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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 36 "The even and odd parts of a funct
ion" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Can
ada" }}{PARA 0 "" 0 "" {TEXT -1 19 "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 22 "Even and odd functions" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 6 "A set " }{TEXT 
306 1 "S" }{TEXT -1 20 " of real numbers is " }{TEXT 259 19 "symmetric
al about 0" }{TEXT -1 14 " if whenever  " }{XPPEDIT 18 0 "x in S" "6#-
%#inG6$%\"xG%\"SG" }{TEXT -1 12 "  then also " }{XPPEDIT 18 0 "-x in S
" "6#-%#inG6$,$%\"xG!\"\"%\"SG" }{TEXT -1 94 ". In particular, the set
 of all real numbers |R is symmetrical about 0 as is any open interval
" }{XPPEDIT 18 0 " ``(-r,r)" "6#-%!G6$,$%\"rG!\"\"F'" }{TEXT -1 20 " o
r closed interval " }{XPPEDIT 18 0 "[-r,r]" "6#7$,$%\"rG!\"\"F%" }
{TEXT -1 8 ", where " }{TEXT 304 1 "r" }{TEXT -1 29 " is a positive re
al number.  " }}{PARA 0 "" 0 "" {TEXT -1 11 "A function " }{XPPEDIT 
18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 47 " whose domain is symmetric
al about 0 is called " }{TEXT 259 4 "even" }{TEXT -1 15 " provided tha
t " }{XPPEDIT 18 0 "f(-x) = f(x);" "6#/-%\"fG6#,$%\"xG!\"\"-F%6#F(" }
{TEXT -1 17 " for all numbers " }{TEXT 292 1 "x" }{TEXT -1 29 " in the
 interval, and called " }{TEXT 259 3 "odd" }{TEXT -1 15 " provided tha
t " }{XPPEDIT 18 0 "f(-x) = -f(x);" "6#/-%\"fG6#,$%\"xG!\"\",$-F%6#F(F
)" }{TEXT -1 17 " for all numbers " }{TEXT 291 1 "x" }{TEXT -1 17 " in
 the interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 3 "If " }{TEXT 297 1 "a" }{TEXT -1 26 " is a positive number,
 an " }{TEXT 259 13 "even function" }{TEXT -1 41 " has the same value \+
at a negative number " }{XPPEDIT 18 0 "-a" "6#,$%\"aG!\"\"" }{TEXT -1 
48 " as it has at the corresponding positive number " }{TEXT 293 1 "a
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 53 "A simple example of a
n even function is the function " }{XPPEDIT 18 0 "f(x) = x^2;" "6#/-%
\"fG6#%\"xG*$F'\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 33 "T
he graph of an even function is " }{TEXT 259 22 "symmetrical about the
 " }{TEXT 294 1 "y" }{TEXT 259 5 " axis" }{TEXT -1 1 "." }}{PARA 256 "
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"" {TEXT -1 15 "Given a point P" }{XPPEDIT 18 0 "``(a,f(a))" "6#-%!G6$
%\"aG-%\"fG6#F&" }{TEXT -1 47 " on the graph, where a is positive, the
 point Q" }{XPPEDIT 18 0 "``(-a,f(a))" "6#-%!G6$,$%\"aG!\"\"-%\"fG6#F'
" }{TEXT -1 31 " is also on the graph, and the " }{TEXT 298 1 "y" }
{TEXT -1 93 " axis is a right-angle bisector of the line QP, that is, \+
QM = MP and angle QMO is 90 degrees." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 296 1 "a" }{TEXT -1 39 " is \+
a positive number, the value of an " }{TEXT 259 12 "odd function" }
{TEXT -1 24 " at the negative number " }{XPPEDIT 18 0 "-a" "6#,$%\"aG!
\"\"" }{TEXT -1 67 " is the negative of its value at the corresponding
 positive number " }{TEXT 295 1 "a" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 52 "A simple example of an odd function is the function " }
{XPPEDIT 18 0 "f(x) = x^3;" "6#/-%\"fG6#%\"xG*$F'\"\"$" }{TEXT -1 1 ".
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etrical about the the origin in the sense indicated by the following p
icture." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 393 475 475 
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n a point P" }{XPPEDIT 18 0 "``(a,f(a))" "6#-%!G6$%\"aG-%\"fG6#F&" }
{TEXT -1 21 " on the graph, where " }{TEXT 299 1 "a" }{TEXT -1 25 " is
 positive, the point Q" }{XPPEDIT 18 0 "``(-a,-f(a))" "6#-%!G6$,$%\"aG
!\"\",$-%\"fG6#F'F(" }{TEXT -1 91 " is also on the graph, and the line
 QP has the origin O as its mid-point, that is, QO = OP." }}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 26 "Examples of even functions" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 282 9 "Example 1" }{TEXT 
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" " }{XPPEDIT 18 0 "f(x) = x^2;" "6#/-%\"fG6#%\"xG*$F'\"\"#" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "plot(x^2,x=-2..2,labels=[x,y]);" }}{PARA 13 "" 1 "" 
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\"17DT^j^8?F07$$!1LL$3WDTL\"F0$\"1O(y=p!*)z<F07$$!1++]d(Q&\\7F0$\"1WY
\\1rMh:F07$$!1nmmc4`i6F0$\"1:$3_Ay9N\"F07$$!1LLLQW*e3\"F0$\"1lK?Jn;z6F
07$$!1,+++()>'***!#;$\"1r\"))\\%))R#***Fbo7$$!1++++0\"*H\"*Fbo$\"1D5!Q
dEbL)Fbo7$$!1++++83&H)Fbo$\"1p4Oxt$3)oFbo7$$!1LLL3k(p`(Fbo$\"1Bt(zL,1o
&Fbo7$$!1nmmmj^NmFbo$\"1@,BXx+.WFbo7$$!1ommm9'=(eFbo$\"1$[s$3d(yW$Fbo7
$$!1,++v#\\N)\\Fbo$\"1.`jPjd$[#Fbo7$$!1pmmmCC(>%Fbo$\"15!*RKWoh<Fbo7$$
!1*****\\FRXL$Fbo$\"1_<l<_\">6\"Fbo7$$!1+++D=/8DFbo$\"1H$*>9#z`J'!#<7$
$!1mmm;a*el\"Fbo$\"1sw$4j*)>u#Fbr7$$!1pmm;Wn(o)Fbr$\"1a/+x'ova(!#=7$$!
1qLLL$eV(>F]s$\"1CHSG34)*Q!#@7$$\"1Mmm;f`@')Fbr$\"1xsBc\")3LuF]s7$$\"1
)****\\nZ)H;Fbo$\"1**GqVMScEFbr7$$\"1lmm;$y*eCFbo$\"1Bo$=Oul/'Fbr7$$\"
1*******R^bJ$Fbo$\"1&>/'3\")G*4\"Fbo7$$\"1'*****\\5a`TFbo$\"12NSD.>D<F
bo7$$\"1(****\\7RV'\\Fbo$\"1u0![HmWY#Fbo7$$\"1'*****\\@fkeFbo$\"1eTe3T
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\"13Cjqg!*=%)Fbo7$$\"1HLLeg`!)**Fbo$\"1z\"p6+56'**Fbo7$$\"1++]#G2A3\"F
0$\"1MgH-E<r6F07$$\"1LLL$)G[k6F0$\"18jd&Q?gN\"F07$$\"1++]7yh]7F0$\"1&G
UH\"\\/k:F07$$\"1nmm')fdL8F0$\"1uR@7\\Uy<F07$$\"1nmm,FT=9F0$\"1K$\\Af%
*=,#F07$$\"1LL$e#pa-:F0$\"1+7LkskdAF07$$\"1+++Sv&)z:F0$\"1^[pY)\\f\\#F
07$$\"1LLLGUYo;F0$\"1XzA\")Gx$y#F07$$\"1nmm1^rZ<F0$\"1y32%43X0$F07$$\"
1++]sI@K=F0$\"1\"*)RIu/qN$F07$$\"1++]2%)38>F0$\"1')3\"\\D2*fOF07$$\"\"
#F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"F*F*-%+AXESLABELSG6$%\"xG%\"yG-%%VIEW
G6$;F(Fhz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT 283 9 "Example 2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = \+
x^4-x^2;" "6#/-%\"fG6#%\"xG,&*$F'\"\"%\"\"\"*$F'\"\"#!\"\"" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 55 "plot(x^4-x^2,x=-1.2..1.2,tickmarks=[3,2],labels=[x,y]
);" }}{PARA 13 "" 1 "" {GLPLOT2D 282 286 286 {PLOTDATA 2 "6&-%'CURVESG
6$7fo7$$!1+++++++7!#:$\"1***********fL'!#;7$$!1++]2<#p=\"F*$\"1m3m]z')
edF-7$$!1+++:M%Q<\"F*$\"1D*zr$4B2_F-7$$!1++]A^wg6F*$\"148^a_X!o%F-7$$!
1+++IooZ6F*$\"1p^()=U\"z<%F-7$$!1++DD\"G\\7\"F*$\"1\")*=rVS$fLF-7$$!1+
+]?%p@5\"F*$\"10tglx14EF-7$$!1++vw[dw5F*$\"17')ztu)H%=F-7$$!1+++L.)40
\"F*$\"1?)e#HQ#\\:\"F-7$$!1+++Kh@D5F*$\"1.S*=*pgn`!#<7$$!1+++5$>X***F-
$!1)4\"Ggnj%4\"!#=7$$!1++]F=5Q(*F-$!1-E2Y'[@!\\Fen7$$!1+++XVo\"[*F-$!1
!o)z1U.y!*Fen7$$!1+++:D&RC*F-$!1]qIS3DV7F-7$$!1*****\\o?i+*F-$!1$pg)pF
/K:F-7$$!1,++0c!R^)F-$!1Xu%3L`V*>F-7$$!1+++D\"H$f#)F-$!1'Q3n[e\"o@F-7$
$!1+++XEv/!)F-$!1hgr'Rk=I#F-7$$!1*****\\f#*4v(F-$!1ZPe4iV)R#F-7$$!1+++
XDB(\\(F-$!1*f\"[zXXhCF-7$$!1++]U\"4iB(F-$!1Gx=TvT%\\#F-7$$!1******Rd=
vpF-$!1,p9th=)\\#F-7$$!1+++&=w_u'F-$!1$)oub(Q(zCF-7$$!1+++ImO:lF-$!1!4
#3Nv*HW#F-7$$!1*****\\#HaciF-$!1eVA:W:#Q#F-7$$!1+++?#>x*fF-$!1iwxaIB.B
F-7$$!1,+++j%zZ&F-$!16eGkdJ+@F-7$$!1+++!y[q(\\F-$!1bDf$H)\\j=F-7$$!1,+
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F-$!1.^=f(or3\"F-7$$!1+++l&H,*HF-$!1y\\#3R#[T\")Fen7$$!1-++![X$=DF-$!1
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%=A#Fen7$$!11+++DPN**Fen$!1x3*)=Gst(*F[o7$$!1(*****\\Yg7_Fen$!13f&p&>u
4FF[o7$$!16,++]h%=\"F[o$!1M(Q*G2J.9!#@7$$\"1)*****\\:#H<&Fen$!1i_iK7vo
EF[o7$$\"1(*****\\g3z(*Fen$!1nj<E/gr%*F[o7$$\"1,++!*pQv9F-$!1D,aMOQH@F
en7$$\"1******R3L*)>F-$!1s*p(*4C3!QFen7$$\"1******HY7#\\#F-$!1:OYo!f\\
#eFen7$$\"1*****\\Z.'yHF-$!1-Bcj3%\\3)Fen7$$\"1+++!Hb(=NF-$!10>%***)e[
3\"F-7$$\"1(*****>d5/SF-$!1.,+yNBY8F-7$$\"1********3KAXF-$!1)RU(H%zoi
\"F-7$$\"1(****\\(3!>*\\F-$!1$yH$or%4(=F-7$$\"1+++?eF0bF-$!11]1svA7@F-
7$$\"1)****\\j@$))fF-$!1;!**pGg+I#F-7$$\"1*****\\m#ySiF-$!1-+)=LRyP#F-
7$$\"1*****\\pVK\\'F-$!1vOk0\"p&QCF-7$$\"1)***\\(\\q+u'F-$!14MuZ=5zCF-
7$$\"1(******H(*o)pF-$!1G5H\"))*f)\\#F-7$$\"1(***\\(3-`C(F-$!1M0(p&zx$
\\#F-7$$\"1)****\\(oq.vF-$!1(>Sf>R-Y#F-7$$\"1****\\(R\"e_xF-$!11e\\B\"
RzR#F-7$$\"1+++?fX,!)F-$!1vbuXrM.BF-7$$\"1+++5iZ5&)F-$!1Y,31c(p*>F-7$$
\"1,++b:G:!*F-$!1!4)3:b&=_\"F-7$$\"1++](R8sC*F-$!1mA.J?(*Q7F-7$$\"1+++
S_9z%*F-$!1:YpA=V;\"*Fen7$$\"1+++0`'\\u*F-$!1Lz#QPt?y%Fen7$$\"1+++P&y5
+\"F*$\"1>M0I_*G;#F[o7$$\"1++]+Q&[-\"F*$\"1)*ztY\\z&G&Fen7$$\"1+++k!H'
[5F*$\"1;dK)Qwa4\"F-7$$\"1++v`%yR2\"F*$\"1(\\#GyRqp<F-7$$\"1++]VyK*4\"
F*$\"1Us!**4I+_#F-7$$\"1+++W/fB6F*$\"1!)))**yOQ8LF-7$$\"1++]WI&y9\"F*$
\"1[nxY;:%=%F-7$$\"1+]P$y*)3;\"F*$\"1hFaL5O&o%F-7$$\"1++DAl#R<\"F*$\"1
BNf4sl5_F-7$$\"1+]7hK'p=\"F*$\"1mxUf4mgdF-7$$\"1+++++++7F*F+-%'COLOURG
6&%$RGBG$\"#5!\"\"\"\"!F]cl-%+AXESLABELSG6$%\"xG%\"yG-%*AXESTICKSG6$\"
\"$\"\"#-%%VIEWG6$;$!#7F\\cl$\"#7F\\cl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 
4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 302 9 "Example 3" }{TEXT -1 1 " \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "f(x) = sqrt(1-x^2);" "6#/-%\"fG6#%\"xG-%%sqrtG6#,&\"\"
\"F,*$F'\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "plot(sqrt(1-x^2),x=-1..1,sca
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" }}{PARA 13 "" 1 "" {GLPLOT2D 229 148 148 {PLOTDATA 2 "6'-%'CURVESG6$
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\"1yD,?$RD/\"F.7$$!1,](=ng#=**F.$\"1*\\#GuA(fF\"F.7$$!1nm;HU,\"*)*F.$
\"1cuB4yNs9F.7$$!1++vV8_O)*F.$\"1Npy'*3z+=F.7$$!1MLLe%G?y*F.$\"1B8`Ih^
w?F.7$$!1++](oUIn*F.$\"1TQYJr=ODF.7$$!1nmm;p0k&*F.$\"1%G&e[NT?HF.7$$!1
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\\iId9(*)F.$\"1'pC*=pM<WF.7$$!1nmmT%p\"e()F.$\"1uy!yHMk#[F.7$$!1mmm\"4
m(G$)F.$\"1*H-C>(eMbF.7$$!1ML$3i.9!zF.$\"18**H-%f#HhF.7$$!1nm;/R=0vF.$
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,]uF.7$$!1++](y$pZiF.$\"1R7%oQ$43yF.7$$!1LLL$yaE\"eF.$\"1%Qd]\")Rr8)F.
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)[oPF.$\"1y-$=zZFE*F.7$$!1MLL$=exJ$F.$\"1IF>)f#eL%*F.7$$!1MLLL2$f$HF.$
\"1V7=H\\If&*F.7$$!1++]PYx\"\\#F.$\"1j56>%yXo*F.7$$!1MLLL7i)4#F.$\"19>
'>!*4tx*F.7$$!1++]P'psm\"F.$\"1Fh>K5.g)*F.7$$!1++]74_c7F.$\"1pBi\"oV2#
**F.7$$!1JLL$3x%z#)F1$\"1\"o%3*=mc'**F.7$$!1MLL3s$QM%F1$\"1du\\%3h0***
F.7$$!1^omm;zr)*!#>$\"0G&QF^******!#:7$$\"1<LLezw5VF1$\"1R2K>Vq!***F.7
$$\"1!****\\PQ#\\\")F1$\"1_4RW'Rn'**F.7$$\"1KLLe\"*[H7F.$\"1'Q!p2+8C**
F.7$$\"1*******pvxl\"F.$\"1s:1(pJ;')*F.7$$\"1)****\\_qn2#F.$\"1GKN_W(>
y*F.7$$\"1)***\\i&p@[#F.$\"1Wv$4mWqo*F.7$$\"1)****\\2'HKHF.$\"1\\tky/U
g&*F.7$$\"1lmmmZvOLF.$\"1c!ez0!)oU*F.7$$\"1+++]2goPF.$\"1D!*\\-?qi#*F.
7$$\"1KL$eR<*fTF.$\"1ILE@Ro$4*F.7$$\"1+++])Hxe%F.$\"1\"f%eX^`&)))F.7$$
\"1lm;H!o-*\\F.$\"1DIn2b'em)F.7$$\"1****\\7k.6aF.$\"1(=*zS\"f&4%)F.7$$
\"1mmm;WTAeF.$\"1&p3sIf,8)F.7$$\"1****\\i!*3`iF.$\"1c[JlKx.yF.7$$\"1ML
LL*zym'F.$\"15V`oXZ_uF.7$$\"1LLL3N1#4(F.$\"1%)Q?)e4+0(F.7$$\"1mm;HYt7v
F.$\"1_Th$[5**f'F.7$$\"1*******p(G**yF.$\"1qyiCh)>8'F.7$$\"1mmmT6KU$)F
.$\"1@,y!QMT^&F.7$$\"1LLLLbdQ()F.$\"1')*oZS?='[F.7$$\"1nm\"zW?)\\*)F.$
\"1e/FE;-hWF.7$$\"1++]i`1h\"*F.$\"1qj$>D\\$4SF.7$$\"1++++PDj$*F.$\"1\"
)RQ\\fL6NF.7$$\"1++]P?Wl&*F.$\"1\"G)fhN(e\"HF.7$$\"1+]7G:3u'*F.$\"1H;B
q;AKDF.7$$\"1++v=5s#y*F.$\"1vO>U4Dt?F.7$$\"1+D1k2/P)*F.$\"1MW(eL^zz\"F
.7$$\"1+]P40O\"*)*F.$\"1H.f%oH+Z\"F.7$$\"1^7.#Q?&=**F.$\"1#>z^*=&RF\"F
.7$$\"1+voa-oX**F.$\"1P%\\8H')3/\"F.7$$\"1\\PMF,%G(**F.$\"1&=hJ*p=ltF1
7$$\"\"\"F*F*-%'COLOURG6&%$RGBG$\"#5F)F*F*-%(SCALINGG6#%,CONSTRAINEDG-
%+AXESLABELSG6$%\"xG%\"yG-%*AXESTICKSG6$\"\"#Fcal-%%VIEWG6$;F(F_`l%(DE
FAULTG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve
 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT 284 9 "Example 4" }{TEXT -1 1 " " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
f(x) = abs(x);" "6#/-%\"fG6#%\"xG-%$absG6#F'" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "pl
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F0$\"1LLL=Kvl;F07$$!1nm;C2G!e\"F0$\"1nm;C2G!e\"F07$$!1LL$3yO5]\"F0$\"1
LL$3yO5]\"F07$$!1++]nU)*=9F0$\"1++]nU)*=9F07$$!1LL$3WDTL\"F0$\"1LL$3WD
TL\"F07$$!1++]d(Q&\\7F0$\"1++]d(Q&\\7F07$$!1nmmc4`i6F0$\"1nmmc4`i6F07$
$!1LLLQW*e3\"F0$\"1LLLQW*e3\"F07$$!1,+++()>'***!#;$\"1,+++()>'***Fbo7$
$!1++++0\"*H\"*Fbo$\"1++++0\"*H\"*Fbo7$$!1++++83&H)Fbo$\"1++++83&H)Fbo
7$$!1LLL3k(p`(Fbo$\"1LLL3k(p`(Fbo7$$!1nmmmj^NmFbo$\"1nmmmj^NmFbo7$$!1o
mmm9'=(eFbo$\"1ommm9'=(eFbo7$$!1,++v#\\N)\\Fbo$\"1,++v#\\N)\\Fbo7$$!1p
mmmCC(>%Fbo$\"1pmmmCC(>%Fbo7$$!1*****\\FRXL$Fbo$\"1*****\\FRXL$Fbo7$$!
1+++D=/8DFbo$\"1+++D=/8DFbo7$$!1mmm;a*el\"Fbo$\"1mmm;a*el\"Fbo7$$!1pmm
;Wn(o)!#<$\"1pmm;Wn(o)Fjr7$$!1.++D^bUWFjr$\"1.++D^bUWFjr7$$!1qLLL$eV(>
!#=$\"1qLLL$eV(>Fes7$$\"1[mmT+07UFjrFis7$$\"1Mmm;f`@')FjrF\\t7$$\"1)**
**\\nZ)H;FboF_t7$$\"1lmm;$y*eCFboFbt7$$\"1*******R^bJ$FboFet7$$\"1'***
**\\5a`TFboFht7$$\"1(****\\7RV'\\FboF[u7$$\"1'*****\\@fkeFboF^u7$$\"1J
LLL&4Nn'FboFau7$$\"1*******\\,s`(FboFdu7$$\"1lmm\"zM)>$)FboFgu7$$\"1**
*****pfa<*FboFju7$$\"1HLLeg`!)**FboF]v7$$\"1++]#G2A3\"F0F`v7$$\"1LLL$)
G[k6F0Fcv7$$\"1++]7yh]7F0Ffv7$$\"1nmm')fdL8F0Fiv7$$\"1nmm,FT=9F0F\\w7$
$\"1LL$e#pa-:F0F_w7$$\"1+++Sv&)z:F0Fbw7$$\"1LLLGUYo;F0Few7$$\"1nmm1^rZ
<F0Fhw7$$\"1++]sI@K=F0F[x7$$\"1++]2%)38>F0F^x7$F+F+-%'COLOURG6&%$RGBG$
\"#5!\"\"F*F*-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6$%\"xG%\"yG-%*A
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1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT 
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{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = \+
cos*x;" "6#/-%\"fG6#%\"xG*&%$cosG\"\"\"F'F*" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 44 "Note that the Maclaurin series expansion of " }
{XPPEDIT 18 0 "cos*x" "6#*&%$cosG\"\"\"%\"xGF%" }{TEXT -1 30 " only in
volves even powers of " }{TEXT 300 1 "x" }{TEXT -1 1 ":" }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*x = 1-x^2/2!+x^4/4!-x^6/6!+x
^8/8!-x^10/10!+` . . . `;" "6#/*&%$cosG\"\"\"%\"xGF&,0F&F&*&F'\"\"#-%*
factorialG6#F*!\"\"F.*&F'\"\"%-F,6#F0F.F&*&F'\"\"'-F,6#F4F.F.*&F'\"\")
-F,6#F8F.F&*&F'\"#5-F,6#F<F.F.%(~.~.~.~GF&" }{TEXT -1 1 " " }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot
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F[_l7$$\"1lmmmiG'*\\F-F^_l7$$\"196662()4hF-Fa_l7$$\"1wxxxwsAsF-Fd_l7$$
\"1TWWW1\\J$)F-Fg_l7$$\"1RWW%pxS'))F-Fj_l7$$\"1OWWWZm'R*F-F]`l7$$\"1;A
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LLLveO=F*$\"1jmmmY7M;F-7$$\"1yxxFh!G%>F*$\"1GAAAsQ>dFbq7$$\"1yxx(\\)e^
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$!1YbbbrX/GF-7$$\"1LLL$[h;R#F*$!1MLLL[h;RF-7$$\"1WWWa]w/DF*$!1WWWW0lZ]
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\"1mmmE3)>;%F*$\"1cmmm#3)>;F-7$$\"1+++!H\"GzUF*$\"1,+++H\"Gz#F-7$$\"1A
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\"1******>Q/3YF*$\"1*)*****>Q/3'F-7$$\"1******4J^>ZF*$\"1%******4J^>(F
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)F-7$$\"1mmm'y0g%\\F*$\"1cmmmy0g%*F-7$$\"1ab0V9us\\F*$\"1TbbIWTF(*F-7$
$\"1WWW*4x%**\\F*$\"1NWW%*4x%***F-7$$\"1LL$ev7i-&F*$\"1rmmTC(yt*F-7$$
\"1@AA7%[H0&F*$\"1&yxx(e^q%*F-7$$\"1bbbl$3,6&F*$\"1]WWWj\"*)*))F-7$$\"
1*)))))=$os;&F*$\"19666oJF$)F-7$$\"1566h$R8G&F*$\"1,*))))Q1m=(F-7$$\"1
cbb&*[s$Q&F*$\"1TWWW5vihF-7$$\"1+++++++bF*$\"1+++++++]F--%'COLOURG6&%$
RGBG$\"#5!\"\"\"\"!Feam-%+AXESLABELSG6$%\"xG%\"yG-%*AXESTICKSG6$%(DEFA
ULTG\"\"$-%%VIEWG6$;$!#bFdam$\"#bFdam;$!#7Fdam$\"#7Fdam" 1 2 0 1 10 0 
2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 36 "The even and odd parts of a function" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 17 "Given a f
unction " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 6 ", th
e " }{TEXT 259 9 "even part" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "phi(x)
" "6#-%$phiG6#%\"xG" }{TEXT -1 17 " is the function " }{XPPEDIT 18 0 "
phi[even](x);" "6#-&%$phiG6#%%evenG6#%\"xG" }{TEXT -1 11 " defined by
" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "phi[even](x) = (
phi(x)+phi(-x))/2;" "6#/-&%$phiG6#%%evenG6#%\"xG*&,&-F&6#F*\"\"\"-F&6#
,$F*!\"\"F/F/\"\"#F3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {XPPEDIT 18 
0 "phi[even](x)" "6#-&%$phiG6#%%evenG6#%\"xG" }{TEXT -1 7 " is an " }
{TEXT 259 13 "even function" }{TEXT -1 8 " because" }}{PARA 256 "" 0 "
" {TEXT -1 2 "  " }{XPPEDIT 18 0 "phi[even](-x) = (phi(-x)+phi(-(-x)))
/2;" "6#/-&%$phiG6#%%evenG6#,$%\"xG!\"\"*&,&-F&6#,$F+F,\"\"\"-F&6#,$,$
F+F,F,F2F2\"\"#F," }{TEXT -1 3 " = " }{XPPEDIT 18 0 "(phi(-x)+phi(x))/
2 = phi[even](x);" "6#/*&,&-%$phiG6#,$%\"xG!\"\"\"\"\"-F'6#F*F,F,\"\"#
F+-&F'6#%%evenG6#F*" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 15 "Similarly, the " }{TEXT 259 8 "odd part
" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }
{TEXT -1 17 " is the function " }{XPPEDIT 18 0 "phi[odd](x)" "6#-&%$ph
iG6#%$oddG6#%\"xG" }{TEXT -1 11 " defined by" }}{PARA 256 "" 0 "" 
{TEXT -1 2 "  " }{XPPEDIT 18 0 "phi[odd](x) = (phi(x)-phi(-x))/2;" "6#
/-&%$phiG6#%$oddG6#%\"xG*&,&-F&6#F*\"\"\"-F&6#,$F*!\"\"F3F/\"\"#F3" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 7 "h is an" }{TEXT 256 1 " \+
" }{TEXT 259 12 "odd function" }{TEXT -1 8 " because" }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi[odd](-x) = (phi(-x)-phi(-(-x))
)/2;" "6#/-&%$phiG6#%$oddG6#,$%\"xG!\"\"*&,&-F&6#,$F+F,\"\"\"-F&6#,$,$
F+F,F,F,F2\"\"#F," }{TEXT -1 3 " = " }{XPPEDIT 18 0 "(phi(-x)-phi(x))/
2 = -(phi(x)-phi(-x))/2;" "6#/*&,&-%$phiG6#,$%\"xG!\"\"\"\"\"-F'6#F*F+
F,\"\"#F+,$*&,&-F'6#F*F,-F'6#,$F*F+F+F,F/F+F+" }{TEXT -1 3 " = " }
{XPPEDIT 18 0 "-phi[odd](x);" "6#,$-&%$phiG6#%$oddG6#%\"xG!\"\"" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 9 "In fact, " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }
{TEXT -1 18 " breaks up as the " }{TEXT 259 3 "sum" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "phi(x) = phi[even](x)+phi[odd](x);" "6#/-%$phiG6#%\"xG,
&-&F%6#%%evenG6#F'\"\"\"-&F%6#%$oddG6#F'F." }{TEXT -1 1 "." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}
}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }
{XPPEDIT 18 0 "g(x) = sqrt(1+x);" "6#/-%\"gG6#%\"xG-%%sqrtG6#,&\"\"\"F
,F'F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 262 9 "Even part" }
{TEXT -1 1 ":" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "g[e
ven](x) = 1/2;" "6#/-&%\"gG6#%%evenG6#%\"xG*&\"\"\"F,\"\"#!\"\"" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "``(sqrt(1+x)+sqrt(1-x));" "6#-%!G6#,&-%
%sqrtG6#,&\"\"\"F+%\"xGF+F+-F(6#,&F+F+F,!\"\"F+" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT 263 8 "Odd part" }{TEXT -1 1 ":" }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g[odd](x) = 1/2;" "6#/-&%\"gG6#%$o
ddG6#%\"xG*&\"\"\"F,\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(sqr
t(1+x)-sqrt(1-x));" "6#-%!G6#,&-%%sqrtG6#,&\"\"\"F+%\"xGF+F+-F(6#,&F+F
+F,!\"\"F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "g := x -> sqrt(1+x);\ng_even := x \+
-> (g(x)+g(-x))/2;\ng_even(x);\ng_odd := x -> (g(x)-g(-x))/2;\ng_odd(x
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%
&arrowGF(-%%sqrtG6#,&\"\"\"F09$F0F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'g_evenGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&#\"\"\"\"\"#F/-
%\"gG6#9$F/F/*&F.F/-F26#,$F4!\"\"F/F/F(F(F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&*&\"\"#!\"\",&\"\"\"F(%\"xGF(#F(F%F(*&F%F&,&F(F(F)F&F
*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&g_oddGf*6#%\"xG6\"6$%)operat
orG%&arrowGF(,&*&#\"\"\"\"\"#F/-%\"gG6#9$F/F/*&#F/F0F/-F26#,$F4!\"\"F/
F:F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"#!\"\",&\"\"\"F(%
\"xGF(#F(F%F(*&F%F&,&F(F(F)F&F*F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 4 "For " }{XPPEDIT 18 0 "-1 < x" "6#2,$\"\"\"!\"\"%\"xG" }
{XPPEDIT 18 0 "``<1" "6#2%!G\"\"\"" }{TEXT -1 42 ", we have the Maclau
rin series expansions:" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 
18 0 "sqrt(1+x) = 1+x/2-x^2/8+x^3/16-5*x^4/128+7*x^5/256-21*x^6/1024+3
3*x^7/2048-429*x^8/32768+` . . . `;" "6#/-%%sqrtG6#,&\"\"\"F(%\"xGF(,6
F(F(*&F)F(\"\"#!\"\"F(*&F)F,\"\")F-F-*&F)\"\"$\"#;F-F(*(\"\"&F(*$F)\"
\"%F(\"$G\"F-F-*(\"\"(F(*$F)F4F(\"$c#F-F(*(\"#@F(*$F)\"\"'F(\"%C5F-F-*
(\"#LF(*$F)F9F(\"%[?F-F(*(\"$H%F(*$F)F/F(\"&oF$F-F-%(~.~.~.~GF(" }
{TEXT -1 12 " ------- (i)" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "sqrt(1-x) = 1-x/2-x^2/8-x^3/16-5*x^4/128-7*x^5/256-21*x
^6/1024-33*x^7/2048-429*x^8/32768+` . . . `;" "6#/-%%sqrtG6#,&\"\"\"F(
%\"xG!\"\",6F(F(*&F)F(\"\"#F*F**&F)F-\"\")F*F**&F)\"\"$\"#;F*F**(\"\"&
F(*$F)\"\"%F(\"$G\"F*F**(\"\"(F(*$F)F4F(\"$c#F*F**(\"#@F(*$F)\"\"'F(\"
%C5F*F**(\"#LF(*$F)F9F(\"%[?F*F**(\"$H%F(*$F)F/F(\"&oF$F*F*%(~.~.~.~GF
(" }{TEXT -1 13 " ------- (ii)" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{XPPEDIT 18 0 "``(sqrt(1
+x)+sqrt(1-x)) = 1-x^2/8-5*x^4/128-21*x^6/1024-429*x^8/32768-` . . . `
;" "6#/-%!G6#,&-%%sqrtG6#,&\"\"\"F,%\"xGF,F,-F)6#,&F,F,F-!\"\"F,,.F,F,
*&F-\"\"#\"\")F1F1*(\"\"&F,*$F-\"\"%F,\"$G\"F1F1*(\"#@F,*$F-\"\"'F,\"%
C5F1F1*(\"$H%F,*$F-F5F,\"&oF$F1F1%(~.~.~.~GF1" }{TEXT -1 15 "  -------
 (iii)" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "1/2" "6#*&
\"\"\"F$\"\"#!\"\"" }{XPPEDIT 18 0 "``(sqrt(1+x)-sqrt(1-x)) = x/2+x^3/
16+7*x^5/256+33*x^7/2048+` . . . `;" "6#/-%!G6#,&-%%sqrtG6#,&\"\"\"F,%
\"xGF,F,-F)6#,&F,F,F-!\"\"F1,,*&F-F,\"\"#F1F,*&F-\"\"$\"#;F1F,*(\"\"(F
,*$F-\"\"&F,\"$c#F1F,*(\"#LF,*$F-F9F,\"%[?F1F,%(~.~.~.~GF," }{TEXT -1 
13 " ------- (iv)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 70 "Adding equations (i) and (ii), and dividing by 2 gives eq
uation (iii)." }}{PARA 0 "" 0 "" {TEXT -1 74 "Subtracting equation (ii
) from equation (i), and dividing by 2 gives (iv)." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "g := x -> s
qrt(1+x):\ng_even := x -> (g(x)+g(-x))/2:\ng_odd := x -> (g(x)-g(-x))/
2:\ng(x)=taylor(g(x),x,9);\ng_even(x)=taylor(g_even(x),x,9);\ng_odd(x)
=taylor(g_odd(x),x,9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$,&\"\"\"F
&%\"xGF&#F&\"\"#+7F'F&\"\"!F(F&#!\"\"\"\")F)#F&\"#;\"\"$#!\"&\"$G\"\"
\"%#\"\"(\"$c#\"\"&#!#@\"%C5\"\"'#\"#L\"%[?F7#!$H%\"&oF$F.-%\"OG6#F&\"
\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&\"\"#!\"\",&\"\"\"F)%\"xGF
)#F)F&F)*&F&F',&F)F)F*F'F+F)+/F*F)\"\"!#F'\"\")F&#!\"&\"$G\"\"\"%#!#@
\"%C5\"\"'#!$H%\"&oF$F1-%\"OG6#F)\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/,&*&\"\"#!\"\",&\"\"\"F)%\"xGF)#F)F&F)*&F&F',&F)F)F*F'F+F'+-F*F
+F)#F)\"#;\"\"$#\"\"(\"$c#\"\"&#\"#L\"%[?F3-%\"OG6#F)\"\"*" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The Maclaurin s
eries (iii) for the even part of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%
\"xG" }{TEXT -1 59 " consists of the terms of (i) which involve even p
owers of " }{TEXT 271 1 "x" }{TEXT -1 52 ", and the Maclaurin series (
iv) for the odd part of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }
{TEXT -1 58 " consists of the terms of (i) which involve odd powers of
 " }{TEXT 272 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "g := x -> sqrt(1+x):\ng_ev
en := x -> (g(x)+g(-x))/2:\ng_odd := x -> (g(x)-g(-x))/2:\nplot([g(x),
g_even(x),g_odd(x)],x=-1..1,y,\n          color=[brown,red,blue],numpo
ints=75);" }}{PARA 13 "" 1 "" {GLPLOT2D 414 358 358 {PLOTDATA 2 "6'-%'
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{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The graph
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\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(sqrt(1+x)-sqrt(1-x));" "6#-%!
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 can be constructed by adding ordinates along the graphs" }}{PARA 0 "
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18 0 "y = -sqrt(1-x)/2;" "6#/%\"yG,$*&-%%sqrtG6#,&\"\"\"F+%\"xG!\"\"F+
\"\"#F-F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
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F`^l$\"3[>R89^52CF/7$Fe^l$\"3Mh&ero7?b#F/7$Fj^l$\"3YW+L&z,>p#F/7$F__l$
\"3O$Q7Iy47%GF/7$Fd_l$\"3#>,Mx'pSyHF/7$Fi_l$\"39Bvn?/aKJF/7$F^`l$\"3%[
@.>_@HG$F/7$Fc`l$\"3sY(pOvSfV$F/7$Fh`l$\"3cM.+\"=#Q*f$F/7$F]al$\"3Cf%z
%[lh`PF/7$Fbal$\"3w9J&*3tw:RF/7$Fgal$\"33_YFj7\\+TF/7$F\\bl$\"3-(R2\")
HlPF%F/7$Fabl$\"3)*es(G**)*zX%F/7$Ffbl$\"3-F*p!p$[Tl%F/7$F[cl$\"3M3goW
)eS%[F/7$F`cl$\"3#*p3#HWS$f]F/7$Fecl$\"3#QhZQ**[-H&F/7$Fjcl$\"3=Lm$3jN
Vc&F/7$F_dl$\"3C*H%3-R`UeF/7$Fddl$\"3Ulxwh\\5ZiF/Fhdl-F^el6&F`elF+F+Fa
el-Fgel6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q\"yF]en-%%VIEWG6$;F(Fidl%(DEFAUL
TG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" 
"Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 
"Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "
" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "g(x) = ln(1+x);" "6#/-%\"gG6#
%\"xG-%#lnG6#,&\"\"\"F,F'F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
264 9 "Even part" }{TEXT -1 1 ":" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " 
}{XPPEDIT 18 0 "g[even](x) = 1/2;" "6#/-&%\"gG6#%%evenG6#%\"xG*&\"\"\"
F,\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(ln(1+x)+ln(1-x));" "6
#-%!G6#,&-%#lnG6#,&\"\"\"F+%\"xGF+F+-F(6#,&F+F+F,!\"\"F+" }{TEXT -1 1 
" " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/2;" "6#/
%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(1-x^2)" "6
#-%#lnG6#,&\"\"\"F'*$%\"xG\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT 265 8 "Odd part" }{TEXT -1 1 ":" }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "g[odd](x) = 1/2;" "6#/-&%\"gG6#%$oddG6#%\"xG*&\"
\"\"F,\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(ln(1+x)-ln(1-x));
" "6#-%!G6#,&-%#lnG6#,&\"\"\"F+%\"xGF+F+-F(6#,&F+F+F,!\"\"F0" }{TEXT 
-1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/2;
" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln((1+x
)/(1-x))" "6#-%#lnG6#*&,&\"\"\"F(%\"xGF(F(,&F(F(F)!\"\"F+" }{TEXT -1 
2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 100 "g := x -> ln(1+x);\ng_even := x -> (g(x)+g(-x))/2;\n
g_even(x);\ng_odd := x -> (g(x)-g(-x))/2;\ng_odd(x);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%#lnG6#,&
\"\"\"F09$F0F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'g_evenGf*6#%
\"xG6\"6$%)operatorG%&arrowGF(,&*&#\"\"\"\"\"#F/-%\"gG6#9$F/F/*&F.F/-F
26#,$F4!\"\"F/F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"
\"\"#F&-%#lnG6#,&F&F&%\"xGF&F&F&*&F%F&-F)6#,&F&F&F,!\"\"F&F&" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%&g_oddGf*6#%\"xG6\"6$%)operatorG%&arrowGF(
,&*&#\"\"\"\"\"#F/-%\"gG6#9$F/F/*&#F/F0F/-F26#,$F4!\"\"F/F:F(F(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"#F&-%#lnG6#,&F&F&%\"xGF
&F&F&*&#F&F'F&-F)6#,&F&F&F,!\"\"F&F2" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 4 "For " }{XPPEDIT 18 0 "-1 < x" "6#2,$\"\"\"!\"\"%\"xG" }
{XPPEDIT 18 0 "``<1" "6#2%!G\"\"\"" }{TEXT -1 42 ", we have the Maclau
rin series expansions:" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 
18 0 "ln(1+x) = x-x^2/2+x^3/3-x^4/4+x^5/5-x^6/6+` . . . `;" "6#/-%#lnG
6#,&\"\"\"F(%\"xGF(,0F)F(*&F)\"\"#F,!\"\"F-*&F)\"\"$F/F-F(*&F)\"\"%F1F
-F-*&F)\"\"&F3F-F(*&F)\"\"'F5F-F-%(~.~.~.~GF(" }{TEXT -1 12 " ------- \+
(i)" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "ln(1-x) = -x-
x^2/2-x^3/3-x^4/4-x^5/5-x^6/6-` . . . `;" "6#/-%#lnG6#,&\"\"\"F(%\"xG!
\"\",0F)F**&F)\"\"#F-F*F**&F)\"\"$F/F*F**&F)\"\"%F1F*F**&F)\"\"&F3F*F*
*&F)\"\"'F5F*F*%(~.~.~.~GF*" }{TEXT -1 13 " ------- (ii)" }}{PARA 256 
"" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(1-x^2) = -x^2/2-x^4/4-x^6/6-x^8/8-` \+
. . . `;" "6#/-%#lnG6#,&\"\"\"F(*$%\"xG\"\"#!\"\",,*&F*F+F+F,F,*&F*\"
\"%F0F,F,*&F*\"\"'F2F,F,*&F*\"\")F4F,F,%(~.~.~.~GF," }{TEXT -1 14 " --
----- (iii)" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "1/2" 
"6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln((1+x)/(1-x
)) = x+x^3/3+x^5/5+x^7/7+` . . . `;" "6#/-%#lnG6#*&,&\"\"\"F)%\"xGF)F)
,&F)F)F*!\"\"F,,,F*F)*&F*\"\"$F/F,F)*&F*\"\"&F1F,F)*&F*\"\"(F3F,F)%(~.
~.~.~GF)" }{TEXT -1 13 " ------- (iv)" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 61 "Adding equations (i) and (ii), and di
viding by 2 gives (iii)." }}{PARA 0 "" 0 "" {TEXT -1 74 "Subtracting e
quation (ii) from equation (i), and dividing by 2 gives (iv)." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
166 "g := x -> ln(1+x);\ng_even := x -> (g(x)+g(-x))/2:\ng_odd := x ->
 (g(x)-g(-x))/2:\ng(x)=taylor(g(x),x,9);\ng_even(x)=taylor(g_even(x),x
,9);\ng_odd(x)=taylor(g_odd(x),x,9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%#lnG6#,&\"\"\"F09$F0F(F(F
(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#lnG6#,&\"\"\"F(%\"xGF(+5F)F(F
(#!\"\"\"\"#F-#F(\"\"$F/#F,\"\"%F1#F(\"\"&F3#F,\"\"'F5#F(\"\"(F7#F,\"
\")F9-%\"OG6#F(\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&#\"\"\"\"
\"#F'-%#lnG6#,&F'F'%\"xGF'F'F'*&F&F'-F*6#,&F'F'F-!\"\"F'F'+-F-#F2F(F(#
F2\"\"%F6#F2\"\"'F8#F2\"\")F:-%\"OG6#F'\"\"*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,&*&#\"\"\"\"\"#F'-%#lnG6#,&F'F'%\"xGF'F'F'*&#F'F(F'-F
*6#,&F'F'F-!\"\"F'F3+-F-F'F'#F'\"\"$F6#F'\"\"&F8#F'\"\"(F:-%\"OG6#F'\"
\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "T
he Maclaurin series (iii) for the even part of " }{XPPEDIT 18 0 "g(x)
" "6#-%\"gG6#%\"xG" }{TEXT -1 59 " consists of the terms of (i) which \+
involve even powers of " }{TEXT 273 1 "x" }{TEXT -1 52 ", and the Macl
aurin series (iv) for the odd part of " }{XPPEDIT 18 0 "g(x)" "6#-%\"g
G6#%\"xG" }{TEXT -1 58 " consists of the terms of (i) which involve od
d powers of " }{TEXT 274 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 244 "p1 := plot(
[ln(1+x),ln(1-x^2)/2,ln((1+x)/(1-x))/2],x=-1.1..1.1,\n   y=-5..5,color
=[brown,red,blue],discont=true):\nasym := plots[implicitplot](\{x=-1,x
=1\},x=-1.1..1.1,\n   y=-5..5,color=black,linestyle=2):\nplots[display
]([p1,asym],tickmarks=[3,5]);" }}{PARA 13 "" 1 "" {GLPLOT2D 317 377 
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{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The graph
 of the even part " }{XPPEDIT 18 0 "y = 1/2" "6#/%\"yG*&\"\"\"F&\"\"#!
\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(ln(1+x)+ln(1-x));" "6#-%!G6#,
&-%#lnG6#,&\"\"\"F+%\"xGF+F+-F(6#,&F+F+F,!\"\"F+" }{TEXT -1 56 " can b
e constructed by adding ordinates along the graphs" }}{PARA 0 "" 0 "" 
{TEXT -1 2 "  " }{XPPEDIT 18 0 "y = ln(1+x)/2;" "6#/%\"yG*&-%#lnG6#,&
\"\"\"F*%\"xGF*F*\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = l
n(1-x)/2;" "6#/%\"yG*&-%#lnG6#,&\"\"\"F*%\"xG!\"\"F*\"\"#F," }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 265 "p1 := plot([ln(1+x)/2,ln(1-x)/2,(ln(1+x)+ln(1-x))/2]
,\n   x=-1..1,y=-3..1,color=[coral,green,red],discont=true,\n   thickn
ess=[1,1,2]):\nasym := plots[implicitplot](\{x=-1,x=1\},x=-1.1..1.1,\n
   y=-3..1,color=black,linestyle=2):\nplots[display]([p1,asym],tickmar
ks=[3,5]);" }}{PARA 13 "" 1 "" {GLPLOT2D 309 369 369 {PLOTDATA 2 "6*-%
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\"&-%+AXESLABELSG6%Q\"x6\"Q\"yF`_p-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F__lF
i]l;FdgnFi]l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 
"Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The graph of the odd part
 " }{XPPEDIT 18 0 "y = 1/2" "6#/%\"yG*&\"\"\"F&\"\"#!\"\"" }{XPPEDIT 
18 0 "``(ln(1+x)-ln(1-x));" "6#-%!G6#,&-%#lnG6#,&\"\"\"F+%\"xGF+F+-F(6
#,&F+F+F,!\"\"F0" }{TEXT -1 56 " can be constructed by adding ordinate
s along the graphs" }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "
y = ln(1+x)/2;" "6#/%\"yG*&-%#lnG6#,&\"\"\"F*%\"xGF*F*\"\"#!\"\"" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = -ln(1-x)/2;" "6#/%\"yG,$*&-%#ln
G6#,&\"\"\"F+%\"xG!\"\"F+\"\"#F-F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 266 "p1 := plot(
[ln(1+x)/2,-ln(1-x)/2,(ln(1+x)-ln(1-x))/2],\n   x=-1..1,y=-3..3,color=
[coral,cyan,blue],\n   discont=true,thickness=[1,1,2]):\nasym := plots
[implicitplot](\{x=-1,x=1\},x=-1.1..1.1,\n   y=-3..3,color=black,lines
tyle=2):\nplots[display]([p1,asym],tickmarks=[3,5]);" }}{PARA 13 "" 1 
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h_oF\\[p7$F`[p7$Fi]l$\"3rMFFFFFFjF17$7$Fi]lF``oFb[p7$Ff[p7$Fi]l$\"3iMF
FFFFF()F17$7$Fi]lFh`oFh[p7$F\\\\p7$Fi]l$\"3Xtsssss76F*7$7$Fi]lF`aoF^\\
p7$Fb\\p7$Fi]l$\"3Wtsssss_8F*7$7$Fi]lFhaoFd\\p7$Fh\\p7$Fi]l$\"3Utsssss
#f\"F*7$7$Fi]lF`boFj\\p7$F^]p7$Fi]l$\"3UtsssssK=F*7$7$Fi]lF[coF`]p7$Fd
]p7$Fi]l$\"3ktssssss?F*7$7$Fi]lFccoFf]p7$Fj]p7$Fi]l$\"3&QFFFFFFJ#F*7$7
$Fi]lF[doF\\^p7$F`^p7$Fi]l$\"31usssss_DF*7$7$Fi]lFcdoFb^p7$Ff^p7$Fi]l$
\"3Gusssss#z#F*7$7$Fi]lF^eoFh^pF`eoFbeo-%*AXESTICKSG6$\"\"$\"\"&-%+AXE
SLABELSG6%Q\"x6\"Q\"yFf_p-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F__lFi]l;Fhgn$
F`_pF[^l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur
ve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 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" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 
0 "g(x) = 1/(1+x);" "6#/-%\"gG6#%\"xG*&\"\"\"F),&F)F)F'F)!\"\"" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 277 9 "Even part" }{TEXT -1 1 
":" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "g[even](x) = 1
/2;" "6#/-&%\"gG6#%%evenG6#%\"xG*&\"\"\"F,\"\"#!\"\"" }{XPPEDIT 18 0 "
``(1/(1+x)+1/(1-x))" "6#-%!G6#,&*&\"\"\"F(,&F(F(%\"xGF(!\"\"F(*&F(F(,&
F(F(F*F+F+F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = 1/(1-x^2);" "6#/%!G*&\"\"\"F&,&F&F&*$%\"xG\"\"#!\"
\"F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 278 8 "Odd part" }{TEXT 
-1 1 ":" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g[odd](x) \+
= 1/2;" "6#/-&%\"gG6#%$oddG6#%\"xG*&\"\"\"F,\"\"#!\"\"" }{XPPEDIT 18 
0 "``(1/(1+x)-1/(1-x));" "6#-%!G6#,&*&\"\"\"F(,&F(F(%\"xGF(!\"\"F(*&F(
F(,&F(F(F*F+F+F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = (-x)/(1-x^2);" "6#/%!G*&,$%\"xG!\"\"\"\"\",&F)F)*$
F'\"\"#F(F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "g := x -> 1/(1+x);\ng_even := x ->
 (g(x)+g(-x))/2;\nsimplify(g_even(x));\ng_odd := x -> (g(x)-g(-x))/2;
\nsimplify(g_odd(x));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%
\"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-,&F-F-9$F-!\"\"F(F(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'g_evenGf*6#%\"xG6\"6$%)operatorG%&a
rrowGF(,&*&#\"\"\"\"\"#F/-%\"gG6#9$F/F/*&F.F/-F26#,$F4!\"\"F/F/F(F(F(
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"#F&-%#lnG6#,&F&F&%
\"xGF&F&F&*&F%F&-F)6#,&F&F&F,!\"\"F&F&" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%&g_oddGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&#\"\"\"\"\"#F/-%
\"gG6#9$F/F/*&#F/F0F/-F26#,$F4!\"\"F/F:F(F(F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&*&#\"\"\"\"\"#F&-%#lnG6#,&F&F&%\"xGF&F&F&*&#F&F'F&-F)
6#,&F&F&F,!\"\"F&F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "For " }
{XPPEDIT 18 0 "-1 < x" "6#2,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "``<1" "
6#2%!G\"\"\"" }{TEXT -1 42 ", we have the Maclaurin series expansions:
" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "1/(1+x) = 1-x+x^
2-x^3+x^4-x^5+` . . . `;" "6#/*&\"\"\"F%,&F%F%%\"xGF%!\"\",0F%F%F'F(*$
F'\"\"#F%*$F'\"\"$F(*$F'\"\"%F%*$F'\"\"&F(%(~.~.~.~GF%" }{TEXT -1 12 "
 ------- (i)" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "1/(1
-x) = 1+x+x^2+x^3+x^4+x^5+` . . . `;" "6#/*&\"\"\"F%,&F%F%%\"xG!\"\"F(
,0F%F%F'F%*$F'\"\"#F%*$F'\"\"$F%*$F'\"\"%F%*$F'\"\"&F%%(~.~.~.~GF%" }
{TEXT -1 13 " ------- (ii)" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "1/(1-x^2) = 1+x^2+x^4+x^6+x^8+` . . . `;" "6#/*&\"\"\"F
%,&F%F%*$%\"xG\"\"#!\"\"F*,.F%F%*$F(F)F%*$F(\"\"%F%*$F(\"\"'F%*$F(\"\"
)F%%(~.~.~.~GF%" }{TEXT -1 14 " ------- (iii)" }}{PARA 256 "" 0 "" 
{TEXT -1 2 "  " }{XPPEDIT 18 0 "(-x)/(1-x^2) = -x-x^3-x^5-x^7-x^9+` . \+
. . `;" "6#/*&,$%\"xG!\"\"\"\"\",&F(F(*$F&\"\"#F'F',.F&F'*$F&\"\"$F'*$
F&\"\"&F'*$F&\"\"(F'*$F&\"\"*F'%(~.~.~.~GF(" }{TEXT -1 13 " ------- (i
v)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Add
ing equations (i) and (ii), and dividing by 2 gives equation (iii)." }
}{PARA 0 "" 0 "" {TEXT -1 74 "Subtracting equation (ii) from equation \+
(i), and dividing by 2 gives (iv)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "g := x -> 1/(1+x);\ng_even \+
:= x -> (g(x)+g(-x))/2:\ng_odd := x -> (g(x)-g(-x))/2:\ng(x)=taylor(g(
x),x,10);\ng_even(x)=taylor(g_even(x),x,10);\ng_odd(x)=taylor(g_odd(x)
,x,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)opera
torG%&arrowGF(*&\"\"\"F-,&F-F-9$F-!\"\"F(F(F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&\"\"\"F%,&F%F%%\"xGF%!\"\"+9F'F%\"\"!F(F%F%\"\"#F(\"
\"$F%\"\"%F(\"\"&F%\"\"'F(\"\"(F%\"\")F(\"\"*-%\"OG6#F%\"#5" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/,&*&\"\"\"F&*&\"\"#F&,&F&F&%\"xGF&F&!\"\"F&
*&F&F&*&F(F&,&F&F&F*F+F&F+F&+/F*F&\"\"!F&F(F&\"\"%F&\"\"'F&\"\")-%\"OG
6#F&\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&\"\"\"F&*&\"\"#F&,&F&
F&%\"xGF&F&!\"\"F&*&F&F&*&F(F&,&F&F&F*F+F&F+F++/F*F+F&F+\"\"$F+\"\"&F+
\"\"(F+\"\"*-%\"OG6#F&\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 60 "Notice that the Maclaurin series (iii) for the \+
even part of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 59 " \+
consists of the terms of (i) which involve even powers of " }{TEXT 
279 1 "x" }{TEXT -1 52 ", and the Maclaurin series (iv) for the odd pa
rt of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 58 " consist
s of the terms of (i) which involve odd powers of " }{TEXT 280 1 "x" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 219 "p1 := plot([1/(1+x),1/(1-x^2),-x/(1-x^2)],x=-1.
8..1.8,\n   y=-8..8,color=[brown,red,blue],discont=true):\nasym := plo
ts[implicitplot](\{x=-1,x=1\},x=-1.8..1.8,\n   y=-8..8,color=black,lin
estyle=2):\nplots[display]([p1,asym]);" }}{PARA 13 "" 1 "" {GLPLOT2D 
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Fees$\"39cbbbbbNkF*7$7$FeesFicsFb_t7$7$Fees$\"3'))***********>nF*7$Fee
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IEWG6$;$Ff^lFbhr$\"#=Fbhr;Fchr$\"\")Fddp" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exampl
e 4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT -1 4 "Let " }{XPPEDIT 18 0 "g(x) = ln(1+exp(x));" "6#/-%\"gG6#%
\"xG-%#lnG6#,&\"\"\"F,-%$expG6#F'F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "
" {TEXT 266 9 "Even part" }{TEXT -1 1 ":" }}{PARA 256 "" 0 "" {TEXT 
-1 2 "  " }{XPPEDIT 18 0 "g[even](x) = 1/2;" "6#/-&%\"gG6#%%evenG6#%\"
xG*&\"\"\"F,\"\"#!\"\"" }{XPPEDIT 18 0 "``(ln(1+exp(x))+ln(1+exp(-x)))
;" "6#-%!G6#,&-%#lnG6#,&\"\"\"F+-%$expG6#%\"xGF+F+-F(6#,&F+F+-F-6#,$F/
!\"\"F+F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "ln(2+exp(x)+exp(-x));" "6#-%#lnG6#,(\"\"#\"\"\"-%$expG6
#%\"xGF(-F*6#,$F,!\"\"F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 267 
8 "Odd part" }{TEXT -1 1 ":" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "g[odd](x) = 1/2;" "6#/-&%\"gG6#%$oddG6#%\"xG*&\"\"\"F,
\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(ln(1+exp(x))-ln(1+exp(-
x)));" "6#-%!G6#,&-%#lnG6#,&\"\"\"F+-%$expG6#%\"xGF+F+-F(6#,&F+F+-F-6#
,$F/!\"\"F+F6" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "ln((1+exp(x))/(1+exp(-x)));" "6#-%#lnG6#*&,&\"\"\"F(-%$
expG6#%\"xGF(F(,&F(F(-F*6#,$F,!\"\"F(F1" }{TEXT -1 1 " " }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!
\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln((exp(x)+exp(2*x))/(exp(x)+1))
 = 1/2;" "6#/-%#lnG6#*&,&-%$expG6#%\"xG\"\"\"-F*6#*&\"\"#F-F,F-F-F-,&-
F*6#F,F-F-F-!\"\"*&F-F-F1F5" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(exp(x)
) =x/2" "6#/-%#lnG6#-%$expG6#%\"xG*&F*\"\"\"\"\"#!\"\"" }{TEXT -1 2 ".
 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "For \+
" }{XPPEDIT 18 0 "-infinity < x;" "6#2,$%)infinityG!\"\"%\"xG" }
{XPPEDIT 18 0 "`` < infinity;" "6#2%!G%)infinityG" }{TEXT -1 42 ", we \+
have the Maclaurin series expansions:" }}{PARA 256 "" 0 "" {TEXT -1 1 
" " }{XPPEDIT 18 0 "ln(1+exp(x)) = ln(2)+x/2+x^2/8-x^4/192+x^6/2880-17
/645120*x^8+` . . . `;" "6#/-%#lnG6#,&\"\"\"F(-%$expG6#%\"xGF(,0-F%6#
\"\"#F(*&F,F(F0!\"\"F(*&F,F0\"\")F2F(*&F,\"\"%\"$#>F2F2*&F,\"\"'\"%!)G
F2F(*(\"#<F(\"'?^kF2F,F4F2%(~.~.~.~GF(" }{TEXT -1 13 " ------- (i) " }
}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(1+exp(-x)) = ln(2
)-x/2+x^2/8-x^4/192+x^6/2880-17/645120*x^8+` . . . `;" "6#/-%#lnG6#,&
\"\"\"F(-%$expG6#,$%\"xG!\"\"F(,0-F%6#\"\"#F(*&F-F(F2F.F.*&F-F2\"\")F.
F(*&F-\"\"%\"$#>F.F.*&F-\"\"'\"%!)GF.F(*(\"#<F(\"'?^kF.F-F5F.%(~.~.~.~
GF(" }{TEXT -1 14 " ------- (ii) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " 
}{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "ln(2+exp(x)+exp(-x)) = ln(2)+x^2/8-x^4/192+x^6/2880-17/
645120*x^8+` . . . `;" "6#/-%#lnG6#,(\"\"#\"\"\"-%$expG6#%\"xGF)-F+6#,
$F-!\"\"F),.-F%6#F(F)*&F-F(\"\")F1F)*&F-\"\"%\"$#>F1F1*&F-\"\"'\"%!)GF
1F)*(\"#<F)\"'?^kF1F-F6F1%(~.~.~.~GF)" }{TEXT -1 15 " ------- (iii) " 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Adding \+
equations (i) and (ii), and dividing by 2 gives equation (iii)." }}
{PARA 0 "" 0 "" {TEXT -1 69 "Subtracting equation (ii) from equation (
i), and dividing by 2 gives " }{XPPEDIT 18 0 "x/2" "6#*&%\"xG\"\"\"\"
\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "g := x -> ln(1+exp(x));\ng_even :=
 x -> (g(x)+g(-x))/2:\ng_odd := x -> (g(x)-g(-x))/2:\ng(x)=taylor(g(x)
,x,10);\ng_even(x)=taylor(g_even(x),x,10);\ng_odd(x)=taylor(g_odd(x),x
,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operato
rG%&arrowGF(-%#lnG6#,&\"\"\"F0-%$expG6#9$F0F(F(F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%#lnG6#,&\"\"\"F(-%$expG6#%\"xGF(+1F,-F%6#\"\"#\"\"!#
F(F0F(#F(\"\")F0#!\"\"\"$#>\"\"%#F(\"%!)G\"\"'#!#<\"'?^kF4-%\"OG6#F(\"
#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&#\"\"\"\"\"#F'-%#lnG6#,&F'F
'-%$expG6#%\"xGF'F'F'*&F&F'-F*6#,&F'F'-F.6#,$F0!\"\"F'F'F'+/F0-F*6#F(
\"\"!#F'\"\")F(#F8\"$#>\"\"%#F'\"%!)G\"\"'#!#<\"'?^kF>-%\"OG6#F'\"#5" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&#\"\"\"\"\"#F'-%#lnG6#,&F'F'-%$
expG6#%\"xGF'F'F'*&#F'F(F'-F*6#,&F'F'-F.6#,$F0!\"\"F'F'F9+'F0F&F'-%\"O
G6#F'\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 60 "Notice that the Maclaurin series (iii) for the even part of " }
{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 59 " consists of the \+
terms of (i) which involve even powers of " }{TEXT 270 1 "x" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 16 "The odd part of " }{XPPEDIT 
18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 67 " consists of the single te
rm of (i) which involves an odd power of " }{TEXT 281 1 "x" }{TEXT -1 
2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 96 "plot([ln(1+exp(x)),ln(2+exp(x)+exp(-x))/2,x/2],x=-6..
6,\n   color=[brown,red,blue],discont=true);" }}{PARA 13 "" 1 "" 
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nu\"F17$Fjw$\"3s***\\(=n#f(=F17$F_x$\"3P+++!)RO+?F17$Fdx$\"30++]_!>w7#
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\"3++++gEd@EF17$F]z$\"39++v3'>$[FF17$Fbz$\"37++D6EjpGF17$Fgz$\"\"$F*-F
\\[l6&F^[lFfelFfelFdel-%+AXESLABELSG6%Q\"x6\"Q!Fc_m-%%FONTG6#%(DEFAULT
G-%%VIEWG6$;F(FgzFh_m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT 
-1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 
0 "g(x) = cos*x+sin*x;" "6#/-%\"gG6#%\"xG,&*&%$cosG\"\"\"F'F+F+*&%$sin
GF+F'F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 268 9 "Even part" 
}{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "
g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 33 " is the sum of the even function
 " }{XPPEDIT 18 0 "cos*x;" "6#*&%$cosG\"\"\"%\"xGF%" }{TEXT -1 22 " an
d the odd function " }{XPPEDIT 18 0 "sin*x;" "6#*&%$sinG\"\"\"%\"xGF%
" }{TEXT -1 18 ", we just recover " }{XPPEDIT 18 0 "cos*x;" "6#*&%$cos
G\"\"\"%\"xGF%" }{TEXT -1 23 " as the even part . . ." }}{PARA 256 "" 
0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "g[even](x) = 1/2;" "6#/-&%\"gG6#%
%evenG6#%\"xG*&\"\"\"F,\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(
cos*x+sin*x+cos(-x)+sin(-x));" "6#-%!G6#,**&%$cosG\"\"\"%\"xGF)F)*&%$s
inGF)F*F)F)-F(6#,$F*!\"\"F)-F,6#,$F*F0F)" }{TEXT -1 1 " " }}{PARA 256 
"" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = cos*x;" "6#/%!G*&%$cosG\"
\"\"%\"xGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 269 8 "Odd part" 
}{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 29 " . . . similarly, we re
cover " }{XPPEDIT 18 0 "sin*x;" "6#*&%$sinG\"\"\"%\"xGF%" }{TEXT -1 
17 " as the odd part." }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 
18 0 "g[odd](x) = 1/2;" "6#/-&%\"gG6#%$oddG6#%\"xG*&\"\"\"F,\"\"#!\"\"
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(cos*x+sin*x-cos(-x)-sin(-x));" "6
#-%!G6#,**&%$cosG\"\"\"%\"xGF)F)*&%$sinGF)F*F)F)-F(6#,$F*!\"\"F0-F,6#,
$F*F0F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "`` = sin*x;" "6#/%!G*&%$sinG\"\"\"%\"xGF'" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 107 "g := x -> cos(x)+sin(x);\ng_even := x -> (g(x)+g(-x))/2;\ng_eve
n(x);\ng_odd := x -> (g(x)-g(-x))/2;\ng_odd(x);\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%$cosG6#9$
\"\"\"-%$sinGF/F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'g_evenGf
*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&#\"\"\"\"\"#F/-%\"gG6#9$F/F/*&F.
F/-F26#,$F4!\"\"F/F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$cosG6
#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&g_oddGf*6#%\"xG6\"6$%)ope
ratorG%&arrowGF(,&*&#\"\"\"\"\"#F/-%\"gG6#9$F/F/*&#F/F0F/-F26#,$F4!\"
\"F/F:F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#%\"xG" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "We have t
he Maclaurin series expansions:" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "cos*x+sin*x = 1+x-x^2/2-x^3/6+x^4/24+x^5/120-x^6/720-x^
7/5040+x^8/40320+` . . . `;" "6#/,&*&%$cosG\"\"\"%\"xGF'F'*&%$sinGF'F(
F'F',6F'F'F(F'*&F(\"\"#F-!\"\"F.*&F(\"\"$\"\"'F.F.*&F(\"\"%\"#CF.F'*&F
(\"\"&\"$?\"F.F'*&F(F1\"$?(F.F.*&F(\"\"(\"%S]F.F.*&F(\"\")\"&?.%F.F'%(
~.~.~.~GF'" }{TEXT -1 12 " ------- (i)" }}{PARA 0 "" 0 "" {TEXT -1 1 "
 " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "cos*x = 1-x^2/2
+x^4/24-x^6/720+x^8/40320+` . . . `;" "6#/*&%$cosG\"\"\"%\"xGF&,.F&F&*
&F'\"\"#F*!\"\"F+*&F'\"\"%\"#CF+F&*&F'\"\"'\"$?(F+F+*&F'\"\")\"&?.%F+F
&%(~.~.~.~GF&" }{TEXT -1 13 " ------- (ii)" }}{PARA 256 "" 0 "" {TEXT 
-1 3 "   " }{XPPEDIT 18 0 "sin*x = x-x^3/6+x^5/120-x^7/5040+` . . . `;
" "6#/*&%$sinG\"\"\"%\"xGF&,,F'F&*&F'\"\"$\"\"'!\"\"F,*&F'\"\"&\"$?\"F
,F&*&F'\"\"(\"%S]F,F,%(~.~.~.~GF&" }{TEXT -1 14 " ------- (iii)" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
94 "cos(x)+sin(x)=taylor(cos(x)+sin(x),x,9);\ncos(x)=taylor(cos(x),x,9
);\nsin(x)=taylor(sin(x),x,9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-
%$cosG6#%\"xG\"\"\"-%$sinGF'F)+7F(F)\"\"!F)F)#!\"\"\"\"#F0#F/\"\"'\"\"
$#F)\"#C\"\"%#F)\"$?\"\"\"&#F/\"$?(F2#F/\"%S]\"\"(#F)\"&?.%\"\")-%\"OG
6#F)\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%\"xG+/F'\"\"\"
\"\"!#!\"\"\"\"#F-#F)\"#C\"\"%#F,\"$?(\"\"'#F)\"&?.%\"\")-%\"OG6#F)\"
\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#%\"xG+-F'\"\"\"F)#!\"
\"\"\"'\"\"$#F)\"$?\"\"\"&#F+\"%S]\"\"(-%\"OG6#F)\"\"*" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The Maclaurin series
 (iii) for the even part of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" 
}{TEXT -1 59 " consists of the terms of (i) which involve even powers \+
of " }{TEXT 275 1 "x" }{TEXT -1 52 ", and the Maclaurin series (iv) fo
r the odd part of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 
58 " consists of the terms of (i) which involve odd powers of " }
{TEXT 276 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "plot([cos(x)+sin(x),cos(x),s
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{TEXT -1 29 ", together with the graph of " }{XPPEDIT 18 0 "f(x)" "6#-
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" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "
" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 41 "Plot graphs of t
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G6#%\"xG-%'arctanG6#,&\"\"\"F,F'F," }{TEXT -1 29 ", together with the \+
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{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 492 "p1 := plot(
x^2,x=-1.4..1.4):\np2 := plot([[-1,0],[-1,1],[1,1],[1,0]],\n       lin
estyle=2,color=COLOR(RGB,.1,.1,.1)):\np3 := plot([[[-1,1],[1,1]]$3],st
yle=point,symbol=[circle,diamond,cross],color=black):\nt1 := plots[tex
tplot]([[1,-0.05,`a`],[-1,-0.05,`-a`],\n   [-0.05,-0.05,`O`],[1.4,-0.0
5,`x`],[-0.05,2,`y`],[-0.05,0.95,`M`],\n   [-1.25,1,`Q(-a,f(a))`],[1.2
3,1,`P(a,f(a))`]],color=black):\nt2 := plots[textplot]([1.15,1.8,`y = \+
f(x)`],color=red):\nplots[display]([p1,p2,p3,t1,t2],tickmarks=[0,0]);
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 515 "p1 := plot(x^3,x=-1.3..1.3):\np2 :
= plot([[-1,0],[-1,-1],[1,1],[1,0]],\n       linestyle=2,color=COLOR(R
GB,.1,.1,.1)):\np3 := plot([[[-1,-1],[1,1]]$3],style=point,symbol=[cir
cle,diamond,cross],color=black):\nt1 := plots[textplot]([[1,-0.1,`a`],
[-1,0.2,`-a`],\n   [-0.05,0.15,`O`],[1.3,-0.1,`x`],[-0.05,2.2,`y`],\n \+
  [-1.35,-0.9,`Q(-a,-f(a))`],[1.35,1,`P(a,f(a))`]]):\nt2 := plots[text
plot]([.9,1.5,`y = f(x)`],color=red):\nplots[display]([p1,p2,p3,t1,t2]
,tickmarks=[0,0],\n                   view=[-1.35..1.35,-2.36..2.36]);
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
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