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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 44 "Using Taylor series to approximat
e integrals" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B
.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 20 "Version:  26.3.2007 " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 72 "Introductory example of using a Taylor \+
series to approximate an integral" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 22 "Consider the function " }{XPPEDIT 18 0 "f
(x) = sqrt(1+2*sin(x));" "6#/-%\"fG6#%\"xG-%%sqrtG6#,&\"\"\"F,*&\"\"#F
,-%$sinG6#F'F,F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "f := x -> sqrt(1+2*sin(x)):
\n'f(x)'=f(x);\nplot(f(x),x=-0.6..Pi,y=0..1.8,tickmarks=[4,4]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*$,&\"\"\"F**&\"\"#F*-%$
sinGF&F*F*#F*F," }}{PARA 13 "" 1 "" {GLPLOT2D 428 255 255 {PLOTDATA 2 
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]l$\"+aEfTJ!\"*;F_]l$\"#=F^]l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 44.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 40 "Maple's explicit analytical formula for \+
" }{XPPEDIT 18 0 "Int(sqrt(1+2*sin(x)),x);" "6#-%$IntG6$-%%sqrtG6#,&\"
\"\"F**&\"\"#F*-%$sinG6#%\"xGF*F*F0" }{TEXT -1 55 " is rather complica
ted and involves elliptic functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Int(sqrt(1+2*sin(x)),x);\n
value(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$-%%sqrtG6#,&
\"\"\"F+-%$sinG6#%\"xG\"\"#F+F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*
&*.-%%sqrtG6#*$)-%$cosG6#%\"xG\"\"#\"\"\"F0-%%csgnG6#F+F0-F'6#,&-%$sin
GF-F/F/F0F0-F'6#,&F7!\"#F/F0F0-F'6#,&F7F<!\"\"F0F0,&-%*EllipticFG6$*$F
4F0#F0F/\"\"$-%*EllipticEGFD!\"%F0F0*&)F+F/F0-F'6#,&F0F0F7F/F0F@FF" }}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "We can o
btain a truncated Taylor series expansion for this integral." }}{PARA 
0 "" 0 "" {TEXT -1 58 "We start by obtaining a Taylor polynomial of de
gree 7 for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 12 " ce
ntred at " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 41 ", that i
s, a truncated Maclaurin series. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "f := x -> sqrt(1+2*sin(x)):
\n'f(x)'=f(x);\nconvert(taylor(f(x),x=0,8),polynom):\np := unapply(%,x
):\n'p(x)'=p(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*$,&
\"\"\"F**&\"\"#F*-%$sinGF&F*F*#F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/-%\"pG6#%\"xG,2\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)!\"\"*&#F)\"\"$F)*
$)F'F2F)F)F)*&#\"#6\"#CF)*$)F'\"\"%F)F)F/*&#\"#>\"#IF)*$)F'\"\"&F)F)F)
*&#\"$h'\"$?(F)*$)F'\"\"'F)F)F/*&#\"$P%\"$:$F)*$)F'\"\"(F)F)F)" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The Taylo
r polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 34 " \+
can be compared graphically with " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%
\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 79 "plot([f(x),p(x)],x=-0.7..1,y=-.1..1.8,col
or=[red,blue],legend=[`f(x)`,`p(x)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 
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<F^q7$$\"3=omTqH(4f)F*$\"3O7t,UQ?e=F^q7$$\"3#RLLL?*yF*)F*$\"3%pW1nRu<'
>F^q7$$\"3S+]7eb!pG*F*$\"3#>)zOC,g)4#F^q7$$\"3$***\\(=tD1j*F*$\"3[th_
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" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Now
 we integrate the Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6
#%\"xG" }{TEXT -1 22 " to give a polynomial " }{XPPEDIT 18 0 "q(x)" "6
#-%\"qG6#%\"xG" }{TEXT -1 89 " of degree 8 which will be an approximat
ion for the (complicated) indefinite integral of " }{XPPEDIT 18 0 "f(x
)" "6#-%\"fG6#%\"xG" }{TEXT -1 19 " given previously. " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Int(p(x
),x);\nq := unapply(value(%),x):\n'q(x)'=q(x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$,2\"\"\"F'%\"xGF'*&#F'\"\"#F'*$)F(F+F'F'!\"\"*
&#F'\"\"$F'*$)F(F1F'F'F'*&#\"#6\"#CF'*$)F(\"\"%F'F'F.*&#\"#>\"#IF'*$)F
(\"\"&F'F'F'*&#\"$h'\"$?(F'*$)F(\"\"'F'F'F.*&#\"$P%\"$:$F'*$)F(\"\"(F'
F'F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG,2F'\"\"\"*&#F
)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)!\"\"*&#F)\"#7F)*$)F'\"
\"%F)F)F)*&#\"#6\"$?\"F)*$)F'\"\"&F)F)F5*&#\"#>\"$!=F)*$)F'F1F)F)F)*&#
\"$h'\"%S]F)*$)F'\"\"(F)F)F5*&#\"$P%\"%?DF)*$)F'\"\")F)F)F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "q(0)=0" 
"6#/-%\"qG6#\"\"!F'" }{TEXT -1 28 ", the value of the function " }
{XPPEDIT 18 0 "q(x);" "6#-%\"qG6#%\"xG" }{TEXT -1 13 " at a number " }
{TEXT 261 1 "x" }{TEXT -1 20 " is an estimate for " }{XPPEDIT 18 0 "In
t(sqrt(1+2*sin(t)),t = 0 .. x);" "6#-%$IntG6$-%%sqrtG6#,&\"\"\"F**&\"
\"#F*-%$sinG6#%\"tGF*F*/F0;\"\"!%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 
"" {TEXT -1 47 "We should obtain a reasonably good estimate if " }
{TEXT 262 1 "x" }{TEXT -1 15 " is close to 0." }}{PARA 0 "" 0 "" 
{TEXT -1 28 "For example, let's estimate " }{XPPEDIT 18 0 "Int(sqrt(1+
2*sin(t)),t = 0 .. 1/8);" "6#-%$IntG6$-%%sqrtG6#,&\"\"\"F**&\"\"#F*-%$
sinG6#%\"tGF*F*/F0;\"\"!*&F*F*\"\")!\"\"" }{TEXT -1 1 "." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "q(1/8
);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"*D:\\C\"\"*'4C&R*
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+s([]K\"!#5" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 4 "Note" }{TEXT -1 47 ": W
e could simply set up the definite integral " }{XPPEDIT 18 0 "Int(p(x)
,x=0..1/8)" "6#-%$IntG6$-%\"pG6#%\"xG/F);\"\"!*&\"\"\"F.\"\")!\"\"" }
{TEXT -1 83 ", evaluate this integral analytically, and then obtain a \+
numerical result by using " }{TEXT 0 5 "evalf" }{TEXT -1 3 ".  " }}
{PARA 0 "" 0 "" {TEXT -1 109 "When this shorter method is used, it is \+
important to obtain the analytical expression first by including the \+
" }{TEXT 258 8 "value(%)" }{TEXT -1 10 "  command." }}{PARA 0 "" 0 "" 
{TEXT -1 87 "If this is not done, Maple will perform numerical integra
tion of the Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG
" }{TEXT -1 77 " which defeats the intention of avoiding such methods \+
in the current context." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Int(p(x),x=0..1/8);\nvalue(%);\neva
lf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,2\"\"\"F'%\"xGF'*&
#F'\"\"#F'*$)F(F+F'F'!\"\"*&#F'\"\"$F'*$)F(F1F'F'F'*&#\"#6\"#CF'*$)F(
\"\"%F'F'F.*&#\"#>\"#IF'*$)F(\"\"&F'F'F'*&#\"$h'\"$?(F'*$)F(\"\"'F'F'F
.*&#\"$P%\"$:$F'*$)F(\"\"(F'F'F'/F(;\"\"!#F'\"\")" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6##\"*D:\\C\"\"*'4C&R*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#$\"+s([]K\"!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 99 "We can compare this value with that obtained from the exp
licit analytical formula for the integral." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Int(sqrt(1+2*sin(x
)),x=0..1/8);\nvalue(%);\nevalf(evalf[15](%));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$*$,&\"\"\"F(*&\"\"#F(-%$sinG6#%\"xGF(F(#F(F*/F
.;\"\"!#F(\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*-%*EllipticFG6$,$
*&#\"\"#\"\"$\"\"\"*&F+#F,F**&,&*&F*F,-%$sinG6##F,\"\")F,F,F,F,F,,&*&F
*F,F2F,F,F*F,!\"\"F.F,F,,$*&F*F9F+F.F,F9-F%6$,$*(F+F9F*F.F+F.F,F:F,-%+
EllipticPiG6%F'#F+\"\"%F:F,-FA6%F>FCF:F9" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"+d([]K\"!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 96 "Maple can also evaluate integrals numerically, without
 first constructing an analytical formula." }}{PARA 0 "" 0 "" {TEXT 
-1 29 "This is achieved by applying " }{TEXT 0 5 "evalf" }{TEXT -1 41 
" to the unevaluated from of the integral." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Int(sqrt(1+2*sin(x
)),x=0..1/8);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*
$,&\"\"\"F(*&\"\"#F(-%$sinG6#%\"xGF(F(#F(F*/F.;\"\"!#F(\"\")" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$\"+d([]K\"!#5" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 232 "Looking back at the value obtaine
d from the truncated series, we see that the last two digits in the va
lue obtained differ from the last two more recent values. If we includ
e extra terms of the Taylor series we get better agreement." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "f
 := x -> sqrt(1+2*sin(x));\nconvert(taylor(f(x),x=0,11),polynom);\nq :
= unapply(int(%,x),x);\nevalf(evalf[15](q(0.125)));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#,&
\"\"\"F0*&\"\"#F0-%$sinG6#9$F0F0F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,8\"\"\"F$%\"xGF$*&#F$\"\"#F$*$)F%F(F$F$!\"\"*&#F$\"\"$F$*$)F%F.
F$F$F$*&#\"#6\"#CF$*$)F%\"\"%F$F$F+*&#\"#>\"#IF$*$)F%\"\"&F$F$F$*&#\"$
h'\"$?(F$*$)F%\"\"'F$F$F+*&#\"$P%\"$:$F$*$)F%\"\"(F$F$F$*&#\"&\">()\"&
?.%F$*$)F%\"\")F$F$F+*&#\"&,$y\"&!oAF$*$)F%\"\"*F$F$F$*&#\")@XQ?\"(+)G
OF$*$)F%\"#5F$F$F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qGf*6#%\"xG6
\"6$%)operatorG%&arrowGF(,89$\"\"\"*&#F.\"\"#F.*$)F-F1F.F.F.*&#F.\"\"'
F.*$)F-\"\"$F.F.!\"\"*&#F.\"#7F.*$)F-\"\"%F.F.F.*&#\"#6\"$?\"F.*$)F-\"
\"&F.F.F:*&#\"#>\"$!=F.*$)F-F6F.F.F.*&#\"$h'\"%S]F.*$)F-\"\"(F.F.F:*&#
\"$P%\"%?DF.*$)F-\"\")F.F.F.*&#\"&\">()\"'!)GOF.*$)F-\"\"*F.F.F:*&#\"&
,$y\"'+oAF.*$)F-\"#5F.F.F.*&#\")@XQ?\")+o\"*RF.*$)F-FCF.F.F:F(F(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+d([]K\"!#5" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 164 "An error arising from a \+
mathematical approximation, such as occurred when the previous degree \+
8 Taylor polynomial was used to approximate the integral, is called a \+
" }{TEXT 264 16 "truncation error" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 62 "More examples of using Taylor series to
 approximate integrals " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";
" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "
" {TEXT 292 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 
13 "(a) Estimate " }{XPPEDIT 18 0 "Int((exp(x)-1)/x,x = -1 .. 1);" "6#
-%$IntG6$*&,&-%$expG6#%\"xG\"\"\"F,!\"\"F,F+F-/F+;,$F,F-F," }{TEXT -1 
50 " using a Taylor polynomial of degree 7 centred at " }{XPPEDIT 18 
0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 16 " to approximate " }{XPPEDIT 18 
0 "f(x) = (exp(x)-1)/x;" "6#/-%\"fG6#%\"xG*&,&-%$expG6#F'\"\"\"F-!\"\"
F-F'F." }{TEXT -1 89 ", and compare this value with the value given by
 Maple's numerical integration procedure " }{TEXT 0 9 "evalf/Int" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 42 "(b) Find the Taylor poly
nomial centred at " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 5 "
 for " }{XPPEDIT 18 0 "f(x) = (exp(x)-1)/x;" "6#/-%\"fG6#%\"xG*&,&-%$e
xpG6#F'\"\"\"F-!\"\"F-F'F." }{TEXT -1 49 " of minimal degree which, wh
en used to calculate " }{XPPEDIT 18 0 "Int((exp(x)-1)/x,x = -1 .. 1);
" "6#-%$IntG6$*&,&-%$expG6#%\"xG\"\"\"F,!\"\"F,F+F-/F+;,$F,F-F," }
{TEXT -1 47 ", gives a value which is correct to 10 digits. " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 291 8 "Solution" }
{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 55 "First we construct a Ta
ylor polynomial of degree 7 for " }{XPPEDIT 18 0 "f(x)=(exp(x)-1)/x" "
6#/-%\"fG6#%\"xG*&,&-%$expG6#F'\"\"\"F-!\"\"F-F'F." }{TEXT -1 7 " abou
t " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT 264 4 "Note" }{TEXT -1 70 ": In order for Maple to construc
t a Taylor polynomial of degree 7 for " }{XPPEDIT 18 0 "f(x)" "6#-%\"f
G6#%\"xG" }{TEXT -1 48 ", a Taylor polynomial of degree 8 is needed fo
r " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 44 ". For thi
s reason it is necessary to supply " }{TEXT 0 6 "taylor" }{TEXT -1 34 
" with the third (order) argument \"" }{TEXT 263 1 "9" }{TEXT -1 3 "\"
. " }}{PARA 0 "" 0 "" {TEXT 258 7 "Warning" }{TEXT -1 224 ": If you re
turn to re-execute these commands after constructing a higher degree T
aylor polynomial from a higher order Maple series structure, you will \+
get a degree 8 Taylor polynomial with the third (order) argument set t
o \"" }{TEXT 263 1 "9" }{TEXT -1 96 "\". A degree 7 Taylor polynomial \+
can then be constructed with the third (order) argument set to \"" }
{TEXT 263 1 "8" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 63 "This \+
happens because Maple \"remembers\" the higher order series." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "f
 := x -> (exp(x)-1)/x:\n'f(x)'=f(x);\ntaylor(f(x),x=0,9):\np := unappl
y(convert(%,polynom),x):\n'p(x)'=p(x);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%\"fG6#%\"xG*&,&-%$expGF&\"\"\"F,!\"\"F,F'F-" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/-%\"pG6#%\"xG,2\"\"\"F)*&#F)\"\"#F)F'F)F)*&#F)\"\"'
F)*$)F'F,F)F)F)*&#F)\"#CF)*$)F'\"\"$F)F)F)*&#F)\"$?\"F)*$)F'\"\"%F)F)F
)*&#F)\"$?(F)*$)F'\"\"&F)F)F)*&#F)\"%S]F)*$)F'F/F)F)F)*&#F)\"&?.%F)*$)
F'\"\"(F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 70 "Now we integrate this polynomial to obtain a polynomial a
pproximation " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 30 " \+
for the indefinite integral  " }{XPPEDIT 18 0 "Int((exp(x)-1)/x,x);" "
6#-%$IntG6$*&,&-%$expG6#%\"xG\"\"\"F,!\"\"F,F+F-F+" }{TEXT -1 2 ". " }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "Int('p(x)',x)=Int(p(x),x);
\nvalue(rhs(%)):\nq := unapply(%,x):\n'q(x)'=q(x);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%$IntG6$-%\"pG6#%\"xGF*-F%6$,2\"\"\"F.*&#F.\"\"#F.F
*F.F.*&#F.\"\"'F.*$)F*F1F.F.F.*&#F.\"#CF.*$)F*\"\"$F.F.F.*&#F.\"$?\"F.
*$)F*\"\"%F.F.F.*&#F.\"$?(F.*$)F*\"\"&F.F.F.*&#F.\"%S]F.*$)F*F4F.F.F.*
&#F.\"&?.%F.*$)F*\"\"(F.F.F.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%
\"qG6#%\"xG,2F'\"\"\"*&#F)\"\"%F)*$)F'\"\"#F)F)F)*&#F)\"#=F)*$)F'\"\"$
F)F)F)*&#F)\"#'*F)*$)F'F,F)F)F)*&#F)\"$+'F)*$)F'\"\"&F)F)F)*&#F)\"%?VF
)*$)F'\"\"'F)F)F)*&#F)\"&!GNF)*$)F'\"\"(F)F)F)*&#F)\"'gDKF)*$)F'\"\")F
)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The polynomial " }
{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 46 " is an approximat
ion for an antiderivative of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "To obtain an estimate
 for  " }{XPPEDIT 18 0 "Int((exp(x)-1)/x,x = -1 .. 1);" "6#-%$IntG6$*&
,&-%$expG6#%\"xG\"\"\"F,!\"\"F,F+F-/F+;,$F,F-F," }{TEXT -1 21 ", we ne
ed to compute " }{XPPEDIT 18 0 "q(1)-q(-1)" "6#,&-%\"qG6#\"\"\"F'-F%6#
,$F'!\"\"F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "q(1)-q(-1);\nest := evalf(evalf[14]
(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"'*\\'=\"&+#))" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%$estG$\"+M6]9@!\"*" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "We can obtain a more accurate v
alue by numerical integration." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "f := x -> (exp(x)-1)/x:\nare
a := evalf(evalf[14](Int(f(x),x=-1..1)));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%areaG$\"+^<]9@!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 27 "The absolute error is . . ." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "ab
serr := abs(est-area);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'abserrG$
\"$<'!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 108 "This error can be reduced by increasing the degree of the Tayl
or polynomial used to approximate thefunction " }{XPPEDIT 18 0 "f(x)=(
exp(x)-1)/x" "6#/-%\"fG6#%\"xG*&,&-%$expG6#F'\"\"\"F-!\"\"F-F'F." }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 43 "Using a degree 8 polyno
mial to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT 
-1 12 " gives . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 231 "f := x -> (exp(x)-1)/x:\n'f(x)'=f(x);\ntay
lor(f(x),x=0,10):\np := unapply(convert(%,polynom),x):\n'p(x)'=p(x);\n
Int('p(x)',x);\nvalue(%):\nq := unapply(%,x):\n'q(x)'=q(x);\nq(1)-q(-1
);\nest2 := evalf(evalf[14](%));\nabserr2 := abs(est2-area);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&,&-%$expGF&\"\"\"F,!\"\"F,F'
F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG,4\"\"\"F)*&#F)\"
\"#F)F'F)F)*&#F)\"\"'F)*$)F'F,F)F)F)*&#F)\"#CF)*$)F'\"\"$F)F)F)*&#F)\"
$?\"F)*$)F'\"\"%F)F)F)*&#F)\"$?(F)*$)F'\"\"&F)F)F)*&#F)\"%S]F)*$)F'F/F
)F)F)*&#F)\"&?.%F)*$)F'\"\"(F)F)F)*&#F)\"'!)GOF)*$)F'\"\")F)F)F)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%\"pG6#%\"xGF)" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG,4F'\"\"\"*&#F)\"\"%F)*$)F'\"\"#F)
F)F)*&#F)\"#=F)*$)F'\"\"$F)F)F)*&#F)\"#'*F)*$)F'F,F)F)F)*&#F)\"$+'F)*$
)F'\"\"&F)F)F)*&#F)\"%?VF)*$)F'\"\"'F)F)F)*&#F)\"&!GNF)*$)F'\"\"(F)F)F
)*&#F)\"'gDKF)*$)F'\"\")F)F)F)*&#F)\"(?fE$F)*$)F'\"\"*F)F)F)" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6##\"*(Q^37\")+O:d" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%est2G$\"+Y<]9@!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%(abserr2G$\"\"&!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 149 "There is no change in the estimated value for the
 integral if the degree of the Taylor polynomial is increased to 9 bec
ause the extra terms added to " }{XPPEDIT 18 0 "q(1)" "6#-%\"qG6#\"\"
\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "q(-1)" "6#-%\"qG6#,$\"\"\"!\"
\"" }{TEXT -1 9 ", namely " }{XPPEDIT 18 0 "``(1/36288000)*`.`*1^10 = \+
1/36288000;" "6#/*(-%!G6#*&\"\"\"F)\")+!)GO!\"\"F)%\".GF)F)\"#5*&F)F)F
*F+" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "``(1/36288000)*`.`*(-1)^10 = \+
1/36288000" "6#/*(-%!G6#*&\"\"\"F)\")+!)GO!\"\"F)%\".GF),$F)F+\"#5*&F)
F)F*F+" }{TEXT -1 48 " respectively, are identical and so cancel out (
" }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 23 " is an even fu
nction). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 231 "f := x -> (exp(x)-1)/x:\n'f(x)'=f(x);\ntaylor(f(x),x
=0,11):\np := unapply(convert(%,polynom),x):\n'p(x)'=p(x);\nInt('p(x)'
,x);\nvalue(%):\nq := unapply(%,x):\n'q(x)'=q(x);\nq(1)-q(-1);\nest3 :
= evalf(evalf[14](%));\nabserr3 := abs(est3-area);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&,&-%$expGF&\"\"\"F,!\"\"F,F'F-" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG,6\"\"\"F)*&#F)\"\"#F)F'
F)F)*&#F)\"\"'F)*$)F'F,F)F)F)*&#F)\"#CF)*$)F'\"\"$F)F)F)*&#F)\"$?\"F)*
$)F'\"\"%F)F)F)*&#F)\"$?(F)*$)F'\"\"&F)F)F)*&#F)\"%S]F)*$)F'F/F)F)F)*&
#F)\"&?.%F)*$)F'\"\"(F)F)F)*&#F)\"'!)GOF)*$)F'\"\")F)F)F)*&#F)\"(+)GOF
)*$)F'\"\"*F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%\"pG6#
%\"xGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG,6F'\"\"\"*&#
F)\"\"%F)*$)F'\"\"#F)F)F)*&#F)\"#=F)*$)F'\"\"$F)F)F)*&#F)\"#'*F)*$)F'F
,F)F)F)*&#F)\"$+'F)*$)F'\"\"&F)F)F)*&#F)\"%?VF)*$)F'\"\"'F)F)F)*&#F)\"
&!GNF)*$)F'\"\"(F)F)F)*&#F)\"'gDKF)*$)F'\"\")F)F)F)*&#F)\"(?fE$F)*$)F'
\"\"*F)F)F)*&#F)\")+!)GOF)*$)F'\"#5F)F)F)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6##\"*(Q^37\")+O:d" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%
est3G$\"+Y<]9@!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(abserr3G$\"\"
&!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 
"Taking a degree 10 polynomial for " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#
%\"xG" }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG
6#%\"xG" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "Int(p(t),t=0..x)=q(
x)" "6#/-%$IntG6$-%\"pG6#%\"tG/F*;\"\"!%\"xG-%\"qG6#F." }{TEXT -1 84 "
 has degree 11, leads to an estimate for the integral which is correct
 to 10 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 231 "f := x -> (exp(x)-1)/x:\n'f(x)'=f(x);\ntaylor(f
(x),x=0,12):\np := unapply(convert(%,polynom),x):\n'p(x)'=p(x);\nInt('
p(x)',x);\nvalue(%):\nq := unapply(%,x):\n'q(x)'=q(x);\nq(1)-q(-1);\ne
st4 := evalf(evalf[14](%));\nabseer4 := abs(est4-area);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&,&-%$expGF&\"\"\"F,!\"\"F,F'F-" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG,8\"\"\"F)*&#F)\"\"#F)
F'F)F)*&#F)\"\"'F)*$)F'F,F)F)F)*&#F)\"#CF)*$)F'\"\"$F)F)F)*&#F)\"$?\"F
)*$)F'\"\"%F)F)F)*&#F)\"$?(F)*$)F'\"\"&F)F)F)*&#F)\"%S]F)*$)F'F/F)F)F)
*&#F)\"&?.%F)*$)F'\"\"(F)F)F)*&#F)\"'!)GOF)*$)F'\"\")F)F)F)*&#F)\"(+)G
OF)*$)F'\"\"*F)F)F)*&#F)\")+o\"*RF)*$)F'\"#5F)F)F)" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%$IntG6$-%\"pG6#%\"xGF)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"qG6#%\"xG,8F'\"\"\"*&#F)\"\"%F)*$)F'\"\"#F)F)F)*&#
F)\"#=F)*$)F'\"\"$F)F)F)*&#F)\"#'*F)*$)F'F,F)F)F)*&#F)\"$+'F)*$)F'\"\"
&F)F)F)*&#F)\"%?VF)*$)F'\"\"'F)F)F)*&#F)\"&!GNF)*$)F'\"\"(F)F)F)*&#F)
\"'gDKF)*$)F'\"\")F)F)F)*&#F)\"(?fE$F)*$)F'\"\"*F)F)F)*&#F)\")+!)GOF)*
$)F'\"#5F)F)F)*&#F)\"*+[3R%F)*$)F'\"#6F)F)F)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6##\",<d.Y#H\",+7<JQ\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%%est4G$\"+^<]9@!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(abseer4G$
\"\"!F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" 
}}{PARA 0 "" 0 "" {TEXT 285 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "
" 0 "" {TEXT -1 34 "(a) Construct a Taylor polynomial " }{XPPEDIT 18 
0 "p(x);" "6#-%\"pG6#%\"xG" }{TEXT -1 66 " of degree 10 as a truncated
 Maclaurin series, which approximates " }{XPPEDIT 18 0 "f(x)=exp(x)/(x
^2+2*x+2)" "6#/-%\"fG6#%\"xG*&-%$expG6#F'\"\"\",(*$F'\"\"#F,*&F/F,F'F,
F,F/F,!\"\"" }{TEXT -1 6 " when " }{TEXT 283 1 "x" }{TEXT -1 11 " is n
ear 0." }}{PARA 0 "" 0 "" {TEXT -1 2 "( " }{TEXT 264 4 "Note" }{TEXT 
-1 60 ": You will need an order 11 Taylor series approximation for " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 3 ". )" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "(b) Plot a graph of \+
the function " }{XPPEDIT 18 0 "f(x)=exp(x)/(x^2+2*x+2)" "6#/-%\"fG6#%
\"xG*&-%$expG6#F'\"\"\",(*$F'\"\"#F,*&F/F,F'F,F,F/F,!\"\"" }{TEXT -1 
6 "  for " }{TEXT 284 1 "x" }{TEXT -1 9 " between " }{XPPEDIT 18 0 "-2
;" "6#,$\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "2;" "6#\"\"#" 
}{TEXT -1 60 " along with the graph of the Taylor polynomial found in \+
(a)." }}{PARA 0 "" 0 "" {TEXT -1 13 "(c) Estimate " }{XPPEDIT 18 0 "In
t(exp(x)/(x^2+2*x+2),x=-1/2..1/2)" "6#-%$IntG6$*&-%$expG6#%\"xG\"\"\",
(*$F*\"\"#F+*&F.F+F*F+F+F.F+!\"\"/F*;,$*&F+F+F.F0F0*&F+F+F.F0" }{TEXT 
-1 30 "  using the Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG
6#%\"xG" }{TEXT -1 15 " found in (a). " }}{PARA 0 "" 0 "" {TEXT -1 13 
"(d) Find the " }{TEXT 263 8 "absolute" }{TEXT -1 5 " and " }{TEXT 
263 14 "relative error" }{TEXT -1 107 " in the estimate found in (c) b
y comparing with the value given by Maple's numerical integration proc
edure " }{TEXT 0 9 "evalf/Int" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" 
{TEXT -1 49 "(e) Increase the degree of the Taylor polynomial " }
{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 20 ", used to estimat
e  " }{XPPEDIT 18 0 "Int(exp(x)/(x^2+2*x+2),x=-1/2..1/2)" "6#-%$IntG6$
*&-%$expG6#%\"xG\"\"\",(*$F*\"\"#F+*&F.F+F*F+F+F.F+!\"\"/F*;,$*&F+F+F.
F0F0*&F+F+F.F0" }{TEXT -1 166 "  until the value of the integral agree
s to 10 digits with that obtained by using Maple's numerical integrati
on.  What is the minimum degree needed for this to occur?" }}{PARA 0 "
" 0 "" {TEXT -1 2 "( " }{TEXT 264 4 "Note" }{TEXT -1 87 ": You may nee
d to use a few guard digits to get agreement in the last decimal place
. ) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 286 8 "S
olution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 41 "(a) We can construct a Taylor polynomial " }
{XPPEDIT 18 0 "p(x);" "6#-%\"pG6#%\"xG" }{TEXT -1 25 " of degree 10 ce
ntred at " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 16 " to appr
oximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 6 " when \+
" }{TEXT 294 1 "x" }{TEXT -1 12 " is near 0. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "f := x -> e
xp(x)/(x^2+2*x+2):\n'f(x)'=f(x);\ntaylor(f(x),x,11);\nconvert(%,polyno
m):\np := unapply(%,x):\n'p(x)'=p(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/-%\"fG6#%\"xG*&-%$expGF&\"\"\",(*$)F'\"\"#F+F+*&F/F+F'F+F+F/F+!\"
\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+7%\"xG#\"\"\"\"\"#\"\"!#F&\"#7
\"\"$#!\"\"\"#;\"\"%#F&\"#S\"\"&#F&\"$W\"\"\"'#!#8\"$s'\"\"(#\"#h\"%SQ
\"\")#!$h\"\"&?f#\"\"*#!$\\$\"'+;?\"#5-%\"OG6#F&\"#6" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/-%\"pG6#%\"xG,4#\"\"\"\"\"#F**&#F*\"#7F**$)F'\"\"$F
*F*F**&#F*\"#;F**$)F'\"\"%F*F*!\"\"*&#F*\"#SF**$)F'\"\"&F*F*F**&#F*\"$
W\"F**$)F'\"\"'F*F*F**&#\"#8\"$s'F**$)F'\"\"(F*F*F8*&#\"#h\"%SQF**$)F'
\"\")F*F*F**&#\"$h\"\"&?f#F**$)F'\"\"*F*F*F8*&#\"$\\$\"'+;?F**$)F'\"#5
F*F*F8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
67 "(b) The following picture shows the graph of the Taylor polynomial
 " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 11 " (drawn in " 
}{TEXT 256 4 "blue" }{TEXT -1 26 ") along with the graph of " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 11 " (drawn in " }
{TEXT 258 3 "red" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "plot([f(x),p(x)],x=-2..2,y=-
1..2,color=[red,blue],legend=[`f(x)`,`p(x)`]);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 449 385 385 {PLOTDATA 2 "6&-%'CURVESG6%7S7$$!\"#\"\"!$\"37N1
$=;knw'!#>7$$!3MLLL$Q6G\">!#<$\"3G(GzLm8\\0)F-7$$!3bmm;M!\\p$=F1$\"3e#
)*49Q#3o$*F-7$$!3MLLL))Qj^<F1$\"3Y%G7I#of36!#=7$$!3ALLL=Kvl;F1$\"3a^**
QE!**)48F>7$$!3wmm;C2G!e\"F1$\"3m5Ara-YS:F>7$$!3OLL$3yO5]\"F1$\"3cz7W:
Ar\"y\"F>7$$!3&*****\\nU)*=9F1$\"39x@@9:Fe?F>7$$!3SLL$3WDTL\"F1$\"3s;;
-(Ro$pBF>7$$!35++]d(Q&\\7F1$\"3/%G_%>\\M)p#F>7$$!3gmmmc4`i6F1$\"3SoS\"
zOhk/$F>7$$!3KLLLQW*e3\"F1$\"3[*fE?pr7N$F>7$$!3w++++()>'***F>$\"3Ex](e
c#>!o$F>7$$!3E++++0\"*H\"*F>$\"3=:\"GQ\\mI)RF>7$$!35++++83&H)F>$\"3%)3
*\\BD4%RUF>7$$!3\\LLL3k(p`(F>$\"3:[*z***y0PWF>7$$!3Anmmmj^NmF>$\"3^*\\
Ku\"H[EYF>7$$!3)zmmmYh=(eF>$\"3o%>x<i/&\\ZF>7$$!3+,++v#\\N)\\F>$\"3eaM
Q/d$Q&[F>7$$!3commmCC(>%F>$\"3BQgT=>s;\\F>7$$!39*****\\FRXL$F>$\"37J\\
+X`bg\\F>7$$!3t*****\\#=/8DF>$\"3SnT)pZjS)\\F>7$$!3=mmm;a*el\"F>$\"3/`
><\\ur&*\\F>7$$!3komm;Wn(o)F-$\"3#4#4DtnT**\\F>7$$!3IqLLL$eV(>!#?$\"3K
spd$*******\\F>7$$\"3)Qjmm\"f`@')F-$\"3\\RJ*)>2]+]F>7$$\"3%z****\\nZ)H
;F>$\"3P7^G^k>.]F>7$$\"3ckmm;$y*eCF>$\"3<1,<pnL5]F>7$$\"3f)******R^bJ$
F>$\"3jMY_#)*[Q-&F>7$$\"3'e*****\\5a`TF>$\"3o5hi=REW]F>7$$\"3'o****\\7
RV'\\F>$\"3Q<`q\\Zgr]F>7$$\"3Y'*****\\@fkeF>$\"3S'[)H)yA96&F>7$$\"3_IL
LL&4Nn'F>$\"3)Q.!=e>=c^F>7$$\"3A*******\\,s`(F>$\"3)=<#=')3x8_F>7$$\"3
%[mm;zM)>$)F>$\"3O\\f;,.*\\F&F>7$$\"3M*******pfa<*F>$\"3a#oO$Q1/_`F>7$
$\"39HLLeg`!)**F>$\"3(eqL1?]WV&F>7$$\"3w****\\#G2A3\"F1$\"3VL%*R0s8JbF
>7$$\"3;LLL$)G[k6F1$\"3t_$*))4YOOcF>7$$\"3#)****\\7yh]7F1$\"3!f$GfzP=e
dF>7$$\"3xmmm')fdL8F1$\"35/AVW,7()eF>7$$\"3bmmm,FT=9F1$\"3joFXW)R6.'F>
7$$\"3FLL$e#pa-:F1$\"3G_0$*f@`'='F>7$$\"3!*******Rv&)z:F1$\"3DMe?@![2M
'F>7$$\"3ILLLGUYo;F1$\"3'R03e+v9`'F>7$$\"3_mmm1^rZ<F1$\"3\"R<aK\"p@:nF
>7$$\"34++]sI@K=F1$\"3ACExAeRDpF>7$$\"34++]2%)38>F1$\"3`Zc5\"yi49(F>7$
$\"\"#F*$\"3'3lI*)4c!*Q(F>-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fb[l-%'
LEGENDG6#%%f(x)G-F$6%7en7$F($\"3fan^@M=GkF17$$!3SLL$e%G?y>F1$\"3)G;7%o
w8oeF17$$!3ymmm\"p0k&>F1$\"31xBKmzw]`F17$$!3&*****\\P&3Y$>F1$\"370;u&
\\PM([F17$F/$\"3R8K+_njLWF17$$!31++v3-)[(=F1$\"3C)*Q,J8v]PF17$F5$\"3%p
He'R4iiJF17$$!3#)***\\7Y\"H%z\"F1$\"3tXP!4,E6g#F17$F:$\"3f)R#oL#z@8#F1
7$$!3<LLL`Np3<F1$\"3y4.uD%*3T<F17$F@$\"38j9Ju>+?9F17$FE$\"3lF,.!\\Dq]*
F>7$FJ$\"33FMTx#))3r'F>7$FO$\"3P#e'H>baT\\F>7$FT$\"33+<-/R:[RF>7$FY$\"
3A]J\"Qu0\"=NF>7$Fhn$\"3Whr#[e%oPMF>7$F]o$\"3l?zL9pzUNF>7$Fbo$\"3#*\\.
ugNHfPF>7$Fgo$\"321$z=1xD,%F>7$F\\p$\"3]p@r39l\\UF>7$Fap$\"3B(z>[jv0W%
F>7$Ffp$\"3GHtn?QKFYF>7$F[q$\"3qR9z\\ir\\ZF>7$F`q$\"3![JrL&)oQ&[F>7$Fe
q$\"3za>X&oEn\"\\F>7$Fjq$\"3'[^h>qb0'\\F>7$F_r$\"3A(QI>\\jS)\\F>7$Fdr$
\"33\"*RJ\\ur&*\\F>7$Fir$\"3SJ5DtnT**\\F>7$F^s$\"3yrpd$*******\\F>7$Fd
s$\"3c\\I*)>2]+]F>7$Fis$\"3oyQ>^k>.]F>7$F^t$\"3ribIhnL5]F>7$Fct$\"34qN
$ey[Q-&F>7$Fht$\"3Q$)R/A<EW]F>7$F]u$\"375zgu+fr]F>7$Fbu$\"3rZSz0qL6^F>
7$Fgu$\"3AKA$=+Ze:&F>7$F\\v$\"3;nvzH&pD@&F>7$Fav$\"3sh!H(*><;F&F>7$Ffv
$\"36lWb4gpU`F>7$F[w$\"3MUi$o%e47aF>7$F`w$\"3/hra\\+`zaF>7$Few$\"3?**R
%oN\"eEbF>7$Fjw$\"3[j&Qy[:(HbF>7$F_x$\"3qGx3g<3YaF>7$Fdx$\"3/ifc'3(4._
F>7$Fix$\"3'3Lmy4%3(p%F>7$F^y$\"3_o,YPgBgQF>7$Fcy$\"31#e-AfIn@#F>7$Fhy
$!3_%)Qy_;)3(=F-7$F]z$!3IUq[#*)3u>%F>7$$\"33+++S2ls=F1$!3=mq#Rl4H$oF>7
$Fbz$!3S#R]$GSj05F17$$\"3/++v.Uac>F1$!3Aa$eM:--V\"F17$Fgz$!3We8pC!e8&>
F1-F\\[l6&F^[lFb[lFb[lF_[l-Fd[l6#%%p(x)G-%+AXESLABELSG6$Q\"x6\"Q\"yF^h
l-%%VIEWG6$;F(Fgz;$!\"\"F*Fgz" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 1 "f(x)" "p(x)" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " }{XPPEDIT 18 0 "Int(p(x),x = -1/
2 .. 1/2);" "6#-%$IntG6$-%\"pG6#%\"xG/F);,$*&\"\"\"F.\"\"#!\"\"F0*&F.F
.F/F0" }{TEXT -1 26 " provides an estimate for " }{XPPEDIT 18 0 "Int(f
(x),x = -1/2 .. 1/2);" "6#-%$IntG6$-%\"fG6#%\"xG/F);,$*&\"\"\"F.\"\"#!
\"\"F0*&F.F.F/F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Int('p(x)',x=-1/2..1/2);\nva
lue(%);\narea_est := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$I
ntG6$-%\"pG6#%\"xG/F);#!\"\"\"\"##\"\"\"F." }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6##\"+$)G1,M\"++sY7o" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%)area_estG$\"+?*4C*\\!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 41 "(d) We can obtain a numerical value for  " }
{XPPEDIT 18 0 "Int(f(x),x = -1/2 .. 1/2);" "6#-%$IntG6$-%\"fG6#%\"xG/F
);,$*&\"\"\"F.\"\"#!\"\"F0*&F.F.F/F0" }{TEXT -1 113 " by Maple's numer
ical integration. Using a few guard digits will ensure that the result
 is correct to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "f := x -> exp(x)/(x^2+2*x+2):\naccu
rate_val := evalf(evalf[14](Int(f(x),x=-1/2..1/2)));" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%-accurate_valG$\"+,#4C*\\!#5" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can now calculate the
 absolute and relative errors in the value obtained by integrating the
 Taylor polynomial. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 82 "abserr := abs(area_est-accurate_val);\nre
lerr := evalf(abserr/abs(accurate_val),4);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'abserrG$\"$>(!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%'relerrG$\"%S9!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 31 "(e) Taking a Taylor polynomial " }{XPPEDIT 18 0 "q(x)" 
"6#-%\"qG6#%\"xG" }{TEXT -1 25 " of degree 16 centred at " }{XPPEDIT 
18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 16 " to approximate " }{XPPEDIT 
18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 63 " gives a value for the int
egral which is correct to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "f := x -> exp(x)/(x^2+2
*x+2):\n'f(x)'=f(x);\ntaylor(f(x),x,17):\nconvert(%,polynom):\nq := un
apply(%,x):\n'q(x)'=q(x);\nInt('q(x)',x=-1/2..1/2);\nvalue(%);\nevalf(
%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&-%$expGF&\"\"\"
,(*$)F'\"\"#F+F+*&F/F+F'F+F+F/F+!\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 
"6#/-%\"qG6#%\"xG,@#\"\"\"\"\"#F**&#F*\"#7F**$)F'\"\"$F*F*F**&#F*\"#;F
**$)F'\"\"%F*F*!\"\"*&#F*\"#SF**$)F'\"\"&F*F*F**&#F*\"$W\"F**$)F'\"\"'
F*F*F**&#\"#8\"$s'F**$)F'\"\"(F*F*F8*&#\"#h\"%SQF**$)F'\"\")F*F*F**&#
\"$h\"\"&?f#F**$)F'\"\"*F*F*F8*&#\"$\\$\"'+;?F**$)F'\"#5F*F*F8*&#\"%\"
e)\"(!3u<F**$)F'\"#6F*F*F**&#\"&t\"p\")S#=u\"F**$)F'F.F*F*F8*&#\"'@IH
\"*+wp)=F**$)F'FGF*F*F**&#\"'Hu'*\"++3MNAF**$)F'\"#9F*F*F**&#\")\\IvM
\",+g4S(GF**$)F'\"#:F*F*F8*&#\")BU_c\",+;wKp&F**$)F'F4F*F*F*" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%\"qG6#%\"xG/F);#!\"\"\"\"##\"\"\"
F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"4Z6*esfTJex\"5+SIm%e3ASb\"" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+,#4C*\\!#5" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT 287 
8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 34 "(a) Const
ruct a Taylor polynomial " }{XPPEDIT 18 0 "p(x);" "6#-%\"pG6#%\"xG" }
{TEXT -1 66 " of degree 10 as a truncated Maclaurin series, which appr
oximates " }{XPPEDIT 18 0 "f(x)=(1+x*tanh(x))/(x^2-x+1)" "6#/-%\"fG6#%
\"xG*&,&\"\"\"F**&F'F*-%%tanhG6#F'F*F*F*,(*$F'\"\"#F*F'!\"\"F*F*F2" }
{TEXT -1 6 " when " }{TEXT 274 1 "x" }{TEXT -1 11 " is near 0." }}
{PARA 0 "" 0 "" {TEXT -1 2 "( " }{TEXT 271 4 "Note" }{TEXT -1 60 ": Yo
u will need an order 11 Taylor series approximation for " }{XPPEDIT 
18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 3 ". )" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "(b) Plot a graph of the f
unction " }{XPPEDIT 18 0 "f(x)=(1+x*tanh(x))/(x^2-x+1)" "6#/-%\"fG6#%
\"xG*&,&\"\"\"F**&F'F*-%%tanhG6#F'F*F*F*,(*$F'\"\"#F*F'!\"\"F*F*F2" }
{TEXT -1 5 " for " }{TEXT 275 1 "x" }{TEXT -1 9 " between " }{XPPEDIT 
18 0 "-2;" "6#,$\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "2;" "6
#\"\"#" }{TEXT -1 60 " along with the graph of the Taylor polynomial f
ound in (a)." }}{PARA 0 "" 0 "" {TEXT -1 13 "(c) Estimate " }{XPPEDIT 
18 0 "Int((1+x*tanh(x))/(x^2-x+1),x=-1/3..1/3)" "6#-%$IntG6$*&,&\"\"\"
F(*&%\"xGF(-%%tanhG6#F*F(F(F(,(*$F*\"\"#F(F*!\"\"F(F(F1/F*;,$*&F(F(\"
\"$F1F1*&F(F(F6F1" }{TEXT -1 29 " using the Taylor polynomial " }
{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 15 " found in (a). " 
}}{PARA 0 "" 0 "" {TEXT -1 13 "(d) Find the " }{TEXT 263 14 "relative \+
error" }{TEXT -1 107 " in the estimate found in (c) by comparing with \+
the value given by Maple's numerical integration procedure " }{TEXT 0 
9 "evalf/int" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 49 "(e) Inc
rease the degree of the Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-
%\"pG6#%\"xG" }{TEXT -1 20 ", used to estimate  " }{XPPEDIT 18 0 "Int(
(1+x*tanh(x))/(x^2-x+1),x=-1/3..1/3)" "6#-%$IntG6$*&,&\"\"\"F(*&%\"xGF
(-%%tanhG6#F*F(F(F(,(*$F*\"\"#F(F*!\"\"F(F(F1/F*;,$*&F(F(\"\"$F1F1*&F(
F(F6F1" }{TEXT -1 124 " until the value of the integral agrees to10 di
gits with that obtained by using Maple's numerical integration.  What \+
is the " }{TEXT 263 14 "minimum degree" }{TEXT -1 26 " needed for this
 to occur?" }}{PARA 0 "" 0 "" {TEXT -1 2 "( " }{TEXT 264 4 "Note" }
{TEXT -1 86 ": You may need to use a few guard digits to get agreement
 in the last decimal place. )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 289 8 "Solution" }{TEXT -1 1 ":" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "(a) We can construct a
 Taylor polynomial " }{XPPEDIT 18 0 "p(x);" "6#-%\"pG6#%\"xG" }{TEXT 
-1 25 " of degree 10 centred at " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!
" }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"
xG" }{TEXT -1 6 " when " }{TEXT 293 1 "x" }{TEXT -1 12 " is near 0. " 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 118 "f := x -> (1+x*tanh(x))/(x^2-x+1):\n'f(x)'=f(x);\ntaylor(f(x),x
,11);\nconvert(%,polynom):\np := unapply(%,x):\n'p(x)'=p(x);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&,&\"\"\"F**&F'F*-%%tanhGF&F*
F*F*,(*$)F'\"\"#F*F*F'!\"\"F*F*F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+
9%\"xG\"\"\"\"\"!F%F%F%\"\"##!\"%\"\"$\"\"%F(\"\"&#F'\"#:\"\"'#\"#AF.
\"\"(#\"$.%\"$:$\"\")#!#fF5\"\"*#!%'4%\"%NG\"#5-%\"OG6#F%\"#6" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG,6\"\"\"F)F'F)*$)F'\"\"#
F)F)*&#\"\"%\"\"$F)*$)F'F/F)F)!\"\"*&#F/F0F)*$)F'\"\"&F)F)F3*&#F,\"#:F
)*$)F'\"\"'F)F)F)*&#\"#AF;F)*$)F'\"\"(F)F)F)*&#\"$.%\"$:$F)*$)F'\"\")F
)F)F)*&#\"#fFHF)*$)F'\"\"*F)F)F3*&#\"%'4%\"%NGF)*$)F'\"#5F)F)F3" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "(b) The f
ollowing picture shows the graph of the Taylor polynomial " }{XPPEDIT 
18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 11 " (drawn in " }{TEXT 271 6 
"purple" }{TEXT -1 26 ") along with the graph of " }{XPPEDIT 18 0 "f(x
)" "6#-%\"fG6#%\"xG" }{TEXT -1 11 " (drawn in " }{TEXT 295 6 "orange" 
}{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 90 "plot([f(x),p(x)],x=-2..2,y=-1..2,color=[coral,
COLOR(RGB,.4,0,.9)],legend=[`f(x)`,`p(x)`]);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 554 334 334 {PLOTDATA 2 "6&-%'CURVESG6%7Y7$$!\"#\"\"!$\"3#Gw
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Prs$H*)4U%F-7$$!3MLLL))Qj^<F1$\"37kl!p[C@b%F-7$$!3ALLL=Kvl;F1$\"3c:a+J
Ra)o%F-7$$!3wmm;C2G!e\"F1$\"3]CF5+&G&G[F-7$$!3OLL$3yO5]\"F1$\"3AG1p1+'
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$\"3M*******pfa<*F-$\"3>/aBr9D,=F17$$\"3Ckm;zy*zd*F-$\"3Q/(RnGOSy\"F17
$$\"39HLLeg`!)**F-$\"3GTi&o+>Fw\"F17$$\"3w****\\#G2A3\"F1$\"3gfIy([\"R
2<F17$$\"3;LLL$)G[k6F1$\"3\"\\G#Gy>2V;F17$$\"3#)****\\7yh]7F1$\"3CF8Us
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3@eR&G,iq$=Fb_l7$$!3+++Drp,B;F1$!3Is4b\"4'3.9Fb_l7$FD$!3=-C>\"esL1\"Fb
_l7$FI$!3?o$3NoU)=i!#;7$FN$!3%Rgxeh.hW$Fhbl7$FS$!3%p2H[Rd&)y\"Fhbl7$FX
$!3]ua``DId()F17$Fgn$!3a/3HC%[X\"QF17$F\\o$!3Gb*Q(H%yrc\"F17$Fao$!3qBi
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F-7$Fep$\"3'o<2%ybOpkF-7$Fjp$\"3m4#)e23H\\nF-7$F_q$\"3wl0\\9M\\MqF-7$F
dq$\"3D%*Q;=Be3tF-7$Fiq$\"3i%39nt7Wm(F-7$F^r$\"3c@YDy`Gy!)F-7$Fcr$\"37
'*H`aM#*4')F-7$Fhr$\"3y7z,bZ,1#*F-7$F^s$\"3I\\FP(R&H!)**F-7$Fds$\"3#32
7)Q&oN4\"F17$Fis$\"3aw>Xu@Y)=\"F17$F^t$\"32Tb%[e9/I\"F17$Fct$\"3=_MP'*
p+@9F17$Fht$\"3(o12Ntij`\"F17$F]u$\"3!4nz')[awj\"F17$Fbu$\"3q<'=sx0+t
\"F17$Fgu$\"3GQM7i85!z\"F17$Fav$\"39V3[!Hct#=F17$Few$\"3lYD7*Q$3J=F17$
F_x$\"3%e4MWWw^x\"F17$Fix$\"3['e%\\5,(ze\"F17$F^y$\"3)\\xtI\\)R^5F17$F
cy$!34-t[z?&=J#F-7$Fhy$!3S\"*R0a$ojE$F17$F]z$!3'o[W1D!4\"Q*F17$Fbz$!3#
)R&>=D#4h@Fhbl7$Fgz$!3vW))fS!GoT%Fhbl7$F\\[l$!3([A./5=P+)Fhbl7$Fa[l$!3
Sd<&>TM#*\\\"Fb_l7$$\"3\"*****\\n'*33<F1$!3CDYyRz%\\&>Fb_l7$Ff[l$!3[D<
+HV<GDFb_l7$$\"3ILLe*3k**y\"F1$!3WS^/[S#yH$Fb_l7$F[\\l$!3]`>)z@KuE%Fb_
l7$$\"35++D1>V_=F1$!3hV%[k)4![\"[Fb_l7$$\"33+++S2ls=F1$!3p7wn9\"HMU&Fb
_l7$$\"33++vt&pG*=F1$!3UI'GUi[#*4'Fb_l7$F`\\l$!3Ct.iF5o[oFb_l7$$\"31+]
i0j\"[$>F1$!3Z-\"*=CP0WxFb_l7$$\"3/++v.Uac>F1$!3!Q]+AG1=u)Fb_l7$$\"3/+
D\"G:3u'>F1$!3EV.aW3@#G*Fb_l7$$\"3-+](=5s#y>F1$!35'H0FDy?&)*Fb_l7$$\"3
-+v$40O\"*)>F1$!31A)Ro-\"GX5F]^l7$Fe\\l$!3rH*>hf'e36F]^l-%&COLORG6&F\\
]l$\"\"%!\"\"Fb]l$\"\"*F^^m-Fd]l6#%%p(x)G-%+AXESLABELSG6$Q\"x6\"Q\"yFh
^m-%%VIEWG6$;F(Fe\\l;$F^^mF*Fe\\l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 1 "f(x)" "p(x)" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " }{XPPEDIT 18 0 "Int(p(x),x=-1/3.
.1/3)" "6#-%$IntG6$-%\"pG6#%\"xG/F);,$*&\"\"\"F.\"\"$!\"\"F0*&F.F.F/F0
" }{TEXT -1 26 " provides an estimate for " }{XPPEDIT 18 0 "Int(f(x),x
 = -1/3 .. 1/3);" "6#-%$IntG6$-%\"fG6#%\"xG/F);,$*&\"\"\"F.\"\"$!\"\"F
0*&F.F.F/F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Int(p(x),x=-1/3..1/3);\nvalu
e(%);\narea_est := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$Int
G6$,6\"\"\"F'%\"xGF'*$)F(\"\"#F'F'*&#\"\"%\"\"$F'*$)F(F.F'F'!\"\"*&#F.
F/F'*$)F(\"\"&F'F'F2*&#F+\"#:F'*$)F(\"\"'F'F'F'*&#\"#AF:F'*$)F(\"\"(F'
F'F'*&#\"$.%\"$:$F'*$)F(\"\")F'F'F'*&#\"#fFGF'*$)F(\"\"*F'F'F2*&#\"%'4
%\"%NGF'*$)F(\"#5F'F'F2/F(;#F2F/#F'F/" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6##\"+ACL2Q\"+&>HV_&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)area_estG$
\"+vh$>*o!#5" }}}{PARA 0 "" 0 "" {TEXT -1 40 "(d) We can obtain a nume
rical value for " }{XPPEDIT 18 0 "Int(f(x),x = -1/3 .. 1/3);" "6#-%$In
tG6$-%\"fG6#%\"xG/F);,$*&\"\"\"F.\"\"$!\"\"F0*&F.F.F/F0" }{TEXT -1 
113 " by Maple's numerical integration. Using a few guard digits will \+
ensure that the result is correct to 10 digits. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "accurate_val
 := evalf(Int(f(x),x=-1/3..1/3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%-accurate_valG$\"+^k$>*o!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 114 "We can now calculate the absolute and re
lative errors in the value obtained by integrating the Taylor polynomi
al. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 82 "abserr := abs(area_est-accurate_val);\nrelerr := eval
f(abserr/abs(accurate_val),5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'a
bserrG$\"$w#!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerrG$\"&Z+%!#
7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "(e)
 Taking a Taylor polynomial " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" 
}{TEXT -1 25 " of degree 16 centred at " }{XPPEDIT 18 0 "x=0" "6#/%\"x
G\"\"!" }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"f
G6#%\"xG" }{TEXT -1 63 " gives a value for the integral which is corre
ct to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 148 "f := x -> (1+x*tanh(x))/(x^2-x+1);\ntaylor(f
(x),x,17):\nconvert(%,polynom):\np := unapply(%,x);\nInt('p(x)',x=-1/3
..1/3);\nevalf[14](value(%));\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,&\"\"\"F.*&9$F.-%%ta
nhG6#F0F.F.F.,(*$)F0\"\"#F.F.F0!\"\"F.F.F8F(F(F(" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,B\"\"\"F-9$F-
*$)F.\"\"#F-F-*&#\"\"%\"\"$F-*$)F.F4F-F-!\"\"*&#F4F5F-*$)F.\"\"&F-F-F8
*&#F1\"#:F-*$)F.\"\"'F-F-F-*&#\"#AF@F-*$)F.\"\"(F-F-F-*&#\"$.%\"$:$F-*
$)F.\"\")F-F-F-*&#\"#fFMF-*$)F.\"\"*F-F-F8*&#\"%'4%\"%NGF-*$)F.\"#5F-F
-F8*&#\"$8(\"$n&F-*$)F.\"#6F-F-F8*&#\"&By#\"'Df:F-*$)F.\"#7F-F-F-*&#\"
')*QAFboF-*$)F.\"#8F-F-F-*&#\"(p(ow\"(v53'F-*$)F.\"#9F-F-F-*&#\"(`K1\"
F_pF-*$)F.F@F-F-F8*&#\"*z=z<*\"*vG^Q'F-*$)F.\"#;F-F-F8F(F(F(" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%\"pG6#%\"xG/F);#!\"\"\"\"$#\"\"\"
F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/*z4XO>*o!#9" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"+^k$>*o!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT 288 8 "Question" }
{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 34 "(a) Construct a Taylor \+
polynomial " }{XPPEDIT 18 0 "p(x);" "6#-%\"pG6#%\"xG" }{TEXT -1 66 " o
f degree 10 as a truncated Maclaurin series, which approximates " }
{XPPEDIT 18 0 "f(x) = exp(x)/(1+sin(x^2));" "6#/-%\"fG6#%\"xG*&-%$expG
6#F'\"\"\",&F,F,-%$sinG6#*$F'\"\"#F,!\"\"" }{TEXT -1 6 " when " }
{TEXT 276 1 "x" }{TEXT -1 11 " is near 0." }}{PARA 0 "" 0 "" {TEXT -1 
2 "( " }{TEXT 271 4 "Note" }{TEXT -1 60 ": You will need an order 11 T
aylor series approximation for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"x
G" }{TEXT -1 3 ". )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 33 "(b) Plot a graph of the function " }{XPPEDIT 18 0 "f(x)
 = exp(x)/(1+sin(x^2));" "6#/-%\"fG6#%\"xG*&-%$expG6#F'\"\"\",&F,F,-%$
sinG6#*$F'\"\"#F,!\"\"" }{TEXT -1 7 " , for " }{TEXT 277 1 "x" }{TEXT 
-1 9 " between " }{XPPEDIT 18 0 "-3/2;" "6#,$*&\"\"$\"\"\"\"\"#!\"\"F(
" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "3/2;" "6#*&\"\"$\"\"\"\"\"#!\"\"
" }{TEXT -1 61 ", along with the graph of the Taylor polynomial found \+
in (a)." }}{PARA 0 "" 0 "" {TEXT -1 13 "(c) Estimate " }{XPPEDIT 18 0 
"Int(exp(x)/(1+sin(x^2)),x = -1/3 .. 1/3);" "6#-%$IntG6$*&-%$expG6#%\"
xG\"\"\",&F+F+-%$sinG6#*$F*\"\"#F+!\"\"/F*;,$*&F+F+\"\"$F2F2*&F+F+F7F2
" }{TEXT -1 30 "  using the Taylor polynomial " }{XPPEDIT 18 0 "p(x)" 
"6#-%\"pG6#%\"xG" }{TEXT -1 15 " found in (a). " }}{PARA 0 "" 0 "" 
{TEXT -1 13 "(d) Find the " }{TEXT 263 14 "relative error" }{TEXT -1 
107 " in the estimate found in (c) by comparing with the value given b
y Maple's numerical integration procedure " }{TEXT 0 9 "evalf/int" }
{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 49 "(e) Increase the degre
e of the Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }
{TEXT -1 20 ", used to estimate  " }{XPPEDIT 18 0 "Int(exp(x)/(1+sin(x
^2)),x = -1/3 .. 1/3);" "6#-%$IntG6$*&-%$expG6#%\"xG\"\"\",&F+F+-%$sin
G6#*$F*\"\"#F+!\"\"/F*;,$*&F+F+\"\"$F2F2*&F+F+F7F2" }{TEXT -1 125 "  u
ntil the value of the integral agrees to10 digits with that obtained b
y using Maple's numerical integration.  What is the " }{TEXT 263 14 "m
inimum degree" }{TEXT -1 26 " needed for this to occur?" }}{PARA 0 "" 
0 "" {TEXT -1 2 "( " }{TEXT 264 4 "Note" }{TEXT -1 86 ": You may need \+
to use a few guard digits to get agreement in the last decimal place. \+
)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 290 8 "Solu
tion" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 41 "(a) We can construct a Taylor polynomial " }{XPPEDIT 
18 0 "p(x);" "6#-%\"pG6#%\"xG" }{TEXT -1 25 " of degree 10 centred at \+
" }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 16 " to approximate \+
" }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 6 " when " }{TEXT 
297 1 "x" }{TEXT -1 12 " is near 0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "f := x -> exp(x)/(1+sin(x
^2)):\n'f(x)'=f(x);\ntaylor(f(x),x,11);\nconvert(%,polynom):\np := una
pply(%,x):\n'p(x)'=p(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%
\"xG*&-%$expGF&\"\"\",&F+F+-%$sinG6#*$)F'\"\"#F+F+!\"\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#+;%\"xG\"\"\"\"\"!F%F%#!\"\"\"\"#F)#!\"&\"\"'\"
\"$#\"#8\"#C\"\"%#\"$,\"\"$?\"\"\"&#!$p#\"$?(F,#!%,M\"%S]\"\"(#\"%TB\"
%k!)\"\")#\"'tW>\"'!)GO\"\"*#!'*3c(\"(+)GO\"#5-%\"OG6#F%\"#6" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG,8\"\"\"F)F'F)*&#F)\"\"#F)*$)F
'F,F)F)!\"\"*&#\"\"&\"\"'F)*$)F'\"\"$F)F)F/*&#\"#8\"#CF)*$)F'\"\"%F)F)
F)*&#\"$,\"\"$?\"F)*$)F'F2F)F)F)*&#\"$p#\"$?(F)*$)F'F3F)F)F/*&#\"%,M\"
%S]F)*$)F'\"\"(F)F)F/*&#\"%TB\"%k!)F)*$)F'\"\")F)F)F)*&#\"'tW>\"'!)GOF
)*$)F'\"\"*F)F)F)*&#\"'*3c(\"(+)GOF)*$)F'\"#5F)F)F/" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "(b) The following pictu
re shows the graph of the Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6
#-%\"pG6#%\"xG" }{TEXT -1 11 " (drawn in " }{TEXT 260 5 "brown" }
{TEXT -1 26 ") along with the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"
fG6#%\"xG" }{TEXT -1 11 " (drawn in " }{TEXT 296 7 "magenta" }{TEXT 
-1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 83 "plot([f(x),p(x)],x=-1.5..1.5,y=-5..8,color=[magenta,b
rown],legend=[`f(x)`,`p(x)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 320 373 
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\"3![(*HE*=&zp)F-7$Fgr$\"3#=6F;g*eH$*F-F\\s7$Fcs$\"3G9(\\g0cB1\"F*7$Fh
s$\"3H2zAXxQ86F*7$F]t$\"3e2EF+t(H;\"F*7$Fbt$\"3O@zTfYo27F*7$Fgt$\"3JMC
7Y\"y[C\"F*7$F\\u$\"3%z)3?6x$\\F\"F*7$Fau$\"3Nom.No:-8F*7$Ffu$\"3\"*)G
9%H+,A8F*7$F[v$\"3*4x>)3'H(R8F*7$F`v$\"3)p?%=4h;a8F*7$Fev$\"3U7(G2ij.P
\"F*7$Fjv$\"3%\\ADv,\"z)Q\"F*7$F_w$\"3-X>)ejzbT\"F*7$Fdw$\"3!*>]!fBB_X
\"F*7$Fiw$\"3<@yXdC`?:F*7$F^x$\"3Iyq3al!)>;F*7$Fcx$\"3%=@\"=COtx<F*7$F
hx$\"3W`'e)*pev,#F*7$F]y$\"3?1y=bG*>M#F*7$Fby$\"3Y(zg&zE?&)GF*7$Fgy$\"
3W/ye&=k8e$F*7$F\\z$\"3\"y]8[/eeh%F*7$Ffz$\"3c>'>(*)>\">(fF*7$F`[l$\"3
A;RpKePQzF*-Fe[l6&Fg[l$\")#)eqkFj[l$\"))eqk\"Fj[lFggl-F^\\l6#%%p(x)G-%
+AXESLABELSG6$Q\"x6\"Q\"yF`hl-%%VIEWG6$;$!#:!\"\"$\"#:Fhhl;$!\"&F\\\\l
$\"\")F\\\\l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 
"f(x)" "p(x)" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 4 "(c) " }{XPPEDIT 18 0 "Int(p(x),x=-1/3..1/3)" "6#-%$IntG6$-
%\"pG6#%\"xG/F);,$*&\"\"\"F.\"\"$!\"\"F0*&F.F.F/F0" }{TEXT -1 26 " pro
vides an estimate for " }{XPPEDIT 18 0 "Int(f(x),x = -1/3 .. 1/3);" "6
#-%$IntG6$-%\"fG6#%\"xG/F);,$*&\"\"\"F.\"\"$!\"\"F0*&F.F.F/F0" }{TEXT 
-1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "Int(p(x),x=-1/3..1/3);\nvalue(%);\narea_est := evalf(
%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,8\"\"\"F'%\"xGF'*&#F'
\"\"#F'*$)F(F+F'F'!\"\"*&#\"\"&\"\"'F'*$)F(\"\"$F'F'F.*&#\"#8\"#CF'*$)
F(\"\"%F'F'F'*&#\"$,\"\"$?\"F'*$)F(F1F'F'F'*&#\"$p#\"$?(F'*$)F(F2F'F'F
.*&#\"%,M\"%S]F'*$)F(\"\"(F'F'F.*&#\"%TB\"%k!)F'*$)F(\"\")F'F'F'*&#\"'
tW>\"'!)GOF'*$)F(\"\"*F'F'F'*&#\"'*3c(\"(+)GOF'*$)F(\"#5F'F'F./F(;#F.F
5#F'F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\".hYz)Q;B\".+[oqb`$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)area_estG$\"+W(o;b'!#5" }}}{PARA 0 
"" 0 "" {TEXT -1 40 "(d) We can obtain a numerical value for " }
{XPPEDIT 18 0 "Int(f(x),x = -1/3 .. 1/3);" "6#-%$IntG6$-%\"fG6#%\"xG/F
);,$*&\"\"\"F.\"\"$!\"\"F0*&F.F.F/F0" }{TEXT -1 113 " by Maple's numer
ical integration. Using a few guard digits will ensure that the result
 is correct to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "accurate_val := evalf(Int(f(x),x=-1
/3..1/3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-accurate_valG$\"+z)o;
b'!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
114 "We can now calculate the absolute and relative errors in the valu
e obtained by integrating the Taylor polynomial. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "abserr := ab
s(area_est-accurate_val);\nrelerr := evalf(abserr/abs(accurate_val),4)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'abserrG$\"$N\"!#5" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%'relerrG$\"%g?!#6" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "(e) Taking a Taylor polynomial \+
" }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 25 " of degree 14 \+
centred at " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 16 " to ap
proximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 63 " giv
es a value for the integral which is correct to 10 digits. " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "f
 := x -> exp(x)/(1+sin(x^2)):\n'f(x)'=f(x);\ntaylor(f(x),x,15):\nconve
rt(%,polynom):\nq := unapply(%,x):\n'q(x)'=q(x);\nInt('q(x)',x=-1/3..1
/3);\nevalf(value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"
xG*&-%$expGF&\"\"\",&F+F+-%$sinG6#*$)F'\"\"#F+F+!\"\"" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG,@\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F
)!\"\"*&#\"\"&\"\"'F)*$)F'\"\"$F)F)F/*&#\"#8\"#CF)*$)F'\"\"%F)F)F)*&#
\"$,\"\"$?\"F)*$)F'F2F)F)F)*&#\"$p#\"$?(F)*$)F'F3F)F)F/*&#\"%,M\"%S]F)
*$)F'\"\"(F)F)F/*&#\"%TB\"%k!)F)*$)F'\"\")F)F)F)*&#\"'tW>\"'!)GOF)*$)F
'\"\"*F)F)F)*&#\"'*3c(\"(+)GOF)*$)F'\"#5F)F)F/*&#\")H_7;\")+o\"*RF)*$)
F'\"#6F)F)F/*&#\")pG(>(\"*+;+z%F)*$)F'\"#7F)F)F)*&#\"*,!*or$\"+gTSX7F)
*$)F'F9F)F)F)*&#\"+(>ksD*\",+7Hyr)F)*$)F'\"#9F)F)F/" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%$IntG6$-%\"qG6#%\"xG/F);#!\"\"\"\"$#\"\"\"F." }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+z)o;b'!#5" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "
Q1" }}{PARA 0 "" 0 "" {TEXT -1 37 "(a) Construct a Taylor polynomial o
f " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 77 " degree 6, a
s a truncated Maclaurin series, which approximates the function  " }
{XPPEDIT 18 0 "f(x) = sin(sqrt(x^2+1));" "6#/-%\"fG6#%\"xG-%$sinG6#-%%
sqrtG6#,&*$F'\"\"#\"\"\"F1F1" }{TEXT -1 7 "  when " }{TEXT 268 1 "x" }
{TEXT -1 13 " is near 0.  " }}{PARA 0 "" 0 "" {TEXT -1 33 "(b) Plot a \+
graph of the function " }{XPPEDIT 18 0 "f(x) = sin(sqrt(x^2+1));" "6#/
-%\"fG6#%\"xG-%$sinG6#-%%sqrtG6#,&*$F'\"\"#\"\"\"F1F1" }{TEXT -1 6 "  \+
for " }{TEXT 266 1 "x" }{TEXT -1 9 " between " }{XPPEDIT 18 0 "-3;" "6
#,$\"\"$!\"\"" }{TEXT -1 66 " and 3 along with the graph of the Taylor
 polynomial found in (a)." }}{PARA 0 "" 0 "" {TEXT -1 66 "(c) Construc
t a Taylor polynomial of degree 7, which approximates " }{XPPEDIT 18 
0 "Int(sin(sqrt(t^2+1)),t = 0 .. x);" "6#-%$IntG6$-%$sinG6#-%%sqrtG6#,
&*$%\"tG\"\"#\"\"\"F0F0/F.;\"\"!%\"xG" }{TEXT -1 7 "  when " }{TEXT 
267 1 "x" }{TEXT -1 11 " is near 0." }}{PARA 0 "" 0 "" {TEXT -1 13 "(d
) Estimate " }{XPPEDIT 18 0 "Int(sin(sqrt(x^2+1)),x = 0 .. 1/4);" "6#-
%$IntG6$-%$sinG6#-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"F0F0/F.;\"\"!*&F0F0\"\"
%!\"\"" }{TEXT -1 43 " using the Taylor polynomial found in (b). " }}
{PARA 0 "" 0 "" {TEXT -1 49 "(e) Increase the degree of the Taylor pol
ynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 19 ", used
 to estimate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 71 " \+
until you obtain an estimate for the numerical value of  the integral \+
" }{XPPEDIT 18 0 "Int(sin(sqrt(x^2+1)),x = 0 .. 1/4);" "6#-%$IntG6$-%$
sinG6#-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"F0F0/F.;\"\"!*&F0F0\"\"%!\"\"" }
{TEXT -1 69 " which is correct to 10 digits. What is the mnimum degree
 needed for " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 20 " f
or this to occur? " }}{PARA 0 "" 0 "" {TEXT -1 34 "___________________
_______________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 34 "(a) Construct a Taylo
r polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 66 " \+
of degree 8, as a truncated Maclaurin series, which approximates " }
{XPPEDIT 18 0 "f(x) = sin(x)/x;" "6#/-%\"fG6#%\"xG*&-%$sinG6#F'\"\"\"F
'!\"\"" }{TEXT -1 7 "  when " }{TEXT 265 1 "x" }{TEXT -1 11 " is near \+
0." }}{PARA 0 "" 0 "" {TEXT -1 33 "(b) Plot a graph of the function " 
}{XPPEDIT 18 0 "f(x) = sin(x)/x;" "6#/-%\"fG6#%\"xG*&-%$sinG6#F'\"\"\"
F'!\"\"" }{TEXT -1 6 "  for " }{TEXT 269 1 "x" }{TEXT -1 9 " between \+
" }{XPPEDIT 18 0 "-6;" "6#,$\"\"'!\"\"" }{TEXT -1 66 " and 6 along wit
h the graph of the Taylor polynomial found in (a)." }}{PARA 0 "" 0 "" 
{TEXT -1 66 "(c) Construct a Taylor polynomial of degree 9, which appr
oximates " }{XPPEDIT 18 0 "Int(sin(t)/t,t = 0 .. x);" "6#-%$IntG6$*&-%
$sinG6#%\"tG\"\"\"F*!\"\"/F*;\"\"!%\"xG" }{TEXT -1 7 "  when " }{TEXT 
270 1 "x" }{TEXT -1 11 " is near 0." }}{PARA 0 "" 0 "" {TEXT -1 13 "(d
) Estimate " }{XPPEDIT 18 0 "Int(sin(x)/x,x = 0 .. 1);" "6#-%$IntG6$*&
-%$sinG6#%\"xG\"\"\"F*!\"\"/F*;\"\"!F+" }{TEXT -1 130 " using the Tayl
or polynomial found in (b), and compare this value with the value give
n by Maple's numerical integration procedure " }{TEXT 0 9 "evalf/Int" 
}{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 49 "(e) Increase the degree of the Taylor polynomial " }
{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 19 ", used to estimat
e " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 71 " until you o
btain an estimate for the numerical value of  the integral " }
{XPPEDIT 18 0 "Int(sin(x)/x,x = 0 .. 1);" "6#-%$IntG6$*&-%$sinG6#%\"xG
\"\"\"F*!\"\"/F*;\"\"!F+" }{TEXT -1 69 " which is correct to 10 digits
. What is the mnimum degree needed for " }{XPPEDIT 18 0 "p(x)" "6#-%\"
pG6#%\"xG" }{TEXT -1 20 " for this to occur? " }}{PARA 0 "" 0 "" 
{TEXT -1 34 "__________________________________" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__
________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "
" {TEXT -1 34 "(a) Construct a Taylor polynomial " }{XPPEDIT 18 0 "p(x
);" "6#-%\"pG6#%\"xG" }{TEXT -1 67 " of degree 10, as a truncated Macl
aurin series, which approximates " }{XPPEDIT 18 0 "f(x) = exp(x)/(1+si
n(x^2));" "6#/-%\"fG6#%\"xG*&-%$expG6#F'\"\"\",&F,F,-%$sinG6#*$F'\"\"#
F,!\"\"" }{TEXT -1 6 " when " }{TEXT 272 1 "x" }{TEXT -1 12 " is near \+
0. " }}{PARA 0 "" 0 "" {TEXT -1 33 "(b) Plot a graph of the function \+
" }{XPPEDIT 18 0 "f(x) = exp(x)/(1+sin(x^2));" "6#/-%\"fG6#%\"xG*&-%$e
xpG6#F'\"\"\",&F,F,-%$sinG6#*$F'\"\"#F,!\"\"" }{TEXT -1 6 "  for " }
{TEXT 273 1 "x" }{TEXT -1 9 " between " }{XPPEDIT 18 0 "-2;" "6#,$\"\"
#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "2;" "6#\"\"#" }{TEXT -1 
60 " along with the graph of the Taylor polynomial found in (a)." }}
{PARA 0 "" 0 "" {TEXT -1 13 "(c) Estimate " }{XPPEDIT 18 0 "Int(exp(x)
/(1+sin(x^2)),x = -1/3 .. 1/3);" "6#-%$IntG6$*&-%$expG6#%\"xG\"\"\",&F
+F+-%$sinG6#*$F*\"\"#F+!\"\"/F*;,$*&F+F+\"\"$F2F2*&F+F+F7F2" }{TEXT 
-1 30 "  using the Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG
6#%\"xG" }{TEXT -1 15 " found in (a). " }}{PARA 0 "" 0 "" {TEXT -1 13 
"(d) Find the " }{TEXT 263 14 "relative error" }{TEXT -1 107 " in the \+
estimate found in (c) by comparing with the value given by Maple's num
erical integration procedure " }{TEXT 0 9 "evalf/Int" }{TEXT -1 3 ".  \+
" }}{PARA 0 "" 0 "" {TEXT -1 49 "(e) Increase the degree of the Taylor
 polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 20 ", \+
used to estimate  " }{XPPEDIT 18 0 "Int(exp(x)/(1+sin(x^2)),x = -1/3 .
. 1/3);" "6#-%$IntG6$*&-%$expG6#%\"xG\"\"\",&F+F+-%$sinG6#*$F*\"\"#F+!
\"\"/F*;,$*&F+F+\"\"$F2F2*&F+F+F7F2" }{TEXT -1 125 "  until the value \+
of the integral agrees to 10 digits with that obtained by using Maple'
s numerical integration. What is the " }{TEXT 263 14 "minimum degree" 
}{TEXT -1 26 " needed for this to occur?" }}{PARA 0 "" 0 "" {TEXT -1 
2 "( " }{TEXT 264 4 "Note" }{TEXT -1 86 ": You may need to use a few g
uard digits to get agreement in the last decimal place. )" }}{PARA 0 "
" 0 "" {TEXT -1 34 "__________________________________" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "f := x \+
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$Q&)p*[!oU\"F17$Ffv$\"3iL$*e#H0l\\\"F17$F[w$\"3yJB%)Rdj:;F17$F`w$\"3y7
s$*ybnP=F17$Few$\"3bO-j\"4:U@#F17$Fjw$\"3/$eZ$QI,yGF17$F_x$\"3dFIbtLQ9
RF17$Fdx$\"3#=9*eD()fkbF17$Fix$\"3\"*oH_q^mF!)F17$F^y$\"3>r2!y*[1E6F`[
l7$Fcy$\"3R$=()\\5ZOk\"F`[l7$Fhy$\"3sp*Q*3?\\pAF`[l7$Fbz$\"3Jd8)>$GJMJ
F`[l7$Fg[l$\"3i8R?78roTF`[l7$Fi^l$\"3#4RM:RM&*\\&F`[l-F^_l6&F`_lFd_lFa
_lFd_l-%+AXESLABELSG6$Q\"x6\"Q\"yFg[m-%%VIEWG6$;F(Fi^l;$Fc_lF*$\"\"%F*
" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C
urve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 34 "__________________________________" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5
" }}{PARA 0 "" 0 "" {TEXT -1 13 "(a) Estimate " }{XPPEDIT 18 0 "Int(1/
(x+exp(x)),x = 0 .. 1/2);" "6#-%$IntG6$*&\"\"\"F',&%\"xGF'-%$expG6#F)F
'!\"\"/F);\"\"!*&F'F'\"\"#F-" }{TEXT -1 54 " by using a Taylor polynom
ial of degree 12 centred at " }{XPPEDIT 18 0 "x = 1/4;" "6#/%\"xG*&\"
\"\"F&\"\"%!\"\"" }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "f(x)
 = 1/(x+exp(x));" "6#/-%\"fG6#%\"xG*&\"\"\"F),&F'F)-%$expG6#F'F)!\"\"
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 87 "(b) Compare this valu
e with the value given by Maple's numerical integration procedure " }
{TEXT 0 9 "evalf/Int" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "(
c) Estimate " }{XPPEDIT 18 0 "Int(1/(x+exp(x)),x = 0 .. 1/2);" "6#-%$I
ntG6$*&\"\"\"F',&%\"xGF'-%$expG6#F)F'!\"\"/F);\"\"!*&F'F'\"\"#F-" }
{TEXT -1 54 " by using a Taylor polynomial of degree 12 centred at " }
{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 16 " to approximate \+
" }{XPPEDIT 18 0 "f(x) = 1/(x+exp(x));" "6#/-%\"fG6#%\"xG*&\"\"\"F),&F
'F)-%$expG6#F'F)!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 264 4 "Note" }{TEXT -1 2 ": " }}{PARA 15 "" 
0 "" {TEXT -1 114 "In part (a) force Maple to use floating point arith
metic by entering the centre for the Taylor series in the form " }
{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 4 ".25." }}{PARA 15 "
" 0 "" {TEXT -1 66 "It is not be sensible to use a Taylor polynomial a
pproximaton for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 
12 " centred at " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 63 " \+
because the radius of convergence of the Maclaurin series for " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " is not much larg
er than " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 2 ".
 " }}{PARA 0 "" 0 "" {TEXT -1 41 "____________________________________
_____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 41 "_______________________________________
__" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 
"" 0 "" {TEXT -1 2 "Q6" }{TEXT 280 1 " " }{TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 34 "(a) Construct a Taylor polynomial " }{XPPEDIT 18 0 "
p(x);" "6#-%\"pG6#%\"xG" }{TEXT -1 66 " of degree 10 as a truncated Ma
claurin series, which approximates " }{XPPEDIT 18 0 "f(x) = cosh(x)/(x
^2-x+1);" "6#/-%\"fG6#%\"xG*&-%%coshG6#F'\"\"\",(*$F'\"\"#F,F'!\"\"F,F
,F0" }{TEXT -1 6 " when " }{TEXT 278 1 "x" }{TEXT -1 11 " is near 0." 
}}{PARA 0 "" 0 "" {TEXT -1 2 "( " }{TEXT 271 4 "Note" }{TEXT -1 60 ": \+
You will need an order 11 Taylor series approximation for " }{XPPEDIT 
18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 3 ". )" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "(b) Plot a graph of the f
unction " }{XPPEDIT 18 0 "f(x) = cosh(x)/(x^2-x+1);" "6#/-%\"fG6#%\"xG
*&-%%coshG6#F'\"\"\",(*$F'\"\"#F,F'!\"\"F,F,F0" }{TEXT -1 6 "  for " }
{TEXT 279 1 "x" }{TEXT -1 9 " between " }{XPPEDIT 18 0 "-2;" "6#,$\"\"
#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "2;" "6#\"\"#" }{TEXT -1 
60 " along with the graph of the Taylor polynomial found in (a)." }}
{PARA 0 "" 0 "" {TEXT -1 13 "(c) Estimate " }{XPPEDIT 18 0 "Int(cosh(x
)/(x^2-x+1),x = -1/3 .. 1/3);" "6#-%$IntG6$*&-%%coshG6#%\"xG\"\"\",(*$
F*\"\"#F+F*!\"\"F+F+F//F*;,$*&F+F+\"\"$F/F/*&F+F+F4F/" }{TEXT -1 30 " \+
 using the Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG
" }{TEXT -1 15 " found in (a). " }}{PARA 0 "" 0 "" {TEXT -1 13 "(d) Fi
nd the " }{TEXT 263 14 "relative error" }{TEXT -1 107 " in the estimat
e found in (c) by comparing with the value given by Maple's numerical \+
integration procedure " }{TEXT 0 9 "evalf/int" }{TEXT -1 3 ".  " }}
{PARA 0 "" 0 "" {TEXT -1 49 "(e) Increase the degree of the Taylor pol
ynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 20 ", used
 to estimate  " }{XPPEDIT 18 0 "Int(cosh(x)/(x^2-x+1),x = -1/3 .. 1/3)
;" "6#-%$IntG6$*&-%%coshG6#%\"xG\"\"\",(*$F*\"\"#F+F*!\"\"F+F+F//F*;,$
*&F+F+\"\"$F/F/*&F+F+F4F/" }{TEXT -1 125 "  until the value of the int
egral agrees to10 digits with that obtained by using Maple's numerical
 integration.  What is the " }{TEXT 263 14 "minimum degree" }{TEXT -1 
26 " needed for this to occur?" }}{PARA 0 "" 0 "" {TEXT -1 2 "( " }
{TEXT 264 4 "Note" }{TEXT -1 86 ": You may need to use a few guard dig
its to get agreement in the last decimal place. )" }}{PARA 0 "" 0 "" 
{TEXT -1 29 "_____________________________" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 29 "_____________________________" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q7" }}
{PARA 0 "" 0 "" {TEXT -1 34 "(a) Construct a Taylor polynomial " }
{XPPEDIT 18 0 "p(x);" "6#-%\"pG6#%\"xG" }{TEXT -1 65 " of degree 8 as \+
a truncated Maclaurin series, which approximates " }{XPPEDIT 18 0 "f(x
) = (1+x*sinh(x))/(x^2-x+1);" "6#/-%\"fG6#%\"xG*&,&\"\"\"F**&F'F*-%%si
nhG6#F'F*F*F*,(*$F'\"\"#F*F'!\"\"F*F*F2" }{TEXT -1 6 " when " }{TEXT 
281 1 "x" }{TEXT -1 11 " is near 0." }}{PARA 0 "" 0 "" {TEXT -1 2 "( \+
" }{TEXT 271 4 "Note" }{TEXT -1 59 ": You will need an order 9 Taylor \+
series approximation for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }
{TEXT -1 3 ". )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 33 "(b) Plot a graph of the function " }{XPPEDIT 18 0 "f(x) =
 (1+x*sinh(x))/(x^2-x+1);" "6#/-%\"fG6#%\"xG*&,&\"\"\"F**&F'F*-%%sinhG
6#F'F*F*F*,(*$F'\"\"#F*F'!\"\"F*F*F2" }{TEXT -1 5 " for " }{TEXT 282 
1 "x" }{TEXT -1 9 " between " }{XPPEDIT 18 0 "-2;" "6#,$\"\"#!\"\"" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "2;" "6#\"\"#" }{TEXT -1 60 " along \+
with the graph of the Taylor polynomial found in (a)." }}{PARA 0 "" 0 
"" {TEXT -1 13 "(c) Estimate " }{XPPEDIT 18 0 "Int((1+x*sinh(x))/(x^2-
x+1),x = -1/4 .. 1/4);" "6#-%$IntG6$*&,&\"\"\"F(*&%\"xGF(-%%sinhG6#F*F
(F(F(,(*$F*\"\"#F(F*!\"\"F(F(F1/F*;,$*&F(F(\"\"%F1F1*&F(F(F6F1" }
{TEXT -1 29 " using the Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-
%\"pG6#%\"xG" }{TEXT -1 15 " found in (a). " }}{PARA 0 "" 0 "" {TEXT 
-1 13 "(d) Find the " }{TEXT 263 14 "relative error" }{TEXT -1 107 " i
n the estimate found in (c) by comparing with the value given by Maple
's numerical integration procedure " }{TEXT 0 9 "evalf/int" }{TEXT -1 
3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 49 "(e) Increase the degree of the \+
Taylor polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 
20 ", used to estimate  " }{XPPEDIT 18 0 "Int((1+x*sinh(x))/(x^2-x+1),
x = -1/4 .. 1/4);" "6#-%$IntG6$*&,&\"\"\"F(*&%\"xGF(-%%sinhG6#F*F(F(F(
,(*$F*\"\"#F(F*!\"\"F(F(F1/F*;,$*&F(F(\"\"%F1F1*&F(F(F6F1" }{TEXT -1 
124 " until the value of the integral agrees to10 digits with that obt
ained by using Maple's numerical integration.  What is the " }{TEXT 
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