFFFFC01FFFFFFC7FFFFF81F8FF8FFFFFFFFFFFFFFF FC7FFFFF81F8FFFFFFFFFFFF007FFFFFFFFF1FFFFFE07E3FE3FFFFFFFFFF FF8FFFFFFFFFFFFFFFFF8FFFFFF03F1FFFFFFFFFFFE00FFFFFFFFFFFFE3F FFFFC0FC7FC0 FFFE3FFFFFC7E3F1FFFFFFFFFFFFC7FF8FFFFFF1F8FC7FC7FFFFFFFFFFF8 FFFFFF1F8FC7FFFFFFFFFFFF1FFFFFE3FFFFFC7E3F1FF1FFFFFFFFFFFFFF E3FFFFFC7E3F1FFFFFFFFFFFFC7FFFFFFFF8FFFFFF1F8FC7FC7FFFFFFFFF FFF1FFFFFFFFFFFFFFFC7FFFFF8FC7E3FFFFFFFFFFFF8FFFFFFFFFFFF1FF FFFE3F1F8FC0 E00E3FE07007E3F1FFFFFFFFFFFFC01F8FF81C01F8FC7FC7FFFFF803FFF8 FF81C01F8FC7FFFFFFFFFFFF007FFFE3FE07007E3F1FF1FFFFFE00FFFFFF E3FE07007E3F1FFFFFFFFFFFFC01FFFFFFF8FF81C01F8FC7FC7FFFFFFFFF FFF1FFFC01FFFFFFFFFC7FC0E00FC7E3FFFFFFFFFFFF803FFFFFFFFFF1FF 03803F1F8FC0 FF81FFFF8E07E3FE3FFFFFFFFFFFF8FC7FFFE381F8FF8FFFFFFFFFE3FFC7 FFFE381F8FF8FFFFFFFFFFFFE3F1FF1FFFF8E07E3FE3FFFFFFFFF8FFFFFF 1FFFF8E07E3FE3FFFFFFFFFFFF8FC7FFFFC7FFFE381F8FF8FFFFFFFFFFFF FFFFFFFFF1FFFFFFFFE3FFFF1C0FC7FC7FFFFFFFFFFFF1F8E3FFFF1F8FFF FC703F1FF1C0 FF81FFFFF1C7E3FE3FFC0000FFFFF8FC7FFFFC71F8FF8FFFFFFFFFE3FFC7 FFFFC71F8FF8FFF00003FFFFE3F1FF1FFFFF1C7E3FE3FFFFFFFFF8FFFFFF 1FFFFF1C7E3FE3FFC0000FFFFF8FC7FFFFC7FFFFC71F8FF8FFFFF8FFF1FF E3FFFFFFF1F8FFFFC7E3FFFFE38FC7FC7FF80001FFFFF1F81FFFFFE38FFF FF8E3F1FF1C0 FFF1FFFFFE381FFE3FFFFFFFFFFFFF1C0FFFFF8E07FF8FFFFFFFFFFC7FC7 FFFFF8E07FF8FFFFFFFFFFFFFC7E3F1FFFFFE381FFE3FFFFFFFFFF1FFFFF 1FFFFFE381FFE3FFFFFFFFFFFFF1F8FFFFC7FFFFF8E07FF8FFFFFFFFFFFF FFFFFFFFFE07FFFFF8E3FFFFFC703FFC7FFFFFFFFFFFFE3F1FF1C0E38FFF FFF1C0FFF1C0 FFF1FFFFF1C7FFFE3FFC0000FFFFFF1C0FFFFC71FFFF8FFFFFFFFFFC7FC7 FFFFC71FFFF8FFF00003FFFFFC7E3F1FFFFF1C7FFFE3FFFFFFFFFF1FFFFF 1FFFFF1C7FFFE3FFC0000FFFFFF1F8FFFFC7FFFFC71FFFF8FFFFFFFFFFFF FFFFFFFFFE07FC7038E3FFFFE38FFFFC7FF80001FFFFFE3F1FF1C7E38FFF FF8E3FFFF1C0 FFF007FFF0381FF1FFFFFFFFFFFFFF000FFFFC0E07FC7FC7FFFFFFFC01F8 FFFFC0E07FC7FFFFFFFFFFFFFC003FE3FFFF0381FF1FF1FFFFFFFF007E3F E3FFFF0381FF1FFFFFFFFFFFFFF000FF8FF8FFFFC0E07FC7FC7FFFFFFFFF FFF1FFFFFE00FC71F8FC7FFFE0703FE3FFFFFFFFFFFFFE001FFE07E3F1FF FF81C0FF8FC0 FFFE3FFFFFFFFFF1FFFFFFFFFFFFFFFF8FFFFFFFFFFC7FFFFFFFFFFF8FF8 FFFFFFFFFFC7FFFFFFFFFFFFFFFE3FE3FFFFFFFFFF1FFFFFFFFFFFE3FE3F E3FFFFFFFFFF1FFFFFFFFFFFFFFFF8FF8FF8FFFFFFFFFFC7FFFFFFFFFFFF FFFFFFFFFFC7FF81F8FC7FFFFFFFFFE3FFFFFFFFFFFFFFFF1FFE3803F1FF FFFFFFFF8FC0 FFFFC0FFFFFFFF8FFFFFFFFFFFFFFFFFF1FFFFFFFFE3FFFFFFFFFFFF8E3F 1FFFFFFFFE3FFFFFFFFFFFFFFFFE3FFC7FFFFFFFF8FFFFFFFFFFFFE38E3F FC7FFFFFFFF8FFFFFFFFFFFFFFFFF8FF8FFF1FFFFFFFFE3FFFFFFFFFFFFF FFFFFFFFFFC71F8E00FF8FFFFFFFFF1FFFFFFFFFFFFFFFFF1FFFC7E3FE3F FFFFFFFC7FC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFC7FFF1F8FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE3FFC71FFFFF FFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFF8FFF1C7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFC0 pop grestore %% End of Graphics MathPictureEnd %% End of picture :[font = text; inactive; preserveAspect] This week in class, we will show that the polynomial with this property can be represented :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 124; pictureWidth = 134; pictureHeight = 34] %! %%Creator: Mathematica MathPictureStart % Start of picture % Scaling calculations 0 1 0 1 [ [ 0.000000 0.000000 0 0 ] [ 1.000000 0.253731 0 0 ] ] MathScale % Start of Graphics 0 setgray 0 setlinewidth gsave 0.000000 0.000000 translate 1.000000 0.253731 scale 1 string 134 34 1 [134 0 0 34 0 0] { currentfile 1 index readhexstring pop } false 3 colorimage FFFFFFFFFFFFFFFFFFFFFFFFFC0007FFFFFF03FFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFF8E3F000FF8FC7FFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFF1C7FFFFF8FC7FFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFF803F000FF8FC7FFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FC7FFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF03FFFFFFFFFFFFFFC01C0E3F FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF1F8FFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF1F8E3F FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE3F03F FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE3F03F FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFE000001FFFFFFFFFFFFFFFFFFFFE3803F FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 007FFF1FFFFF1FFFFFFFFFFFFFFE0000003FFFFFFFFFFFFFFFFFFFFFFE3F FFFFFFFFFFC7FFFFFFFFFFFFFFFFFFE07E3FFFFFC0 FC7FF8FFFFFFE3FFFFFFFFFFFFFE07FFFFC7FFFFFFFFFFFFFFFFFFFFFE3F FFFFFFFFFE3FFFFFFFFFFFFFFFFFFF1F8FC7FFFFC0 FC01F8FF81C0E3FFFFFFFFFFFFFFC0FFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFE3FE0703FFFFFFFFF81C01F8FC7FFFFC0 FF8FC7FFFE38FC7FFFFFFFFFFFFFF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFF1FFFF8E3FFFFFFFFFFE381F8FF8FFFFC0 FF8FC7FFFFC7FC7FF80001FFFFFFFF03FFFFFFFFFF000000000000000000 0000000001FFFFF1FFFFFFFFFFFFC71F8FF8FFFFC0 FFF1C0FFFFF8FC7FFFFFFFFFFFFFFFE07FFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFF1FFFFFE3FE0003FFFFFF8E07FF8FFFFC0 FFF1C0FFFFC71C7FF80001FFFFFFFFFC0FFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFF1FFFFF1C7FFFFFFFFFFC71FFFF8FC0000 FFF000FFFFC0E07FFFFFFFFFFFFFFFE07FFFFFFFFFFFFFFFFFFFFFFC7FFF FF81F8FFFE3FFFF0381FFFFFFFFFC0E07FC7FF8E00 FFFFF8FFFFFFE3FFFFFFFFFFFFFFFF03FFFFFFFFFFFFFFFFFFFFFFE3FFFF FC7E3F1FFE3FFFFFFFFFFFFFFFFFFFFFFFC7FFF1C0 FFFFFF1FFFFF1FFFFFFFFFFFFFFFF81FFFFFFFFFFFE00FFFFFFFFFE3FE07 007E3F1FFFC7FFFFFFFFFFFFFFFFFFFFFE3FFF8000 FFFFFFFFFFFFFFFFFFFFFFFFFFFFC0FFFFC7FFFFFFFF8FFFFFFFFF1FFFF8 E07E3FE3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFE0000003FFFFFFFFF8FC7FFFE3F1FFFFF 1C7E3FE3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF03FFFFFC71FFFFF E381FFE3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF03FE000071FFFFF 1C7FFFE3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF007FC71C7E3FFFF 0381FF1FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE381FFFFFFFFFFFE3FFF8E07E3FFFF FFFFFF1FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE38FFFFFFFFFFFFE38FC01C7FC7FFF FFFFF8FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0FFFFFFFFFFFFE3FFFFFC7FFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC703FFFFFFFFFFFC7FFFE3FFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 pop grestore %% End of Graphics MathPictureEnd %% End of picture :[font = text; inactive; preserveAspect] With that thought in mind, we make the following definition. Definition: If f (x) has derivatives of all orders at a point x0, the Taylor series for f about x0 is ;[s] 17:0,0;62,1;72,0;78,2;79,0;81,2;82,0;125,2;126,3;127,0;133,2;146,0;151,2;152,0;159,2;160,3;161,0;165,-1; 4:8,16,12,Chicago,0,12,0,0,0;1,16,12,Chicago,1,12,0,0,0;6,16,12,Chicago,2,12,0,0,0;2,18,12,Chicago,64,9,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 141; pictureTop = 1; pictureWidth = 101; pictureHeight = 36] %! %%Creator: Mathematica MathPictureStart % Start of picture % Scaling calculations 0 1 0 1 [ [ 0.000000 0.000000 0 0 ] [ 1.000000 0.356436 0 0 ] ] MathScale % Start of Graphics 0 setgray 0 setlinewidth gsave 0.000000 0.000000 translate 1.000000 0.356436 scale 1 string 101 36 1 [101 0 0 36 0 0] { currentfile 1 index readhexstring pop } false 3 colorimage E0003FFFFFF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE FC71F8007FC7E3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE FF8E3FFFFFC7E3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE FC01F8007FC7E3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE FFFFFFFFFFC7E3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE FFFFFFFFFFF81FFFFFFFFFFFFFFE00E071FFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE FFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FC7FFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE FFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FC71FFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE FFFFFFFFFFFFFFFFFFFFFFFFFFFFFF1F81FFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE FFFFFFFFFFFFFFFFFFFFFFFFFFFFFF1F81FFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE FFF000000FFFFFFFFFFFFFFFFFFFFF1C01FFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE FFF0000001FFFFFFFFFFFFFFFFFFFFFFF1FFFFFFFFFFFE3FFFFFFFFFFFFF FFFFFF03F1FFFFFE FFF03FFFFE3FFFFFFFFFFFFFFFFFFFFFF1FFFFFFFFFFF1FFFFFFFFFFFFFF FFFFF8FC7E3FFFFE FFFE07FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF1FF0381FFFFFFFF FC0E00FC7E3FFFFE FFFFC0FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FFFFC71FFFFFFFF FFF1C0FC7FC7FFFE FFFFF81FFFFFFFFFF8000000000000000000000000000FFFFF8FFFFFFFFF FFFE38FC7FC7FFFE FFFFFF03FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FFFFFF1FF0001FF FFFFC703FFC7FFFE FFFFFFE07FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FFFFF8E3FFFFFFF FFFE38FFFFC7E000 FFFFFF03FFFFFFFFFFFFFFFFFFFFFFE3FFFFFC0FC7FFF1FFFF81C0FFFFFF FFFE0703FE3FFC70 FFFFF81FFFFFFFFFFFFFFFFFFFFFFF1FFFFFE3F1F8FFF1FFFFFFFFFFFFFF FFFFFFFFFE3FFF8E FFFFC0FFFFFFFFFFFF007FFFFFFFFF1FF03803F1F8FFFE3FFFFFFFFFFFFF FFFFFFFFF1FFFC00 FFFE07FFFE3FFFFFFFFC7FFFFFFFF8FFFFC703F1FF1FFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE FFF0000001FFFFFFFFFC7E3FFFF1F8FFFFF8E3F1FF1FFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE FFFFFFFFFFFFFFFFFFFF81FFFFFE38FFFFFF1C0FFF1FFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE FFFFFFFFFFFFFFFFFFFF81FF000038FFFFF8E3FFFF1FFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE FFFFFFFFFFFFFFFFFFFF803FE38E3F1FFFF81C0FF8FFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE FFFFFFFFFFFFFFFFFFFFF1FFFC703F1FFFFFFFFFF8FFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE FFFFFFFFFFFFFFFFFFFFF1C7E00E3FE3FFFFFFFFC7FFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE FFFFFFFFFFFFFFFFFFFFF1FFFFFE3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE FFFFC0E07FFFFFFFFFFFFE3FFFF1FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE FFFFC71C7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE FFFFC0E07FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFE pop grestore %% End of Graphics MathPictureEnd %% End of picture :[font = text; inactive; preserveAspect] In the case that x0 = 0, the Taylor series is also called a Maclaurin series. ;[s] 6:0,0;17,1;18,2;19,0;60,1;76,0;78,-1; 3:3,16,12,Chicago,0,12,0,0,0;2,16,12,Chicago,2,12,0,0,0;1,18,12,Chicago,64,9,0,0,0; :[font = text; inactive; preserveAspect] Notice that the partial sums of the Taylor (Maclaurin) series for a function f are simply the polynomials you worked with in the last lab. Mathematica has a command for calculating these polynomials. The format of the command is Normal[Series[f[x], {x, x0, n}]], where x0 is the point about which we want to approximate f[x] and n is the degree of the approximating polynomial. (Note: The Series command is really doing all of the work here; Normal merely puts the polynomial in an appropriate form.) ;[s] 23:0,0;77,1;78,0;140,1;151,0;231,2;245,3;249,2;255,3;257,2;259,3;260,2;263,0;271,3;273,0;322,3;326,0;331,3;332,0;393,2;399,0;446,2;452,0;505,-1; 4:9,16,12,Chicago,0,12,0,0,0;2,16,12,Chicago,2,12,0,0,0;6,13,10,Courier,1,12,0,0,0;6,13,10,Courier,0,12,0,0,0; :[font = text; inactive; preserveAspect] In this part of the lab, we will explore the convergence of the Maclaurin series for the function cosx. The command ;[s] 3:0,0;101,1;102,0;117,-1; 2:2,16,12,Chicago,0,12,0,0,0;1,16,12,Chicago,2,12,0,0,0; :[font = input; preserveAspect] f[x_] = Normal[Series[Cos[x], {x, 0, 4}]] :[font = text; inactive; preserveAspect] will compute the fourth Maclaurin polynomial for the cosine function. For technical reasons, it is critical that you use immediate assignment (=) here rather than delayed assignment (:=). Then use ;[s] 5:0,0;144,1;145,0;184,1;186,0;199,-1; 2:3,16,12,Chicago,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] Plot[{f[x],Cos[x]},{x,-2Pi,2Pi},PlotStyle->{Red,Blue}, PlotRange->{-2,2}] :[font = text; inactive; preserveAspect] to examine the accuracy of the approximation. (The chosen PlotRange is reasonable since we know that all of the values of cosx lie between Ð1 and 1.) On what interval is this polynomial a good approximation of cosine? ;[s] 5:0,0;59,1;68,0;126,2;127,0;220,-1; 3:3,16,12,Chicago,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0;1,16,12,Chicago,2,12,0,0,0; :[font = text; inactive; preserveAspect] :[font = text; inactive; preserveAspect] By computing several additional Maclaurin polynomials for the cosine function and comparing their graphs to that of cosx, try to determine on what interval the Maclaurin series for cosx converges to cosx. Explain your conclusion. (Write this up on a separate sheet. You may want to attach printouts of one or more of the plots you used to decide on your answer. If so, be sure to label the plots and give a clear written description of how they helped you answer the question.) ;[s] 7:0,0;119,1;120,0;184,1;185,0;202,1;203,0;482,-1; 2:4,16,12,Chicago,0,12,0,0,0;3,16,12,Chicago,2,12,0,0,0; ^*)