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0 0 1 1 1 2 2 2 2 0 0 0 1 } {CSTYLE "LaTeX" -1 32 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "Help Menus" -1 36 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "Prompt" -1 1 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "Help Underlined" -1 44 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Italic" -1 43 "Times" 1 12 0 0 0 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "_cstyle3" -1 217 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold" -1 5 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle256" -1 218 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic" -1 3 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }} {SECT 0 {EXCHG {PARA 204 "" 0 "" {TEXT 215 46 "Preparation for Fourier Series, Taylor Series:" }{TEXT 215 0 "" }}{PARA 210 "" 0 "" {TEXT 204 34 "a visual approach to approximation" }{TEXT 204 0 "" }}{PARA 202 "" 0 "" {TEXT 217 57 "\251Mike May, S.J., maymk@slu.edu, Saint Lou is University" }{TEXT 217 0 "" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 207 0 "" }}}{SECT 1 {PARA 212 "" 0 "" {TEXT 208 27 "Background and i ntroduction" }{TEXT 208 0 "" }}{EXCHG {PARA 211 "" 0 "" {TEXT 216 53 " Why we look to Calculus while studying Linear Algebra" }{TEXT 206 0 "" }}{PARA 208 "" 0 "" {TEXT 210 251 "One of the challenging topics of l inear algebra is the use of inner product spaces to project onto a sub space. The underlying concept that is often missed is that the proces s gives the best approximation of a vector with a vector in a given su bspace." }}{PARA 208 "" 0 "" {TEXT 210 0 "" }}{PARA 208 "" 0 "" {TEXT 210 325 "This process has a nice visualization with Fourier series, bu t the computations are challenging enough that students often miss the point of approximation. Since most students in linear algebra have h ad at least a year of calculus, it is useful to refer back to Taylor s eries where approximation has a similar visualization." }}{PARA 208 "" 0 "" {TEXT 210 0 "" }}}{EXCHG {PARA 211 "" 0 "" {TEXT 216 35 "A fast \+ review of Taylor Polynomials" }{TEXT 206 0 "" }}{PARA 206 "" 0 "" {TEXT 202 81 "One of the ongoing themes of calculus is polynomial appr oximation of functions. " }{TEXT 202 0 "" }}{PARA 206 "" 0 "" {TEXT 202 0 "" }}{PARA 206 "" 0 "" {TEXT 202 95 "The tangent line can be des cribed as the \"best linear approximation\" to a curve at a point. " }{TEXT 202 0 "" }}{PARA 206 "" 0 "" {TEXT 202 0 "" }}{PARA 206 "" 0 "" {TEXT 202 211 "When we first introduced second derivatives, we looked at the \"best quadratic approximation\" of a curve at a point. This \+ was a quadratic polynomial that p(x) that approximated f(x) at a given point x=c, with " }{TEXT 202 0 "" }}{PARA 214 "" 0 "" {TEXT 209 49 "p (c) = f(x), p'(c) = f'(c), and p''(c) = f''(c)." }{TEXT 209 0 "" }} {PARA 206 "" 0 "" {TEXT 202 0 "" }}{PARA 206 "" 0 "" {TEXT 202 520 "We have continued this theme by studying Taylor series, which comes at \+ the end of the chapter on sequences and series. In the flurry of comp utations on sequences and series, it is all to easy to loose sight of \+ the idea that a Taylor polynomial is the \"best nth degree polynomial \+ approximation\" to a function at a point. A visual approach makes thi s clear. (If a picture is worth 1000 words, how much is a movie worth ?) The visual approach also makes concepts like the interval of conve rgence easier to understand." }{TEXT 202 0 "" }}{PARA 206 "" 0 "" {TEXT 202 0 "" }}}{SECT 1 {PARA 212 "" 0 "" {TEXT 208 0 "" }}{EXCHG {PARA 206 "" 0 "" {TEXT 202 141 "We start with the technical details. \+ Put your cursor in the red section below, and hit the return key. Th is should load the plots routines." }{TEXT 202 0 "" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 207 12 "with(plots):" }{MPLTEXT 1 207 0 "" }} }}}{EXCHG {PARA 206 "" 0 "" {TEXT 202 193 "We next view an example. T he example we look at is the sin function expanded about the point x=0 . We look at the plot of sin(x) plotted on the same axis with the nth degree Taylor polynomial " }{XPPEDIT 18 0 "T[n];" "6#&%\"TG6#%\"nG" } {TEXT 202 33 ", where n goes from 2 to 20 by 2." }{TEXT 202 0 "" }}} {SECT 1 {PARA 212 "" 0 "" {TEXT 208 72 "The first example, an animatio n on a nice sequence of Taylor polynomials" }{TEXT 208 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 207 15 "func := sin(x);" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 2 "\n" }{MPLTEXT 1 207 37 "minx := -3: maxx \+ := 10: aboutx := 1:" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 2 "\n" } {MPLTEXT 1 207 25 "miny := -1.25: maxy := 2:" }{MPLTEXT 1 207 0 "" } {MPLTEXT 1 207 2 "\n" }{MPLTEXT 1 207 39 "mindeg := 2: degsteps := 12: bydeg :=2:" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 2 "\n" }{MPLTEXT 1 207 17 "A := display(seq(" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 2 "\n" }{MPLTEXT 1 207 63 "plot(convert(taylor(func,x=aboutx, mindeg + bydeg *i), polynom)," }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 2 "\n" }{MPLTEXT 1 207 38 "x=minx..maxx,y=miny..maxy, color=blue," }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 2 "\n" }{MPLTEXT 1 207 6 "title=" }{MPLTEXT 1 207 10 "substring(" }{MPLTEXT 1 207 34 "\"T\"||(mindeg + bydeg*i)||\" = \"" } {MPLTEXT 1 207 83 "||(convert(convert(taylor(func,x=aboutx, mindeg + b ydeg*i),polynom),string)),1..60)" }{MPLTEXT 1 207 2 ", " }{MPLTEXT 1 207 2 "\n" }{MPLTEXT 1 207 25 "titlefont=[HELVETICA, 14]" }{MPLTEXT 1 207 3 "), " }{MPLTEXT 1 207 2 "\n" }{MPLTEXT 1 207 33 "i=0..degsteps), insequence=true):" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 2 "\n" } {MPLTEXT 1 207 74 "B := animate(func,x=minx..maxx,y=miny..maxy,frames= degsteps+1, color=red):" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 2 "\n" } {MPLTEXT 1 207 65 "print(` `||(convert(func, string))||` vs its Taylor polynomial`);" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 2 "\n" }{MPLTEXT 1 207 42 "display(A,B,view=[minx..maxx,miny..maxy]);" }{MPLTEXT 1 207 0 "" }}}{PARA 212 "" 0 "" {TEXT 208 20 "To run the animation" }{TEXT 208 0 "" }}{EXCHG {PARA 206 "" 0 "" {TEXT 202 527 "To run the animatio n, click once on the picture above. A box should appear around the pi cture. If you see the box, you should see a series of buttons, like y ou might see on a tape recorder. From left to right, the buttons are: back one frame, stop, start, advance one frame; a bow and slider for \+ the current frame; a drop down choice of forward, down and back, or b ackwards for direction; a drop down choice of stop after one cycleor r un in a loop, and a ontrols for the speed. Click the on button and wa tch the animation." }{TEXT 202 0 "" }}}}{SECT 1 {PARA 212 "" 0 "" {TEXT 208 47 "Taylor polynomials and interval of convergence." }{TEXT 208 0 "" }}{SECT 1 {PARA 212 "" 0 "" {TEXT 208 46 "An example with a f inite radius of convergence" }{TEXT 208 0 "" }}{EXCHG {PARA 206 "" 0 " " {TEXT 202 313 "The first animation we looked at has a Taylor series \+ that works everywhere. To make the Taylor polynomial work as an appro ximation on a bigger interval we simply increased the degree of the po lynomial. We now consider a case where the approximation is only good in a small region, no matter how high the degree." }{TEXT 202 0 "" }} {PARA 206 "" 0 "" {TEXT 202 32 "Execute the following animation:" } {TEXT 202 0 "" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 207 20 "func : = 1/(1 + x^2);" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 2 "\n" }{MPLTEXT 1 207 40 "minx := -0.5: maxx := 2.5: aboutx := 1:" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 2 "\n" }{MPLTEXT 1 207 26 "miny := -0.2: maxy := 1. 2:" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 2 "\n" }{MPLTEXT 1 207 39 "mi ndeg := 2: degsteps := 15: bydeg :=8:" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 2 "\n" }{MPLTEXT 1 207 58 "for i from mindeg by bydeg to mindeg \+ + bydeg * degsteps do" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 2 "\n" } {MPLTEXT 1 207 55 "T[i] :=convert(taylor(func, x=aboutx, i), polynom): od:" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 2 "\n" }{MPLTEXT 1 207 42 " A := display(seq(plot(T[mindeg + bydeg*i]," }{MPLTEXT 1 207 0 "" } {MPLTEXT 1 207 2 "\n" }{MPLTEXT 1 207 38 "x=minx..maxx,y=miny..maxy, c olor=blue," }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 2 "\n" }{MPLTEXT 1 207 135 "title=substring(\"T\"||(mindeg + bydeg*i)||\" = \"||(convert( convert(taylor(func,x=aboutx, mindeg + bydeg*i),polynom),string)),1..6 0), " }{MPLTEXT 1 207 2 "\n" }{MPLTEXT 1 207 28 "titlefont=[HELVETICA, 14]), " }{MPLTEXT 1 207 2 "\n" }{MPLTEXT 1 207 33 "i=0..degsteps), in sequence=true):" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 2 "\n" }{MPLTEXT 1 207 74 "B := animate(func,x=minx..maxx,y=miny..maxy,frames=degsteps +1, color=red):" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 2 "\n" }{MPLTEXT 1 207 65 "print(` `||(convert(func, string))||` vs its Taylor polynom ial`);" }{MPLTEXT 1 207 0 "" }{MPLTEXT 1 207 2 "\n" }{MPLTEXT 1 207 42 "display(A,B,view=[minx..maxx,miny..maxy]);" }{MPLTEXT 1 207 0 "" } }}{EXCHG {PARA 206 "" 0 "" {TEXT 202 168 "Notice that the Taylor polyn omial approximation of 1/(1 + x^2) centered at x = 1.5 only gives a go od approximation for x between 0 and 3, no matter how high the degree. " }{TEXT 202 0 "" }}}}}{EXCHG {PARA 212 "" 0 "" {TEXT 208 19 "Final In structions:" }{TEXT 208 0 "" }}{PARA 206 "" 0 "" {TEXT 202 61 "When yo u complete the worksheet, print it out and hand it in." }{TEXT 202 0 " " }}{PARA 206 "" 0 "" {TEXT 202 121 "(One way to print is to use the \+ \"Export as...RTF\" option under the file menu. RTF format means it i s a Word document.)" }{TEXT 202 0 "" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 207 0 "" }}}{PARA 203 "" 0 "" {TEXT 219 0 "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }