{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 29 "Taylor Series Experimenta tion" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "One Variable Case" }}{EXCHG {PARA 256 "" 0 "" {TEXT -1 58 "Taylor polynomials of degrees 1 through 4 for f about x=1." }{TEXT 256 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "f:=1/(1+x+x^2);\ngr0:=plot(f,x=-2..2):\nfor i from 1 to 5 do\n f || i:=mtaylor(f,[x=1],i+1);\nod;" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 108 "The first graph compares f with these Taylor polynomials for x between 0 and 2.\nThe second between -2 and 2." }}{PARA 256 "" 0 "" {TEXT -1 135 "Do you see a difference in the quality of the approximat ion?\nWhat do you think would happen over [-2,2] with higher degree po lynomials?" }}{PARA 256 "" 0 "" {TEXT -1 71 "Would they eventually giv e a good approximation on the interval [-2,2]?" }{TEXT 257 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 366 "colors:=[red,yellow,green,blue,vio let]:\ngr0:=plot(f,x=-2..2,color=colors[1]):\nfor i from 1 to 4 do\n \+ gr || i := plot(f || i,x=-2..2,color=colors[i+1]):\nod:\ndisplay([gr0 ,gr1,gr2,gr3,gr4],thickness=2,title=`Taylor polynomials about x=1; f \+ is in red`,view=[0 .. 2,0..1]);\ndisplay([gr0,gr1,gr2,gr3,gr4],thickne ss=2,title=`Taylor polynomials about x=1; f is in red`);\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 28 "A Basic Two Variable Example" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 74 "The 3 below in the mtaylor call below mea ns degree 3-1=2 Taylor polynomial" }{TEXT -1 1 "." }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 41 "g:=1/(y^2+x);\ng2:=mtaylor(g,[x=1,y=1],3);" }}} {EXCHG {PARA 256 "" 0 "" {TEXT -1 97 "The first graph shows g and its \+ second Taylor polynomial over the x_range and y_range specified. " }} {PARA 256 "" 0 "" {TEXT -1 72 "The second graphs the difference, which may be interpreted as an error. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 91 "Try rotating the graphs (especially the first) to examine the quality of the approximation." }}{PARA 256 "" 0 "" {TEXT -1 62 "Try some other ranges besides the 0.5 .. 1.5 specifi ed below. " }}{PARA 256 "" 0 "" {TEXT -1 69 "Try at least some ranges \+ close to 1 (e.g 0.9 .. 1.1 or 0.75 .. 1.25) " }}{PARA 256 "" 0 "" {TEXT -1 59 "as well as some bigger ranges (e.g. 0 ..2 or 0.25 .. 1.75 )." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 77 "U se the second graph to compare maximal errors with the width of the x_ range." }}{PARA 0 "" 0 "" {TEXT 258 73 "(e.g. compare the largest erro rs in the cases 0.9 .. 1.1 and 0.95..1.05) " }}{PARA 256 "" 0 "" {TEXT -1 75 "How much smaller has the error become as the width of th e interval halved?" }}{PARA 256 "" 0 "" {TEXT -1 112 "Can you explain \+ approximately why this is so?\nDo bigger intervals also work this way when their size is halved?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 264 "x_r ange:=0.5 .. 1.5;\ny_range:=x_range;\ngr5:=plot3d(g,x=x_range,y=y_rang e,color=blue):\ngr6:=plot3d(g2,x=x_range,y=y_range,color=red):\ndispla y([gr5,gr6],axes=normal,title=`g is in blue,p2 in red`);\nplot3d(g-g2, x=x_range,y=y_range,axes=normal,title=`error using p2`);" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 71 "Do you see something fundamentally diff erent about the approximation on" }}{PARA 258 "" 0 "" {TEXT -1 98 "a b ig range vs. a small one? You could try changing the degree of the Tay lor \napproximation above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 12 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }