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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 31 "\011\011\011\011Multivar
iable Taylor Series" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 7 "Summary" }
}{EXCHG {PARA 256 "" 0 "" {TEXT 257 433 "This worksheet\n\n\011shows h
ow to compute Taylor series in Maple.\n\ncompares the function 1/(x+y^
2) with various \nasociated Taylor polynomials.\n\nshows how maximum e
rrors tend to depend on the size of the region\nbeing evaluated.\n\nsh
ows how to use estimates on the size of appropriate \n\011partial deri
vatives to bound the maximal error in using the\nTaylor polynomial. (T
hese size estimates are generated by graphs\nof the partial derivative
s.)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with(plots): with(plottools)
:" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has b
een redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 43 "Warning, the name arro
w has been redefined\n" }}}}{EXCHG {PARA 7 "" 0 "" {TEXT 258 82 "We ne
ed to load the mtaylor function before doing any multivariable Taylor \+
series." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "readlib(mtaylor);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6\"6*%\"fG%\"kG%\"vG%\"mG%\"nG%\"sG%
\"tG%\"wG6#%aoCopyright~(c)~1991~by~the~University~of~Waterloo.~All~ri
ghts~reserved.GF$C2>8$&9\"6#\"\"\">8&&F46#\"\"#@&-%%typeG6$F8%$setG>F8
7#-%#opG6#F84-F>6$F8%%listG>F87#F8@$4-F>6$F8-FI6#<$%%nameG/FS%*algebra
icGYQGinvalid~2nd~argument~(expansion~point)F$>8)-%$mapG6$f*6#%\"xGF$F
$F$@%-F>6$9$%\"=G-%$rhsG6#F]o\"\"!F$F$F$F8>F8-Fen6$f*FhnF$F$F$@%F[o-%$
lhsGFaoF]oF$F$F$F8>8'-%%nopsGFE@$0F[p-F]p6#<#FCYQHvariables~(2nd~argum
ent)~must~be~uniqueF$@%/9#F;>8(\"\"'>Fip&F46#\"\"$@%/Fgp\"\"%>8+&F46#F
aq>Fcq7#-%\"$G6$F6F[p@$4-F>6$F8<$-FI6#FS-F@FarYQQ2nd~argument~(the~var
iable(s))~must~be~a~name(s)F$@$34-F>6$Fip%*nonnegintG0Fip%)infinityGYQ
X3rd~argument~(the~order)~must~be~a~non-negative~integerF$@$54-F>6$Fcq
-FI6#%'posintG0-F]p6#FcqF[pYQen4th~argument~(weights)~must~be~a~list~o
f~positive~integersF$>F2-%%subsG6$7#-%$seqG6$/&F86#8%,&*&FetF6)8*&FcqF
ftF6F6&FYFftF6/Fgt;F6F[pF2>F2-Fen6&%(collectG-F^t6$/-%\"OGF5Fbo-%'tayl
orG6%F2F[uFipF8.%,distributedG>F2-F^t6$7#-Fbt6$/Fet,&FetF6F]u!\"\"F^u-
F^t6$/F[uF6F2F$F$F$" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 15 "A Basic E
xample" }}{EXCHG {PARA 0 "" 0 "" {TEXT 273 100 "(It is worth noting he
re how the quality of approximation relates to singularities of the fu
nction.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "expn := 1/(x+y^
2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%expnG*&\"\"\"F&,&%\"xGF&*$)%
\"yG\"\"#F&F&!\"\"" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 259 77 "Compute \+
the Taylor series to order 2 (one less than the argument 3) for expn \+
" }}{PARA 256 "" 0 "" {TEXT 272 61 " with respect to the variables x a
nd y about the point (1,1)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "poly
2 :=mtaylor(expn,[x=1,y=1],3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&p
oly2G,.#\"\"&\"\"%\"\"\"*&#F)F(F)%\"xGF)!\"\"*&#F)\"\"#F)%\"yGF)F-*&#F
)\"\")F)),&F,F)F)F-F0F)F)*(#F)F0F)F6F),&F1F)F)F-F)F)*&#F)F(F))F9F0F)F)
" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 260 55 "Convert all the coefficien
ts to floating point numbers." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "po
ly2n := map(evalf,poly2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'poly2n
G,.$\"++++]7!\"*\"\"\"*&$\"+++++D!#5F)%\"xGF)!\"\"*&$\"+++++]F-F)%\"yG
F)F/*&$F'F-F)),&F.F)$F)\"\"!F/\"\"#F)F)*($\"+++++]F-F)F7F),&F3F)$F)F9F
/F)F)*&$\"+++++DF-F))F>F:F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 57 "gr2 := plot3d(poly2n,x=.5 .. 1.5,y= .5 .. 1.5,color=red):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "gr3 := plot3d(expn,x= .5 .. \+
1.5,y= .5 .. 1.5,color=blue):" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 261 
127 "Display both the original function (blue) and the second order Ta
ylor Series (red).\nThese are tangent to second order at (1,1)." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "display3d(\{gr3,gr2\},axes=framed);
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{EXCHG {PARA 256 "" 0 "" {TEXT 263 191 "Reducing the size of the recta
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ast a factor of 2^3=8." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "gr5 := pl
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{SECT 0 {PARA 3 "" 0 "" {TEXT 274 57 "Estimating the Error For the \n
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ve 1" }}}}{EXCHG {PARA 256 "" 0 "" {TEXT 267 142 "Plot the second part
ial with respect to y and find that it appears to be bounded by 3\nin \+
absolute value for the rectangle|x-1| < .5, |y-1|<.5." }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 76 " plot3d(diff(expn,y,y),x=0.5.. 1.5,y= 0.5 .. 1.5
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0 "Curve 1" }}}}{EXCHG {PARA 256 "" 0 "" {TEXT 268 154 "Using the form
ula involving derivatives at  an unknown point for the the error in th
e\nlinear approximation , we estimate that the error should be at most
:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "(1/2) * ( 5* (.5)^2 + 2*5 * (.
5)^2 + 3* (.5)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++++]A!\"*" }
}}{EXCHG {PARA 256 "" 0 "" {TEXT 269 226 "The actual error appears to \+
be quite a bit smaller than this maximum bound.\n(The partials were mo
stly just big near (.5,.5) so it is not surprising\n that the actual e
rrrors never got as big as the conservative estimate above.)" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 88 "plot3d(expn-mtaylor(expn,\{x=1,y=1\},2),x
= .5 .. 1.5,y=.5 ..1.5,axes=framed,shading=zhue);" }}{PARA 13 "" 1 "" 
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