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{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 "" }}}}
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