,
** * +
* * "!$#&%'(
)
-.0/0132547698;:<>=@?&ABC!EDGFH'IFHJ
?KB?ML'I?&NO$FHJP(PDGFPDGFHQIFH'IJ
!$A(R'S(
TU?&'IJP(M(&BIBC%J
!ED(PL?&'3%J
VL(RFRNBC??d W?&J
?&#FHX AFH?PYFHZ\$[ ] FHO^"BC%BCLFHABC?LFHOE(&B3DG%(&BSFHZGQI'IFHJ
!ED(&DGFRB_^"BC?&N`(M(V&a (Rb ?cDGFPBCFH%cL'?&[ L'I!$?
efgUhjki ml fn
o ] !EDGFHAGQIFHJ
FHAGQIF&^qpK%(RADG?c(RQI!$A#&!$'3?FRpK%!$OC\ [ N'I!$?^"? V (RN? ] (R!5r V (R'H^@QI?GDG?cFHO$FK^ V ?&AGQI!EDG?FHJs%J
LOE(RA?^?
LOE(RA? ] FH'IQI! V (RO"pK%F3L(&BIBI(
L?&'SBC%(&BDG%(&B7FHZGQI'IFHJ
!ED(&DGFRB_d7"! V (RJ
?KB_^FHAGQ (Rb ?^ V ?&Jt%J
L'I?&NO$FHJP(SA?
LOE(RA?d7=@?&O$?GpK%FHA?KB_^uAFRBCQIFSLOE(RA?^u%JtBC!EBCQIFHJP(3DGF V ?K?&'DGFHA(&D(&B7FHJvpK%FS?0FH!$ZG?
wxBCFCy( ] FH'IQI! V (ROFjL(&BIBCFLFHO$?L?&AGQI?3JP(R!EB"N(R!$ZG? zG DG? V (RN?d5{`TU?&'&V a (SDGFjQIFHAB (Rb ?cF [ ] (R'I!R( [ ] FHO
(R?3O$?&A#&?0DG? V (RN?d5{|QIFHAB (Rb ?3FHJ}%JL?&AGQI?
~DG? V (RN?0DGFHLFHADGFK^KFHAGQI'IFj?&%QI'(&B5#&'(RADGFHR(&B_^DG?
LFRBC?
D(3L?&'&V&a (Rb ?
DGF V (RN?
( V !$JP(SDG?0L?&AGQI?P~Sd
=@?&ABC!EDGFH'IFHJ
?KB7%JQI'IF W?PDG? (RN?^@DGF ?&J
L'I!$J
FHAKQI? ^"FHAGQI'IF3?ML?&AGQI?DGF ?K?&'DGFHA(&D(&B
Fj?SL?&AGQI?SJP(R!EB"N(R!$ZGV ?SDG? V (RNV ?d5o@BCQIFjQIV 'IF V W?FRBCQ(7[ FHJ} FRpK%!$OC\ [ N'I!$?7BC?&NP((V&a (Rb ?3V DGF77TU?&'&V a (&B_
?
BCFH%MLFRBC?^u(3QIFHAB (Rb ?
jA?0L?&AGQI?
JP(R!EB5N(R!$ZG?3F3(3QIFHAB (Rb ?
|A?0L?&AGQI?
JP(R!EB(RO$QI?dT(RQI?
DG?
VDG(R!$'INF?Sa (RBC?7FH'"QY(RAFHZ#&FH\$[ ] AGFHQIOKFSBC(F
FH%ZGL' 'I(GFRBIdqBIC(jBCQIJP?^&(RLQIFH?&JP'pK(R%QIFj! V A (RJ0(R?7FHWjAKQI(j[ F@TUDG?&'&!$Ha FH(&AB"DG!$AG?QIpKFH%'IAF(&(B_^&TU?&? '&V a (R(NDG?F5A QI(RFH?AB ?R(Rb TU?FH'IQIF FHJxF5ABCFHFHJ
AWGL%'IJPF5((
'IFRBC!EBCV&QRb FHX A V !E(P( V %' ] (R'CV CBCF
] A(MDG!$'IFV&a (Rb ?D(PQIFHb AB (Rb ?dP{V BC?&JP(MDGFRBCQ(&BS
V TU?&'&V a (&BSb pK%F
(R#&V FHJBC?&N'IF
?
QI'IF V W? V ?&ABC!EDGFH'(&DG?
DG? V (RN?`F[ AG%OE(Gd5=@?&ABC!EDGFH'(RADG?P(&B V ?&J
L?&AFHAGQIFRB@W?&'I!$H?&AGQ(R!EB_^QIFHJ
?KB
)
| V ?KB FK^G!$#&%(ROE(RADG?P( V ?&J
L?&AFHAGQIFRB ] FH'IQI! V (ROuDGFS(R?0LFRBC?
DG?3QI'IF V W?^QIFHJ
?KB
K
MBCFHA @ ?&ADGF0FS[ (0DGFHABC!ED(&DGFO$!$AFR(R'DG? V (RN?d
j! ] !EDG!$ADG? K L?&' ) ^?&NQIFHJ
?KB
Q(RA @
o@BCQ(RJ
?KBL'I? V %'(RADG?M(STU%AqV&a (Rb ? u Kp %F3D(0[ (3TU?&'IJP(3(&BIBC%J
!ED(3LFHO$? V (RN?dj{ V ?&ADG!IV&a (Rb ?
A?KBjDG!$pK%F
@
G
j?&QIFMpK%F u FP[ TU%AqV&a (Rb ?cDGF d{ !$#&%(ROED(&DGF( V !$JP(PA (Rb ?}FM[ (R!$AD(%JP(Mo@S^@L?&!EB0
] (R'I!R( [ ] FH!EB@FRBCQ (Rb ?0FH A ] ?&O ] !ED(&B_d q(R'(0BC%LFH'(R'jFRBCQ(0DG! r V %OED(&DGFK^DGFH'I! ] (RJ
?KB FHJ'IFHOE(V&a (Rb ?
( ^
@
K
5(RJ
?KB7(&pK%!@(RN'I!$'%JtL(R'RFHX AGQIFRBCFRB7L(R'(
'IF ] !EBI(R'7%JP(3TK?&[ 'IJ0%OE(0DG? V (R[ O V %O$?^(0TK?&[ 'IJ0%OE( K
(RN(R!$ZG?d
{|r#&%'(
(R?0OE(&DG?0J
?KBCQI'(3%J LFRpK%FHA?
BCFH#&J
FHAGQI?3DGF V ?&J
L'I!$J
F_AGQI? (3L(R'IQI!$'7DG?0L?&AGQI? d
%FH'IFHJ
?KB"FHZGL'IFRBIBI(R'5?SLFRpK%FHA? ?&J
L'I!$J
FHAGQI?jDGF(R' ? ^KJ
FRDG!EDG?BC?&N'IF( %' (Gd@{!ED5FH[ !E(F [
(RL'I?RZG!$JP(R' LFHO$? V ?&J
L'I!$J
FHAGQIV?jJ
FRDG!EDG?SBC?&N'IFS('IFHV Q(Q(R A#&FHAGQIFKd5=@W(RJP(RADGV ?0DG] F ?(RX A#&%O$?
FHAGQI'IF7FRBCQ(3Q(RA #&FHAGQIFF?0?3FH!$ZG? c^QIFHJ
?KB
V ?KB
?&#&?
) |BCF V ) cQ(RA V ?KB
(RHFHADG? z ^?&NQIFHJ
?KB
K
) %NBCQI!$QI%!$ADG? K FHJ K ^?&NQIFHJ
?KB
@ ) C &
$
"!
#
% %
(
'&
% *)
pK%FPFj[ %JP(o@|DGF7BCFH#&%AD(?&'DGFHJ}'IFRDG%QR\$[ ] FHO(
L'I!$J
FH!$'(?&'DGFHJdq=@W(RJP(RADG?S ^KQIFHJ
?KB
K
) ?&ADGF 3 @ d@{9o@ K FS[ BCFHL(R'( [ ] FHO^ FHL(R'(RADG?
(&B ] (R'I!R( [ ] FH!EB@F!$AGQIFH#&'(RADG?^QIFHJ
?KB
)
(RHFHADG?0(0BC%NBCQI!$QI%!IV&a (Rb ? 3`Q(RA u^ 3|BCF V u^
BCF V BCF V
BCF V )
%
(
$
`O$A Q(RA
>BCF V K`O$A )
?&#&?
(
$O A ) 7BI(RADG?7pK%F5(Q(RA#&FHAGQIF5A?jL?&AGQI?7JP(R!EBN(R!$ZG?7D( V %' ] (
F5[ W?&'I!$H?&AGQ(RO^R!EBCQI?PFK[ ^ zK zK z ^
DGFRDG%H!$J
?KB5pK%F@ z d ?&#&?
O$A ) ?&%BCFCy(G^
) FA?KB V ?&A V FHAGQI'(R'IJ
?KB@A?0OE(&DG?
DG!$'IFH!$QI?3DG? V (RN?^?&ADGF 3 z ^QIFHJ
?KB
) "?GDGFHJ
?KB5!EBC?&OE(R' ^
) ) > ) > ) !
?&#&?
!
(
%
(
(
$
%
(
%
(
F
! ) V ?KBCW u
=@?&A V O$%"\ [ J
?KB5pK%FS(3L?KBC!IV&a (Rb ?
DGFFRpK%!$OC\ [ N'I!$?SDG? V (RN?`FS[ D(&D(3L?&'
V ?KBCW u !EBCQI?^jFK[ ^u%JP(3QI'(RABCOE(V&a (Rb ? ] FH'IQI! V (RO"D( V %' ] ( V ?KBCW u ^upK%FSA(&D(
JP(R!EB
F3[ DG?MpK%FS%JP(
J0%D(RAqV a (SDGFFRB V (ROE( %JP(S(RJ
LO$!E(V&a (Rb ?S?&%M'IFRDG%qV&a (Rb ? D( V %' ] (
V ?KBCW y(0[ pK%F7L'I? ] FH[ JvDGFHOE(3(RQI'( ] FR[ BD(3QI'(RAGTU?&'IJP(V&a (Rb ? d
cosh(x)
10
8
y
6
4
2
–3
–2
–1
1 x 2
3
"!$#&%'IF0)&@%AqV&a (Rb ? V ?KBCW u
{ V %' ] ( V ?KBCW V W(RJP(HCBCF kKm
m Um ^DG?cOE(RQI!$J&^jpK%FBC!$#&A! r V ( V (&DGFH!E(?&%
V ?&'I'IFHAGQIFKdq{ V (RQIFHAj(R[ 'I!E( ^GL?&'IQ(RAGQI?^FS[ J
(FHTUA?&?K'IBJPDG(SF(&%BIJBC%J
T(R!EQID?&('LDG?&FS'(R%J
J LO$V !E(R(Na (R?3?3Y?&FH%Z\$[ 'I] FRFHDGO%qBC%a (RBCL?dFHA"!BC$?Z^(H(
DG?KB L?&AGQI?KB F V ?&J
?3A(rV&#&b %'(
(R?0OE(&DGV&?b ^ V ?&ABC!EDK
FH'IFHJ
?KB V (RN?KB@YFHZ\$[ ] FH!EB@DGF V ?&J
L'I!$J
FHAGQC?KB@DG! TUFH'IFHAGQIFRB
BC%BCLFHABC?KBPL?&'PFRBCQIFRB L?&AGQI?KB_d{tTU?&'IJP(c(&BIBC%J
!ED(
LFHO$? V (RN?DGFJ
FHA?&' V ?&J
L'I!$J
FHAKQI?^pK%FMr V ?&%JP(R!EB
( V !$JP(`FP[ (DGF
%JLFRpK%FHA?MQI'IF V W?MD( V (RQIFHAj(R[ 'I!E( ^
NFHJ9L'?&[ ZG!$J
?j(R?7L?&AGQI?7JP(R!EBuN(R!$ZG?^KBC%NJ
FHQI!EDG?j(j%JP(
#&'(RADGFP(RJ
LO$!E(V&a (Rb ?d @(M[ (
TU?&'IJP(P(&BIBC%J
!ED(
LFHO$? V (RN?
DGFjJP(R!$?&' V ?&J
L'I!$J
FHAGQI?MF[ (SDGF7%JQI'IF V W?J0%!$QI?7JP(R!EB
O$?&A#&?MD( V (RQIFHAj(R[ 'I!E( ^BC%RyIFH!$QI?M(
%JP(3LFRpK%FHA(P(RJ0
LO$!E(V&a (Rb ?3?&%(RQF[ J
FRBCJ
?S(3%JP('IFRDG%qV&a (Rb ?d
- .0/0132547698u<5jFHQIFH'IJ
!$AF(jTU?&'IJP(j(&BIBC%J
!ED(7L?&'@%J V (RN?3DGFjBC%BCQIFHAGQ(V&a (Rb ?3DGF%J}L?&AGQIF
L"FHX ABC!$OjDGFDGFHABC!ED(&DGFPW?&'I!$H?&AGQ(RO V ?&ABCQ(RAGQIFK^BC%L?&ADG?>(JP(&BIBI(DG? V (RN?cDGFRBCL'IFHG\$[ ] FHO@T( V FM(
JP(&BIBI(3D(3L?&AGQIFSpK%FFHO$FBC%BCQIFHAGQ(Gd
efgUhjki ml fn
=@?&ABC!EDGFH'IFHJ
?KB_^"A? (RJ
FHAGQIF0A(M"!$#&%'(c)&^@(&B3PTU?&'&a (&B3(R#&!$ADG?A?QI'IF W?MDG? (RN?FHAGQI'IFPBCFH%
L?&AGQI?
L(R!EBjN(R!$ZG?0FS] ?
L?&AGQI? d{ %[ A! V (
DG! TUFH'IFHV AqV a (cF3[ pK%F3(R#&?&'(
FHV J O$%#K(R'V DGF V ?&ABC!EDGFH'(R'
?PLFRBC?DG?MQIF V W?PDGF V (RN?^qDGF ] FHJ
?KB V ?&ABC!EDGFH'(R'?PLFRBC?DG?PQI'IF V W?MD(
L?&AGQIFPpK%F3FRBCQ(MBC?&N?
QI'IF V W?0DG? V (RN?djoqJ ?&%QI'(&BjL(ROE( ] '(&B_^u( V ?&ADG!IV&a (Rb ? ) V ?&AGQI!$AG%(3(3J
FRBCJP(G^KJP(&B5FHJ O$%#K(R'7DGF
K DGF ] FHJ
?KB V ?&ABC!EDGFH'(R'j(R#&?&'(
( V ?&ADG!IV&a (Rb ?
) zK
MBCFHA @ ?&ADGF0PDGFRBC!$#&A(0(R#&?&'(
(3DGFHABC!ED(&DGFSW?&'I!$H?&AGQ(ROD(3L?&AGQIFKd5j! ] !EDG!$ADG? ) zK L?&' ) ^
)&) Q(RA @ ;
?&#&?P(STU%AqV&a (Rb ? u pK%FSD(0[ (TU?&'IJP(3DG? V (RN?PBI(RQI!EBT(RS(3FRpK%(V&a (Rb ?
DG! TUFH'IFHA V !E(RO
@ ;
'(RQ(HCBCFcDGFM%JP(FRpK%(V&a (Rb ?>DG! TUFH'IFHA V !E(ROjDGFML'I!$J
FH!$'(M?&'DGFHJ BCFHL(R'( [ ] FHOjJ0%!$QI?BC!$J
LO$FRB_^ V %Ry(
BC?&O$%qV&a (Rb ?0#&FH'(ROF [
@
= = {b S jJ V (RN?0DGF7BC%BCQIFHAGQ(V&a (Rb ?0DGFj%JP(7L?&AGQIFjL"FHX ABC!$Ou(&BIBC%J
F7(7TU?&'IJP(DGFj%J(R' V ?
DGFL(R'(R[ N?&OE(Gd
(
$