(*^ ::[ 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, groupLikeTitle, center, M7, e8, 24, "B Univers 65 Bold"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, e6, 18, "B Garamond Bold"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, e6, 14, "I Garamond LightItalic"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, a20, 18, "B Univers 65 Bold"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, a15, 14, "AGaramond Semibold"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12, 12, "Garamond"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "AGaramond"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "AGaramond"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-5, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L-5, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B65535, L-5, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7, 12, "Garamond"; fontset = leftheader, inactive, L2, 12, "Garamond"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, 12, "Garamond"; fontset = leftfooter, inactive, L2, 12, "Garamond"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, N55, 12, "Garamond"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special4, 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 9 Week of March 29, 1993 Name: _____________________________ Lab Partner: ___________________________ Consulted with: ____________________________________________________________ :[font = smalltext; inactive; preserveAspect; right] © Lafayette College, 1994. :[font = title; inactive; preserveAspect] Approximating Functions with Polynomials :[font = subsubtitle; inactive; preserveAspect; left] Problem: Given a function f(x) and a point x = a, find a polynomial, p(x), which approximates f(x) near x = a. ;[s] 17:0,0;44,1;45,0;48,1;49,0;70,1;71,0;72,1;73,0;95,1;96,0;97,1;98,0;105,1;106,0;109,1;110,0;112,-1; 2:9,17,11,I Garamond LightItalic,0,14,0,0,0;8,17,11,I Garamond LightItalic,2,14,0,0,0; :[font = text; inactive; preserveAspect] The general idea behind our approach is that if f (x) is a given function and the polynomial p(x) is chosen so that the function and all its derivatives have the same value as f and its derivatives at x = a, then p(x) should agree pretty well with f(x). After all, we will have p(a) = f (a), the graphs of p and f will be tangent to the same line at x = a, the graphs will have the same concavity at x = a, and so forth. ;[s] 43:0,0;48,1;50,0;51,1;52,0;93,1;94,0;95,1;96,0;176,1;178,0;201,1;202,0;205,1;206,0;213,1;214,0;215,1;216,0;248,1;249,0;250,1;251,0;279,1;280,0;281,1;282,0;286,1;288,0;289,1;290,0;307,1;308,0;313,1;314,0;351,1;352,0;355,1;356,0;401,1;402,0;405,1;406,0;422,-1; 2:22,14,10,AGaramond,0,12,0,0,0;21,14,10,AGaramond,2,12,0,0,0; :[font = text; inactive; preserveAspect] Since we are interested in approximating our function f (x) for values of x near a, it makes sense to express our approximating polynomial in powers of x Ð a, rather than powers of x. Thus our polynomial will be of the form ;[s] 15:0,0;54,1;56,0;57,1;58,0;74,1;75,0;81,1;82,0;152,1;153,0;156,1;157,0;181,1;182,0;225,-1; 2:8,14,10,AGaramond,0,12,0,0,0;7,14,10,AGaramond,2,12,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 62; pictureWidth = 260; pictureHeight = 16] %! %%Creator: Mathematica MathPictureStart % Start of picture % Scaling calculations 0 1 0 1 [ [ 0.000000 0.000000 0 0 ] [ 1.000000 0.061539 0 0 ] ] MathScale % Start of Graphics 0 setgray 0 setlinewidth gsave 0.000000 0.000000 translate 1.000000 0.061539 scale 1 string 260 16 1 [260 0 0 16 0 0] { currentfile 1 index readhexstring pop } false 3 colorimage FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFF0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFF0 007FFF1FFFFF1FFFFFFFFFFFFFFFF8000E3FFFFFFFFFFFFFFFFFC7FFFFFF FFFFFFFFFFFFFC0007FFF007FC7FFFFFFFFFFFFFFFFF8FFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC01FF1FFFFFFFFFFFFFFFFFE3FFFF FFFFFFFFFFFF03F0 FC7FF8FFFFFFE3FFFFFFFFFFFFFFFF1C71FFFFFFFFFFFFFFFFFFF8FFFFFF FFFFFFFFFFFFFF8E3FFFFE3FE3FFFFFFFFFFFFFFFFFFF1FFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FF8FFFFFFFFFFFFFFFFFFFC7FFF FFFFFFFFFFF8FC70 FC01F8FF81C0E3FFFFFFFFFFFF81FFE381FF0381FFFFFFFFFF81C0FFFFFF FFFFFFFFFFC0FFF1C003FE3FE3FE0703FFFFFFFFFF0381FFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFC0FF8FF8FF81C0FFFFFFFFFFC0E07FFF FFFFFFFFF038FC70 FF8FC7FFFE38FC7FFFFFFFFFFF8FC7000FFFFC71FFFFFFFFFF8FC71FFFFF FFFE3FFFFFC7E3803FFFFE3F1FFFF8E3FFFFFFFFFF1F8E3FFFFFFFFFFFFF FFF1FFFFFFFFFFFFFFFFFE3FFFFFC7E38FC7FFFE38FFFFFFFFFFC7E38FFF FF8FFFFFF1F8FC70 FF8FC7FFFFC7FC7FF80001FFFF8FFFFF8FFFFF8FFFFFFFFFFF8FC71FFFFF FFFE3FFFFFC7FFFFFFFFF03F1FFFFF1FFFFFFFFFFF1F8E3FFFFFFFFFFFFF FFF1FFFF8FFF1FFE3FFFFE3FFFFFC7FC0FC7FFFFC7FFFFFFFFFFC7E38FFF FF8FFFFFF1F8FC70 FFF1C0FFFFF8FC7FFFFFFFFFFFF1FFFF8FFFFFF1FF0001FFFFFE001FFFFF FF8000FFFFF8FFFFFFFFFE3F1FFFFFE3FE0003FFFFFC003FFFFFFFFFFFFF FC0007FFFFFFFFFFFFFF8000FFFFF8FF8FC7FFFFF8FF8000FFFFFF000FFF E0003FFFFE3F03F0 FFF1C0FFFFC71C7FF80001FFFFF1F8FF8FFFFF8E3FFFFFFFFFF1F81F8000 FFFE3FFFFFF8FC7FFFFFFFFF1FFFFF1C7FFFFFFFFFE3F03F0001FFFC01FF FFF1FFFFFFFFFFFFFFFFFE3FFFFFF8FC7FC7FFFFC71FFFFFFFFFF8FC0FFF FF8FFFFFFE3F1FF0 FFF000FFFFC0E07FFFFFFFFFFFFFC01FF1FFFF81C0FFFFFFFFFFC0FFF1C7 FFFE3FFFFFFFE00FFFFFFFFFE3FFFF0381FFFFFFFFFF81FFE38FFFFF8FFF FFF1FFFFFFFFFFFFFFFFFE3FFFFFFFE00FF8FFFFC0E07FFFFFFFFFE07FFF FF8FFFFFFFF803F0 FFFFF8FFFFFFE3FFFFFFFFFFFFFFFFFFF1FFFFFFFFFFFFFFFFFFF8FFFE38 FFFFFFFFFFFFFFFFFFFFFFFFE3FFFFFFFFFFFFFFFFFFF1FFFC7000FF8FFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FFFFFFFFFFFFFFFFFFFC7FFF FFFFFFFFFFFFFFF0 FFFFFF1FFFFF1FFFFFFFFFFFFFFFFFFFFE3FFFFFFFFFFFFFFFFFC7FFF007 FFFFFFFFFFFFFFFFFFFFFFFFFC7FFFFFFFFFFFFFFFFF8FFFE00FFFFF8FFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF1FFFFFFFFFFFFFFFFFE3FFFF FFFFFFFFFFFFFFF0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0FFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFF0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFF0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFF0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFF0 pop grestore %% End of Graphics MathPictureEnd %% End of picture :[font = text; inactive; preserveAspect] All we have to do is determine the values of the coefficients c0, c1, c2, ..., cn so that the polynomial and its derivatives have the same value as the function at a. ;[s] 16:0,0;62,1;63,2;64,0;66,1;67,2;68,0;70,1;71,2;72,0;79,1;80,3;81,1;82,0;164,1;165,0;167,-1; 4:6,16,12,Chicago,0,12,0,0,0;6,16,12,Chicago,2,12,0,0,0;3,18,12,Chicago,64,9,0,0,0;1,18,12,Chicago,66,9,0,0,0; :[font = section; inactive; preserveAspect] Part 1: Approximate with quadratic polynomials :[font = text; inactive; preserveAspect] Define the function :[font = input; preserveAspect] f[x_] := Sqrt[x] :[font = text; inactive; preserveAspect] We will approximate f near x = 1 with a 2nd degree polynomial. Based on the discussion above, we should find an approximating polynomial of the form ;[s] 5:0,1;20,2;21,1;27,2;28,1;150,-1; 3:0,18,12,Chicago,66,9,0,0,0;3,16,12,Chicago,0,12,0,0,0;2,16,12,Chicago,2,12,0,0,0; :[font = input; preserveAspect] poly[x_] := c[0] + c[1] (x-1) + c[2] (x-1)^2 :[font = text; inactive; preserveAspect] where the c's are chosen so that the following equations hold: ;[s] 3:0,0;10,1;11,0;63,-1; 2:2,16,12,Chicago,0,12,0,0,0;1,16,12,Chicago,2,12,0,0,0; :[font = input; preserveAspect] equations := {f[1]==poly[1],f'[1]==poly'[1],f''[1]==poly''[1]} :[font = text; inactive; preserveAspect] We wish to solve these equations for the c's, so we will define them to be the solutions: ;[s] 4:0,1;1,0;42,1;43,0;91,-1; 2:2,16,12,Chicago,0,12,0,0,0;2,16,12,Chicago,2,12,0,0,0; :[font = input; preserveAspect] solutions := {c[0], c[1], c[2]} :[font = text; inactive; preserveAspect] To find the appropriate values, we can use the Solve command: ;[s] 3:0,0;47,1;52,0;62,-1; 2:2,16,12,Chicago,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] Solve[equations, solutions] :[font = text; inactive; preserveAspect] To define our approximating polynomial we can enter :[font = input; preserveAspect] p2[x_] = poly[x] /. % :[font = text; inactive; preserveAspect] Write down the definition of p2(x) below, along with the values of the function and its first and second derivatives at x = 1. Compare them with the corresponding values of f. Do they actually agree? ;[s] 10:0,0;29,1;30,2;31,0;32,1;33,0;120,1;121,0;174,1;175,0;202,-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] p2(x) = p2(1) = f (1) = p'2(1) = f '(1) = p''2(1) = f ''(1) = ;[s] 20:0,1;2,2;3,0;4,1;5,0;12,1;13,2;14,0;76,1;77,0;85,1;87,2;88,0;149,1;150,0;162,1;165,2;166,0;225,1;226,0;240,-1; 3:8,16,12,Chicago,0,12,0,0,0;8,16,12,Chicago,2,12,0,0,0;4,18,12,Chicago,64,9,0,0,0; :[font = text; inactive; preserveAspect] To see how well p2(x) approximates f (x) for x near 1, plot a graph: ;[s] 12:0,0;16,1;17,2;18,0;19,1;20,0;35,1;36,0;38,1;39,0;45,1;46,0;69,-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 = input; preserveAspect] Plot[{f[x],p2[x]}, {x,0,2}, PlotStyle -> {Red,Blue}] :[font = text; inactive; preserveAspect] Based on your graph, over what interval is the polynomial a good approximation for the square root function? :[font = text; inactive; preserveAspect] :[font = section; inactive; preserveAspect] Part 2: Approximate with bigger polynomials :[font = text; inactive; preserveAspect] A higher degree polynomial should give a better approximation of our function. Implementing this is not conceptually harder than Part 1, but using the procedure in Part 1 to create an approximating polynomial of degree 30, say, would involve a lot of typing. We will cut down on that typing by using Mathematica's Sum and Table commands. First, ;[s] 7:0,0;302,1;313,0;316,2;319,0;324,2;329,0;348,-1; 3:4,16,12,Chicago,0,12,0,0,0;1,16,12,Chicago,2,12,0,0,0;2,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] Clear[poly, equations, solutions] :[font = text; inactive; preserveAspect] so that we will be ready to start a new approximation. Now, define :[font = input; preserveAspect] poly[x_] := Sum[ c[j] (x-a)^j, {j, 0, n}] :[font = text; inactive; preserveAspect] so our approximating polynomial will have the form shown on the first page. The system of equations which determine the coefficients in poly[x] is ;[s] 3:0,0;137,1;144,0;148,-1; 2:2,16,12,Chicago,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] equations := Table[ (D[f[x],{x, k}] /. x->a)==(D[poly[x],{x, k}] /. x->a), {k, 0, n}] :[font = text; inactive; preserveAspect] with solutions :[font = input; preserveAspect] solutions := Table[c[m], {m, 0, n}] :[font = text; inactive; preserveAspect] From here, we should be able to choose a value a to approximate at, and a degree n for the aproximating polynomial and then use Solve to create the approximating polynomial. Check the new procedure by letting ;[s] 7:0,0;47,1;48,0;81,1;82,0;128,2;133,0;210,-1; 3:4,16,12,Chicago,0,12,0,0,0;2,16,12,Chicago,2,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = input; preserveAspect] a = 1 :[font = input; preserveAspect] n = 2 :[font = text; inactive; preserveAspect] Repeat the procedure of Part 1: :[font = input; preserveAspect] Solve[equations, solutions] :[font = input; preserveAspect] p2[x_] = poly[x] /. % :[font = text; inactive; preserveAspect] Now, let n = 3 to generate a 3rd degree approximating polynomial p3(x). Write your result below: ;[s] 8:0,0;9,1;10,0;65,1;66,2;67,0;68,1;69,0;98,-1; 3:4,16,12,Chicago,0,12,0,0,0;3,16,12,Chicago,2,12,0,0,0;1,18,12,Chicago,64,9,0,0,0; :[font = text; inactive; preserveAspect] p3(x) = ;[s] 6:0,0;1,1;2,2;3,0;4,1;5,0;11,-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] Now, follow the same steps to generate a 4th-degree polynomial which approximates our function. Write it below: :[font = text; inactive; preserveAspect] p4(x) = ;[s] 6:0,0;1,1;2,2;3,0;4,1;5,0;11,-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] To compare all these polynomials, plot their graphs together: :[font = input; preserveAspect] Plot[{f[x], p2[x], p3[x], p4[x]}, {x, 0, 3}, PlotStyle -> {Red, Blue, Green, Black}] :[font = text; inactive; preserveAspect] Write a couple of sentences describing the apparent accuracy (or inaccuracy) of the approximations. :[font = text; inactive; preserveAspect] :[font = input; preserveAspect] Join[{{"x"," ","Errors"," "},{"Value","p2","p3","p4"}}, Table[{x, f[x]-p2[x], f[x]-p3[x], f[x]-p4[x]}, {x,0.5,1.5,0.1}]] :[font = text; inactive; preserveAspect] Notice that for x = 1, all of the errors are equal to 0. Explain why this is the case. For x = 1.2, approximately how much extra accuracy do we gain when we increase the degree of the approximating polynomial by one? ;[s] 5:0,0;16,1;17,0;93,1;94,0;219,-1; 2:3,16,12,Chicago,0,12,0,0,0;2,16,12,Chicago,2,12,0,0,0; :[font = text; inactive; preserveAspect] :[font = section; inactive; preserveAspect] Part 3: Approximate the arctan function :[font = text; inactive; preserveAspect] Let :[font = input; preserveAspect] f[x_] := ArcTan[x] :[font = text; inactive; preserveAspect] We will find polynomials which approximate this function using the same procedure we used in Part 2. Because of the way we defined poly, equations, and solutions, we do not need to Clear or redefine these here. ;[s] 9:0,0;133,1;137,0;139,1;148,0;154,1;163,0;183,1;188,0;213,-1; 2:5,16,12,Chicago,0,12,0,0,0;4,13,10,Courier,1,12,0,0,0; :[font = text; inactive; preserveAspect] Create the 7th degree approximation of arctan near x = 0. For what values of x does this polynomial seem to give a good approximation of f (x)? (You might look at a plot over the interval [Ð2,2].) Explain your answer. ;[s] 9:0,0;51,1;52,0;78,1;79,0;138,1;139,0;141,1;142,0;221,-1; 2:5,16,12,Chicago,0,12,0,0,0;4,16,12,Chicago,2,12,0,0,0; :[font = text; inactive; preserveAspect] :[font = text; inactive; preserveAspect] How good is this polynomial's approximation of f(-0.5)? of f(1)? of f(1.5)? Explain your answers, being as quantitative as possible. :[font = text; inactive; preserveAspect] :[font = text; inactive; preserveAspect] Obtain a 7th degree approximation of f near x = 1 and use it to estimate tanÐ1(1.5). What is the error in this estimate? ;[s] 7:0,0;37,1;38,0;44,1;45,0;76,2;78,0;122,-1; 3:4,16,12,Chicago,0,12,0,0,0;2,16,12,Chicago,2,12,0,0,0;1,18,12,Chicago,32,9,0,0,0; :[font = text; inactive; preserveAspect] tanÐ1(1.5) is about: error in this estimate is: ;[s] 3:0,0;4,1;6,0;96,-1; 2:2,16,12,Chicago,0,12,0,0,0;1,18,12,Chicago,32,9,0,0,0; :[font = text; inactive; preserveAspect] Does this polynomial give a good approximation of tanÐ1(Ð0.5)? What is the error in this estimate? ;[s] 3:0,0;53,1;55,0;100,-1; 2:2,16,12,Chicago,0,12,0,0,0;1,18,12,Chicago,32,9,0,0,0; :[font = text; inactive; preserveAspect] :[font = text; inactive; preserveAspect] If you wanted to estimate tanÐ1(Ð0.5), would you use the first or second polynomial? Which polynomial would you use to estimate tanÐ1(1.5)? ;[s] 5:0,0;29,1;31,0;132,1;134,0;141,-1; 2:3,16,12,Chicago,0,12,0,0,0;2,18,12,Chicago,32,9,0,0,0; :[font = text; inactive; preserveAspect] ^*)