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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 66 "Click on the [+] sign below to see the contents of this workshe
et." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
-1 31 "12.4  Quadratic  Approximations" }}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 5 "Goals" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 129 "In this section we illustrate how to use Maple V to appr
oximate a function of two variables with a Taylor polynomial of degree
 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 47 "Mathematical Concepts Discussed in This Section" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "If you wish to go d
irectly to some of the mathematical concepts that we discuss in this s
ection then click on the following bookmark:   " }{HYPERLNK 17 "Taylor
 polynomials of  degree 2" 1 "" "Taylor polynomials of  degree 2" }
{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 
0 "" {TEXT -1 10 "Discussion" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "Taylor polynomials of  degree 2" {TEXT -1 42 "Just as o
ne can use the Maple V procedure " }{TEXT 257 7 "mtaylor" }{TEXT -1 
141 " to linearly approximate functions of more than one variable,  on
e can also find higher order approximations of these same functions.  \+
Using " }{TEXT 258 7 "mtaylor" }{TEXT -1 153 " it is easy to obtain Ta
ylor polynomials of  degree 2.  The following Maple V segment shows ho
w one can find the quadratic approximation for the function" }}{PARA 
0 "" 0 "" {TEXT -1 42 "                                          " }
{XPPEDIT 18 0 "f(x,y) =  sin(x) * cos(y)" "/-%\"fG6$%\"xG%\"yG*&-%$sin
G6#F&\"\"\"-%$cosG6#F'F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
52 "near the point  (0,0).  First we make the procedure " }{TEXT 259 
7 "mtaylor" }{TEXT -1 26 " available with a call to " }{TEXT 260 7 "re
adlib" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "readlib(mtaylor);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#:6\"6*%\"fG%\"kG%\"vG%\"mG%\"nG%\"sG%\"tG%\"wG6#%aoC
opyright~(c)~1991~by~the~University~of~Waterloo.~All~rights~reserved.G
F$C2>8$&9\"6#\"\"\">8&&F46#\"\"#@&-%%typeG6$F8%$setG>F87#-%#opG6#F84-F
>6$F8%%listG>F87#F8@$4-F>6$F8-FI6#<$%%nameG/FS%*algebraicG-%&ERRORG6#%
Ginvalid~2nd~argument~(expansion~point)G>8)-%$mapG6$:6#%\"xGF$F$F$@%-F
>6$9$%\"=G-%$rhsG6#F_o\"\"!F$F$F8>F8-Fgn6$:FjnF$F$F$@%F]o-%$lhsGFcoF_o
F$F$F8>8'-%%nopsGFE@$0F]p-F_p6#<#FC-FW6#%Hvariables~(2nd~argument)~mus
t~be~uniqueG@%/9#F;>8(\"\"'>F\\q&F46#\"\"$@%/Fjp\"\"%>8+&F46#Fdq>Ffq7#
-%\"$G6$F6F]p@$4-F>6$F8<$-FI6#FS-F@Fdr-FW6#%O2nd~argument~(the~variabl
e(s))~must~be~a~namesG@$34-F>6$F\\q%*nonnegintG0F\\q%)infinityG-FW6#%X
3rd~argument~(the~order)~must~be~a~non-negative~integerG@$54-F>6$Ffq-F
I6#%'posintG0-F_p6#FfqF]p-FW6#%en4th~argument~(weights)~must~be~a~list
~of~positive~integersG>F2-%%subsG6$7#-%$seqG6$/&F86#8%,&*&F[uF6)8*&Ffq
F\\uF6F6&FenF\\uF6/F]u;F6F]pF2>F2-Fgn6&%(collectG-Fdt6$/-%\"OGF5Fdo-%'
taylorG6%F2FauF\\qF8.%,distributedG>F2-Fdt6$7#-Fht6$/F[u,&F[uF6Fcu!\"
\"Fdu-Fdt6$/FauF6F2F$F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 28 "Now define the function  f. " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f :=
 (x,y)-> sin(x)*sin(y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6$%
\"xG%\"yG6\"6$%)operatorG%&arrowGF)*&-%$sinG6#9$\"\"\"-F/6#9%F2F)F)" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "
Now invoke the procedure and assign the approximation to the variable \+
f2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "mtaylo
r" {MPLTEXT 1 0 35 "f2 :=  mtaylor(f(x,y),[x=0,y=0],3);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#f2G*&%\"xG\"\"\"%\"yGF'" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Next convert the e
xpression  f2  to a function with the procedure " }{TEXT 261 7 "unappl
y" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 22 "f2 := unapply(f2,x,y);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#f2G:6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)*&9$\"\"\"
9%F/F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 174 "Some graphical comparisons of the function  f with its q
uadratic approximation f2 will now be made.  Plots of  f   and  f2  ar
e created by the following two Maple V commands." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "P1 := plot3
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OURG-%+PROJECTIONG6%$\"$q#\"\"!F_`m\"\"\"" 3 252 252 252 7 0 1 0 2 1 
0 1 2 1.000000 0.000000 270.000000 0 0 0 0 0 0 0 0 0 0 0 83 90 235 83 
90 235 1 0 0 0 0 0 0 0 0 0 0 0 1 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 190 "Observe that the contours of f(x,y) have several singula
r points (points where the gradient of f(x,y) is the zero vector) but \+
the one at the origin looks like the singular point for f2(x,y)." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Some of The Maple V Co
mmands Used in This Section" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 135 "Examples of Maple V commands, or options,  or \+
 variables used in this section can be obtained by clicking on the fol
lowing bookmarks:  " }{HYPERLNK 17 "mtaylor" 1 "" "mtaylor" }{TEXT -1 
2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 14 "Exercises 12.4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 4 "Use " }{TEXT 256 7 "mtaylor" }{TEXT -1 135 "  to
 find linear and quadratic approximations near the origin for each of \+
the following functions.  Make graphical comparisons of each." }}
{PARA 14 "" 0 "" {TEXT -1 4 "1.  " }{XPPEDIT 18 0 " f(x,y) = exp(3*x^2
)*cos(y)" "/-%\"fG6$%\"xG%\"yG*&-%$expG6#*&\"\"$\"\"\"*$F&\"\"#F.F.-%$
cosG6#F'F." }{TEXT -1 2 "  " }}{PARA 14 "" 0 "" {TEXT -1 4 "2.  " }
{XPPEDIT 18 0 "f(x,y) = sin(x-y)" "/-%\"fG6$%\"xG%\"yG-%$sinG6#,&F&\"
\"\"F'!\"\"" }{TEXT -1 1 " " }}{PARA 14 "" 0 "" {TEXT -1 4 "3.  " }
{XPPEDIT 18 0 "f(x,y) = 1/(1-x+y)" "/-%\"fG6$%\"xG%\"yG*&\"\"\"F),(F)F
)F&!\"\"F'F)F+" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}}{MARK "1" 0 }
{VIEWOPTS 1 1 0 1 1 1803 }