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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 11 "Calculus II" }}{PARA 256 
"" 0 "" {TEXT -1 63 "Lesson 22:  Approximating General Functions with \+
Taylor Series\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "In the previo
us lesson, we saw how to approximate the sine and cosine functions arb
itrarily accurately near " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }
{TEXT -1 187 " by using polynomials.   In this worksheet, we will see \+
that the same approach can be used to approximate other functions, and
 that it can be used to get approximations near other points." }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 21 "Approximations near 0" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 181 "Recall how the method works: we find an approximation to
 our function near 0, then we subtract that approximation from the fun
ction, and see how quickly the difference goes to 0 as " }{XPPEDIT 18 
0 "proc (x) options operator, arrow; 0 end;" "6#R6#%\"xG7\"6$%)operato
rG%&arrowG6\"\"\"!F*F*F*" }{TEXT -1 26 " by dividing by powers of " }
{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 22 "f := x -> sqrt(1 + x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#,&\"
\"\"F09$F0F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "plot(f,
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45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Since \+
" }{XPPEDIT 18 0 "f(0) = 1;" "6#/-%\"fG6#\"\"!\"\"\"" }{TEXT -1 85 ", \+
our first approximation should be the constant function (0-th degree p
olynomial) 1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "p0 := x ->
 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p0G\"\"\"" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 25 "limit(f(x) - p0(x), x=0);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "As e
xpected, our approximation is good enough that the difference goes to \+
0 as " }{XPPEDIT 18 0 "proc (x) options operator, arrow; 0 end;" "6#R6
#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"!F*F*F*" }{TEXT -1 16 " , but how
 fast?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "limit((f(x) - p0(
x))/x, x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"#" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "This limit shows that " }{XPPEDIT 
18 0 "f(x)-p0(x);" "6#,&-%\"fG6#%\"xG\"\"\"-%#p0G6#F'!\"\"" }{TEXT -1 
12 " looks like " }{XPPEDIT 18 0 "x/2;" "6#*&%\"xG\"\"\"\"\"#!\"\"" }
{TEXT -1 6 " when " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 40 " is sm
all, so our next approximation is " }{XPPEDIT 18 0 "p0(x)+x/2;" "6#,&-
%#p0G6#%\"xG\"\"\"*&F'F(\"\"#!\"\"F(" }{TEXT -1 2 " :" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "p1 := x -> 1 + (1/2)*x;" }}{PARA 
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\"\"F-*&#F-\"\"#F-9$F-F-F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 
"Before computing further, let's plot our original function " }
{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 42 " , together with the linear
 approximation " }{XPPEDIT 18 0 "p1;" "6#%#p1G" }{TEXT -1 2 " :" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(\{f,p1\}, -1..1);" }}
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0 "" 0 "" {TEXT -1 94 "Observe that the (graph of the) linear approxim
ation is just the tangent line to the graph of " }{XPPEDIT 18 0 "f;" "
6#%\"fG" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }
{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "limit((f(
x) - p1(x))/x, x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "limit((f(x) - p1(x))/x^2, x=
0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"\"\"\")" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 181 "These limits show that we have improved the appr
oximation, and tell us what the next term should be.  We can repeat th
e process as many times as we like.  Here are a few more steps." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "p2 := x -> 1 + (1/2)*x - (1/
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orG%&arrowGF(,(\"\"\"F-*&#F-\"\"#F-9$F-F-*&#F-\"\")F-*$)F1F0F-F-!\"\"F
(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "limit((f(x) - p2(x
))/x^3, x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"#;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "p3 := x -> p2(x) + (1/16)*x^
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{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"&\"$G\"" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 31 "p4 := x -> p3(x) - (5/128)*x^4;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#p4GR6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%#p3G6#9$
\"\"\"*&#\"\"&\"$G\"F1*$)F0\"\"%F1F1!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> \+
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{TEXT -1 31 "How well has the method worked?" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 23 "plot(\{f,p2,p4\}, -1..1);" }}{PARA 13 "" 1 "" 
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "There is nothing special about the
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e method will work with any reasonably nice function.  Here is one tha
t is important in statistics (it is used to describe the so-called " }
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0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "g(x) = 1;" "6#/-%\"gG6#%\"xG\"\"\"" }
{TEXT -1 42 ", so that will be our first approximation." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "q0 := x -> 1;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#q0G\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
25 "limit(g(x) - q0(x), x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "limit((g(x) - q0(x))/x,
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" 1 "" {XPPMATH 20 "6##!\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
102 "These limits show that the constant 1 is also the linear approxim
ation, as is clear from the graph of " }{XPPEDIT 18 0 "g;" "6#%\"gG" }
{TEXT -1 47 ".  The quadratic approximation adds a new term." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "q2 := x -> q0(x) - (1/2)*x^2
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "q2(x);" }}{PARA 11 "" 1 "" 
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{PARA 0 "" 0 "" {TEXT -1 28 "In fact, the only powers of " }{XPPEDIT 
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" }{XPPEDIT 18 0 "g;" "6#%\"gG" }{TEXT -1 6 " near " }{XPPEDIT 18 0 "x
 = 0;" "6#/%\"xG\"\"!" }{TEXT -1 95 " are even (why?).  We will take a
dvantage of this fact to avoid unnecessary computations below." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "limit((g(x) - q2(x))/x^4, x=
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"" {TEXT -1 39 "Approximations near points other than 0" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 78 "As you can see from the examples above, our approximation
s are very good near " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT 
-1 63 ", but not near other points.  Suppose we wanted to approximate \+
" }{XPPEDIT 18 0 "sqrt(1+x);" "6#-%%sqrtG6#,&\"\"\"F'%\"xGF'" }{TEXT 
-1 6 " near " }{XPPEDIT 18 0 "x = 3;" "6#/%\"xG\"\"$" }{TEXT -1 27 ", \+
for example.  Of course, " }{TEXT 258 2 "at" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "x = 3;" "6#/%\"xG\"\"$" }{TEXT -1 92 " this expression \+
is exactly equal to 2, so this should certainly be our first approxima
tion." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f := x -> sqrt(1 +
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{XPPEDIT 18 0 "proc (x) options operator, arrow; 3 end;" "6#R6#%\"xG7
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%\"xG" }{TEXT -1 28 " .  Now we are working near " }{XPPEDIT 18 0 "x =
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Unfortunately, " }{TEXT 259 5 "Map
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***\\(QIKH@Ff_l$\"3K-')Qq#*)*o<Ff_l7$$\"3#****\\7:xWC#Ff_l$\"3xo,zBKC,
=Ff_l7$$\"37++]Zn%)oBFf_l$\"3USw&)R=WN=Ff_l7$$\"3y******4FL(\\#Ff_l$\"
3%4/#G%p:,(=Ff_l7$$\"3#)****\\d6.BEFf_l$\"3CA0(G;EM!>Ff_l7$$\"3(****\\
(o3lWFFf_l$\"3([tv//5^$>Ff_l7$$\"3!*****\\A))ozGFf_l$\"3s5^bqDpp>Ff_l7
$$\"3e******Hk-,IFf_l$\"3@f-/\"fc-+#Ff_l7$$\"36+++D-eIJFf_l$\"3J%H0L*G
QK?Ff_l7$$\"3u***\\(=_(zC$Ff_l$\"3c)p=Fnh51#Ff_l7$$\"3M+++b*=jP$Ff_l$
\"3gmd7L`'>4#Ff_l7$$\"3g***\\(3/3(\\$Ff_l$\"3$p2[w2K17#Ff_l7$$\"33++vB
4JBOFf_l$\"3k$R['y&)=]@Ff_l7$$\"3u*****\\KCnu$Ff_l$\"3W;:y]ypy@Ff_l7$$
\"3s***\\(=n#f(QFf_l$\"3nR53n+:3AFf_l7$$\"3P+++!)RO+SFf_l$\"3=,4Uk$\\h
B#Ff_l7$$\"30++]_!>w7%Ff_l$\"3Sz)zDmCWE#Ff_l7$$\"3O++v)Q?QD%Ff_l$\"3].
2/t87#H#Ff_l7$$\"3G+++5jypVFf_l$\"3ep=xP*zsJ#Ff_l7$$\"3<++]Ujp-XFf_l$
\"3KEeirEyXBFf_l7$$\"3++++gEd@YFf_l$\"3iUbJ)e&)4P#Ff_l7$$\"39++v3'>$[Z
Ff_l$\"3]!Qc-NlvR#Ff_l7$$\"37++D6Ejp[Ff_l$\"3OA\"[zYKFU#Ff_l7$$FczF][l
$\"3)y<$yU(*[\\CFf_l-Fgz6&FizF\\[lFjzF\\[l-%+AXESLABELSG6$Q!6\"F^\\m-%
%VIEWG6$;Fb[lFf[m%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 52 "Here are approximations for our other function near " }
{XPPEDIT 18 0 "x = -2;" "6#/%\"xG,$\"\"#!\"\"" }{TEXT -1 164 ".  The c
oefficients are not very nice, but you can see from the plots that the
 method is still working.  (If you need decimal approximations, you ca
n of course use " }{TEXT 260 5 "evalf" }{TEXT -1 82 ", but you should \+
be sure to compute the \"exact approximations\" first and only use " }
{TEXT 261 5 "evalf" }{TEXT -1 35 " at the very end of the procedure.)
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "g := x -> exp(-x^2/2);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operatorG%&ar
rowGF(-%$expG6#,$*$)9$\"\"#\"\"\"#!\"\"F3F(F(F(" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 24 "q0 := x -> g(-2); q0(x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#q0GR6#%\"xG6\"6$%)operatorG%&arrowGF(-%\"gG6#!\"#F(F
(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#!\"#" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 26 "limit(g(x) - q0(x), x=-2);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
42 "l1 := limit((g(x) - q0(x))/(x + 2), x=-2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#l1G,$-%$expG6#!\"#\"\"#" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 54 "Notice that the approximations will be in powers of  (" }
{XPPEDIT 18 0 "x-(-2);" "6#,&%\"xG\"\"\",$\"\"#!\"\"F(" }{TEXT -1 5 ")
 = (" }{XPPEDIT 18 0 "x+2;" "6#,&%\"xG\"\"\"\"\"#F%" }{TEXT -1 3 ") .
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "q1 := x -> q0(x) + l1*(
x+2); q1(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q1GR6#%\"xG6\"6$%)o
peratorG%&arrowGF(,&-%#q0G6#9$\"\"\"*&%#l1GF1,&F0F1\"\"#F1F1F1F(F(F(" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#!\"#\"\"\"*(\"\"#F(F$F(,&
%\"xGF(F*F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "As always, the
 linear approximation gives us the tangent line:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 20 "plot(\{g,q1\}, -4..0);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!\"%\"\"!$!33\"Q
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$\"3?zi@=')p$4&Fc\\l7$Fgn$\"30\\M()*)=bKnFc\\l7$F\\o$\"3sY=FnL'Qb)Fc\\
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q$\"3[emohWz6WF_p7$Ffq$\"3yw:)>l%*GN&F_p7$F[r$\"31n*HFc.5d'F_p7$F`r$\"
3;#y3(=BjKzF_p7$Fer$\"3Y*)\\?uQy&e*F_p7$Fjr$\"3])ppR(R@L6F-7$F_s$\"3![
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$\"3W9'f<\"RQb]F-7$Ffv$\"33D<z2lImbF-7$F[w$\"39LvkM6]`gF-7$F`w$\"3!*3c
!=&QxilF-7$Few$\"3!o]^3$zh`qF-7$Fjw$\"3g4]:4I*=b(F-7$F_x$\"3xP(o@Cp'3!
)F-7$Fdx$\"3lY.7,(fSW)F-7$Fix$\"3)y\"3(fz&=O))F-7$F^y$\"33!HWE`H_:*F-7
$Fcy$\"33Tyej\"\\_Y*F-7$Fhy$\"3?7sF`@x'o*F-7$F]z$\"3[Z^V.QAg)*F-7$Fbz$
\"3+<DI2JIi**F-7$Fgz$\"\"\"F*-F[[l6&F][lFgzF^[lFgz-%+AXESLABELSG6$Q!6
\"F]el-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 34 "limit((g(x) - q1(x))/(x+2), x=-2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
42 "l2 := limit((g(x) - q1(x))/(x+2)^2, x=-2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#l2G,$-%$expG6#!\"##\"\"$\"\"#" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 37 "q2 := x -> q1(x) + l2*(x+2)^2; q2(x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q2GR6#%\"xG6\"6$%)operatorG%&arrowG
F(,&-%#q1G6#9$\"\"\"*&%#l2GF1),&F0F1\"\"#F1F6F1F1F(F(F(" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,(-%$expG6#!\"#\"\"\"*(\"\"#F(F$F(,&%\"xGF(F*F(F
(F(*(#\"\"$F*F(F$F()F+F*F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 42 "l3 := limit((g(x) - q2(x))/(x+2)^3, x=-2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#l3G,$-%$expG6#!\"##\"\"\"\"\"$" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 37 "q3 := x -> q2(x) + l3*(x+2)^3; q3(x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q3GR6#%\"xG6\"6$%)operatorG%&arrowG
F(,&-%#q2G6#9$\"\"\"*&%#l3GF1),&F0F1\"\"#F1\"\"$F1F1F(F(F(" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,*-%$expG6#!\"#\"\"\"*(\"\"#F(F$F(,&%\"xGF(F
*F(F(F(*(#\"\"$F*F(F$F()F+F*F(F(*(#F(F/F(F$F()F+F/F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "l4 := limit((g(x) - q3(x))/(x+2)^4,
 x=-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#l4G,$-%$expG6#!\"##!\"&
\"#C" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "q4 := x -> q3(x) + \+
l4*(x+2)^4; q4(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q4GR6#%\"xG6
\"6$%)operatorG%&arrowGF(,&-%#q3G6#9$\"\"\"*&%#l4GF1),&F0F1\"\"#F1\"\"
%F1F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,-%$expG6#!\"#\"\"\"*(
\"\"#F(F$F(,&%\"xGF(F*F(F(F(*(#\"\"$F*F(F$F()F+F*F(F(*(#F(F/F(F$F()F+F
/F(F(*&#\"\"&\"#CF(*&F$F()F+\"\"%F(F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 20 "plot(\{g,q4\}, -4..0);" }}{PARA 13 "" 1 "" 
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6WFR7$Ffq$\"3yw:)>l%*GN&FR7$F[r$\"31n*HFc.5d'FR7$F`r$\"3;#y3(=BjKzFR7$
Fer$\"3Y*)\\?uQy&e*FR7$Fjr$\"3])ppR(R@L6F-7$F_s$\"3![HV_t;![8F-7$Fds$
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#o$F-7$Fgu$\"3?NryoX([6%F-7$F\\v$\"3#p$3Gx9k*f%F-7$Fav$\"3W9'f<\"RQb]F
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\"3!o]^3$zh`qF-7$Fjw$\"3g4]:4I*=b(F-7$F_x$\"3xP(o@Cp'3!)F-7$Fdx$\"3lY.
7,(fSW)F-7$Fix$\"3)y\"3(fz&=O))F-7$F^y$\"33!HWE`H_:*F-7$Fcy$\"33Tyej\"
\\_Y*F-7$Fhy$\"3?7sF`@x'o*F-7$F]z$\"3[Z^V.QAg)*F-7$Fbz$\"3+<DI2JIi**F-
7$Fgz$\"\"\"F*-F[[l6&F][lFgzF^[lFgz-%+AXESLABELSG6$Q!6\"F]el-%%VIEWG6$
;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 
0 0 "Curve 1" "Curve 2" }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 4 "The " }{TEXT 256 6 "Taylor" }
{TEXT -1 8 " command" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "Now that \+
we have seen how to produce the approximating polynomials for any func
tion at any point, it is time to admit that " }{TEXT 262 5 "Maple" }
{TEXT -1 92 " has a built-in command to do the computation.  The polyn
omials we are producing are called " }{TEXT 263 20 "Taylor polynomials
, " }{TEXT -1 17 " and the command " }{TEXT 264 6 "taylor" }{TEXT -1 
174 " will produce them.   To use this command, you have to give an ex
pression to be expanded, the point near which you want the expansion, \+
and the order of expansion required.  (" }{TEXT 265 5 "Maple" }{TEXT 
-1 198 " has rules about how many terms it gives in the expansion, and
 they are not always what you might expect, but if you ask for more te
rms you should get more terms.)  You can omit the last parameter; " }
{TEXT 266 5 "Maple" }{TEXT -1 27 " will then use its default " }{TEXT 
267 5 "Order" }{TEXT -1 43 " variable to determine the number of terms
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f := x -> sqrt(1 + x);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&ar
rowGF(-%%sqrtG6#,&\"\"\"F09$F0F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "taylor(f(x), x=0, 6);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#+1%\"xG\"\"\"\"\"!#F%\"\"#F%#!\"\"\"\")F(#F%\"#;\"\"$#!\"&\"$G\"
\"\"%#\"\"(\"$c#\"\"&-%\"OG6#F%\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "taylor(f(x), x=0, 12);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#+=%\"xG\"\"\"\"\"!#F%\"\"#F%#!\"\"\"\")F(#F%\"#;\"\"$#!\"&\"$G\"
\"\"%#\"\"(\"$c#\"\"&#!#@\"%C5\"\"'#\"#L\"%[?F4#!$H%\"&oF$F+#\"$:(\"&O
b'\"\"*#!%JC\"'W@E\"#5#\"%*>%\"')GC&\"#6-%\"OG6#F%\"#7" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "taylor(f(x), x=0);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#+1%\"xG\"\"\"\"\"!#F%\"\"#F%#!\"\"\"\")F(#F%\"#;\"
\"$#!\"&\"$G\"\"\"%#\"\"(\"$c#\"\"&-%\"OG6#F%\"\"'" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 18 "taylor(f(x), x=3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#+1,&%\"xG\"\"\"\"\"$!\"\"\"\"#\"\"!#F&\"\"%F&#F(\"#kF)#
F&\"$7&F'#!\"&\"&%Q;F,#\"\"(\"'s58\"\"&-%\"OG6#F&\"\"'" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "g := x -> exp(-x^2/2);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%$expG
6#,$*$)9$\"\"#\"\"\"#!\"\"F3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "taylor(g(x), x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#++%\"xG\"\"\"\"\"!#!\"\"\"\"#F)#F%\"\")\"\"%-%\"OG6#F%\"\"'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "taylor(g(x), x=-2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#+1,&%\"xG\"\"\"\"\"#F&-%$expG6#!\"#\"
\"!,$F(F'F&,$F(#\"\"$F'F',$F(#F&F0F0,$F(#!\"&\"#C\"\"%,$F(#!\"$\"#?\"
\"&-%\"OG6#F&\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "Notice tha
t these are the same series we produced earlier in the worksheet.  Her
e are a few more that should already be familiar to you." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "taylor(sin(x), x=0, 10);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#+/%\"xG\"\"\"F%#!\"\"\"\"'\"\"$#F%\"$?\"\"\"
&#F'\"%S]\"\"(#F%\"'!)GO\"\"*-%\"OG6#F%\"#5" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 24 "taylor(cos(x), x=0, 10);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#+/%\"xG\"\"\"\"\"!#!\"\"\"\"#F)#F%\"#C\"\"%#F(\"$?(\"\"
'#F%\"&?.%\"\")-%\"OG6#F%\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 24 "taylor(exp(x), x=0, 10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+9%
\"xG\"\"\"\"\"!F%F%#F%\"\"#F(#F%\"\"'\"\"$#F%\"#C\"\"%#F%\"$?\"\"\"&#F
%\"$?(F*#F%\"%S]\"\"(#F%\"&?.%\"\")#F%\"'!)GO\"\"*-%\"OG6#F%\"#5" }}}}
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