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1300 Math Formulas ==============================
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Preface ====qÜáë= Ü~åÇÄççâ= áë= ~= ÅçãéäÉíÉ= ÇÉëâíçé= êÉÑÉêÉåÅÉ= Ñçê= ëíì-ÇÉåíë= ~åÇ= ÉåÖáåÉÉêëK= fí= Ü~ë= ÉîÉêóíÜáåÖ= Ñêçã= ÜáÖÜ= ëÅÜççä=ã~íÜ=íç=ã~íÜ=Ñçê=~Çî~åÅÉÇ=ìåÇÉêÖê~Çì~íÉë=áå=ÉåÖáåÉÉêáåÖI=ÉÅçåçãáÅëI=éÜóëáÅ~ä=ëÅáÉåÅÉëI=~åÇ=ã~íÜÉã~íáÅëK=qÜÉ=ÉÄççâ=Åçåí~áåë= ÜìåÇêÉÇë= çÑ= Ñçêãìä~ëI= í~ÄäÉëI= ~åÇ= ÑáÖìêÉë= Ñêçã=kìãÄÉê= pÉíëI= ^äÖÉÄê~I= dÉçãÉíêóI= qêáÖçåçãÉíêóI= j~íêáÅÉë=~åÇ= aÉíÉêãáå~åíëI= sÉÅíçêëI= ^å~äóíáÅ= dÉçãÉíêóI= `~äÅìäìëI=aáÑÑÉêÉåíá~ä=bèì~íáçåëI=pÉêáÉëI=~åÇ=mêçÄ~Äáäáíó=qÜÉçêóK==qÜÉ= ëíêìÅíìêÉÇ= í~ÄäÉ= çÑ= ÅçåíÉåíëI= äáåâëI= ~åÇ= ä~óçìí= ã~âÉ=ÑáåÇáåÖ= íÜÉ= êÉäÉî~åí= áåÑçêã~íáçå= èìáÅâ= ~åÇ= é~áåäÉëëI= ëç= áí=Å~å=ÄÉ=ìëÉÇ=~ë=~å=ÉîÉêóÇ~ó=çåäáåÉ=êÉÑÉêÉåÅÉ=ÖìáÇÉK===
==
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Contents ====
1 krj_bo=pbqp= NKN= pÉí=fÇÉåíáíáÉë==1= NKO= pÉíë=çÑ=kìãÄÉêë==5= NKP= _~ëáÅ=fÇÉåíáíáÉë==7= NKQ= `çãéäÉñ=kìãÄÉêë==8= =2 ^idb_o^= OKN= c~ÅíçêáåÖ=cçêãìä~ë==12= OKO= mêçÇìÅí=cçêãìä~ë==13= OKP= mçïÉêë==14= OKQ= oççíë==15= OKR= içÖ~êáíÜãë==16= OKS= bèì~íáçåë==18= OKT= fåÉèì~äáíáÉë==19= OKU= `çãéçìåÇ=fåíÉêÉëí=cçêãìä~ë==22= =3 dbljbqov= PKN= oáÖÜí=qêá~åÖäÉ==24= PKO= fëçëÅÉäÉë=qêá~åÖäÉ==27= PKP= bèìáä~íÉê~ä=qêá~åÖäÉ==28= PKQ= pÅ~äÉåÉ=qêá~åÖäÉ==29= PKR= pèì~êÉ==33= PKS= oÉÅí~åÖäÉ==34= PKT= m~ê~ääÉäçÖê~ã==35= PKU= oÜçãÄìë==36= PKV= qê~éÉòçáÇ==37= PKNM= fëçëÅÉäÉë=qê~éÉòçáÇ==38= PKNN= fëçëÅÉäÉë=qê~éÉòçáÇ=ïáíÜ=fåëÅêáÄÉÇ=`áêÅäÉ==40= PKNO= qê~éÉòçáÇ=ïáíÜ=fåëÅêáÄÉÇ=`áêÅäÉ==41=
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PKNP= háíÉ==42= PKNQ= `óÅäáÅ=nì~Çêáä~íÉê~ä==43= PKNR= q~åÖÉåíá~ä=nì~Çêáä~íÉê~ä==45= PKNS= dÉåÉê~ä=nì~Çêáä~íÉê~ä==46= PKNT= oÉÖìä~ê=eÉñ~Öçå==47= PKNU= oÉÖìä~ê=mçäóÖçå==48= PKNV= `áêÅäÉ==50= PKOM= pÉÅíçê=çÑ=~=`áêÅäÉ==53= PKON= pÉÖãÉåí=çÑ=~=`áêÅäÉ==54= PKOO= `ìÄÉ==55= PKOP= oÉÅí~åÖìä~ê=m~ê~ääÉäÉéáéÉÇ==56= PKOQ= mêáëã==57= PKOR= oÉÖìä~ê=qÉíê~ÜÉÇêçå==58= PKOS= oÉÖìä~ê=móê~ãáÇ==59= PKOT= cêìëíìã=çÑ=~=oÉÖìä~ê=móê~ãáÇ==61= PKOU= oÉÅí~åÖìä~ê=oáÖÜí=tÉÇÖÉ==62= PKOV= mä~íçåáÅ=pçäáÇë==63= PKPM= oáÖÜí=`áêÅìä~ê=`óäáåÇÉê==66= PKPN= oáÖÜí=`áêÅìä~ê=`óäáåÇÉê=ïáíÜ=~å=lÄäáèìÉ=mä~åÉ=c~ÅÉ==68= PKPO= oáÖÜí=`áêÅìä~ê=`çåÉ==69= PKPP= cêìëíìã=çÑ=~=oáÖÜí=`áêÅìä~ê=`çåÉ==70= PKPQ= péÜÉêÉ==72= PKPR= péÜÉêáÅ~ä=`~é==72= PKPS= péÜÉêáÅ~ä=pÉÅíçê==73= PKPT= péÜÉêáÅ~ä=pÉÖãÉåí==74= PKPU= péÜÉêáÅ~ä=tÉÇÖÉ==75= PKPV= bääáéëçáÇ==76= PKQM= `áêÅìä~ê=qçêìë==78= = =4 qofdlkljbqov= QKN= o~Çá~å=~åÇ=aÉÖêÉÉ=jÉ~ëìêÉë=çÑ=^åÖäÉë==80= QKO= aÉÑáåáíáçåë=~åÇ=dê~éÜë=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==81= QKP= páÖåë=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==86= QKQ= qêáÖçåçãÉíêáÅ=cìåÅíáçåë=çÑ=`çããçå=^åÖäÉë==87= QKR= jçëí=fãéçêí~åí=cçêãìä~ë==88=
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QKS= oÉÇìÅíáçå=cçêãìä~ë==89= QKT= mÉêáçÇáÅáíó=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==90= QKU= oÉä~íáçåë=ÄÉíïÉÉå=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==90= QKV= ^ÇÇáíáçå=~åÇ=pìÄíê~Åíáçå=cçêãìä~ë==91= QKNM= açìÄäÉ=^åÖäÉ=cçêãìä~ë==92= QKNN= jìäíáéäÉ=^åÖäÉ=cçêãìä~ë==93= QKNO= e~äÑ=^åÖäÉ=cçêãìä~ë==94= QKNP= e~äÑ=^åÖäÉ=q~åÖÉåí=fÇÉåíáíáÉë==94= QKNQ= qê~åëÑçêãáåÖ=çÑ=qêáÖçåçãÉíêáÅ=bñéêÉëëáçåë=íç=mêçÇìÅí==95= QKNR= qê~åëÑçêãáåÖ=çÑ=qêáÖçåçãÉíêáÅ=bñéêÉëëáçåë=íç=pìã==97=== QKNS= mçïÉêë=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==98= QKNT= dê~éÜë=çÑ=fåîÉêëÉ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==99= QKNU= mêáåÅáé~ä=s~äìÉë=çÑ=fåîÉêëÉ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==102= QKNV= oÉä~íáçåë=ÄÉíïÉÉå=fåîÉêëÉ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==103= QKOM= qêáÖçåçãÉíêáÅ=bèì~íáçåë==106= QKON= oÉä~íáçåë=íç=eóéÉêÄçäáÅ=cìåÅíáçåë==106= = =5 j^qof`bp=^ka=abqbojfk^kqp= RKN= aÉíÉêãáå~åíë==107= RKO= mêçéÉêíáÉë=çÑ=aÉíÉêãáå~åíë==109= RKP= j~íêáÅÉë==110= RKQ= léÉê~íáçåë=ïáíÜ=j~íêáÅÉë==111= RKR= póëíÉãë=çÑ=iáåÉ~ê=bèì~íáçåë==114= = =6 sb`qlop= SKN= sÉÅíçê=`ççêÇáå~íÉë==118= SKO= sÉÅíçê=^ÇÇáíáçå==120= SKP= sÉÅíçê=pìÄíê~Åíáçå==122= SKQ= pÅ~äáåÖ=sÉÅíçêë==122= SKR= pÅ~ä~ê=mêçÇìÅí==123= SKS= sÉÅíçê=mêçÇìÅí==125= SKT= qêáéäÉ=mêçÇìÅí=127= = =7 ^k^ivqf`=dbljbqov= TKN= låÉ=-aáãÉåëáçå~ä=`ççêÇáå~íÉ=póëíÉã==130=
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TKO= qïç=-aáãÉåëáçå~ä=`ççêÇáå~íÉ=póëíÉã==131= TKP= píê~áÖÜí=iáåÉ=áå=mä~åÉ==139= TKQ= `áêÅäÉ==149= TKR= bääáéëÉ==152= TKS= eóéÉêÄçä~==154= TKT= m~ê~Äçä~==158= TKU= qÜêÉÉ=-aáãÉåëáçå~ä=`ççêÇáå~íÉ=póëíÉã==161= TKV= mä~åÉ==165= TKNM= píê~áÖÜí=iáåÉ=áå=pé~ÅÉ==175= TKNN= nì~ÇêáÅ=pìêÑ~ÅÉë==180= TKNO= péÜÉêÉ==189= = =8 afccbobkqf^i=`^i`rirp= UKN= cìåÅíáçåë=~åÇ=qÜÉáê=dê~éÜë==191= UKO= iáãáíë=çÑ=cìåÅíáçåë==208= UKP= aÉÑáåáíáçå=~åÇ=mêçéÉêíáÉë=çÑ=íÜÉ=aÉêáî~íáîÉ==209= UKQ= q~ÄäÉ=çÑ=aÉêáî~íáîÉë==211= UKR= eáÖÜÉê=lêÇÉê=aÉêáî~íáîÉë==215= UKS= ^ééäáÅ~íáçåë=çÑ=aÉêáî~íáîÉ==217= UKT= aáÑÑÉêÉåíá~ä==221= UKU= jìäíáî~êá~ÄäÉ=cìåÅíáçåë==222= UKV= aáÑÑÉêÉåíá~ä=léÉê~íçêë==225= = =9 fkqbdo^i=`^i`rirp= VKN= fåÇÉÑáåáíÉ=fåíÉÖê~ä==227= VKO= fåíÉÖê~äë=çÑ=o~íáçå~ä=cìåÅíáçåë==228= VKP= fåíÉÖê~äë=çÑ=fêê~íáçå~ä=cìåÅíáçåë==231= VKQ= fåíÉÖê~äë=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==237= VKR= fåíÉÖê~äë=çÑ=eóéÉêÄçäáÅ=cìåÅíáçåë==241= VKS= fåíÉÖê~äë=çÑ=bñéçåÉåíá~ä=~åÇ=içÖ~êáíÜãáÅ=cìåÅíáçåë==242= VKT= oÉÇìÅíáçå=cçêãìä~ë==243= VKU= aÉÑáåáíÉ=fåíÉÖê~ä==247= VKV= fãéêçéÉê=fåíÉÖê~ä==253= VKNM= açìÄäÉ=fåíÉÖê~ä==257= VKNN= qêáéäÉ=fåíÉÖê~ä==269=
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VKNO= iáåÉ=fåíÉÖê~ä==275= VKNP= pìêÑ~ÅÉ=fåíÉÖê~ä==285= = =10 afccbobkqf^i=bnr^qflkp= NMKN= cáêëí=lêÇÉê=lêÇáå~êó=aáÑÑÉêÉåíá~ä=bèì~íáçåë==295= NMKO= pÉÅçåÇ=lêÇÉê=lêÇáå~êó=aáÑÑÉêÉåíá~ä=bèì~íáçåë==298= NMKP= pçãÉ=m~êíá~ä=aáÑÑÉêÉåíá~ä=bèì~íáçåë==302= = =11 pbofbp= NNKN= ^êáíÜãÉíáÅ=pÉêáÉë==304= NNKO= dÉçãÉíêáÅ=pÉêáÉë==305= NNKP= pçãÉ=cáåáíÉ=pÉêáÉë==305= NNKQ= fåÑáåáíÉ=pÉêáÉë==307= NNKR= mêçéÉêíáÉë=çÑ=`çåîÉêÖÉåí=pÉêáÉë==307= NNKS= `çåîÉêÖÉåÅÉ=qÉëíë==308= NNKT= ^äíÉêå~íáåÖ=pÉêáÉë==310= NNKU= mçïÉê=pÉêáÉë==311= NNKV= aáÑÑÉêÉåíá~íáçå=~åÇ=fåíÉÖê~íáçå=çÑ=mçïÉê=pÉêáÉë==312= NNKNM= q~óäçê=~åÇ=j~Åä~ìêáå=pÉêáÉë==313= NNKNN= mçïÉê=pÉêáÉë=bñé~åëáçåë=Ñçê=pçãÉ=cìåÅíáçåë==314= NNKNO= _áåçãá~ä=pÉêáÉë==316= NNKNP= cçìêáÉê=pÉêáÉë==316= = =12 mol_^_fifqv= NOKN= mÉêãìí~íáçåë=~åÇ=`çãÄáå~íáçåë==318= NOKO= mêçÄ~Äáäáíó=cçêãìä~ë==319= = = =
==
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Chapter 1
Number Sets ====
1.1 Set Identities =pÉíëW=^I=_I=`=råáîÉêë~ä=ëÉíW=f=`çãéäÉãÉåí=W= ^′ =mêçéÉê=ëìÄëÉíW= _^⊂ ==bãéíó=ëÉíW=∅ =råáçå=çÑ=ëÉíëW= _^∪ =fåíÉêëÉÅíáçå=çÑ=ëÉíëW= _^∩ =aáÑÑÉêÉåÅÉ=çÑ=ëÉíëW= _y^ ==
=1. f^⊂ =
=2. ^^⊂ =
=3. _^ = =áÑ= _^⊂ =~åÇ= ^_⊂ .=
=4. bãéíó=pÉí=
^⊂∅ ==
5. råáçå=çÑ=pÉíë=={ }_ñçê^ñöñ_^` ∈∈=∪= =
=
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2
===== ==
Figure 1. =
6. `çããìí~íáîáíó=^__^ ∪=∪ =
=7. ^ëëçÅá~íáîáíó=
( ) ( ) `_^`_^ ∪∪=∪∪ ==8. fåíÉêëÉÅíáçå=çÑ=pÉíë=
{ }_ñ~åÇ^ñöñ_^` ∈∈=∪= = ==
===== ==
Figure 2. =
9. `çããìí~íáîáíó=^__^ ∩=∩ =
=10. ^ëëçÅá~íáîáíó=
( ) ( ) `_^`_^ ∩∩=∩∩ ==
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3
11. aáëíêáÄìíáîáíó=( ) ( ) ( )`^_^`_^ ∪∩∪=∩∪ I=( ) ( ) ( )`^_^`_^ ∩∪∩=∪∩ K=
=12. fÇÉãéçíÉåÅó=
^^^ =∩ I==^^^ =∪ =
=13. açãáå~íáçå=
∅=∅∩^ I=ff^ =∪ =
=14. fÇÉåíáíó=
^^ =∅∪ I==^f^ =∩
=15. `çãéäÉãÉåí=
{ }^ñöfñ^ ∉∈=′ =
16. `çãéäÉãÉåí=çÑ=fåíÉêëÉÅíáçå=~åÇ=råáçå f^^ =′∪ I==∅=′∩^^ =
=17. aÉ=jçêÖ~å∞ë=i~ïë
( ) _^_^ ′∩′=′∪ I==
( ) _^_^ ′∪′=′∩ ==
18. aáÑÑÉêÉåÅÉ=çÑ=pÉíë { }^ñ~åÇ_ñöñ^y_` ∉∈== =
=
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4
===== ==
Figure 3. =
19. ( )_^y_^y_ ∩= =
20. ^_^y_ ′∩= =
21. ∅=^y^ =
22. ^_y^ = =áÑ= ∅=∩_^ . =
===== ==
Figure 4. =
23. ( ) ( ) ( )`_y`^`_y^ ∩∩=∩ 24. ^yf^ =′ 25. `~êíÉëá~å=mêçÇìÅí
( ){ }_ó~åÇ^ñöóIñ_^` ∈∈=×= ==
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5
1.2 Sets of Numbers =k~íìê~ä=åìãÄÉêëW=k=tÜçäÉ=åìãÄÉêëW= Mk =
fåíÉÖÉêëW=w=mçëáíáîÉ=áåíÉÖÉêëW= +w =kÉÖ~íáîÉ=áåíÉÖÉêëW= −w =o~íáçå~ä=åìãÄÉêëW=n=oÉ~ä=åìãÄÉêëW=o==`çãéäÉñ=åìãÄÉêëW=`====
26. k~íìê~ä=kìãÄÉêë `çìåíáåÖ=åìãÄÉêëW { }KIPIOINk = K=
27. tÜçäÉ=kìãÄÉêë
`çìåíáåÖ=åìãÄÉêë=~åÇ=òÉêçW= { }KIPIOINIMkM = K=
=28. fåíÉÖÉêë
tÜçäÉ=åìãÄÉêë=~åÇ=íÜÉáê=çééçëáíÉë=~åÇ=òÉêçW={ }KIPIOINkw ==+ I=
{ }NIOIPIw −−−=− K I=
{ } { }KK IPIOINIMINIOIPIwMww −−−=∪∪= +− K==29. o~íáçå~ä=kìãÄÉêë
oÉéÉ~íáåÖ=çê=íÉêãáå~íáåÖ=ÇÉÅáã~äëW==
≠∈∈== MÄ~åÇwÄ~åÇw~~åÇ
Ä
~ñöñn K=
=30. fêê~íáçå~ä=kìãÄÉêë
kçåêÉéÉ~íáåÖ=~åÇ=åçåíÉêãáå~íáåÖ=ÇÉÅáã~äëK =
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6
31. oÉ~ä=kìãÄÉêë==råáçå=çÑ=ê~íáçå~ä=~åÇ=áêê~íáçå~ä=åìãÄÉêëW=oK=
=32. `çãéäÉñ=kìãÄÉêë
{ }oó~åÇoñöáóñ` ∈∈+= I==ïÜÉêÉ=á=áë=íÜÉ=áã~Öáå~êó=ìåáíK
=33. `onwk ⊂⊂⊂⊂ =
=
=== ==
Figure 5. ======
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7
1.3 Basic Identities =oÉ~ä=åìãÄÉêëW=~I=ÄI=Å=
==
34. ^ÇÇáíáîÉ=fÇÉåíáíó=~M~ =+ =
=35. ^ÇÇáíáîÉ=fåîÉêëÉ=
( ) M~~ =−+ ==
36. `çããìí~íáîÉ=çÑ=^ÇÇáíáçå=~ÄÄ~ +=+ =
=37. ^ëëçÅá~íáîÉ=çÑ=^ÇÇáíáçå=
( ) ( )ÅÄ~ÅÄ~ ++=++ ==38. aÉÑáåáíáçå=çÑ=pìÄíê~Åíáçå=
( )Ä~Ä~ −+=− ==
39. jìäíáéäáÅ~íáîÉ=fÇÉåíáíó=~N~ =⋅ =
=40. jìäíáéäáÅ~íáîÉ=fåîÉêëÉ=
N~
N~ =⋅ I= M~ ≠
=41. jìäíáéäáÅ~íáçå=qáãÉë=M
MM~ =⋅ =
42. `çããìí~íáîÉ=çÑ=jìäíáéäáÅ~íáçå=~ÄÄ~ ⋅=⋅
==
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8
43. ^ëëçÅá~íáîÉ=çÑ=jìäíáéäáÅ~íáçå=( ) ( )ÅÄ~ÅÄ~ ⋅⋅=⋅⋅
=44. aáëíêáÄìíáîÉ=i~ï=
( ) ~Å~ÄÅÄ~ +=+ ==45. aÉÑáåáíáçå=çÑ=aáîáëáçå=
Ä
N~
Ä
~⋅= =
===
1.4 Complex Numbers =k~íìê~ä=åìãÄÉêW=å=fã~Öáå~êó=ìåáíW=á=`çãéäÉñ=åìãÄÉêW=ò=oÉ~ä=é~êíW=~I=Å=fã~Öáå~êó=é~êíW=ÄáI=Çá=jçÇìäìë=çÑ=~=ÅçãéäÉñ=åìãÄÉêW=êI= Nê I= Oê =^êÖìãÉåí=çÑ=~=ÅçãéäÉñ=åìãÄÉêW=ϕ I= Nϕ I= Oϕ =
==
ááN = = ááR = = áá NåQ =+ =NáO −= = NáS −= = Ná OåQ −=+ =ááP −= = ááT −= = áá PåQ −=+ =
46.
NáQ = = NáU = = Ná åQ = ==47. Äá~ò += ==48. `çãéäÉñ=mä~åÉ=
=
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9
===== ==
Figure 6. =
49. ( ) ( ) ( ) ( )áÇÄÅ~ÇáÅÄá~ +++=+++ ==50. ( ) ( ) ( ) ( )áÇÄÅ~ÇáÅÄá~ −+−=+−+ ==51. ( )( ) ( ) ( )áÄÅ~ÇÄÇ~ÅÇáÅÄá~ ++−=++ ==
52. áÇÅ
~ÇÄÅ
ÇÅ
ÄÇ~Å
ÇáÅ
Äá~OOOO⋅
+−
+++
=++
=
=53. `çåàìÖ~íÉ=`çãéäÉñ=kìãÄÉêë=
Äá~Äá~|||||||
−=+ ==54. ϕ= Åçëê~ I= ϕ= ëáåêÄ ==
=
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10
==
Figure 7. =
55. mçä~ê=mêÉëÉåí~íáçå=çÑ=`çãéäÉñ=kìãÄÉêë=( )ϕ+ϕ=+ ëáåáÅçëêÄá~ =
=56. jçÇìäìë=~åÇ=^êÖìãÉåí=çÑ=~=`çãéäÉñ=kìãÄÉê=
fÑ= Äá~ + =áë=~=ÅçãéäÉñ=åìãÄÉêI=íÜÉå=OO Ä~ê += =EãçÇìäìëFI==
~
Ä~êÅí~å=ϕ =E~êÖìãÉåíFK=
=57. mêçÇìÅí=áå=mçä~ê=oÉéêÉëÉåí~íáçå=
( ) ( )OOONNNON ëáåáÅçëêëáåáÅçëêòò ϕ+ϕ⋅ϕ+ϕ=⋅ =( ) ( )[ ]ONONON ëáåáÅçëêê ϕ+ϕ+ϕ+ϕ= =
=58. `çåàìÖ~íÉ=kìãÄÉêë=áå=mçä~ê=oÉéêÉëÉåí~íáçå=
( ) ( ) ( )[ ]ϕ−+ϕ−=ϕ+ϕ ëáåáÅçëêëáåáÅçëê|||||||||||||||||||||
==59. fåîÉêëÉ=çÑ=~=`çãéäÉñ=kìãÄÉê=áå=mçä~ê=oÉéêÉëÉåí~íáçå=
( ) ( ) ( )[ ]ϕ−+ϕ−=ϕ+ϕ
ëáåáÅçëê
N
ëáåáÅçëê
N=
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CHAPTER 1. NUMBER SETS
11
60. nìçíáÉåí=áå=mçä~ê=oÉéêÉëÉåí~íáçå=( )( ) ( ) ( )[ ]ONON
O
N
OOO
NNN
O
N ëáåáÅçëê
ê
ëáåáÅçëê
ëáåáÅçëê
ò
òϕ−ϕ+ϕ−ϕ=
ϕ+ϕϕ+ϕ
= =
=61. mçïÉê=çÑ=~=`çãéäÉñ=kìãÄÉê=
( )[ ] ( ) ( )[ ]ϕ+ϕ=ϕ+ϕ= åëáåáåÅçëêëáåáÅçëêò ååå ==62. cçêãìä~=±aÉ=jçáîêÉ≤=
( ) ( ) ( )ϕ+ϕ=ϕ+ϕ åëáåáåÅçëëáåáÅçë å ==63. kíÜ=oççí=çÑ=~=`çãéäÉñ=kìãÄÉê=
( )
π+ϕ
+π+ϕ
=ϕ+ϕ=å
âOëáåá
å
âOÅçëêëáåáÅçëêò ååå I==
ïÜÉêÉ==NåIIOINIMâ −= K K==
=64. bìäÉê∞ë=cçêãìä~=
ñëáåáñÅçëÉáñ += ===
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12
Chapter 2
Algebra ====
2.1 Factoring Formulas =oÉ~ä=åìãÄÉêëW=~I=ÄI=Å==k~íìê~ä=åìãÄÉêW=å=
==
65. ( )( )Ä~Ä~Ä~ OO −+=− ==
66. ( )( )OOPP Ä~Ä~Ä~Ä~ ++−=− ==67. ( )( )OOPP Ä~Ä~Ä~Ä~ +−+=+ ==68. ( )( ) ( )( )( )OOOOOOQQ Ä~Ä~Ä~Ä~Ä~Ä~ ++−=+−=− ==
69. ( )( )QPOOPQRR Ä~ÄÄ~Ä~~Ä~Ä~ ++++−=− ==70. ( )( )QPOOPQRR Ä~ÄÄ~Ä~~Ä~Ä~ +−+−+=+ ==71. fÑ=å=áë=çÇÇI=íÜÉå=
( )( )NåOåOPåOåNååå Ä~ÄÄ~Ä~~Ä~Ä~ −−−−− +−−+−+=+ K K===72. fÑ=å=áë=ÉîÉåI=íÜÉå==
( )( )NåOåOPåOåNååå Ä~ÄÄ~Ä~~Ä~Ä~ −−−−− +++++−=− K I==
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CHAPTER 2. ALGEBRA
13
( )( )NåOåOPåOåNååå Ä~ÄÄ~Ä~~Ä~Ä~ −−−−− −+−+−+=+ K K====
2.2 Product Formulas oÉ~ä=åìãÄÉêëW=~I=ÄI=Å==tÜçäÉ=åìãÄÉêëW=åI=â===
73. ( ) OOO Ä~ÄO~Ä~ +−=− ==
74. ( ) OOO Ä~ÄO~Ä~ ++=+ ==
75. ( ) POOPP Ä~ÄPÄ~P~Ä~ −+−=− ==
76. ( ) POOPP Ä~ÄPÄ~P~Ä~ +++=+ ==
77. ( ) QPOOPQQ Ä~ÄQÄ~SÄ~Q~Ä~ +−+−=− ==
78. ( ) QPOOPQQ Ä~ÄQÄ~SÄ~Q~Ä~ ++++=+ ==79. _áåçãá~ä=cçêãìä~=
( ) IÄ`~Ä`Ä~`Ä~`~`Ä~ åå
åNåNå
åOOåO
åNåN
ååM
åå +++++=+ −−
−− K
ïÜÉêÉ= ( )>âå>â
>å`â
å
−= =~êÉ=íÜÉ=Äáåçãá~ä=ÅçÉÑÑáÅáÉåíëK=
=
80. ( ) ÄÅO~ÅO~ÄOÅÄ~ÅÄ~ OOOO +++++=++ ==
81. ( ) ++++++=+++++ OOOOOO îìÅÄ~îìÅÄ~ KK =( )ìîÄîÄìÄÅ~î~ì~Å~ÄO +++++++++++ KKK =
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CHAPTER 2. ALGEBRA
14
2.3 Powers =_~ëÉë=EéçëáíáîÉ=êÉ~ä=åìãÄÉêëFW=~I=Ä==mçïÉêë=Eê~íáçå~ä=åìãÄÉêëFW=åI=ã=
==
82. åãåã ~~~ += ==
83. åã
å
ã
~~
~ −= =
=
84. ( ) ããã Ä~~Ä = ==
85. ã
ãã
Ä
~
Ä
~=
=
=
86. ( ) ãååã ~~ = ==87. N~M = I= M~ ≠ ==88. N~N = ==
89. ã
ã
~
N~ =− =
=
90. å ãå
ã
~~ = ======
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CHAPTER 2. ALGEBRA
15
2.4 Roots =_~ëÉëW=~I=Ä==mçïÉêë=Eê~íáçå~ä=åìãÄÉêëFW=åI=ã=
MÄI~ ≥ =Ñçê=ÉîÉå=êççíë=E âOå = I= kâ∈ F===
91. ååå Ä~~Ä = ==
92. åã åããå Ä~Ä~ = ==
93. å
å
å
Ä
~
Ä
~= I= MÄ ≠ =
=
94. åãå
ã
åã å
åã ã
ã
å
Ä
~
Ä
~
Ä
~== I= MÄ ≠ K=
=
95. ( ) å ãéé
å ã ~~ = ==
96. ( ) ~~åå = =
=
97. åé ãéå ã ~~ = ==
98. å
ãå ã ~~ = =
=
99. ãåã å ~~ = ==
100. ( ) å ããå ~~ = =
=
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CHAPTER 2. ALGEBRA
16
101. ~
~
~
N å Nå
å
−
= I= M~ ≠ K=
=
102. O
Ä~~
O
Ä~~Ä~
OO −−±
−+=± =
=
103. Ä~
Ä~
Ä~
N
−=
±m
=
===
2.5 Logarithms =mçëáíáîÉ=êÉ~ä=åìãÄÉêëW=ñI=óI=~I=ÅI=â=k~íìê~ä=åìãÄÉêW=å====
104. aÉÑáåáíáçå=çÑ=içÖ~êáíÜã=ñäçÖó ~= =áÑ=~åÇ=çåäó=áÑ= ó~ñ = I= M~ > I= N~ ≠ K=
=105. MNäçÖ~ = =
=106. N~äçÖ~ = =
=
107.
<∞+>∞−
=N~áÑ
N~áÑMäçÖ~ =
=108. ( ) óäçÖñäçÖñóäçÖ ~~~ += =
=
109. óäçÖñäçÖó
ñäçÖ ~~~ −= =
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CHAPTER 2. ALGEBRA
17
110. ( ) ñäçÖåñäçÖ ~å
~ = =
=
111. ñäçÖå
NñäçÖ ~
å~ = =
=
112. ÅäçÖñäçÖ~äçÖ
ñäçÖñäçÖ ~Å
Å
Å~ ⋅== I= MÅ > I= NÅ ≠ K=
=
113. ~äçÖ
NÅäçÖ
Å~ = =
=114. ñäçÖ~~ñ = =
=115. içÖ~êáíÜã=íç=_~ëÉ=NM=
ñäçÖñäçÖNM = =
=116. k~íìê~ä=içÖ~êáíÜã=
ñäåñäçÖÉ = I==
ïÜÉêÉ= KTNUOUNUOUKOâ
NNäáãÉ
â
â=
+=
∞→=
=
117. ñäåQPQOVQKMñäåNMäå
NñäçÖ == =
=
118. ñäçÖPMORURKOñäçÖÉäçÖ
Nñäå == =
=====
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CHAPTER 2. ALGEBRA
18
2.6 Equations =oÉ~ä=åìãÄÉêëW=~I=ÄI=ÅI=éI=èI=ìI=î=pçäìíáçåëW= Nñ I= Oñ I= Nó I= Oó I= Pó =
==
119. iáåÉ~ê=bèì~íáçå=áå=låÉ=s~êá~ÄäÉ=
MÄ~ñ =+ I=~
Äñ −= K==
=120. nì~Çê~íáÅ=bèì~íáçå=
MÅÄñ~ñO =++ I=~O
~ÅQÄÄñ
O
OIN
−±−= K=
=121. aáëÅêáãáå~åí=
~ÅQÄa O −= ==
122. sáÉíÉ∞ë=cçêãìä~ë=
fÑ= MèéññO =++ I=íÜÉå==
=−=+
èññ
éññ
ON
ON K=
=
123. MÄñ~ñO =+ I= MñN = I=~
ÄñO −= K=
=
124. MÅ~ñ O =+ I=~
Åñ OIN −±= K=
=125. `ìÄáÅ=bèì~íáçåK=`~êÇ~åç∞ë=cçêãìä~K==
MèéóóP =++ I==
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CHAPTER 2. ALGEBRA
19
îìóN += I= ( ) ( ) áîìO
Pîì
O
Nó PIO +±+−= I==
ïÜÉêÉ==
P
OO
P
é
O
è
O
èì
+
+−= I= P
OO
P
é
O
è
O
èî
+
−−= K==
==
2.7 Inequalities s~êá~ÄäÉëW=ñI=óI=ò=
oÉ~ä=åìãÄÉêëW=
åPON ~II~I~I~
ÇIÅIÄI~
KI=ãI=å=
aÉíÉêãáå~åíëW=aI= ña I= óa I= òa ==
==
126. fåÉèì~äáíáÉëI=fåíÉêî~ä=kçí~íáçåë=~åÇ=dê~éÜë===
fåÉèì~äáíó= fåíÉêî~ä=kçí~íáçå= dê~éÜ=Äñ~ ≤≤ = [ ]ÄI~ =
=Äñ~ ≤< = ( ]ÄI~ =
=Äñ~ <≤ = [ )ÄI~ =
=Äñ~ << = ( )ÄI~ =
=Äñ ≤<∞− I=
Äñ ≤ =( ]ÄI∞− =
=Äñ <<∞− I=
Äñ < =( )ÄI∞− =
=∞<≤ ñ~ I=
~ñ ≥ =[ )∞I~ =
=∞<< ñ~ I=
~ñ > =( )∞I~ =
=
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CHAPTER 2. ALGEBRA
20
127. fÑ= Ä~ > I=íÜÉå= ~Ä < K==128. fÑ= Ä~ > I=íÜÉå= MÄ~ >− =çê= M~Ä <− K==129. fÑ= Ä~ > I=íÜÉå= ÅÄÅ~ +>+ K==130. fÑ= Ä~ > I=íÜÉå= ÅÄÅ~ −>− K==131. fÑ= Ä~ > =~åÇ= ÇÅ > I=íÜÉå= ÇÄÅ~ +>+ K==132. fÑ= Ä~ > =~åÇ= ÇÅ > I=íÜÉå= ÅÄÇ~ −>− K==133. fÑ= Ä~ > =~åÇ= Mã > I=íÜÉå= ãÄã~ > K==
134. fÑ= Ä~ > =~åÇ= Mã > I=íÜÉå=ã
Ä
ã
~> K=
=135. fÑ= Ä~ > =~åÇ= Mã < I=íÜÉå= ãÄã~ < K==
136. fÑ= Ä~ > =~åÇ= Mã < I=íÜÉå=ã
Ä
ã
~< K=
=137. fÑ= Ä~M << =~åÇ= Må > I=íÜÉå= åå Ä~ < K==138. fÑ= Ä~M << =~åÇ= Må < I=íÜÉå= åå Ä~ > K==
139. fÑ= Ä~M << I=íÜÉå= åå Ä~ < K==
140. O
Ä~~Ä
+≤ I==
ïÜÉêÉ= M~ > =I= MÄ > X=~å=Éèì~äáíó=áë=î~äáÇ=çåäó=áÑ= Ä~ = K===
141. O~
N~ ≥+ I=ïÜÉêÉ= M~ > X=~å=Éèì~äáíó=í~âÉë=éä~ÅÉ=çåäó=~í= N~ = K=
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CHAPTER 2. ALGEBRA
21
142. å
~~~~~~ åONå
åON
+++≤
KK I=ïÜÉêÉ= M~II~I~ åON >K K=
=
143. fÑ= MÄ~ñ >+ =~åÇ= M~ > I=íÜÉå=~
Äñ −> K=
=
144. fÑ= MÄ~ñ >+ =~åÇ= M~ < I=íÜÉå=~
Äñ −< K==
=145. MÅÄñ~ñ O >++ ==
= M~ > = M~ < ====
Ma> =
=
=
Nññ < I= Oññ > =
=
==
ON ñññ << =
===
Ma= =
=
ññN < I= Nññ > =
==∅∈ñ =
===
Ma< =
=
=∞<<∞− ñ =
===∅∈ñ =
=
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CHAPTER 2. ALGEBRA
22
146. Ä~Ä~ +≤+ =
=147. fÑ= ~ñ < I=íÜÉå= ~ñ~ <<− I=ïÜÉêÉ= M~ > K=
=148. fÑ= ~ñ > I=íÜÉå= ~ñ −< =~åÇ= ~ñ > I=ïÜÉêÉ= M~ > K=
=
149. fÑ= ~ñO < I=íÜÉå= ~ñ < I=ïÜÉêÉ= M~ > K=
=
150. fÑ= ~ñO > I=íÜÉå= ~ñ > I=ïÜÉêÉ= M~ > K=
=
151. fÑ=( )( ) MñÖ
ñÑ> I=íÜÉå=
( ) ( )( )
≠>⋅
MñÖ
MñÖñÑK=
=
152. ( )( ) MñÖ
ñÑ< I=íÜÉå=
( ) ( )( )
≠<⋅
MñÖ
MñÖñÑK=
===
2.8 Compound Interest Formulas =cìíìêÉ=î~äìÉW=^=fåáíá~ä=ÇÉéçëáíW=`=^ååì~ä=ê~íÉ=çÑ=áåíÉêÉëíW=ê=kìãÄÉê=çÑ=óÉ~êë=áåîÉëíÉÇW=í=kìãÄÉê=çÑ=íáãÉë=ÅçãéçìåÇÉÇ=éÉê=óÉ~êW=å=
==
153. dÉåÉê~ä=`çãéçìåÇ=fåíÉêÉëí=cçêãìä~=åí
å
êN`^
+= =
=
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CHAPTER 2. ALGEBRA
23
154. páãéäáÑáÉÇ=`çãéçìåÇ=fåíÉêÉëí=cçêãìä~=fÑ=áåíÉêÉëí=áë=ÅçãéçìåÇÉÇ=çåÅÉ=éÉê=óÉ~êI=íÜÉå=íÜÉ=éêÉîáçìë=Ñçêãìä~=ëáãéäáÑáÉë=íçW=
( )íêN`^ += K==155. `çåíáåìçìë=`çãéçìåÇ=fåíÉêÉëí=
fÑ=áåíÉêÉëí=áë=ÅçãéçìåÇÉÇ=Åçåíáåì~ääó=E ∞→å FI=íÜÉå==êí`É^ = K=
==
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24
Chapter 3
Geometry ====
3.1 Right Triangle =iÉÖë=çÑ=~=êáÖÜí=íêá~åÖäÉW=~I=Ä=eóéçíÉåìëÉW=Å=^äíáíìÇÉW=Ü=jÉÇá~åëW= ~ã I= Äã I= Åã =
^åÖäÉëW=α Iβ =o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=^êÉ~W=p===
==
Figure 8. =
156. °=β+α VM ==
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CHAPTER 3. GEOMETRY
25
157. β==α ÅçëÅ
~ëáå =
=
158. β==α ëáåÅ
ÄÅçë =
=
159. β==α ÅçíÄ
~í~å =
=
160. β==α í~å~
ÄÅçí =
=
161. β==α ÉÅÅçëÄ
ÅëÉÅ =
=
162. β==α ëÉÅ~
ÅÉÅÅçë =
=163. móíÜ~ÖçêÉ~å=qÜÉçêÉã=
OOO ÅÄ~ =+ ==
164. ÑÅ~O = I= ÖÅÄO = I==
ïÜÉêÉ= Ñ= ~åÇ= Å= ~êÉ= éêçàÉÅíáçåë= çÑ= íÜÉ= äÉÖë= ~= ~åÇ= ÄI= êÉëéÉÅ-íáîÉäóI=çåíç=íÜÉ=ÜóéçíÉåìëÉ=ÅK==
===== ==
Figure 9. =
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CHAPTER 3. GEOMETRY
26
165. ÑÖÜO = I===ïÜÉêÉ=Ü=áë=íÜÉ=~äíáíìÇÉ=Ñêçã=íÜÉ=êáÖÜí=~åÖäÉK==
=
166. Q
~Äã
OOO
~ −= I=Q
Ä~ã
OOO
Ä −= I===
ïÜÉêÉ= ~ã =~åÇ= Äã =~êÉ=íÜÉ=ãÉÇá~åë=íç=íÜÉ=äÉÖë=~=~åÇ=ÄK==
=
==
Figure 10. =
167. O
ÅãÅ = I==
ïÜÉêÉ= Åã =áë=íÜÉ=ãÉÇá~å=íç=íÜÉ=ÜóéçíÉåìëÉ=ÅK=
=
168. ÅãO
Åo == =
=
169. ÅÄ~
~Ä
O
ÅÄ~ê
++=
−+= =
=170. ÅÜ~Ä = =
==
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CHAPTER 3. GEOMETRY
27
171. O
ÅÜ
O
~Äp == =
===
3.2 Isosceles Triangle =_~ëÉW=~=iÉÖëW=Ä=_~ëÉ=~åÖäÉW=β =sÉêíÉñ=~åÖäÉW=α =^äíáíìÇÉ=íç=íÜÉ=Ä~ëÉW=Ü=mÉêáãÉíÉêW=i=^êÉ~W=p=
==
==
Figure 11. =
172. O
VMα
−°=β =
=
173. Q
~ÄÜ
OOO −= =
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CHAPTER 3. GEOMETRY
28
174. ÄO~i += ==
175. α== ëáåO
Ä
O
~Üp
O
=
===
3.3 Equilateral Triangle =páÇÉ=çÑ=~=Éèìáä~íÉê~ä=íêá~åÖäÉW=~=^äíáíìÇÉW=Ü=o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=mÉêáãÉíÉêW=i=^êÉ~W=p===
==
Figure 12. =
176. O
P~Ü = =
=
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CHAPTER 3. GEOMETRY
29
177. P
P~Ü
P
Oo == =
=
178. O
o
S
P~Ü
P
Nê === =
=179. ~Pi = =
=
180. Q
P~
O
~Üp
O
== =
===
3.4 Scalene Triangle E^=íêá~åÖäÉ=ïáíÜ=åç=íïç=ëáÇÉë=Éèì~äF=
==páÇÉë=çÑ=~=íêá~åÖäÉW=~I=ÄI=Å=
pÉãáéÉêáãÉíÉêW=O
ÅÄ~é
++= ==
^åÖäÉë=çÑ=~=íêá~åÖäÉW= γβα II =^äíáíìÇÉë=íç=íÜÉ=ëáÇÉë=~I=ÄI=ÅW= ÅÄ~ ÜIÜIÜ =
jÉÇá~åë=íç=íÜÉ=ëáÇÉë=~I=ÄI=ÅW= ÅÄ~ ãIãIã =
_áëÉÅíçêë=çÑ=íÜÉ=~åÖäÉë= γβα II W= ÅÄ~ íIíIí =
o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=^êÉ~W=p===
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CHAPTER 3. GEOMETRY
30
===== ==
Figure 13. =
181. °=γ+β+α NUM ==
182. ÅÄ~ >+ I==~ÅÄ >+ I==ÄÅ~ >+ K=
=183. ÅÄ~ <− I==
~ÅÄ <− I==
ÄÅ~ <− K=
=184. jáÇäáåÉ=
O
~è = I= ~ööè K=
=
===== ==
Figure 14. =
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CHAPTER 3. GEOMETRY
31
185. i~ï=çÑ=`çëáåÉë=
α−+= ÅçëÄÅOÅÄ~ OOO I=
β−+= Åçë~ÅOÅ~Ä OOO I=
γ−+= Åçë~ÄOÄ~Å OOO K==
186. i~ï=çÑ=páåÉë=
oOëáå
Å
ëáå
Ä
ëáå
~=
γ=
β=
αI==
ïÜÉêÉ=o=áë=íÜÉ=ê~Çáìë=çÑ=íÜÉ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉK===
187. pQ
~ÄÅ
ÜO
~Ä
ÜO
~Å
ÜO
ÄÅ
ëáåO
Å
ëáåO
Ä
ëáåO
~o
ÅÄ~
====γ
=β
=α
= =
=
188. ( )( )( )é
ÅéÄé~éêO −−−= I==
ÅÄ~ Ü
N
Ü
N
Ü
N
ê
N++= K=
=
189. ( )( )ÄÅ
ÅéÄé
Oëáå
−−=
αI=
( )ÄÅ
~éé
OÅçë
−=
αI=
( )( )( )~éé
ÅéÄé
Oí~å
−−−
=α
K=
=
190. ( )( )( )ÅéÄé~éé~
OÜ~ −−−= I=
( )( )( )ÅéÄé~ééÄ
OÜÄ −−−= I=
( )( )( )ÅéÄé~ééÅ
OÜÅ −−−= K=
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CHAPTER 3. GEOMETRY
32
191. β=γ= ëáåÅëáåÄÜ~ I=
α=γ= ëáåÅëáå~ÜÄ I=
α=β= ëáåÄëáå~ÜÅ K=
=
192. Q
~
O
ÅÄã
OOOO~ −
+= I==
Q
Ä
O
Å~ã
OOOOÄ −
+= I==
Q
Å
O
Ä~ã
OOOOÅ −
+= K=
=
===== ==
Figure 15. =
193. ~ãP
O^j = I= Äã
P
O_j = I= Åã
P
O`j = =EcáÖKNRFK=
=
194. ( )( )O
O~
ÅÄ
~éÄÅéQí
+−
= I==
( )( )O
OÄ
Å~
Äé~ÅéQí
+−
= I==
( )( )O
OÅ
Ä~
Åé~ÄéQí
+−
= K=
=
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CHAPTER 3. GEOMETRY
33
195. O
ÅÜ
O
ÄÜ
O
~Üp ÅÄ~ === I==
O
ëáåÄÅ
O
ëáå~Å
O
ëáå~Äp
α=
β=
γ= I==
( )( )( )ÅéÄé~éép −−−= =EeÉêçå∞ë=cçêãìä~FI=
éêp = I==
oQ
~ÄÅp = I=
γβα= ëáåëáåëáåoOp O I=
Oí~å
Oí~å
Oí~åép O γβα
= K=
===
3.5 Square páÇÉ=çÑ=~=ëèì~êÉW=~=aá~Öçå~äW=Ç=o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=mÉêáãÉíÉêW=i=^êÉ~W=p==
==
Figure 16.
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CHAPTER 3. GEOMETRY
34
196. O~Ç = ===
197. O
O~
O
Ço == =
=
198. O
~ê = =
=199. ~Qi = =
=
200. O~p = ====
3.6 Rectangle =páÇÉë=çÑ=~=êÉÅí~åÖäÉW=~I=Ä=aá~Öçå~äW=Ç=o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=mÉêáãÉíÉêW=i=^êÉ~W=p===
==
Figure 17. =
201. OO Ä~Ç += ==
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CHAPTER 3. GEOMETRY
35
202. O
Ço = =
=203. ( )Ä~Oi += =
=204. ~Äp = =
===
3.7 Parallelogram =páÇÉë=çÑ=~=é~ê~ääÉäçÖê~ãW=~I=Ä=aá~Öçå~äëW= ON ÇIÇ =`çåëÉÅìíáîÉ=~åÖäÉëW= βαI =^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ =^äíáíìÇÉW=Ü==mÉêáãÉíÉêW=i=^êÉ~W=p===
===== ==
Figure 18. =
205. °=β+α NUM ==
206. ( )OOOO
ON Ä~OÇÇ +=+ =
=
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CHAPTER 3. GEOMETRY
36
207. β=α= ëáåÄëáåÄÜ ==
208. ( )Ä~Oi += ==
209. α== ëáå~Ä~Üp I==
ϕ= ëáåÇÇO
Np ON K=
===
3.8 Rhombus =páÇÉ=çÑ=~=êÜçãÄìëW=~=aá~Öçå~äëW= ON ÇIÇ =`çåëÉÅìíáîÉ=~åÖäÉëW= βαI =^äíáíìÇÉW=e=o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=mÉêáãÉíÉêW=i=^êÉ~W=p===
===== ==
Figure 19. =
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CHAPTER 3. GEOMETRY
37
210. °=β+α NUM ==
211. OOO
ON ~QÇÇ =+ =
=
212. ~O
ÇÇëáå~Ü ON=α= =
=
213. O
ëáå~
~Q
ÇÇ
O
Üê ON α
=== =
=214. ~Qi = =
=
215. α== ëáå~~Üp O I==
ONÇÇO
Np = K=
===
3.9 Trapezoid =_~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä=jáÇäáåÉW=è=^äíáíìÇÉW=Ü=^êÉ~W=p===
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CHAPTER 3. GEOMETRY
38
==
Figure 20. =
216. O
Ä~è
+= =
=
217. èÜÜO
Ä~p =⋅
+= =
===
3.10 Isosceles Trapezoid =_~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä=iÉÖW=Å=jáÇäáåÉW=è=^äíáíìÇÉW=Ü=aá~Öçå~äW=Ç=o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=^êÉ~W=p===
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CHAPTER 3. GEOMETRY
39
==
Figure 21. =
218. O
Ä~è
+= =
=
219. OÅ~ÄÇ += ==
220. ( )OO ~ÄQ
NÅÜ −−= =
=
221. ( )( )Ä~ÅOÄ~ÅO
Å~ÄÅo
O
−++−+
= =
=
222. èÜÜO
Ä~p =⋅
+= =
======
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CHAPTER 3. GEOMETRY
40
3.11 Isosceles Trapezoid with Inscribed Circle
=_~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä=iÉÖW=Å=jáÇäáåÉW=è=^äíáíìÇÉW=Ü=aá~Öçå~äW=Ç=o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=o=o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=ê=mÉêáãÉíÉêW=i=^êÉ~W=p===
==
Figure 22. =
223. ÅOÄ~ =+ ==
224. ÅO
Ä~è =
+= =
=225. OOO ÅÜÇ += =
=
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CHAPTER 3. GEOMETRY
41
226. O
~Ä
O
Üê == =
=
227. ~
ÄS
Ä
~
U
Ä~ÅÜ
ÜO
Å
~Ä
ÅN
O
Å
êQ
ÅÇ
ÜO
ÅÇo OO
O
+++
=+=+=== =
=228. ( ) ÅQÄ~Oi =+= =
=
229. ( )O
iêÅÜèÜ
O
~ÄÄ~Ü
O
Ä~p ===
+=⋅
+= ==
===
3.12 Trapezoid with Inscribed Circle =_~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä=i~íÉê~ä=ëáÇÉëW=ÅI=Ç=jáÇäáåÉW=è=^äíáíìÇÉW=Ü=aá~Öçå~äëW= ON ÇIÇ =^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ =o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=mÉêáãÉíÉêW=i=^êÉ~W=p==
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CHAPTER 3. GEOMETRY
42
==
Figure 23. =
230. ÇÅÄ~ +=+ ==
231. O
ÇÅ
O
Ä~è
+=
+= =
=232. ( ) ( )ÇÅOÄ~Oi +=+= =
=
233. èÜÜO
ÇÅÜ
O
Ä~p =⋅
+=⋅
+= I==
ϕ= ëáåÇÇO
Np ON K=
===
3.13 Kite =páÇÉë=çÑ=~=âáíÉW=~I=Ä=aá~Öçå~äëW= ON ÇIÇ =^åÖäÉëW= γβα II =mÉêáãÉíÉêW=i=^êÉ~W=p===
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CHAPTER 3. GEOMETRY
43
==
Figure 24. =
234. °=γ+β+α PSMO ==
235. ( )Ä~Oi += ==
236. O
ÇÇp ON= =
===
3.14 Cyclic Quadrilateral páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç=aá~Öçå~äëW= ON ÇIÇ =^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ =fåíÉêå~ä=~åÖäÉëW= δγβα III =o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=mÉêáãÉíÉêW=i=pÉãáéÉêáãÉíÉêW=é==^êÉ~W=p=
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CHAPTER 3. GEOMETRY
44
==
Figure 25. =
237. °=δ+β=γ+α NUM ==
238. míçäÉãó∞ë=qÜÉçêÉã=
ONÇÇÄÇ~Å =+ ==
239. ÇÅÄ~i +++= ==
240. ( )( )( )( )( )( )( )ÇéÅéÄé~é
ÅÇ~ÄÄÅ~ÇÄÇ~Å
Q
No
−−−−+++
= I==
ïÜÉêÉ=O
ié = K=
=
241. ϕ= ëáåÇÇO
Np ON I==
( )( )( )( )ÇéÅéÄé~ép −−−−= I==
ïÜÉêÉ=O
ié = K=
===
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CHAPTER 3. GEOMETRY
45
3.15 Tangential Quadrilateral =páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç=aá~Öçå~äëW= ON ÇIÇ =^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ =o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=mÉêáãÉíÉêW=i=pÉãáéÉêáãÉíÉêW=é==^êÉ~W=p===
==
Figure 26. =
242. ÇÄÅ~ +=+ ==
243. ( ) ( )ÇÄOÅ~OÇÅÄ~i +=+=+++= ==
244. ( ) ( )
éO
éÄ~Ä~ÇÇê
OOOO
ON −+−−
= I==
ïÜÉêÉ=O
ié = K==
=
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CHAPTER 3. GEOMETRY
46
245. ϕ== ëáåÇÇO
Néêp ON =
===
3.16 General Quadrilateral =páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç=aá~Öçå~äëW= ON ÇIÇ =^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ =fåíÉêå~ä=~åÖäÉëW= δγβα III =mÉêáãÉíÉêW=i=^êÉ~W=p===
======= ==
Figure 27. =
246. °=δ+γ+β+α PSM ==
247. ÇÅÄ~i +++= ==
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CHAPTER 3. GEOMETRY
47
248. ϕ= ëáåÇÇO
Np ON =
===
3.17 Regular Hexagon =páÇÉW=~=fåíÉêå~ä=~åÖäÉW=α =pä~åí=ÜÉáÖÜíW=ã=o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=mÉêáãÉíÉêW=i=pÉãáéÉêáãÉíÉêW=é==^êÉ~W=p===
==
Figure 28. =
249. °=α NOM ==
250. O
P~ãê == =
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CHAPTER 3. GEOMETRY
48
251. ~o = ==
252. ~Si = ==
253. O
PP~éêp
O
== I==
ïÜÉêÉ=O
ié = K=
===
3.18 Regular Polygon =páÇÉW=~=kìãÄÉê=çÑ=ëáÇÉëW=å=fåíÉêå~ä=~åÖäÉW=α =pä~åí=ÜÉáÖÜíW=ã=o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=mÉêáãÉíÉêW=i=pÉãáéÉêáãÉíÉêW=é==^êÉ~W=p===
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CHAPTER 3. GEOMETRY
49
==
Figure 29. =
254. °⋅−
=α NUMO
Oå=
=
255. °⋅−
=α NUMO
Oå=
=
256.
åëáåO
~o
π= =
=
257. Q
~o
åí~åO
~ãê
OO −=
π== =
=258. å~i = =
=
259. å
Oëáå
O
åop
O π= I==
Q
~oééêp
OO −== I==
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CHAPTER 3. GEOMETRY
50
ïÜÉêÉ=O
ié = K==
===
3.19 Circle =o~ÇáìëW=o=aá~ãÉíÉêW=Ç=`ÜçêÇW=~=pÉÅ~åí=ëÉÖãÉåíëW=ÉI=Ñ=q~åÖÉåí=ëÉÖãÉåíW=Ö=`Éåíê~ä=~åÖäÉW=α =fåëÅêáÄÉÇ=~åÖäÉW=β =mÉêáãÉíÉêW=i=^êÉ~W=p===
260. O
ëáåoO~α
= =
=
==
Figure 30. =
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CHAPTER 3. GEOMETRY
51
261. ONON ÄÄ~~ = ==
==
Figure 31. =
262. NN ÑÑÉÉ = ==
===== ==
Figure 32. =
263. NO ÑÑÖ = =
=
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CHAPTER 3. GEOMETRY
52
===== ==
Figure 33. =
264. O
α=β =
=
==
Figure 34. =
265. ÇoOi π=π= ==
266. O
io
Q
Çop
OO =
π=π= ==
=
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CHAPTER 3. GEOMETRY
53
3.20 Sector of a Circle =o~Çáìë=çÑ=~=ÅáêÅäÉW=o=^êÅ=äÉåÖíÜW=ë=`Éåíê~ä=~åÖäÉ=Eáå=ê~Çá~åëFW=ñ=`Éåíê~ä=~åÖäÉ=Eáå=ÇÉÖêÉÉëFW=α =mÉêáãÉíÉêW=i=^êÉ~W=p===
==
Figure 35. =
267. oñë = ==
268. °απ
=NUM
oë =
=269. oOëi += =
=
270. °απ
===PSM
o
O
ño
O
oëp
OO
==
==
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CHAPTER 3. GEOMETRY
54
3.21 Segment of a Circle =o~Çáìë=çÑ=~=ÅáêÅäÉW=o=^êÅ=äÉåÖíÜW=ë=`ÜçêÇW=~=`Éåíê~ä=~åÖäÉ=Eáå=ê~Çá~åëFW=ñ=`Éåíê~ä=~åÖäÉ=Eáå=ÇÉÖêÉÉëFW=α =eÉáÖÜí=çÑ=íÜÉ=ëÉÖãÉåíW=Ü=mÉêáãÉíÉêW=i=^êÉ~W=p===
==
Figure 36. =
271. OÜÜoOO~ −= ==
272. OO ~oQO
NoÜ −−= I= oÜ < =
=273. ~ëi += =
=
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CHAPTER 3. GEOMETRY
55
274. ( )[ ] ( )ñëáåñO
oëáå
NUMO
oÜo~ëo
O
Np
OO
−=
α−
°απ
=−−= I==
Ü~P
Op ≈ K=
===
3.22 Cube =bÇÖÉW=~==aá~Öçå~äW=Ç=o~Çáìë=çÑ=áåëÅêáÄÉÇ=ëéÜÉêÉW=ê=o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ëéÜÉêÉW=ê=pìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===
=== ==
Figure 37. =
275. P~Ç = ==
276. O
~ê = =
=
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CHAPTER 3. GEOMETRY
56
277. O
P~o = =
=278. O~Sp = =
=279. P~s = ==
===
3.23 Rectangular Parallelepiped =bÇÖÉëW=~I=ÄI=Å==aá~Öçå~äW=Ç=pìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===
===== ==
Figure 38. =
280. OOO ÅÄ~Ç ++= ==
281. ( )ÄÅ~Å~ÄOp ++= ==
282. ~ÄÅs = ==
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CHAPTER 3. GEOMETRY
57
3.24 Prism =i~íÉê~ä=ÉÇÖÉW=ä=eÉáÖÜíW=Ü=i~íÉê~ä=~êÉ~W= ip =^êÉ~=çÑ=Ä~ëÉW= _p =qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===
===== ==
Figure 39. =
283. _i pOpp += K===
284. i~íÉê~ä=^êÉ~=çÑ=~=oáÖÜí=mêáëã=( )ä~~~~p åPONi ++++= K =
=285. i~íÉê~ä=^êÉ~=çÑ=~å=lÄäáèìÉ=mêáëã=
éäpi = I==ïÜÉêÉ=é=áë=íÜÉ=éÉêáãÉíÉê=çÑ=íÜÉ=Åêçëë=ëÉÅíáçåK=
=
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CHAPTER 3. GEOMETRY
58
286. Üps _= ==
287. `~î~äáÉêáDë=mêáåÅáéäÉ==dáîÉå=íïç=ëçäáÇë= áåÅäìÇÉÇ=ÄÉíïÉÉå=é~ê~ääÉä=éä~åÉëK=fÑ=ÉîÉêó=éä~åÉ=Åêçëë=ëÉÅíáçå=é~ê~ääÉä=íç=íÜÉ=ÖáîÉå=éä~åÉë=Ü~ë=íÜÉ=ë~ãÉ=~êÉ~=áå=ÄçíÜ=ëçäáÇëI=íÜÉå=íÜÉ=îçäìãÉë=çÑ=íÜÉ=ëçäáÇë=~êÉ=Éèì~äK====
3.25 Regular Tetrahedron =qêá~åÖäÉ=ëáÇÉ=äÉåÖíÜW=~=eÉáÖÜíW=Ü=^êÉ~=çÑ=Ä~ëÉW= _p =pìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===
==
Figure 40. =
288. ~P
OÜ = =
=
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CHAPTER 3. GEOMETRY
59
289. Q
~Pp
O
_ = =
=
290. O~Pp = ==
291. OS
~Üp
P
Ns
P
_ == K==
===
3.26 Regular Pyramid =páÇÉ=çÑ=Ä~ëÉW=~=i~íÉê~ä=ÉÇÖÉW=Ä=eÉáÖÜíW=Ü=pä~åí=ÜÉáÖÜíW=ã==kìãÄÉê=çÑ=ëáÇÉëW=å==pÉãáéÉêáãÉíÉê=çÑ=Ä~ëÉW=é=o~Çáìë=çÑ=áåëÅêáÄÉÇ=ëéÜÉêÉ=çÑ=Ä~ëÉW=ê=^êÉ~=çÑ=Ä~ëÉW= _p =i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= ip =qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===
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CHAPTER 3. GEOMETRY
60
==
Figure 41. =
292. Q
~Äã
OO −= =
=
293.
åëáåO
~å
ëáåÄQÜ
OOO
π
−π
= =
=
294. éã~ÄQå~Q
Nå~ã
O
Np OO
i =−== =
=295. éêp_ = =
=296. i_ ppp += =
=
297. éêÜP
NÜp
P
Ns _ == ==
===
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CHAPTER 3. GEOMETRY
61
3.27 Frustum of a Regular Pyramid =
_~ëÉ=~åÇ=íçé=ëáÇÉ=äÉåÖíÜëW=
åPON
åPON
ÄIIÄIÄIÄ
~II~I~I~
K
K=
eÉáÖÜíW=Ü=pä~åí=ÜÉáÖÜíW=ã==^êÉ~=çÑ=Ä~ëÉëW= Np I= Op =i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= ip =mÉêáãÉíÉê=çÑ=Ä~ëÉëW= Nm I= Om =pÅ~äÉ=Ñ~ÅíçêW=â=qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===
==
Figure 42. =
298. â~
Ä
~
Ä
~
Ä
~
Ä
~
Ä
å
å
P
P
O
O
N
N ====== K =
=
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CHAPTER 3. GEOMETRY
62
299. O
N
O âp
p= =
=
300. ( )O
mmãp ON
i
+= =
=301. ONi pppp ++= =
=
302. ( )OONN ppppP
Üs ++= =
=
303. [ ]ON
O
N ââNP
Üp
~
Ä
~
ÄN
P
Üps ++=
++= =
===
3.28 Rectangular Right Wedge =páÇÉë=çÑ=Ä~ëÉW=~I=Ä=qçé=ÉÇÖÉW=Å=eÉáÖÜíW=Ü=i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= ip =^êÉ~=çÑ=Ä~ëÉW= _p =qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===
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CHAPTER 3. GEOMETRY
63
==
Figure 43. =
304. ( ) ( )OOOOi Å~ÜÄÄÜQÅ~
O
Np −++++= =
=305. ~Äp_ = =
=306. i_ ppp += =
=
307. ( )Å~OS
ÄÜs += =
===
3.29 Platonic Solids =bÇÖÉW=~=o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=pìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===
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CHAPTER 3. GEOMETRY
64
308. cáîÉ=mä~íçåáÅ=pçäáÇë=qÜÉ= éä~íçåáÅ= ëçäáÇë= ~êÉ= ÅçåîÉñ= éçäóÜÉÇê~= ïáíÜ= Éèìáî~äÉåí=Ñ~ÅÉë=ÅçãéçëÉÇ=çÑ=ÅçåÖêìÉåí=ÅçåîÉñ=êÉÖìä~ê=éçäóÖçåëK==
=pçäáÇ= kìãÄÉê=
çÑ=sÉêíáÅÉëkìãÄÉê=çÑ=bÇÖÉë=
kìãÄÉê=çÑ=c~ÅÉë=
pÉÅíáçå=
qÉíê~ÜÉÇêçå== Q= S= Q= PKOR=`ìÄÉ= U= NO= S= PKOO=
lÅí~ÜÉÇêçå= S= NO= U= PKOT=fÅçë~ÜÉÇêçå= NO= PM= OM= PKOT=
açÇÉÅ~ÜÉÇêçå= OM= PM= NO= PKOT===
Octahedron =
==
Figure 44. =
309. S
S~ê = =
=
310. O
O~o = =
=
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CHAPTER 3. GEOMETRY
65
311. P~Op O= ==
312. P
O~s
P
= =
==
Icosahedron =
==
Figure 45. =
313. ( )NO
RPP~ê
+= =
=
314. ( )RROQ
~o += =
=
315. P~Rp O= ==
316. ( )NO
RP~Rs
P += =
==
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CHAPTER 3. GEOMETRY
66
Dodecahedron =
==
Figure 46. =
317. ( )O
RNNORNM~ê
+= =
=
318. ( )Q
RNP~o
+= =
=
319. ( )RORR~Pp O += =
=
320. ( )Q
RTNR~s
P += =
===
3.30 Right Circular Cylinder =o~Çáìë=çÑ=Ä~ëÉW=o=aá~ãÉíÉê=çÑ=Ä~ëÉW=Ç=
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CHAPTER 3. GEOMETRY
67
eÉáÖÜíW=e=i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= ip =^êÉ~=çÑ=Ä~ëÉW= _p =qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===
===== ==
Figure 47. =
321. oeOpi π= ==
322. ( )
+π=+π=+=
O
ÇeÇoeoOpOpp _i =
=
323. eoeps O_ π== =
===
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CHAPTER 3. GEOMETRY
68
3.31 Right Circular Cylinder with an Oblique Plane Face
=o~Çáìë=çÑ=Ä~ëÉW=o=qÜÉ=ÖêÉ~íÉëí=ÜÉáÖÜí=çÑ=~=ëáÇÉW= NÜ =qÜÉ=ëÜçêíÉëí=ÜÉáÖÜí=çÑ=~=ëáÇÉW= OÜ =i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= ip =^êÉ~=çÑ=éä~åÉ=ÉåÇ=Ñ~ÅÉëW= _p =qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===
==
Figure 48. =
324. ( )ONi ÜÜop +π= ==
325. O
ONOO_ O
ÜÜooop
−
+π+π= =
=
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CHAPTER 3. GEOMETRY
69
326.
−
++++π=+=O
ONOON_i O
ÜÜooÜÜoppp =
=
327. ( )ON
O
ÜÜO
os +
π= =
===
3.32 Right Circular Cone o~Çáìë=çÑ=Ä~ëÉW=o=aá~ãÉíÉê=çÑ=Ä~ëÉW=Ç=eÉáÖÜíW=e=pä~åí=ÜÉáÖÜíW=ã=i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= ip =^êÉ~=çÑ=Ä~ëÉW= _p =qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===
==
Figure 49.
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CHAPTER 3. GEOMETRY
70
328. OO oãe −= ==
329. O
ãÇoãpi
π=π= =
=
330. O_ op π= =
=
331. ( )
+π=+π=+=
O
ÇãÇ
O
Noãoppp _i =
=
332. eoP
Nep
P
Ns O
_ π== =
===
3.33 Frustum of a Right Circular Cone =o~Çáìë=çÑ=Ä~ëÉëW=oI=ê=eÉáÖÜíW=e=pä~åí=ÜÉáÖÜíW=ã=pÅ~äÉ=Ñ~ÅíçêW=â=^êÉ~=çÑ=Ä~ëÉëW= Np I= Op =i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= ip =qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===
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CHAPTER 3. GEOMETRY
71
==
Figure 50. =
333. ( )OO êoãe −−= =
=
334. âê
o= =
=
335. O
O
O
N
O âê
o
p
p== =
=336. ( )êoãpi +π= ==337. ( )[ ]êoãêopppp OO
iON +++π=++= ==
338. ( )OONN ppppP
Üs ++= =
=
339. [ ]ON
O
N ââNP
Üp
ê
o
ê
oN
P
Üps ++=
++= =
===
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72
3.34 Sphere =o~ÇáìëW=o=aá~ãÉíÉêW=Ç=pìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s==
==
Figure 51. =
340. OoQp π= ==
341. poP
NÇ
S
Neo
P
Qs PP =π=π= =
===
3.35 Spherical Cap o~Çáìë=çÑ=ëéÜÉêÉW=o=o~Çáìë=çÑ=Ä~ëÉW=ê=eÉáÖÜíW=Ü=^êÉ~=çÑ=éä~åÉ=Ñ~ÅÉW= _p =^êÉ~=çÑ=ëéÜÉêáÅ~ä=Å~éW= `p =
qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s=
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CHAPTER 3. GEOMETRY
73
==
Figure 52. =
342. ÜO
Üêo
OO += =
=343. O
_ êp π= ==344. ( )OO
` êÜp +π= =
=345. ( ) ( )OOO
`_ êoÜOêOÜppp +π=+π=+= =
=
346. ( ) ( )OOO ÜêPÜS
ÜoPÜS
s +π
=−π
= =
===
3.36 Spherical Sector =o~Çáìë=çÑ=ëéÜÉêÉW=o=o~Çáìë=çÑ=Ä~ëÉ=çÑ=ëéÜÉêáÅ~ä=Å~éW=ê=eÉáÖÜíW=Ü=qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s==
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CHAPTER 3. GEOMETRY
74
====== === ==
Figure 53. =
347. ( )êÜOop +π= ==
348. ÜoP
Os Oπ= =
=kçíÉW=qÜÉ=ÖáîÉå= Ñçêãìä~ë=~êÉ=ÅçêêÉÅí=ÄçíÜ= Ñçê=±çéÉå≤= ~åÇ=±ÅäçëÉÇ≤=ëéÜÉêáÅ~ä=ëÉÅíçêK====
3.37 Spherical Segment =o~Çáìë=çÑ=ëéÜÉêÉW=o=o~Çáìë=çÑ=Ä~ëÉëW= Nê I= Oê =eÉáÖÜíW=Ü=^êÉ~=çÑ=ëéÜÉêáÅ~ä=ëìêÑ~ÅÉW= pp =
^êÉ~=çÑ=éä~åÉ=ÉåÇ=Ñ~ÅÉëW= Np I= Op =qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s==
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CHAPTER 3. GEOMETRY
75
===== ==
Figure 54. =
349. oÜOpp π= =
=350. ( )O
OO
NONp êêoÜOpppp ++π=++= =
=
351. ( )OOO
ON ÜêPêPÜ
S
Ns ++π= =
===
3.38 Spherical Wedge =o~ÇáìëW=o=aáÜÉÇê~ä=~åÖäÉ=áå=ÇÉÖêÉÉëW=ñ=aáÜÉÇê~ä=~åÖäÉ=áå=ê~Çá~åëW=α =^êÉ~=çÑ=ëéÜÉêáÅ~ä=äìåÉW= ip =qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===
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CHAPTER 3. GEOMETRY
76
==
Figure 55. =
352. ñoOVM
op O
O
i =απ
= =
=
353. ñoOoVM
oop OO
OO +π=α
π+π= =
=
354. ñoP
O
OTM
os P
P
=απ
= =
===
3.39 Ellipsoid =pÉãá-~ñÉëW=~I=ÄI=Å=sçäìãÉW=s=
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CHAPTER 3. GEOMETRY
77
======= ==
Figure 56. =
355. ~ÄÅP
Qs π= =
===
Prolate Spheroid =pÉãá-~ñÉëW=~I=ÄI=Ä=E Ä~ > F=pìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===
356.
+π=
É
É~êÅëáå~ÄÄOp I==
ïÜÉêÉ=~
Ä~É
OO −= K=
=
357. ~ÄP
Qs Oπ= =
=
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CHAPTER 3. GEOMETRY
78
Oblate Spheroid =pÉãá-~ñÉëW=~I=ÄI=Ä=E Ä~ < F=pìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===
358.
+π=~LÄÉ
~
ÄÉ~êÅëáåÜ~
ÄÄOp I==
ïÜÉêÉ=Ä
~ÄÉ
OO −= K=
=
359. ~ÄP
Qs Oπ= =
===
3.40 Circular Torus =j~àçê=ê~ÇáìëW=o=jáåçê=ê~ÇáìëW=ê=pìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s==
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CHAPTER 3. GEOMETRY
79
== =Picture 57.
=360. oêQp Oπ= ==361. OOoêOs π= =
==
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80
Chapter 4
Trigonometry ====^åÖäÉëW=α I=β =oÉ~ä=åìãÄÉêë=EÅççêÇáå~íÉë=çÑ=~=éçáåíFW=ñI=ó==tÜçäÉ=åìãÄÉêW=â===
4.1 Radian and Degree Measures of Angles =
362. ?QRDNTRTNUM
ê~ÇN °≈π°
= =
=
363. ê~ÇMNTQRPKMê~ÇNUM
N ≈π
=° =
=
364. ê~ÇMMMOVNKMê~ÇSMNUM
DN ≈⋅π
= =
=
365. ê~ÇMMMMMRKMê~ÇPSMMNUM
?N ≈⋅π
= =
=366. ==
^åÖäÉ=EÇÉÖêÉÉëF=
M= PM= QR= SM= VM= NUM= OTM= PSM=
^åÖäÉ=Eê~Çá~åëF= M=
S
π=
Q
π=
P
π=
O
π= π =
O
Pπ= πO =
===
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CHAPTER 4. TRIGONOMETRY
81
4.2 Definitions and Graphs of Trigonometric Functions
=
= ==
Figure 58. =
367. ê
óëáå =α =
=
368. ê
ñÅçë =α =
=
369. ñ
óí~å =α =
=
370. ó
ñÅçí =α =
=
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CHAPTER 4. TRIGONOMETRY
82
371. ñ
êëÉÅ =α =
=
372. ó
êÅçëÉÅ =α =
=373. páåÉ=cìåÅíáçå=
ñëáåó = I= NñëáåN ≤≤− K==
=
Figure 59. =
374. `çëáåÉ=cìåÅíáçå==ñÅçëó = I= NñÅçëN ≤≤− K=
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CHAPTER 4. TRIGONOMETRY
83
==
Figure 60. =
375. q~åÖÉåí=cìåÅíáçå=
ñí~åó = I= ( )O
NâOñπ
+≠ I= Kñí~å ∞≤≤∞− =
=
==
Figure 61. =
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CHAPTER 4. TRIGONOMETRY
84
376. `çí~åÖÉåí=cìåÅíáçå==ñÅçíó = I= π≠ âñ I== ∞≤≤∞− ñÅçí K=
=
==
Figure 62. =
377. pÉÅ~åí=cìåÅíáçå=
ñëÉÅó = I= ( )O
NâOñπ
+≠ K=
==
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CHAPTER 4. TRIGONOMETRY
85
==
Figure 63. =
378. `çëÉÅ~åí=cìåÅíáçå==ñÉÅÅçëó = I= π≠ âñ K=
=
Figure 64.
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CHAPTER 4. TRIGONOMETRY
86
4.3. Signs of Trigonometric Functions 379. ==
nì~Çê~åí=páåα =
`çëα =
q~åα =
`çíα =
pÉÅα =
`çëÉÅ=α =
f= H= H= H= H= H= H=ff= H= �= �= �= �= H=fff= �= �= H= H= �= �=fs= �= H= �= �= H= �==
==
380. ==
=
Figure 65.
==========
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CHAPTER 4. TRIGONOMETRY
87
4.4 Trigonometric Functions of Common Angles 381. =°α = ê~Çα = αëáå = αÅçë = αí~å = αÅçí αëÉÅ = αÅçëÉÅ =
M= M= M= N= M= ∞ = N= ∞ =
PM=S
π=
O
N=
O
P=
P
N= P =
P
O= O=
QR=Q
π=
O
O=
O
O= N= N= O = O =
SM=P
π=
O
P=
O
N= P =
P
N= O=
P
O=
VM=O
π= N= M= ∞ = M= ∞ = N=
NOM=P
Oπ=
O
P=
O
N− = P− =
P
N− � O− =
P
O=
NUM= π = M= N− = M= ∞ = N− = ∞ =
OTM=O
Pπ= N− = M= ∞ = M= ∞ = N− =
PSM= πO = M= N= M= ∞ = N= ∞ ==============
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CHAPTER 4. TRIGONOMETRY
88
382. =°α = ê~Çα = αëáå = αÅçë = αí~å = αÅçí =
NR=NO
π=
Q
OS −=
Q
OS += PO− = PO+ =
NU=NM
π=
Q
NR −=
Q
RONM+R
ROR−= ROR+ =
PS=R
π=
Q
RONM−Q
NR +=
NR
RONM
+−
RONM
NR
−
+
=
RQ=NM
Pπ=
Q
NR +=
Q
RONM−RONM
NR
−
+NR
RONM
+−
=
TO=R
Oπ=
Q
RONM+Q
NR −= ROR+ = R
ROR−
=
TR=NO
Rπ=
Q
OS +=
Q
OS −= PO+ = PO− =
===
4.5 Most Important Formulas =
383. NÅçëëáå OO =α+α ==
384. Ní~åëÉÅ OO =α−α ==
385. NÅçíÅëÅ OO =α−α ==
386. αα
=αÅçë
ëáåí~å =
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CHAPTER 4. TRIGONOMETRY
89
387. αα
=αëáå
ÅçëÅçí =
=388. NÅçíí~å =α⋅α =
=
389. α
=αÅçë
NëÉÅ =
=
390. α
=αëáå
NÅçëÉÅ =
===
4.6 Reduction Formulas =
391. ==
β = βëáå = βÅçë = βí~å = βÅçí =
α− = α− ëáå = α+ Åçë = α− í~å = α− Åçí =
α−°VM = α+ Åçë = α+ ëáå = α+ Åçí = α+ í~å =
α+°VM = α+ Åçë = α− ëáå = α− Åçí = α− í~å =
α−°NUM α+ ëáå = α− Åçë = α− í~å = α− Åçí =
α+°NUM α− ëáå = α− Åçë = α+ í~å = α+ Åçí =
α−°OTM α− Åçë = α− ëáå = α+ Åçí = α+ í~å =
α+°OTM α− Åçë = α+ ëáå = α− Åçí = α− í~å =
α−°PSM α− ëáå = α+ Åçë = α− í~å = α− Åçí =
α+°PSM α+ ëáå = α+ Åçë = α+ í~å = α+ Åçí ==
=====
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CHAPTER 4. TRIGONOMETRY
90
4.7 Periodicity of Trigonometric Functions =
392. ( ) α=π±α ëáååOëáå I=éÉêáçÇ= πO =çê= °PSM K==
393. ( ) α=π±α ÅçëåOÅçë I=éÉêáçÇ= πO =çê= °PSM K==
394. ( ) α=π±α í~ååí~å I=éÉêáçÇ=π =çê= °NUM K==
395. ( ) α=π±α ÅçíåÅçí I=éÉêáçÇ=π =çê= °NUM K====
4.8 Relations between Trigonometric Functions
=
396. ( ) NQO
ÅçëOOÅçëNO
NÅçëNëáå OO −
π
−α
=α−±=α−±=α =
=
Oí~åN
Oí~åO
O α+
α
= =
=
397. ( ) NO
ÅçëOOÅçëNO
NëáåNÅçë OO −
α=α+±=α−±=α =
=
Oí~åN
Oí~åN
O
O
α+
α−
= =
=
398. αα−
=α+
α=−α±=
αα
=αOëáå
OÅçëN
OÅçëN
OëáåNëÉÅ
Åçë
ëáåí~å O =
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CHAPTER 4. TRIGONOMETRY
91
=
Oí~åN
Oí~åO
OÅçëN
OÅçëN
O α+
α
=α+α−
±= =
=
399. α−
α=
αα+
=−α±=αα
=αOÅçëN
Oëáå
Oëáå
OÅçëNNÅëÅ
ëáå
ÅçëÅçí O =
=
Oí~åO
Oí~åN
OÅçëN
OÅçëNO
α
α−
=α−α+
±= =
=
400.
Oí~åN
Oí~åN
í~åNÅçë
NëÉÅ
O
O
O
α−
α+
=α+±=α
=α =
=
401.
Oí~åO
Oí~åN
ÅçíNëáå
NÅëÅ
O
O
α
α+
=α+±=α
=α =
===
4.9 Addition and Subtraction Formulas =
402. ( ) αβ+βα=β+α ÅçëëáåÅçëëáåëáå ==
403. ( ) αβ−βα=−α ÅçëëáåÅçëëáåóëáå ==
404. ( ) βα−βα=β+α ëáåëáåÅçëÅçëÅçë ==
405. ( ) βα+βα=β−α ëáåëáåÅçëÅçëÅçë =
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CHAPTER 4. TRIGONOMETRY
92
406. ( )βα−β+α
=β+αí~åí~åN
í~åí~åí~å =
=
407. ( )βα+β−α
=β−αí~åí~åN
í~åí~åí~å =
=
408. ( )β+αβα−
=β+αí~åí~å
í~åí~åNÅçí =
=
409. ( )β−αβα+
=β−αí~åí~å
í~åí~åNÅçí =
===
4.10 Double Angle Formulas =
410. α⋅α=α ÅçëëáåOOëáå ==
411. NÅçëOëáåONëáåÅçëOÅçë OOOO −α=α−=α−α=α ==
412. α−α
=α−α
=αí~åÅçí
O
í~åN
í~åOOí~å
O=
=
413. O
í~åÅçí
ÅçíO
NÅçíOÅçí
O α−α=
α−α
=α =
======
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CHAPTER 4. TRIGONOMETRY
93
4.11 Multiple Angle Formulas =
414. α−α⋅α=α−α=α POP ëáåëáåÅçëPëáåQëáåPPëáå ==
415. α⋅α−α⋅α=α ÅçëëáåUÅçëëáåQQëáå P ==
416. α+α−α=α RP ëáåNSëáåOMëáåRRëáå ==
417. α⋅α−α=α−α=α OPP ëáåÅçëPÅçëÅçëPÅçëQPÅçë ==
418. NÅçëUÅçëUQÅçë OQ +α−α=α ==
419. α+α−α=α ÅçëRÅçëOMÅçëNSRÅçë PR ==
420. α−α−α
=αO
P
í~åPN
í~åí~åPPí~å =
=
421. α+α−α−α
=αQO
P
í~åí~åSN
í~åQí~åQQí~å =
=
422. α+α−α+α−α
=αQO
PR
í~åRí~åNMN
í~åRí~åNMí~åRí~å =
=
423. NÅçíP
ÅçíPÅçíPÅçí
O
P
−αα−α
=α =
=
424. α−αα+α−
=αP
QO
í~åQí~åQ
í~åí~åSNQÅçí ==
=
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CHAPTER 4. TRIGONOMETRY
94
425. α+α−α
α+α−=α
í~åRí~åNMí~å
í~åRí~åNMNRÅçí
PR
QO
=
===
4.12 Half Angle Formulas =
426. O
ÅçëN
Oëáå
α−±=
α=
=
427. O
ÅçëN
OÅçë
α+±=
α=
=
428. α−α=αα−
=α+
α=
α+α−
±=α
ÅçíÅëÅëáå
ÅçëN
ÅçëN
ëáå
ÅçëN
ÅçëN
Oí~å =
=
429. α+α=αα+
=α−
α=
α−α+
±=α
ÅçíÅëÅëáå
ÅçëN
ÅçëN
ëáå
ÅçëN
ÅçëN
OÅçí =
===
4.13 Half Angle Tangent Identities =
430.
Oí~åN
Oí~åO
ëáåO α+
α
=α =
=
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CHAPTER 4. TRIGONOMETRY
95
431.
Oí~åN
Oí~åN
ÅçëO
O
α+
α−
=α =
=
432.
Oí~åN
Oí~åO
í~åO α−
α
=α =
=
433.
Oí~åO
Oí~åN
Åçí
O
α
α−
=α =
===
4.14 Transforming of Trigonometric Expressions to Product
=
434. O
ÅçëO
ëáåOëáåëáåβ−αβ+α
=β+α =
=
435. O
ëáåO
ÅçëOëáåëáåβ−αβ+α
=β−α =
=
436. O
ÅçëO
ÅçëOÅçëÅçëβ−αβ+α
=β+α =
=
437. O
ëáåO
ëáåOÅçëÅçëβ−αβ+α
−=β−α =
=
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CHAPTER 4. TRIGONOMETRY
96
438. ( )β⋅α
β+α=β+α
ÅçëÅçë
ëáåí~åí~å =
=
439. ( )β⋅α
β−α=β−α
ÅçëÅçë
ëáåí~åí~å =
=
440. ( )β⋅α
α+β=β+α
ëáåëáå
ëáåÅçíÅçí =
=
441. ( )β⋅α
α−β=β−α
ëáåëáå
ëáåÅçíÅçí =
=
442.
α+π
=
α−π
=α+αQ
ëáåOQ
ÅçëOëáåÅçë =
=
443.
α+π
=
α−π
=α−αQ
ÅçëOQ
ëáåOëáåÅçë =
=
444. ( )β⋅α
β−α=β+α
ëáåÅçë
ÅçëÅçíí~å =
=
445. ( )β⋅α
β+α−=β−α
ëáåÅçë
ÅçëÅçíí~å =
=
446. O
ÅçëOÅçëN O α=α+ =
=
447. O
ëáåOÅçëN O α=α− =
=
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CHAPTER 4. TRIGONOMETRY
97
448.
α
−π
=α+OQ
ÅçëOëáåN O =
=
449.
α
−π
=α−OQ
ëáåOëáåN O =
===
4.15 Transforming of Trigonometric Expressions to Sum
=
450. ( ) ( )O
ÅçëÅçëëáåëáå
β+α−β−α=β⋅α =
=
451. ( ) ( )O
ÅçëÅçëÅçëÅçë
β+α+β−α=β⋅α =
=
452. ( ) ( )O
ëáåëáåÅçëëáå
β+α+β−α=β⋅α =
=
453. β+αβ+α
=β⋅αÅçíÅçí
í~åí~åí~åí~å =
=
454. β+αβ+α
=β⋅αí~åí~å
ÅçíÅçíÅçíÅçí =
=
455. β+αβ+α
=β⋅αí~åÅçí
Åçíí~åÅçíí~å =
===
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CHAPTER 4. TRIGONOMETRY
98
4.16 Powers of Trigonometric Functions =
456. O
OÅçëNëáåO α−
=α =
=
457. Q
PëáåëáåPëáåP α−α
=α =
=
458. U
POÅçëQQÅçëëáåQ +α−α
=α =
=
459. NS
RëáåPëáåRëáåNMëáåR α+α−α
=α =
=
460. PO
SÅçëQÅçëSOÅçëNRNMëáåS α−α+α−
=α =
=
461. O
OÅçëNÅçëO α+
=α =
=
462. Q
PÅçëÅçëPÅçëP α+α
=α =
=
463. U
POÅçëQQÅçëÅçëQ +α+α
=α =
=
464. NS
RÅçëPëáåRÅçëNMÅçëR α+α+α
=α =
=
465. PO
SÅçëQÅçëSOÅçëNRNMÅçëS α+α+α+
=α =
=
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99
4.17 Graphs of Inverse Trigonometric Functions
=466. fåîÉêëÉ=páåÉ=cìåÅíáçå==
ñ~êÅëáåó = I= NñN ≤≤− I=O
ñ~êÅëáåO
π≤≤
π− K=
=
==
Figure 66. =
467. fåîÉêëÉ=`çëáåÉ=cìåÅíáçå==ñ~êÅÅçëó = I= NñN ≤≤− I= π≤≤ ñ~êÅÅçëM K=
=
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CHAPTER 4. TRIGONOMETRY
100
==
Figure 67. =
468. fåîÉêëÉ=q~åÖÉåí=cìåÅíáçå==
ñ~êÅí~åó = I= ∞≤≤∞− ñ I=O
ñ~êÅí~åO
π<<
π− K=
=
===== ==
Figure 68.
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CHAPTER 4. TRIGONOMETRY
101
469. fåîÉêëÉ=`çí~åÖÉåí=cìåÅíáçå==ñÅçí~êÅó = I= ∞≤≤∞− ñ I= π<< ñÅçí~êÅM K=
===== =
Figure 69. =
470. fåîÉêëÉ=pÉÅ~åí=cìåÅíáçå==
( ] [ ) KIOO
IMñëÉÅ~êÅIINNIñIñ=~êÅëÉÅó
ππ
∪
π
∈∞∪−∞−∈=
=
Figure 70.
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CHAPTER 4. TRIGONOMETRY
102
471. fåîÉêëÉ=`çëÉÅ~åí=cìåÅíáçå==
( ] [ ) KO
IMMIO
ñÅëÅ~êÅIINNIñIñ~êÅÅëÅó
π
∪
π−∈∞∪−∞−∈=
==
Figure 71. ==
4.18 Principal Values of Inverse Trigonometric Functions 472. ñ = M=
O
N=
O
O=
O
PN=
ñ~êÅëáå = °M = °PM = °QR = °SM °VMñ~êÅÅçë = °VM °SM = °QR = °PM °M =
ñ = O
N−
O
O−
O
P− N− = =
ñ~êÅëáå =°−PM
=°− QR °− SM
°− VM=
=
ñ~êÅÅçë =°NOM
=°NPR = °NRM =
°NUM=
=
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CHAPTER 4. TRIGONOMETRY
103
473. ñ = M=
P
PN= P =
P
P− N− = P− =
ñ~êÅí~å = °M = °PM °QR °SM °−PM°− QR
=°− SM =
ñÅçí~êÅ = °VM °SM °QR °PM °NOM =°NPR
=°NRM =
===
4.19 Relations between Inverse Trigonometric Functions
=474. ( ) ñ~êÅëáåñ~êÅëáå −=− =
=
475. ñ~êÅÅçëO
ñ~êÅëáå −π
= =
=
476. OñN~êÅÅçëñ~êÅëáå −= I= NñM ≤≤ K==
477. OñN~êÅÅçëñ~êÅëáå −−= I= MñN ≤≤− K==
478. OñN
ñ~êÅí~åñ~êÅëáå
−= I= NñO < K=
=
479. ñ
ñNÅçí~êÅñ~êÅëáå
O−= I= NñM ≤< K=
=
480. π−−
=ñ
ñNÅçí~êÅñ~êÅëáå
O
I= MñN <≤− K=
=481. ( ) ñ~êÅÅçëñ~êÅÅçë −π=− =
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104
482. ñ~êÅëáåO
ñ~êÅÅçë −π
= =
=
483. OñN~êÅëáåñ~êÅÅçë −= I= NñM ≤≤ K==
484. OñN~êÅëáåñ~êÅÅçë −−π= I= MñN ≤≤− K==
485. ñ
ñN~êÅí~åñ~êÅÅçë
O−= I= NñM ≤< K=
=
486. ñ
ñN~êÅí~åñ~êÅÅçë
O−+π= I= MñN <≤− K=
=
487. OñN
ñÅçí~êÅñ~êÅÅçë
−= I= NñN ≤≤− K=
=488. ( ) ñ~êÅí~åñ~êÅí~å −=− =
=
489. ñÅçí~êÅO
ñ~êÅí~å −π
= =
=
490. OñN
ñ~êÅëáåñ~êÅí~å
+= =
=
491. OñN
N~êÅÅçëñ~êÅí~å
+= I= Mñ ≥ K=
=
492. OñN
N~êÅÅçëñ~êÅí~å
+−= I= Mñ ≤ K=
=
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CHAPTER 4. TRIGONOMETRY
105
493. ñ
N~êÅí~å
Oñ~êÅí~å −
π= I= Mñ > K=
=
494. ñ
N~êÅí~å
Oñ~êÅí~å −
π−= I= Mñ < K=
=
495. ñ
NÅçí~êÅñ~êÅí~å = I= Mñ > K=
=
496. π−=ñ
NÅçí~êÅñ~êÅí~å I= Mñ < K=
=497. ( ) ñÅçí~êÅñÅçí~êÅ −π=− =
=
498. ñ~êÅí~åO
ñÅçí~êÅ −π
= =
=
499. OñN
N~êÅëáåñÅçí~êÅ
+= I= Mñ > K=
=
500. OñN
N~êÅëáåñÅçí~êÅ
+−π= I= Mñ < K=
=
501. OñN
ñ~êÅÅçëñÅçí~êÅ
+= =
=
502. ñ
N~êÅí~åñÅçí~êÅ = I= Mñ > K=
=
503. ñ
N~êÅí~åñÅçí~êÅ +π= I= Mñ < K=
==
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106
4.20 Trigonometric Equations =tÜçäÉ=åìãÄÉêW=å===
504. ~ñëáå = I= ( ) å~~êÅëáåNñ å π+−= ==
505. ~ñÅçë = I= åO~~êÅÅçëñ π+±= ==
506. ~ñí~å = I= å~~êÅí~åñ π+= ==
507. ~ñÅçí = I= å~Åçí~êÅñ π+= ====
4.21 Relations to Hyperbolic Functions =fã~Öáå~êó=ìåáíW=á===
508. ( ) ñëáåÜááñëáå = ==
509. ( ) ñí~åÜááñí~å = ==
510. ( ) ñÅçíÜááñÅçí −= ==
511. ( ) ñëÉÅÜáñëÉÅ = ==
512. ( ) ñÅëÅÜááñÅëÅ −= ====
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107
Chapter 5
Matrices and Determinants ====j~íêáÅÉëW=^I=_I=`=bäÉãÉåíë=çÑ=~=ã~íêáñW= á~ I= áÄ I= áà~ I= áàÄ I= áàÅ =
aÉíÉêãáå~åí=çÑ=~=ã~íêáñW= ^ÇÉí =jáåçê=çÑ=~å=ÉäÉãÉåí= áà~ W= áàj =
`çÑ~Åíçê=çÑ=~å=ÉäÉãÉåí= áà~ W= áà` =
qê~åëéçëÉ=çÑ=~=ã~íêáñW= q^ I= ^ú
=^Çàçáåí=çÑ=~=ã~íêáñW= ^~Çà =qê~ÅÉ=çÑ=~=ã~íêáñW= ^íê =
fåîÉêëÉ=çÑ=~=ã~íêáñW= N^− =oÉ~ä=åìãÄÉêW=â=oÉ~ä=î~êá~ÄäÉëW= áñ =k~íìê~ä=åìãÄÉêëW=ãI=å=====
5.1 Determinants =
513. pÉÅçåÇ=lêÇÉê=aÉíÉêãáå~åí=
NOONOO
NN Ä~Ä~Ä~
Ä~^ÇÉí −== =
=====
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CHAPTER 5. MATRICES AND DETERMINANTS
108
514. qÜáêÇ=lêÇÉê=aÉíÉêãáå~åí=
−++== POONNPPNOPNOPPOONN
PPPOPN
OPOOON
NPNONN
~~~~~~~~~
~~~
~~~
~~~
^ÇÉí =
PNOONPPPONNOPOOPNN ~~~~~~~~~ −−− =
=515. p~êêìë=oìäÉ=E^êêçï=oìäÉF=
==
Figure 72. =
516. k-íÜ=lêÇÉê=aÉíÉêãáå~åí=
åååàOåNå
áåáàOáNá
åOàOOOON
åNàNNONN
~~~~
~~~~
~~~~
~~~~
^ÇÉí
KK
KKKKKK
KK
KKKKKK
KK
KK
= =
=517. jáåçê=
qÜÉ=ãáåçê= áàj =~ëëçÅá~íÉÇ=ïáíÜ=íÜÉ=ÉäÉãÉåí= áà~ =çÑ=å-íÜ=çêÇÉê=
ã~íêáñ= ^= áë= íÜÉ= ( )Nå− -íÜ= çêÇÉê= ÇÉíÉêãáå~åí= ÇÉêáîÉÇ= Ñêçã=
íÜÉ=ã~íêáñ=^=Äó=ÇÉäÉíáçå=çÑ=áíë=á-íÜ=êçï=~åÇ=à-íÜ=ÅçäìãåK====
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CHAPTER 5. MATRICES AND DETERMINANTS
109
518. `çÑ~Åíçê=
( ) áààá
áà jN` +−= =
=519. i~éä~ÅÉ=bñé~åëáçå=çÑ=å-íÜ=lêÇÉê=aÉíÉêãáå~åí=
i~éä~ÅÉ=Éñé~åëáçå=Äó=ÉäÉãÉåíë=çÑ=íÜÉ=á-íÜ=êçï=
∑=
=å
Nàáàáà`~^ÇÉí I= åIIOINá K= K=
i~éä~ÅÉ=Éñé~åëáçå=Äó=ÉäÉãÉåíë=çÑ=íÜÉ=à-íÜ=Åçäìãå=
∑=
=å
Nááàáà`~^ÇÉí I= åIIOINà K= K==
===
5.2 Properties of Determinants =
520. qÜÉ==î~äìÉ==çÑ=~=ÇÉíÉêãáå~åí=êÉã~áåë==ìåÅÜ~åÖÉÇ=áÑ=êçïë=~êÉ=ÅÜ~åÖÉÇ=íç=Åçäìãåë=~åÇ=Åçäìãåë=íç=êçïëK=
=OO
NN
ON
ON
Ä~
Ä~
ÄÄ
~~= ==
=521. fÑ=íïç==êçïë==Eçê=íïç=ÅçäìãåëF=~êÉ==áåíÉêÅÜ~åÖÉÇI=íÜÉ=ëáÖå=çÑ=
íÜÉ=ÇÉíÉêãáå~åí=áë=ÅÜ~åÖÉÇK=
NN
OO
OO
NN
Ä~
Ä~
Ä~
Ä~−= =
=522. fÑ=íïç=êçïë==Eçê=íïç=ÅçäìãåëF=~êÉ==áÇÉåíáÅ~äI=íÜÉ=î~äìÉ=çÑ=íÜÉ=
ÇÉíÉêãáå~åí=áë=òÉêçK=
M~~
~~
OO
NN = =
=
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CHAPTER 5. MATRICES AND DETERMINANTS
110
523. fÑ==íÜÉ===ÉäÉãÉåíë==çÑ==~åó=êçï==Eçê=ÅçäìãåF=~êÉ=ãìäíáéäáÉÇ=Äó=====~==Åçããçå==Ñ~ÅíçêI==íÜÉ==ÇÉíÉêãáå~åí==áë==ãìäíáéäáÉÇ==Äó==íÜ~í=Ñ~ÅíçêK=
OO
NN
OO
NN
Ä~
Ä~â
Ä~
âÄâ~= =
=524. fÑ==íÜÉ==ÉäÉãÉåíë==çÑ==~åó==êçï==Eçê==ÅçäìãåF=~êÉ=áåÅêÉ~ëÉÇ=Eçê=
ÇÉÅêÉ~ëÉÇFÄó=Éèì~ä=ãìäíáéäÉë=çÑ=íÜÉ=ÅçêêÉëéçåÇáåÖ=ÉäÉãÉåíë=çÑ=~åó=çíÜÉê=êçï==Eçê=ÅçäìãåFI==íÜÉ=î~äìÉ=çÑ=íÜÉ=ÇÉíÉêãáå~åí=áë=ìåÅÜ~åÖÉÇK=
OO
NN
OOO
NNN
Ä~
Ä~
ÄâÄ~
ÄâÄ~=
++
=
===
5.3 Matrices =
525. aÉÑáåáíáçå=^å= åã× =ã~íêáñ=^=áë=~=êÉÅí~åÖìä~ê=~êê~ó=çÑ=ÉäÉãÉåíë=Eåìã-ÄÉêë=çê=ÑìåÅíáçåëF=ïáíÜ=ã=êçïë=~åÇ=å=ÅçäìãåëK==
[ ]
==
ãåOãNã
åOOOON
åNNONN
áà
~~~
~~~
~~~
~^
K
MMM
K
K
==
=526. pèì~êÉ=ã~íêáñ=áë=~=ã~íêáñ=çÑ=çêÇÉê= åå× K==
=527. ^=ëèì~êÉ=ã~íêáñ== [ ]áà~ ==áë==ëóããÉíêáÅ==áÑ== àááà ~~ = I==áKÉK==áí==áë=
ëóããÉíêáÅ=~Äçìí=íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~äK===
528. ^=ëèì~êÉ=ã~íêáñ= [ ]áà~ =áë=ëâÉï-ëóããÉíêáÅ=áÑ= àááà ~~ −= K==
=
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CHAPTER 5. MATRICES AND DETERMINANTS
111
529. aá~Öçå~ä=ã~íêáñ==áë==~=ëèì~êÉ==ã~íêáñ=ïáíÜ=~ää==ÉäÉãÉåíë==òÉêç=ÉñÅÉéí=íÜçëÉ=çå=íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~äK===
530. råáí=ã~íêáñ==áë==~=Çá~Öçå~ä==ã~íêáñ==áå=ïÜáÅÜ=íÜÉ=ÉäÉãÉåíë=çå=íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~ä=~êÉ=~ää=ìåáíóK=qÜÉ=ìåáí=ã~íêáñ=áë===========ÇÉåçíÉÇ=Äó=fK===
531. ^=åìää=ã~íêáñ=áë=çåÉ=ïÜçëÉ=ÉäÉãÉåíë=~êÉ=~ää=òÉêçK====
5.4 Operations with Matrices =
532. qïç=ã~íêáÅÉë=^=~åÇ=_=~êÉ=Éèì~ä=áÑI=~åÇ=çåäó=áÑI=íÜÉó=~êÉ=ÄçíÜ=çÑ==íÜÉ==ë~ãÉ==ëÜ~éÉ== åã× ==~åÇ=ÅçêêÉëéçåÇáåÖ=ÉäÉãÉåíë=~êÉ=Éèì~äK==
533. qïç=ã~íêáÅÉë==^=~åÇ=_==Å~å=ÄÉ=~ÇÇÉÇ=Eçê=ëìÄíê~ÅíÉÇF=çÑI=~åÇ=çåäó=áÑI=íÜÉó=Ü~îÉ=íÜÉ=ë~ãÉ=ëÜ~éÉ= åã× K=fÑ==
[ ]
==
ãåOãNã
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áà
~~~
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K
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áà
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=====
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CHAPTER 5. MATRICES AND DETERMINANTS
112
íÜÉå==
+++
++++++
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qïç= ã~íêáÅÉë= Å~å= ÄÉ= ãìäíáéäáÉÇ= íçÖÉíÜÉê= çåäó= ïÜÉå= íÜÉ=åìãÄÉê= çÑ= Åçäìãåë= áå= íÜÉ= Ñáêëí= áë= Éèì~ä= íç= íÜÉ= åìãÄÉê= çÑ=êçïë=áå=íÜÉ=ëÉÅçåÇK===fÑ=
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CHAPTER 5. MATRICES AND DETERMINANTS
113
íÜÉå==
==
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∑=λ
λλ=+++=å
NàáåàáåàOOáàNNááà Ä~Ä~Ä~Ä~Å K =
E ãIIOINá K= X âIIOINà K= FK===qÜìë=áÑ=
[ ]
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OPOOON
NPNONNáà ~~~
~~~~^ I= [ ]
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P
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=536. qê~åëéçëÉ=çÑ=~=j~íêáñ=
fÑ=íÜÉ=êçïë=~åÇ=Åçäìãåë=çÑ=~=ã~íêáñ=~êÉ=áåíÉêÅÜ~åÖÉÇI=íÜÉå=íÜÉ=åÉï=ã~íêáñ=áë=Å~ääÉÇ=íÜÉ=íê~åëéçëÉ=çÑ=íÜÉ=çêáÖáå~ä=ã~íêáñK===fÑ= ^= áë= íÜÉ= çêáÖáå~ä= ã~íêáñI= áíë= íê~åëéçëÉ= áë= ÇÉåçíÉÇ= q^ = çê=
^ú
K===
537. qÜÉ=ã~íêáñ=^=áë=çêíÜçÖçå~ä=áÑ= f^^q = K===
538. fÑ=íÜÉ=ã~íêáñ=éêçÇìÅí=^_=áë=ÇÉÑáåÉÇI=íÜÉå==
( ) qqq ^_^_ = K===
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CHAPTER 5. MATRICES AND DETERMINANTS
114
539. ^Çàçáåí=çÑ=j~íêáñ=fÑ=^=áë=~=ëèì~êÉ= åå× ã~íêáñI=áíë=~ÇàçáåíI=ÇÉåçíÉÇ=Äó= ^~Çà I=áë=íÜÉ=íê~åëéçëÉ=çÑ=íÜÉ=ã~íêáñ=çÑ=ÅçÑ~Åíçêë= áà` =çÑ=^W=
[ ]qáà`^~Çà = K==
=540. qê~ÅÉ=çÑ=~=j~íêáñ=
fÑ= ^= áë= ~= ëèì~êÉ= åå× ã~íêáñI= áíë= íê~ÅÉI= ÇÉåçíÉÇ= Äó= ^íê I= áë=ÇÉÑáåÉÇ=íç=ÄÉ==íÜÉ=ëìã=çÑ==íÜÉ=íÉêãë=çå=íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~äW=
ååOONN ~~~^íê +++= K K==541. fåîÉêëÉ=çÑ=~=j~íêáñ=
fÑ=^=áë=~=ëèì~êÉ= åå× ã~íêáñ=ïáíÜ=~=åçåëáåÖìä~ê=ÇÉíÉêãáå~åí=^ÇÉí I=íÜÉå=áíë=áåîÉêëÉ= N^− =áë=ÖáîÉå=Äó=
^ÇÉí
^~Çà^ N =− K=
=542. fÑ=íÜÉ=ã~íêáñ=éêçÇìÅí=^_=áë=ÇÉÑáåÉÇI=íÜÉå==
( ) NNN ^_^_ −−− = K==
543. fÑ==^==áë=~=ëèì~êÉ=== åå× ==ã~íêáñI==íÜÉ==ÉáÖÉåîÉÅíçêë==u===ë~íáëÑó=íÜÉ=Éèì~íáçå=
u^u λ= I==ïÜáäÉ=íÜÉ=ÉáÖÉåî~äìÉë=λ =ë~íáëÑó=íÜÉ=ÅÜ~ê~ÅíÉêáëíáÅ=Éèì~íáçå=
Mf^ =λ− K===
===
5.5 Systems of Linear Equations ==s~êá~ÄäÉëW=ñI=óI=òI= Nñ I= KIñO =oÉ~ä=åìãÄÉêëW= KI~I~IÄI~I~I~ NONNNPON =
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CHAPTER 5. MATRICES AND DETERMINANTS
115
aÉíÉêãáå~åíëW=aI= ña I= óa I= òa ==
j~íêáÅÉëW=^I=_I=u===
544.
=+=+
OOO
NNN
ÇóÄñ~
ÇóÄñ~I==
a
añ ñ= I=
a
aó ó= =E`ê~ãÉê∞ë=êìäÉFI==
ïÜÉêÉ==
NOONOO
NN Ä~Ä~Ä~
Ä~a −== I==
NOONOO
NNñ ÄÇÄÇ
ÄÇ
ÄÇa −== I==
NOONOO
NNó Ç~Ç~
Ç~
Ç~a −== K==
=545. fÑ= Ma≠ I=íÜÉå=íÜÉ=ëóëíÉã=Ü~ë=~=ëáåÖäÉ=ëçäìíáçåW==
a
añ ñ= I=
a
aó ó= K=
fÑ= Ma= = ~åÇ= Mañ ≠ Eçê= Maó ≠ FI= íÜÉå= íÜÉ= ëóëíÉã= Ü~ë= = åç==
ëçäìíáçåK=fÑ= Maaa óñ === I= íÜÉå= íÜÉ= ëóëíÉã= Ü~ë= = áåÑáåáíÉäó= = ã~åó==
ëçäìíáçåëK==
546.
=++=++=++
PPPP
OOOO
NNNN
ÇòÅóÄñ~
ÇòÅóÄñ~
=ÇòÅóÄñ~
I==
a
añ ñ= I=
a
aó ó= I=
a
aò ò= =E`ê~ãÉê∞ë=êìäÉFI==
=
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CHAPTER 5. MATRICES AND DETERMINANTS
116
ïÜÉêÉ==
PPP
OOO
NNN
ÅÄ~
ÅÄ~
ÅÄ~
a= I=
PPP
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NNN
ñ
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a = I=
PPP
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ó
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a = I=
PPP
OOO
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ò
ÇÄ~
ÇÄ~
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a = K==
=547. fÑ= Ma≠ I=íÜÉå=íÜÉ=ëóëíÉã=Ü~ë=~=ëáåÖäÉ=ëçäìíáçåW==
a
añ ñ= I=
a
aó ó= I=
a
aò ò= K=
fÑ= Ma= =~åÇ= Mañ ≠ Eçê= Maó ≠ =çê= Maò ≠ FI=íÜÉå=íÜÉ=ëóëíÉã=
Ü~ë=åç=ëçäìíáçåK=fÑ= Maaaa òóñ ==== I= íÜÉå= íÜÉ= ëóëíÉã= Ü~ë= áåÑáåáíÉäó=
ã~åó=ëçäìíáçåëK==
548. j~íêáñ=cçêã=çÑ=~=póëíÉã=çÑ=å=iáåÉ~ê=bèì~íáçåë=áå=================å=råâåçïåë=qÜÉ=ëÉí=çÑ=äáåÉ~ê=Éèì~íáçåë==
=+++
=+++=+++
ååååOOåNNå
OååOOOONON
NååNONONNN
Äñ~ñ~ñ~
Äñ~ñ~ñ~
Äñ~ñ~ñ~
K
KKKKKKKKKKKK
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=
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=
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å
O
N
ååOåNå
åOOOON
åNNONN
Ä
Ä
Ä
ñ
ñ
ñ
~~~
~~~
~~~
MM
K
MMM
K
K
I==
áKÉK==_u^ =⋅ I==
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CHAPTER 5. MATRICES AND DETERMINANTS
117
ïÜÉêÉ==
=
ååOåNå
åOOOON
åNNONN
~~~
~~~
~~~
^
K
MMM
K
K
I=
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=
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K==
=549. pçäìíáçå=çÑ=~=pÉí=çÑ=iáåÉ~ê=bèì~íáçåë= åå× =
_^u N ⋅= − I==ïÜÉêÉ= N^− =áë=íÜÉ=áåîÉêëÉ=çÑ=^K===
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118
Chapter 6
Vectors ====
sÉÅíçêëW= ìr
I= îr
I= ïr
I= êr
I=→
^_ I=£=sÉÅíçê=äÉåÖíÜW= ì
rI= îr
I=£=
råáí=îÉÅíçêëW= ár
I= àr
I= âr
=
kìää=îÉÅíçêW= Mr
=`ççêÇáå~íÉë=çÑ=îÉÅíçê= ì
rW= NNN wIvIu =
`ççêÇáå~íÉë=çÑ=îÉÅíçê= îr
W= OOO wIvIu =pÅ~ä~êëW=λ Iµ =aáêÉÅíáçå=ÅçëáåÉëW= αÅçë I= βÅçë I= γÅçë =^åÖäÉ=ÄÉíïÉÉå=íïç=îÉÅíçêëW=θ ===
6.1 Vector Coordinates =
550. råáí=sÉÅíçêë=
( )MIMINá =r
I=
( )MINIMà =r
I=
( )NIMIMâ =r
I=
Nâàá ===rrr
K=
=
551. ( ) ( ) ( )âòòàóóáññ^_ê MNMNMN
rrrr−+−+−==
→
=
=
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CHAPTER 6. VECTORS
119
======= ==
Figure 73. =
552. ( ) ( ) ( )OMNO
MNO
MN òòóóññ^_ê −+−+−==→r
=
=
553. fÑ= ê^_r
=→
I=íÜÉå= ê_^r
−=→
K==
==
Figure 74. =
554. α= Åçëêur
I=
β= Åçëêvr
I=
γ= Åçëêwr
K=
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CHAPTER 6. VECTORS
120
===== ==
Figure 75. =
555. fÑ= ( ) ( )NNNN wIvIuêwIvIuêrr
= I=íÜÉå==
Nuu = I= Nvv = I= Nww = K=====
6.2 Vector Addition =
556. îìïrrr
+= ==
== ==
Figure 76.
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CHAPTER 6. VECTORS
121
== ==
Figure 77. =
557. åPON ììììïr
Krrrr
++++= =
=
== ==
Figure 78. =
558. `çããìí~íáîÉ=i~ï=ìîîìrrrr
+=+ ==
559. ^ëëçÅá~íáîÉ=i~ï=( ) ( )ïîìïîì
rrrrrr++=++ =
=560. ( )ONONON wwIvvIuuîì +++=+
rr=
======
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CHAPTER 6. VECTORS
122
6.3 Vector Subtraction =
561. îìïrrr
−= =áÑ= ìïîrrr
=+ K==
==
Figure 79. =
== ==
Figure 80. =
562. ( )îìîìrrrr
−+=− ==
563. ( )MIMIMMìì ==−rrr
==
564. MM =r
=
=565. ( )ONONON wwIvvIuuîì −−−=−
rrI==
===
6.4 Scaling Vectors =
566. ìïrr
λ= =
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CHAPTER 6. VECTORS
123
==
Figure 81. =
567. ìïrr
⋅λ= =
=568. ( )wIvIuì λλλ=λ
r=
=569. λ=λ ìì
rr=
=570. ( ) ììì
rrrµ+λ=µ+λ =
=571. ( ) ( ) ( )ììì
rrrλµ=λµ=µλ =
=572. ( ) îìîì
rrrrλ+λ=+λ =
===
6.5 Scalar Product =
573. pÅ~ä~ê=mêçÇìÅí=çÑ=sÉÅíçêë= ìr
=~åÇ îr
=θ⋅⋅=⋅ Åçëîìîì
rrrrI==
ïÜÉêÉ=θ =áë=íÜÉ=~åÖäÉ=ÄÉíïÉÉå=îÉÅíçêë= ìr
=~åÇ îr
K=====
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CHAPTER 6. VECTORS
124
= ==
Figure 82. =
574. pÅ~ä~ê=mêçÇìÅí=áå=`ççêÇáå~íÉ=cçêã=fÑ= ( )NNN wIvIuì =r
I= ( )OOO wIvIuî =r
I=íÜÉå==
ONONON wwvvuuîì ++=⋅rr
K==
575. ^åÖäÉ=_ÉíïÉÉå=qïç=sÉÅíçêë==fÑ= ( )NNN wIvIuì =r
I= ( )OOO wIvIuî =r
I=íÜÉå==
OO
OO
OO
ON
ON
ON
ONONON
wvuwvu
wwvvuuÅçë
++++
++=θ K=
=576. `çããìí~íáîÉ=mêçéÉêíó=
ìîîìrrrr⋅=⋅ =
=577. ^ëëçÅá~íáîÉ=mêçéÉêíó=
( ) ( ) îìîìrrrr⋅λµ=µ⋅λ =
=578. aáëíêáÄìíáîÉ=mêçéÉêíó=
( ) ïìîìïîìrrrrrrr⋅+⋅=+⋅ =
=
579. Mîì =⋅rr
=áÑ= ìr
I îr
=~êÉ=çêíÜçÖçå~ä=EO
π=θ FK=
=
580. Mîì >⋅rr
=áÑ=O
Mπ
<θ< K=
=
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CHAPTER 6. VECTORS
125
581. Mîì <⋅rr
=áÑ= π<θ<πO
K=
=582. îìîì
rrrr⋅≤⋅ =
=583. îìîì
rrrr⋅=⋅ =áÑ= ì
rI îr
=~êÉ=é~ê~ääÉä=E M=θ FK=
=584. fÑ= ( )NNN wIvIuì =
rI=íÜÉå==
ON
ON
ON
OO wvuìììì ++===⋅rrrr
K=
=
585. Nââààáá =⋅=⋅=⋅rrrrrr
==
586. Máââààá =⋅=⋅=⋅rrrrrr
====
6.6 Vector Product =
587. sÉÅíçê=mêçÇìÅí=çÑ=sÉÅíçêë= ìr
=~åÇ îr
=ïîìrrr
=× I=ïÜÉêÉ==
• θ⋅⋅= ëáåîìïrrr
I=ïÜÉêÉ=O
Mπ
≤θ≤ X=
• ìïrr
⊥ = ~åÇ= îïrr
⊥ X=• =sÉÅíçêë= ì
rI= îr
I= ïr
=Ñçêã=~=êáÖÜí-Ü~åÇÉÇ=ëÅêÉïK==
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CHAPTER 6. VECTORS
126
======= ==
Figure 83. =
588. OOO
NNN
wvu
wvu
âàá
îìï
rrr
rrr=×= =
=
589.
−=×=
OO
NN
OO
NN
OO
NN
vu
vuI
wu
wuI
wv
wvîìïrrr
=
=590. θ⋅⋅=×= ëáåîìîìp
rrrr=EcáÖKUPF=
=591. ^åÖäÉ=_ÉíïÉÉå=qïç=sÉÅíçêë=EcáÖKUPF=
îì
îìëáå rr
rr
⋅×
=θ =
=592. kçåÅçããìí~íáîÉ=mêçéÉêíó=
( )ìîîìrrrr
×−=× ===
593. ^ëëçÅá~íáîÉ=mêçéÉêíó=( ) ( ) îìîì
rrrr×λµ=µ×λ =
==
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CHAPTER 6. VECTORS
127
594. aáëíêáÄìíáîÉ=mêçéÉêíó=( ) ïìîìïîì
rrrrrrr×+×=+× =
=595. Mîì
rrr=× =áÑ= ì
r=~åÇ= î
r=~êÉ=é~ê~ääÉä=E M=θ FK=
=
596. Mââààáárrrrrrr
=×=×=× ==
597. âàárrr
=× I= áâàrrr
=× I= àáârrr
=× ====
6.7 Triple Product =
598. pÅ~ä~ê=qêáéäÉ=mêçÇìÅí=[ ] ( ) ( ) ( )îìïìïîïîìïîì
rrrrrrrrrrrr×⋅=×⋅=×⋅= =
=599. [ ] [ ] [ ] [ ] [ ] [ ]îïììîïïìîìïîîìïïîì
rrrrrrrrrrrrrrrrrr−=−=−=== =
=600. ( ) [ ]ïîìâïîìâ
rrrrrr=×⋅ =
=601. pÅ~ä~ê=qêáéäÉ=mêçÇìÅí=áå=`ççêÇáå~íÉ=cçêã=
( )PPP
OOO
NNN
wvu
wvu
wvu
ïîì =×⋅rrr
I==
ïÜÉêÉ==( )NNN wIvIuì =
rI= ( )OOO wIvIuî =r
I= ( )PPP wIvIuï =r
K==
=602. sçäìãÉ=çÑ=m~ê~ääÉäÉéáéÉÇ=
( )ïîìsrrr
×⋅= =
=
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CHAPTER 6. VECTORS
128
============ ==
Figure 84. =
603. sçäìãÉ=çÑ=móê~ãáÇ=
( )ïîìS
Ns
rrr×⋅= =
=
==
Figure 85. =
604. fÑ== ( ) Mïîì =×⋅rrr
I=íÜÉå=íÜÉ=îÉÅíçêë== ìr
I= îr
I=~åÇ= ïr
=~êÉ=äáåÉ~êäó=ÇÉéÉåÇÉåí=I=ëç= îìï
rrrµ+λ= =Ñçê=ëçãÉ=ëÅ~ä~êë=λ =~åÇ=µ K==
=605. fÑ== ( ) Mïîì ≠×⋅
rrrI=íÜÉå=íÜÉ=îÉÅíçêë== ì
rI= îr
I=~åÇ= ïr
=~êÉ=äáåÉ~êäó=áåÇÉéÉåÇÉåíK==
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CHAPTER 6. VECTORS
129
606. sÉÅíçê=qêáéäÉ=mêçÇìÅí=( ) ( ) ( )ïîìîïìïîì
rrrrrrrrr⋅−⋅=×× ==
========
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130
Chapter 7
Analytic Geometry ====
7.1 One-Dimensional Coordinate System =mçáåí=ÅççêÇáå~íÉëW= Mñ I= Nñ I= Oñ I= Mó I= Nó I= Oó =
oÉ~ä=åìãÄÉêW=λ ==aáëí~åÅÉ=ÄÉíïÉÉå=íïç=éçáåíëW=Ç===
607. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë=
ONNO ññññ^_Ç −=−== =
=
==
Figure 86. =
608. aáîáÇáåÖ=~=iáåÉ=pÉÖãÉåí=áå=íÜÉ=o~íáç=λ =
λ+λ+
=N
ñññ ON
M I=`_
^`=λ I= N−≠λ K=
=
======== ==
Figure 87.
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CHAPTER 7. ANALYTIC GEOMETRY
131
609. jáÇéçáåí=çÑ=~=iáåÉ=pÉÖãÉåí=
O
ñññ ON
M
+= I= N=λ K=
===
7.2 Two-Dimensional Coordinate System =mçáåí=ÅççêÇáå~íÉëW= Mñ I= Nñ I= Oñ I= Mó I= Nó I= Oó =
mçä~ê=ÅççêÇáå~íÉëW= ϕIê =oÉ~ä=åìãÄÉêW=λ ==mçëáíáîÉ=êÉ~ä=åìãÄÉêëW=~I=ÄI=ÅI==aáëí~åÅÉ=ÄÉíïÉÉå=íïç=éçáåíëW=Ç=^êÉ~W=p===
610. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë=
( ) ( )ONOO
NO óóññ^_Ç −+−== =
=
==
Figure 88.
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CHAPTER 7. ANALYTIC GEOMETRY
132
611. aáîáÇáåÖ=~=iáåÉ=pÉÖãÉåí=áå=íÜÉ=o~íáç=λ =
λ+λ+
=N
ñññ ON
M I=λ+λ+
=N
óóó ON
M I==
`_
^`=λ I= N−≠λ K=
=
======= ==
Figure 89. ==
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CHAPTER 7. ANALYTIC GEOMETRY
133
======= ==
Figure 90. =
612. jáÇéçáåí=çÑ=~=iáåÉ=pÉÖãÉåí=
O
ñññ ON
M
+= I=
O
óóó ON
M
+= I= N=λ K=
=613. `ÉåíêçáÇ=EfåíÉêëÉÅíáçå=çÑ=jÉÇá~åëF=çÑ=~=qêá~åÖäÉ=
P
ññññ PON
M
++= I=
P
óóóó PON
M
++= I==
ïÜÉêÉ== ( )NN óIñ^ I== ( )OO óIñ_ I==~åÇ== ( )PP óIñ` ==~êÉ=îÉêíáÅÉë=çÑ=
íÜÉ=íêá~åÖäÉ= ^_` K= ==
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CHAPTER 7. ANALYTIC GEOMETRY
134
========= ==
Figure 91. =
614. fåÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=^åÖäÉ=_áëÉÅíçêëF=çÑ=~=qêá~åÖäÉ=
ÅÄ~
ÅñÄñ~ññ PON
M ++++
= I=ÅÄ~
ÅóÄó~óó PON
M ++++
= I==
ïÜÉêÉ= _`~ = I= `^Ä = I= ^_Å = K===
======== ==
Figure 92.
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CHAPTER 7. ANALYTIC GEOMETRY
135
615. `áêÅìãÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=íÜÉ=páÇÉ=mÉêéÉåÇáÅìä~ê======================_áëÉÅíçêëF=çÑ=~=qêá~åÖäÉ=
Nóñ
Nóñ
Nóñ
O
Nóóñ
Nóóñ
Nóóñ
ñ
PP
OO
NN
POP
OP
OOO
OO
NON
ON
M
+++
= I=
Nóñ
Nóñ
Nóñ
O
Nóññ
Nóññ
Nóññ
ó
PP
OO
NN
OP
OPP
OO
OOO
ON
ONN
M
+++
= =
=
======== ===
Figure 93. =======
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CHAPTER 7. ANALYTIC GEOMETRY
136
616. lêíÜçÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=^äíáíìÇÉëF=çÑ=~=qêá~åÖäÉ=
Nóñ
Nóñ
Nóñ
Nóññó
Nóññó
Nóññó
ñ
PP
OO
NN
OPONP
OONPO
ONPON
M
+++
= I=
Nóñ
Nóñ
Nóñ
Nñóóñ
Nñóóñ
Nñóóñ
ó
PP
OO
NN
PONOP
ONPOO
NPOON
M
+++
= =
=
====== ==
Figure 94. =
617. ^êÉ~=çÑ=~=qêá~åÖäÉ=
( ) ( )NPNP
NONO
PP
OO
NN
óóññ
óóññ
O
N
Nóñ
Nóñ
Nóñ
O
Np
−−−−
±=±= =
===
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CHAPTER 7. ANALYTIC GEOMETRY
137
618. ^êÉ~=çÑ=~=nì~Çêáä~íÉê~ä=
( ) ( )( ) ( )( )[ ++−++−±= POPOONON óóññóóññO
Np =
( )( ) ( )( )]NQNQQPQP óóññóóññ +−++−+ =
=
=== ==
Figure 95. =kçíÉW=få=Ñçêãìä~ë=SNTI=SNU=ïÉ=ÅÜççëÉ=íÜÉ=ëáÖå=EHF=çê=E¥F=ëç=íÜ~í=íç=ÖÉí=~=éçëáíáîÉ=~åëïÉê=Ñçê=~êÉ~K==
=619. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë=áå=mçä~ê=`ççêÇáå~íÉë=
( )NOONOO
ON ÅçëêêOêê^_Ç ϕ−ϕ−+== =
=
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CHAPTER 7. ANALYTIC GEOMETRY
138
==
Figure 96. =
620. `çåîÉêíáåÖ=oÉÅí~åÖìä~ê=`ççêÇáå~íÉë=íç=mçä~ê=`ççêÇáå~íÉë=ϕ= Åçëêñ I= ϕ= ëáåêó K=
=
==
Figure 97. =
621. `çåîÉêíáåÖ=mçä~ê=`ççêÇáå~íÉë=íç=oÉÅí~åÖìä~ê=`ççêÇáå~íÉë=
OO óñê += I=ñ
óí~å =ϕ K=
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CHAPTER 7. ANALYTIC GEOMETRY
139
7.3 Straight Line in Plane =mçáåí=ÅççêÇáå~íÉëW=uI=vI=ñI= Mñ I= Nñ I== Mó I= Nó I= N~ I= O~ I=£==
oÉ~ä=åìãÄÉêëW=âI=~I=ÄI=éI=íI=^I=_I=`I= N^ I= O^ I=£=^åÖäÉëW=α I=β =^åÖäÉ=ÄÉíïÉÉå=íïç=äáåÉëW=ϕ =kçêã~ä=îÉÅíçêW= å
r=
mçëáíáçå=îÉÅíçêëW= êr
I= ~r
I= Är
===
622. dÉåÉê~ä=bèì~íáçå=çÑ=~=píê~áÖÜí=iáåÉ=M`_ó^ñ =++ =
=623. kçêã~ä=sÉÅíçê=íç=~=píê~áÖÜí=iáåÉ=
qÜÉ=îÉÅíçê= ( )_I^år
=áë=åçêã~ä=íç=íÜÉ=äáåÉ= M`_ó^ñ =++ K==
==
Figure 98. =
624. bñéäáÅáí=bèì~íáçå=çÑ=~=píê~áÖÜí=iáåÉ=EpäçéÉ-fåíÉêÅÉéí=cçêãF=Äâñó += K==
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CHAPTER 7. ANALYTIC GEOMETRY
140
qÜÉ=Öê~ÇáÉåí=çÑ=íÜÉ=äáåÉ=áë= α= í~åâ K==
==
Figure 99. =
625. dê~ÇáÉåí=çÑ=~=iáåÉ==
NO
NO
ññ
óóí~åâ
−−
=α= =
=
==
Figure 100.
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CHAPTER 7. ANALYTIC GEOMETRY
141
626. bèì~íáçå=çÑ=~=iáåÉ=dáîÉå=~=mçáåí=~åÇ=íÜÉ=dê~ÇáÉåí=( )MM ññâóó −+= I==
ïÜÉêÉ=â=áë=íÜÉ=Öê~ÇáÉåíI= ( )MM óIñm =áë=~=éçáåí=çå=íÜÉ=äáåÉK=
=
==
Figure 101. =
627. bèì~íáçå=çÑ=~=iáåÉ=qÜ~í=m~ëëÉë=qÜêçìÖÜ=qïç=mçáåíë=
NO
N
NO
N
ññ
ññ
óó
óó
−−
=−−
==
çê=
M
Nóñ
Nóñ
Nóñ
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ñÅçëÜ
NñÜëÉÅó −+
=== I= oñ∈ K=
=Figure 167.
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CHAPTER 8. DIFFERENTIAL CALCULUS
203
742. eóéÉêÄçäáÅ=`çëÉÅ~åí=cìåÅíáçå==
ñÅëÅÜó = I=ññ ÉÉ
O
ñëáåÜ
NñÅëÅÜó −−
=== I= oñ∈ I= Mñ ≠ K=
=
====== ==
Figure 168. =
743. fåîÉêëÉ=eóéÉêÄçäáÅ=páåÉ=cìåÅíáçå==ñ~êÅëáåÜó = I= oñ∈ K=
=
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CHAPTER 8. DIFFERENTIAL CALCULUS
204
===== ==
Figure 169. =
744. fåîÉêëÉ=eóéÉêÄçäáÅ=`çëáåÉ=cìåÅíáçå==ñ~êÅÅçëÜó = I= [ )∞∈ INñ K=
=
===== ==
Figure 170. =
745. fåîÉêëÉ=eóéÉêÄçäáÅ=q~åÖÉåí=cìåÅíáçå==ñ~êÅí~åÜó = I= ( )NINñ −∈ K=
=
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CHAPTER 8. DIFFERENTIAL CALCULUS
205
===== ==
Figure 171. =
746. fåîÉêëÉ=eóéÉêÄçäáÅ=`çí~åÖÉåí=cìåÅíáçå==ñ~êÅÅçíÜó = I= ( ) ( )∞∪−∞−∈ INNIñ K==
=
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CHAPTER 8. DIFFERENTIAL CALCULUS
206
===== ==
Figure 172. =
747. fåîÉêëÉ=eóéÉêÄçäáÅ=pÉÅ~åí=cìåÅíáçå==ñ~êÅëÉÅÜó = I= ( ]NIMñ∈ K=
=
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CHAPTER 8. DIFFERENTIAL CALCULUS
207
=Figure 173.
=748. fåîÉêëÉ=eóéÉêÄçäáÅ=`çëÉÅ~åí=cìåÅíáçå==
ñ~êÅÅëÅÜó = I= oñ∈ I= Mñ ≠ K==
= Figure 174.
=
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CHAPTER 8. DIFFERENTIAL CALCULUS
208
8.2 Limits of Functions =cìåÅíáçåëW= ( )ñÑ I= ( )ñÖ =^êÖìãÉåíW=ñ=oÉ~ä=Åçåëí~åíëW=~I=â===
749. ( ) ( )[ ] ( ) ( )ñÖäáãñÑäáãñÖñÑäáã~ñ~ñ~ñ →→→
+=+ =
=750. ( ) ( )[ ] ( ) ( )ñÖäáãñÑäáãñÖñÑäáã
~ñ~ñ~ñ →→→−=− =
=751. ( ) ( )[ ] ( ) ( )ñÖäáãñÑäáãñÖñÑäáã
~ñ~ñ~ñ →→→⋅=⋅ =
=
752. ( )( )
( )( )ñÖäáã
ñÑäáã
ñÖ
ñÑäáã
~ñ
~ñ
~ñ→
→
→= I=áÑ= ( ) MñÖäáã
~ñ≠
→K=
=753. ( )[ ] ( )ñÑäáãâñâÑäáã
~ñ~ñ →→= =
=754. ( )( ) ( )( )ñÖäáãÑñÖÑäáã
~ñ~ñ →→= =
=755. ( ) ( )~ÑñÑäáã
~ñ=
→I=áÑ=íÜÉ=ÑìåÅíáçå= ( )ñÑ =áë=Åçåíáåìçìë=~í= ~ñ = K=
=
756. Nñ
ñëáåäáã
Mñ=
→=
=
757. Nñ
ñí~åäáã
Mñ=
→=
=
758. Nñ
ñëáåäáã
N
Mñ=
−
→=
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CHAPTER 8. DIFFERENTIAL CALCULUS
209
759. Nñ
ñí~åäáã
N
Mñ=
−
→=
=
760. ( )N
ñ
ñNäåäáã
Mñ=
+→
=
=
761. Éñ
NNäáã
ñ
ñ=
+
∞→=
=
762. âñ
ñÉ
ñ
âNäáã =
+
∞→=
=763. N~äáã ñ
Mñ=
→=
===
8.3 Definition and Properties of the Derivative =cìåÅíáçåëW=ÑI=ÖI=óI=ìI=î=fåÇÉéÉåÇÉåí=î~êá~ÄäÉW=ñ=oÉ~ä=Åçåëí~åíW=â=^åÖäÉW=α ===
764. ( ) ( ) ( )Çñ
Çó
ñ
óäáã
ñ
ñÑññÑäáãñó
MñMñ=
∆∆
=∆
−∆+=′
→∆→∆==
=
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CHAPTER 8. DIFFERENTIAL CALCULUS
210
== ==
Figure 175. =
765. α= í~åÇñ
Çó==
=
766. ( )Çñ
Çî
Çñ
Çì
Çñ
îìÇ+=
+=
=
767. ( )Çñ
Çî
Çñ
Çì
Çñ
îìÇ−=
−=
=
768. ( )Çñ
Çìâ
Çñ
âìÇ= =
=769. mêçÇìÅí=oìäÉ=
( )Çñ
Çîìî
Çñ
Çì
Çñ
îìÇ⋅+⋅=
⋅==
==
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CHAPTER 8. DIFFERENTIAL CALCULUS
211
770. nìçíáÉåí=oìäÉ=
OîÇñ
Çîìî
Çñ
Çì
î
ì
Çñ
Ç ⋅−⋅=
=
=771. `Ü~áå=oìäÉ=
( )( )ñÖÑó = I= ( )ñÖì = I==
Çñ
Çì
Çì
Çó
Çñ
Çó⋅= K=
=772. aÉêáî~íáîÉ=çÑ=fåîÉêëÉ=cìåÅíáçå=
Çó
ÇñN
Çñ
Çó= I==
ïÜÉêÉ= ( )óñ áë=íÜÉ=áåîÉêëÉ=ÑìåÅíáçå=çÑ= ( )ñó K===
773. oÉÅáéêçÅ~ä=oìäÉ=
OóÇñ
Çó
ó
N
Çñ
Ç−=
=
=774. içÖ~êáíÜãáÅ=aáÑÑÉêÉåíá~íáçå=
( )ñÑó = I= ( )ñÑäåóäå = I==
( ) ( )[ ]ñÑäåÇñ
ÇñÑ
Çñ
Çó⋅= K=
==
8.4 Table of Derivatives =fåÇÉéÉåÇÉåí=î~êá~ÄäÉW=ñ=oÉ~ä=Åçåëí~åíëW=`I=~I=ÄI=Å=k~íìê~ä=åìãÄÉêW=å=
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CHAPTER 8. DIFFERENTIAL CALCULUS
212
775. ( ) M`Çñ
Ç= =
=
776. ( ) NñÇñ
Ç= =
=
777. ( ) ~Ä~ñÇñ
Ç=+ =
=
778. ( ) Ä~ñÅÄñ~ñÇñ
Ç O +=++ =
=
779. ( ) Nåå åññÇñ
Ç −= =
=
780. ( )Nå
å
ñ
åñ
Çñ
Ç+
− −= =
=
781. Oñ
N
ñ
N
Çñ
Ç−=
=
=
782. ( )ñO
Nñ
Çñ
Ç= =
=
783. ( )å Nå
å
ñå
Nñ
Çñ
Ç−
= =
=
784. ( )ñ
Nñäå
Çñ
Ç= =
=
785. ( )~äåñ
NñäçÖ
Çñ
Ç~ = I= M~ > I= N~ ≠ K=
=
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CHAPTER 8. DIFFERENTIAL CALCULUS
213
786. ( ) ~äå~~Çñ
Ç ññ = I= M~ > I= N~ ≠ K=
=
787. ( ) ññ ÉÉÇñ
Ç= =
=
788. ( ) ñÅçëñëáåÇñ
Ç= =
=
789. ( ) ñëáåñÅçëÇñ
Ç−= =
=
790. ( ) ñëÉÅñÅçë
Nñí~å
Çñ
Ç O
O== =
=
791. ( ) ñÅëÅñëáå
NñÅçí
Çñ
Ç O
O−=−= =
=
792. ( ) ñëÉÅñí~åñëÉÅÇñ
Ç⋅= =
=
793. ( ) ñÅëÅñÅçíñÅëÅÇñ
Ç⋅−= =
=
794. ( )OñN
Nñ~êÅëáå
Çñ
Ç
−= =
=
795. ( )OñN
Nñ~êÅÅçë
Çñ
Ç
−−= =
=
796. ( )OñN
Nñ~êÅí~å
Çñ
Ç
+= =
=
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CHAPTER 8. DIFFERENTIAL CALCULUS
214
797. ( )OñN
NñÅçí~êÅ
Çñ
Ç
+−= =
=
798. ( )Nññ
NñëÉÅ~êÅ
Çñ
ÇO −
= =
=
799. ( )Nññ
NñÅëÅ~êÅ
Çñ
ÇO −
−= =
=
800. ( ) ñÅçëÜñëáåÜÇñ
Ç= =
=
801. ( ) ñëáåÜñÅçëÜÇñ
Ç= =
=
802. ( ) ñëÉÅÜñÅçëÜ
Nñí~åÜ
Çñ
Ç O
O== =
=
803. ( ) ñÅëÅÜñëáåÜ
NñÅçíÜ
Çñ
Ç O
O−=−= =
=
804. ( ) ñí~åÜñëÉÅÜñëÉÅÜÇñ
Ç⋅−= =
=
805. ( ) ñÅçíÜñÅëÅÜñÅëÅÜÇñ
Ç⋅−= =
=
806. ( )Nñ
Nñ=~êÅëáåÜ
Çñ
ÇO +
= =
=
807. ( )Nñ
Nñ=~êÅÅçëÜ
Çñ
ÇO −
= =
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CHAPTER 8. DIFFERENTIAL CALCULUS
215
808. ( )OñN
Nñ=~êÅí~åÜ
Çñ
Ç
−= I= Nñ < K=
=
809. ( )Nñ
Nñ=~êÅÅçíÜ
Çñ
ÇO −
−= I= Nñ > K=
=
810. ( )Çñ
Çîìäåì
Çñ
Çìîìì
Çñ
Ç îNîî ⋅+⋅= − =
===
8.5 Higher Order Derivatives =cìåÅíáçåëW=ÑI=óI=ìI=î=fåÇÉéÉåÇÉåí=î~êá~ÄäÉW=ñ=k~íìê~ä=åìãÄÉêW=å===
811. pÉÅçåÇ=ÇÉêáî~íáîÉ=
( )O
O
Çñ
óÇ
Çñ
Çó
Çñ
Ç
Çñ
ÇóÑÑ =
=′
=′′=′′ =
=812. eáÖÜÉê-lêÇÉê=ÇÉêáî~íáîÉ=
( ) ( ) ( )( ) ′=== −Nåå
å
åå Ñó
Çñ
óÇÑ =
=
813. ( )( ) ( ) ( )ååå îìîì +=+ ==
814. ( )( ) ( ) ( )ååå îìîì −=− ==
815. iÉáÄåáíò∞ë=cçêãìä~ë=
( ) îìîìOîììî ′′+′′+′′=′′ =
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CHAPTER 8. DIFFERENTIAL CALCULUS
216
( ) îìîìPîìPîììî ′′′+′′′+′′′+′′′=′′′ =
( )( ) ( ) ( ) ( ) ( ) ( )åOåNååå ìîîìON
Nååîåìîììî ++′′
⋅−
+′+= −− K =
=
816. ( )( )
( )åãåã ñ
>åã
>ãñ −
−= =
=
817. ( )( )>åñ
åå = ==
818. ( )( ) ( ) ( )~äåñ
>NåNñäçÖ
å
Nåå
~
−−=
−
=
=
819. ( )( ) ( ) ( )å
Nåå
ñ
>NåNñäå
−−=
−
=
=
820. ( )( )~äå~~ åñåñ = =
=
821. ( )( ) ñåñ ÉÉ = ==
822. ( )( )~äå~ã~ åãñååãñ = =
=
823. ( )( )
π
+=O
åñëáåñëáå å =
=
824. ( )( )
π
+=O
åñÅçëñÅçë å =
===
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CHAPTER 8. DIFFERENTIAL CALCULUS
217
8.6 Applications of Derivative =cìåÅíáçåëW=ÑI=ÖI=ó=mçëáíáçå=çÑ=~å=çÄàÉÅíW=ë==sÉäçÅáíóW=î=^ÅÅÉäÉê~íáçåW=ï=fåÇÉéÉåÇÉåí=î~êá~ÄäÉW=ñ=qáãÉW=í=k~íìê~ä=åìãÄÉêW=å===
825. sÉäçÅáíó=~åÇ=^ÅÅÉäÉê~íáçå=( )íÑë = =áë=íÜÉ=éçëáíáçå=çÑ=~å=çÄàÉÅí=êÉä~íáîÉ=íç=~=ÑáñÉÇ=
ÅççêÇáå~íÉ=ëóëíÉã=~í=~=íáãÉ=íI==( )íÑëî ′=′= =áë=íÜÉ=áåëí~åí~åÉçìë=îÉäçÅáíó=çÑ=íÜÉ=çÄàÉÅíI=
( )íÑëîï ′′=′′=′= =áë=íÜÉ=áåëí~åí~åÉçìë=~ÅÅÉäÉê~íáçå=çÑ=íÜÉ=çÄàÉÅíK===
826. q~åÖÉåí=iáåÉ=( )( )MMM ñññÑóó −′=− =
=
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CHAPTER 8. DIFFERENTIAL CALCULUS
218
==
Figure 176. =
827. kçêã~ä=iáåÉ=
( )( )M
M
M ñññÑ
Nóó −
′−=− =EcáÖ=NTSF=
=828. fåÅêÉ~ëáåÖ=~åÇ=aÉÅêÉ~ëáåÖ=cìåÅíáçåëK==
fÑ= ( ) MñÑ M >′ I=íÜÉå=ÑEñF=áë=áåÅêÉ~ëáåÖ=~í= Mñ K=EcáÖ=NTTI= Nññ < I=
ññO < FI=fÑ= ( ) MñÑ M <′ I=íÜÉå=ÑEñF=áë=ÇÉÅêÉ~ëáåÖ=~í= Mñ K=EcáÖ=NTTI=
ON ñññ << FI=fÑ= ( )MñÑ ′ =ÇçÉë=åçí=Éñáëí=çê=áë=òÉêçI=íÜÉå=íÜÉ=íÉëí=Ñ~áäëK==
=
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CHAPTER 8. DIFFERENTIAL CALCULUS
219
==
Figure 177. =
829. içÅ~ä=ÉñíêÉã~=^=ÑìåÅíáçå=ÑEñF=Ü~ë=~=äçÅ~ä=ã~ñáãìã=~í= Nñ =áÑ=~åÇ=çåäó=áÑ=íÜÉêÉ=Éñáëíë=ëçãÉ=áåíÉêî~ä=Åçåí~áåáåÖ= Nñ =ëìÅÜ=íÜ~í=( ) ( )ñÑñÑ N ≥ =Ñçê=~ää=ñ=áå=íÜÉ=áåíÉêî~ä=EcáÖKNTTFK==
=^=ÑìåÅíáçå=ÑEñF=Ü~ë=~=äçÅ~ä=ãáåáãìã=~í= Oñ =áÑ=~åÇ=çåäó=áÑ=íÜÉêÉ=Éñáëíë=ëçãÉ=áåíÉêî~ä=Åçåí~áåáåÖ= Oñ =ëìÅÜ=íÜ~í=( ) ( )ñÑñÑ O ≤ =Ñçê=~ää=ñ=áå=íÜÉ=áåíÉêî~ä=EcáÖKNTTFK=
=830. `êáíáÅ~ä=mçáåíë=
^=ÅêáíáÅ~ä=éçáåí=çå=ÑEñF=çÅÅìêë=~í= Mñ =áÑ=~åÇ=çåäó=áÑ=ÉáíÜÉê=
( )MñÑ ′ =áë=òÉêç=çê=íÜÉ=ÇÉêáî~íáîÉ=ÇçÉëå∞í=ÉñáëíK=
=831. cáêëí=aÉêáî~íáîÉ=qÉëí=Ñçê=içÅ~ä=bñíêÉã~K=
fÑ=ÑEñF=áë==áåÅêÉ~ëáåÖ==E ( ) MñÑ >′ F=Ñçê==~ää==ñ==áå==ëçãÉ==áåíÉêî~ä=( ]NñI~ ==~åÇ==ÑEñF==áë==ÇÉÅêÉ~ëáåÖ==E ( ) MñÑ <′ F==Ñçê=~ää==ñ=áå=ëçãÉ=áåíÉêî~ä= = [ )ÄIñN I= = íÜÉå= ÑEñF= Ü~ë= ~= = äçÅ~ä= ã~ñáãìã= = ~í= = Nñ =EcáÖKNTTFK==
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CHAPTER 8. DIFFERENTIAL CALCULUS
220
832. fÑ=ÑEñF=áë=ÇÉÅêÉ~ëáåÖ=E ( ) MñÑ <′ F=Ñçê=~ää=ñ=áå=ëçãÉ=áåíÉêî~ä=( ]OñI~ =~åÇ=ÑEñF=áë=áåÅêÉ~ëáåÖ=E ( ) MñÑ >′ F=Ñçê=~ää=ñ=áå=ëçãÉ=áåíÉêî~ä= [ )ÄIñ O I=íÜÉå=ÑEñF=Ü~ë=~=äçÅ~ä=ãáåáãìã=~í= Oñ K==EcáÖKNTTFK==
833. pÉÅçåÇ=aÉêáî~íáîÉ=qÉëí=Ñçê=içÅ~ä=bñíêÉã~K=fÑ= ( ) MñÑ N =′ =~åÇ= ( ) MñÑ N <′′ I=íÜÉå=ÑEñF=Ü~ë=~=äçÅ~ä=ã~ñáãìã=~í== Nñ K=fÑ= ( ) MñÑ O =′ =~åÇ= ( ) MñÑ O >′′ I=íÜÉå=ÑEñF=Ü~ë=~=äçÅ~ä=ãáåáãìã=~í= Oñ K=EcáÖKNTTF==
834. `çåÅ~îáíóK==ÑEñF= áë= = ÅçåÅ~îÉ= ìéï~êÇ= ~í= = Mñ = = áÑ= = ~åÇ= = çåäó= = áÑ= = ( )ñÑ ′ = áë============
áåÅêÉ~ëáåÖ=~í= Mñ =EcáÖKNTTI= ññ P < FK===
ÑEñF= áë= =ÅçåÅ~îÉ= =Ççïåï~êÇ=~í= = Mñ = = áÑ=~åÇ=çåäó= áÑ= = ( )ñÑ ′ = = áë===============
ÇÉÅêÉ~ëáåÖ=~í= Mñ K=EcáÖKNTTI= Pññ < FK===
=835. pÉÅçåÇ=aÉêáî~íáîÉ=qÉëí=Ñçê=`çåÅ~îáíóK==
fÑ= ( ) MñÑ M >′′ I=íÜÉå=ÑEñF=áë=ÅçåÅ~îÉ=ìéï~êÇ=~í= Mñ K==
fÑ= ( ) MñÑ M <′′ I=íÜÉå=ÑEñF=áë=ÅçåÅ~îÉ=Ççïåï~êÇ=~í= Mñ K=
fÑ= ( )ñÑ ′′ =ÇçÉë=åçí=Éñáëí=çê=áë=òÉêçI=íÜÉå=íÜÉ=íÉëí=Ñ~áäëK==
836. fåÑäÉÅíáçå=mçáåíë=fÑ= = ( )PñÑ ′ = =Éñáëíë==~åÇ== ( )ñÑ ′′ = =ÅÜ~åÖÉë=ëáÖå=~í= Pññ = I= =íÜÉå=
íÜÉ= éçáåí= ( )( )PP ñÑIñ = áë= ~å= áåÑäÉÅíáçå= éçáåí= çÑ= íÜÉ= Öê~éÜ= çÑ=
( )ñÑ K=fÑ= ( )PñÑ ′′ =Éñáëíë=~í=íÜÉ=áåÑäÉÅíáçå=éçáåíI=íÜÉå= ( ) MñÑ P =′′ =
EcáÖKNTTFK==
837. i∞eçéáí~ä∞ë=oìäÉ=
( )( )
( )( )ñÖ
ñÑäáã
ñÖ
ñÑäáã
ÅñÅñ ′′
=→→
=áÑ= ( ) ( )∞
==→→
MñÖäáãñÑäáã
ÅñÅñK==
=
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CHAPTER 8. DIFFERENTIAL CALCULUS
221
8.7 Differential =cìåÅíáçåëW=ÑI=ìI=î=fåÇÉéÉåÇÉåí=î~êá~ÄäÉW=ñ=aÉêáî~íáîÉ=çÑ=~=ÑìåÅíáçåW= ( )ñó′ I= ( )ñÑ ′ =oÉ~ä=Åçåëí~åíW=`=aáÑÑÉêÉåíá~ä=çÑ=ÑìåÅíáçå= ( )ñÑó = W=Çó=aáÑÑÉêÉåíá~ä=çÑ=ñW=Çñ=pã~ää=ÅÜ~åÖÉ=áå=ñW= ñ∆ =pã~ää=ÅÜ~åÖÉ=áå=óW= ó∆ ===
838. ÇñóÇó ′= ==
839. ( ) ( ) ( ) ññÑñÑññÑ ∆′+=∆+ ==
==
Figure 178.
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CHAPTER 8. DIFFERENTIAL CALCULUS
222
840. pã~ää=`Ü~åÖÉ=áå=ó=( ) ( )ñÑññÑó −∆+=∆ =
=841. ( ) ÇîÇìîìÇ +=+ =
=842. ( ) ÇîÇìîìÇ −=− =
=843. ( ) `Çì`ìÇ = =
=844. ( ) ìÇîîÇììîÇ += =
=
845. Oî
ìÇîîÇì
î
ìÇ
−=
=
===
8.8 Multivariable Functions =cìåÅíáçåë=çÑ=íïç=î~êá~ÄäÉëW= ( )óIñò I= ( )óIñÑ I= ( )óIñÖ I= ( )óIñÜ ==^êÖìãÉåíëW=ñI=óI=í=pã~ää=ÅÜ~åÖÉë=áå=ñI=óI=òI=êÉëéÉÅíáîÉäóW= ñ∆ I= ó∆ I= ò∆ K===
846. cáêëí=lêÇÉê=m~êíá~ä=aÉêáî~íáîÉë=qÜÉ=é~êíá~ä=ÇÉêáî~íáîÉ=ïáíÜ=êÉëéÉÅí=íç=ñ=
ñÑñ
Ñ=
∂∂
=E~äëç= ñòñ
ò=
∂∂
FI=
qÜÉ=é~êíá~ä=ÇÉêáî~íáîÉ=ïáíÜ=êÉëéÉÅí=íç=ó=
óÑó
Ñ=
∂∂
=E~äëç= óòó
ò=
∂∂
FK=
==
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CHAPTER 8. DIFFERENTIAL CALCULUS
223
847. pÉÅçåÇ=lêÇÉê=m~êíá~ä=aÉêáî~íáîÉë=
ññO
O
Ññ
Ñ
ñ
Ñ
ñ=
∂∂
=
∂∂
∂∂
I==
óóO
O
Ñó
Ñ
ó
Ñ
ó=
∂∂
=
∂∂
∂∂
I==
ñó
O
Ññó
Ñ
ñ
Ñ
ó=
∂∂∂
=
∂∂
∂∂
I==
óñ
O
Ñóñ
Ñ
ó
Ñ
ñ=
∂∂∂
=
∂∂
∂∂
K==
fÑ=íÜÉ=ÇÉêáî~íáîÉë=~êÉ=ÅçåíáåìçìëI=íÜÉå==
óñ
Ñ
ñó
Ñ OO
∂∂∂
=∂∂
∂K==
=848. `Ü~áå=oìäÉë==
fÑ= ( ) ( )( )óIñÜÖóIñÑ = =EÖ=áë=~=ÑìåÅíáçå=çÑ=çåÉ=î~êá~ÄäÉ=ÜFI=íÜÉå==
( )( )ñ
ÜóIñÜÖ
ñ
Ñ
∂∂′=
∂∂
I= ( )( )ó
ÜóIñÜÖ
ó
Ñ
∂∂′=
∂∂
K==
=
fÑ= ( ) ( ) ( )( )íóIíñÑíÜ = I=íÜÉå= ( )Çí
Çó
ó
Ñ
Çí
Çñ
ñ
ÑíÜ
∂∂
+∂∂
=′ K==
=fÑ= ( ) ( )( )îIìóIîIìñÑò = I=íÜÉå==
ì
ó
ó
Ñ
ì
ñ
ñ
Ñ
ì
ò
∂∂
∂∂
+∂∂
∂∂
=∂∂
I=î
ó
ó
Ñ
î
ñ
ñ
Ñ
î
ò
∂∂
∂∂
+∂∂
∂∂
=∂∂
K==
=849. pã~ää=`Ü~åÖÉë=
óó
Ññ
ñ
Ñò ∆
∂∂
+∆∂∂
≈∆ =
==
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CHAPTER 8. DIFFERENTIAL CALCULUS
224
850. içÅ~ä=j~ñáã~=~åÇ=jáåáã~=( )óIñÑ =Ü~ë=~=äçÅ~ä=ã~ñáãìã=~í= ( )MM óIñ =áÑ= ( ) ( )MM óIñÑóIñÑ ≤ =
Ñçê=~ää= ( )óIñ =ëìÑÑáÅáÉåíäó=ÅäçëÉ=íç= ( )MM óIñ K==
=( )óIñÑ =Ü~ë=~=äçÅ~ä=ãáåáãìã=~í= ( )MM óIñ =áÑ= ( ) ( )MM óIñÑóIñÑ ≥ =
Ñçê=~ää= ( )óIñ =ëìÑÑáÅáÉåíäó=ÅäçëÉ=íç= ( )MM óIñ K=
=851. pí~íáçå~êó=mçáåíë=
Mó
Ñ
ñ
Ñ=
∂∂
=∂∂
K=
içÅ~ä=ã~ñáã~=~åÇ=äçÅ~ä=ãáåáã~=çÅÅìê=~í=ëí~íáçå~êó=éçáåíëK===
852. p~ÇÇäÉ=mçáåí=^=ëí~íáçå~êó= =éçáåí= =ïÜáÅÜ= = áë= =åÉáíÜÉê= =~= = äçÅ~ä= =ã~ñáãìã=åçê=~=äçÅ~ä=ãáåáãìã==
853. pÉÅçåÇ=aÉêáî~íáîÉ=qÉëí=Ñçê=pí~íáçå~êó=mçáåíë=
iÉí= ( )MM óIñ =ÄÉ=~=ëí~íáçå~êó=éçáåí=E Mó
Ñ
ñ
Ñ=
∂∂
=∂∂
FK==
( ) ( )( ) ( )MMóóMMóñ
MMñóMMññ
óIñÑóIñÑ
óIñÑóIñÑa = K==
=fÑ= Ma> I= ( ) MóIñÑ MMññ > I== ( )MM óIñ ==áë=~=éçáåí=çÑ=äçÅ~ä=ãáåáã~K=
fÑ= Ma> I= ( ) MóIñÑ MMññ < I== ( )MM óIñ ==áë=~=éçáåí=çÑ=äçÅ~ä=ã~ñáã~K=
fÑ= Ma< I= ( )MM óIñ =áë=~=ë~ÇÇäÉ=éçáåíK=
fÑ= Ma= I=íÜÉ=íÉëí=Ñ~áäëK==
854. q~åÖÉåí=mä~åÉ=qÜÉ=Éèì~íáçå=çÑ=íÜÉ=í~åÖÉåí=éä~åÉ=íç=íÜÉ=ëìêÑ~ÅÉ= ( )óIñÑò = =~í= ( )MMM òIóIñ =áë==
( )( ) ( )( )MMMóMMMñM óóóIñÑññóIñÑòò −+−=− K=
=
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CHAPTER 8. DIFFERENTIAL CALCULUS
225
855. kçêã~ä=íç=pìêÑ~ÅÉ=qÜÉ=Éèì~íáçå=çÑ=íÜÉ=åçêã~ä=íç=íÜÉ=ëìêÑ~ÅÉ= ( )óIñÑò = =~í=( )MMM òIóIñ =áë==
( ) ( ) N
òò
óIñÑ
óó
óIñÑ
ññ M
MMó
M
MMñ
M
−−
=−
=−
K=
===
8.9 Differential Operators =
råáí=îÉÅíçêë=~äçåÖ=íÜÉ=ÅççêÇáå~íÉ=~ñÉëW= ár
I= àr
I= âr
=pÅ~ä~ê=ÑìåÅíáçåë=EëÅ~ä~ê=ÑáÉäÇëFW= ( )òIóIñÑ I= ( )åON ñIIñIñì K =dê~ÇáÉåí=çÑ=~=ëÅ~ä~ê=ÑáÉäÇW= ìÖê~Ç I= ì∇ =
aáêÉÅíáçå~ä=ÇÉêáî~íáîÉW=ä
Ñ
∂∂
=
sÉÅíçê=ÑìåÅíáçå=EîÉÅíçê=ÑáÉäÇFW= ( )oInImcr
=
aáîÉêÖÉåÅÉ=çÑ=~=îÉÅíçê=ÑáÉäÇW= cÇáîr
I= cr⋅∇ =
`ìêä=çÑ=~=îÉÅíçê=ÑáÉäÇW= cÅìêär
I= cr
×∇ =
i~éä~Åá~å=çéÉê~íçêW= O∇ ===
856. dê~ÇáÉåí=çÑ=~=pÅ~ä~ê=cìåÅíáçå=
∂∂
∂∂
∂∂
=∇=ò
ÑI
ó
ÑI
ñ
ÑÑÑÖê~Ç I==
∂∂
∂∂
∂∂
=∇=åON ñ
ìII
ñ
ìI
ñ
ìììÖê~Ç K K=
=857. aáêÉÅíáçå~ä=aÉêáî~íáîÉ=
γ∂∂
+β∂∂
+α∂∂
=∂∂
Åçëò
ÑÅçë
ó
ÑÅçë
ñ
Ñ
ä
ÑI==
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CHAPTER 8. DIFFERENTIAL CALCULUS
226
ïÜÉêÉ=íÜÉ=ÇáêÉÅíáçå=áë=ÇÉÑáåÉÇ=Äó=íÜÉ=îÉÅíçê=
( )γβα ÅçëIÅçëIÅçëär
I= NÅçëÅçëÅçë OOO =γ+β+α K===
858. aáîÉêÖÉåÅÉ=çÑ=~=sÉÅíçê=cáÉäÇ=
ò
o
ó
n
ñ
mccÇáî
∂∂
+∂∂
+∂∂
=⋅∇=rr
=
=859. `ìêä=çÑ=~=sÉÅíçê=cáÉäÇ=
onmñññ
âàá
ccÅìêä∂∂
∂∂
∂∂
=×∇=
rrr
rr=
âó
m
ñ
nà
ñ
o
ò
má
ò
n
ó
o rrr
∂∂
−∂∂
+
∂∂
−∂∂
+
∂∂
−∂∂
= =
=860. i~éä~Åá~å=léÉê~íçê=
O
O
O
O
O
OO
ò
Ñ
ó
Ñ
ñ
ÑÑ
∂∂
+∂∂
+∂∂
=∇ =
=
861. ( ) ( ) MccÅìêäÇáî ≡×∇⋅∇=rr
==
862. ( ) ( ) MÑÑÖê~ÇÅìêä ≡∇×∇= ==
863. ( ) ( ) ÑÑÑÖê~ÇÇáî O∇=∇⋅∇= ==
864. ( ) ( ) ( ) ccccÇáîÖê~ÇcÅìêäÅìêä OOrrrrr
∇−⋅∇∇=∇−= ===
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227
Chapter 9
Integral Calculus ====cìåÅíáçåëW=ÑI=ÖI=ìI=î=fåÇÉéÉåÇÉåí=î~êá~ÄäÉëW=ñI=íI=ξ =
fåÇÉÑáåáíÉ=áåíÉÖê~ä=çÑ=~=ÑìåÅíáçåW= ( )∫ ÇññÑ I= ( )∫ ÇññÖ I=£=
aÉêáî~íáîÉ=çÑ=~=ÑìåÅíáçåW= ( )ñó′ I= ( )ñÑ ′ I= ( )ñc′ I=£=oÉ~ä=Åçåëí~åíëW=`I=~I=ÄI=ÅI=ÇI=â=k~íìê~ä=åìãÄÉêëW=ãI=åI=áI=à===
9.1 Indefinite Integral =
865. ( ) ( ) `ñcÇññÑ +=∫ =áÑ= ( ) ( )ñÑñc =′ K=
=
866. ( )( ) ( )ñÑÇññÑ =′∫ =
=
867. ( ) ( )∫∫ = ÇññÑâÇññâÑ =
=
868. ( ) ( )[ ] ( ) ( )∫∫∫ +=+ ÇññÖÇññÑÇññÖñÑ =
=
869. ( ) ( )[ ] ( ) ( )∫∫∫ −=− ÇññÖÇññÑÇññÖñÑ =
=
870. ( ) ( ) `~ñc~
NÇñ~ñÑ +=∫ =
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CHAPTER 9. INTEGRAL CALCULUS
228
871. ( ) ( ) `Ä~ñc~
NÇñÄ~ñÑ ++=+∫ =
=
872. ( ) ( ) ( ) `ñÑO
NÇññÑñÑ O +=′∫ =
=
873. ( )( ) ( ) `ñÑäåÇññÑ
ñÑ+=
′∫ =
=874. jÉíÜçÇ=çÑ=pìÄëíáíìíáçå=
( ) ( )( ) ( )∫∫ ′= ÇííìíìÑÇññÑ =áÑ= ( )íìñ = K=
=875. fåíÉÖê~íáçå=Äó=m~êíë=
∫∫ −= îÇììîìÇî I==
ïÜÉêÉ= ( )ñì I= ( )ñî =~êÉ=ÇáÑÑÉêÉåíá~ÄäÉ=ÑìåÅíáçåëK=====
9.2 Integrals of Rational Functions =
876. `~ñ~Çñ +=∫ =
=
877. `O
ññÇñ
O
+=∫ =
=
878. `P
ñÇññ
PO +=∫ =
=
879. `Né
ñÇññ
Néé +
+=
+
∫ I= Né −≠ K=
=
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CHAPTER 9. INTEGRAL CALCULUS
229
880. ( ) ( )( ) `
Nå~
Ä~ñÇñÄ~ñ
Nåå +
++
=++
∫ I= Nå −≠ K=
=
881. `ñäåñ
Çñ+=∫ =
=
882. `Ä~ñäå~
N
Ä~ñ
Çñ++=
+∫ =
=
883. `ÇÅñäåÅ
~ÇÄÅñ
Å
~Çñ
ÇÅñ
Ä~ñO
++−
+=++
∫ =
=
884. ( )( ) `~ñ
Äñäå
Ä~
N
Äñ~ñ
Çñ+
++
−=
++∫ I= Ä~ ≠ K=
=
885. ( ) `Äñ~äå~Äñ~Ä
N
Äñ~
ñÇñO
++−+=+∫ =
=
886. ( ) ( ) `Äñ~äå~Äñ~~OÄñ~O
N
Ä
N
Äñ~
Çññ OO
P
O
+
+++−+=
+∫ =
=
887. ( ) `ñ
Äñ~äå
~
N
Äñ~ñ
Çñ+
+=
+∫ =
=
888. ( ) `ñ
Äñ~äå
~
Ä
~ñ
N
Äñ~ñ
ÇñOO
++
+−=+∫ =
=
889. ( )
`Äñ~
~Äñ~äå
Ä
N
Äñ~
ñÇñOO +
+++=
+∫ =
=
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CHAPTER 9. INTEGRAL CALCULUS
230
890. ( )
`Äñ~
~Äñ~äå~OÄñ~
Ä
N
Äñ~
Çññ O
PO
O
+
+
−+−+=+∫ =
=
891. ( ) ( ) `
ñ
Äñ~äå
~
N
Äñ~~
N
Äñ~ñ
ÇñOO
++
++
=+∫ =
=
892. `Nñ
Nñäå
O
N
Nñ
ÇñO
++−
=−∫ =
=
893. `ñN
ñNäå
O
N
ñN
ÇñO
+−+
=−∫ =
=
894. `ñ~
ñ~äå
~O
N
ñ~
ÇñOO
+−+
=−∫ =
=
895. `~ñ
~ñäå
~O
N
~ñ
ÇñOO
++−
=−∫ =
=
896. `ñí~åñN
Çñ N
O+=
+−∫ =
=
897. `~
ñí~å
~
N
ñ~
Çñ N
OO+=
+−∫ =
=
898. ( ) `~ñäåO
N
~ñ
ñÇñ OO
OO++=
+∫ =
=
899. `~
Äñ~êÅí~å
~Ä
N
Äñ~
ÇñO
+
=
+∫ I= M~Ä > K=
=
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CHAPTER 9. INTEGRAL CALCULUS
231
900. `Ä
~ñäå
ÄO
N
Äñ~
ñÇñ O
O++=
+∫ =
=
901. ( ) `Äñ~
ñäå
~O
N
Äñ~ñ
ÇñO
O
O+
+=
+∫ =
=
902. `Äñ~
Äñ~äå
~ÄO
N
ñÄ~
ÇñOOO
+−+
=−∫ =
=
903. `~ÅQÄÄ~ñO
~ÅQÄÄ~ñOäå
~ÅQÄ
N
ÅÄñ~ñ
ÇñO
O
OO+
−++
−−+
−=
++∫ I=
M~ÅQÄO >− K==
904. `Ä~ÅQ
Ä~ñO~êÅí~å
Ä~ÅQ
O
ÅÄñ~ñ
ÇñOOO+
−
+
−=
++∫ I=
M~ÅQÄO <− K====
9.3 Integrals of Irrational Functions =
905. `Ä~ñ~
O
Ä~ñ
Çñ++=
+∫ =
=
906. ( ) `Ä~ñ~P
OÇñÄ~ñ O
P++=+∫ =
=
907. ( )`Ä~ñ
~P
ÄO~ñO
Ä~ñ
ñÇñO
++−
=+∫ =
=
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CHAPTER 9. INTEGRAL CALCULUS
232
908. ( )( ) `Ä~ñ~NR
ÄO~ñPOÇñÄ~ññ O
P
O++
−=+∫ =
=
909. ( )
`~ÅÄÄ~ñ
~ÅÄÄ~ñäå
~ÅÄ
N
Ä~ñÅñ
Çñ+
−++−−+
−=
++∫ I==
M~ÅÄ >− K==
910. ( )
`Ä~Å
Ä~ñ~êÅí~å
Ä~Å
N
Ä~ñÅñ
Çñ+
−+
−=
++∫ I==
M~ÅÄ <− K==
911. ( )( ) −++=++
∫ ÇÅñÄ~ñÅ
NÇñ
ÇÅñ
Ä~ñ=
( ) ( ) `Ä~ñÅÇÅñ~äå~ÅÅ
ÄÅ~Ç++++
−− I= M~ > K=
=
912. ( )( ) −++=++
∫ ÇÅñÄ~ñÅ
NÇñ
ÇÅñ
Ä~ñ=
( )( ) `
Ä~ñÅ
ÇÅñ~~êÅí~å
~ÅÅ
ÄÅ~Ç+
++−
− I=E M~ < I= MÅ > FK==
=
913. ( ) ( ) `Äñ~ÄNMR
ñÄNR~ÄñNO~UOÇñÄñ~ñ P
P
OOOO ++
+−=+∫ =
=
914. ( )`Äñ~
ÄNR
ñÄP~ÄñQ~UO
Äñ~
ÇññP
OOOO
+++−
=+∫ =
=
915. `~Äñ~
~Äñ~äå
~
N
Äñ~ñ
Çñ+
++−+
=+∫ I= M~ > K=
=
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CHAPTER 9. INTEGRAL CALCULUS
233
916. `~
Äñ~~êÅí~å
~
O
Äñ~ñ
Çñ+
−+
−=
+∫ I= M~ < K=
=
917. ( )( ) ( ) `Ä~
Äñ~êÅëáåÄ~ñÄñ~Çñ
ñÄ
ñ~+
++
+++−=+−
∫ =
=
918. ( )( ) ( ) `Ä~
ñÄ~êÅëáåÄ~ñÄñ~Çñ
ñÄ
ñ~+
+−
+−−+−=−+
∫ =
=
919. `ñ~êÅëáåñNÇññN
ñN O ++−−=−+
∫ =
=
920. ( )( )
`~Ä
~ñ~êÅëáåO
~Ä~ñ
Çñ+
−−
=−−∫ =
=
921. +−+−
=−+∫ OO ÅñÄñ~ÅQ
ÄÅñOÇñÅñÄñ~ =
`~ÅQÄ
ÄÅñO~êÅëáå
ÅU
~ÅQÄOP
O
++
−−+ =
=
922. ( ) `ÅÄñ~ñ~OÄ~ñOäå~
N
ÅÄñ~ñ
Çñ O
O+++++=
++∫ I==
M~ > K==
923. `~ÅQÄ~Q
Ä~ñO~êÅëáå
~
N
ÅÄñ~ñ
Çñ O
O+−
+−=
++∫ I= M~ < K=
=
924. `~ññäåO
~~ñ
O
ñÇñ~ñ OO
OOOOO +++++=+∫ =
=
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CHAPTER 9. INTEGRAL CALCULUS
234
925. ( ) `~ñP
NÇñ~ññ O
POOOO ++=+∫ =
=
926. ( ) −++=+∫ OOOOOOO ~ñ~ñOU
ñÇñ~ññ =
`~ññäåU
~ OOQ
+++− =
=
927. `~ññäåñ
~ñÇñ
ñ
~ñ OOOO
O
OO
+++++
−=+
∫ =
=
928. `~ññäå~ñ
Çñ OO
OO+++=
+∫ =
=
929. `~ñ~
ñäå~~ñÇñ
ñ
~ñOO
OOOO
+++
++=+
∫ =
=
930. `~ñ~ñ
ñÇñ OO
OO++=
+∫ =
=
931. `~ññäåO
~~ñ
O
ñ
~ñ
Çññ OOO
OO
OO
O
+++−+=+∫ =
=
932. `~ñ~
ñäå
~
N
~ññ
ÇñOOOO+
++=
+∫ =
=
933. `~ññäåO
~~ñ
O
ñÇñ~ñ OO
OOOOO +−+−−=−∫ =
=
934. ( ) `~ñP
NÇñ~ññ O
POOOO +−=−∫ =
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CHAPTER 9. INTEGRAL CALCULUS
235
935. `ñ
~~êÅëáå~~ñÇñ
ñ
~ñ OOOO
++−=−
∫ =
=
936. `~ññäåñ
~ñÇñ
ñ
~ñ OOOO
O
OO
+−++−
−=−
∫ =
=
937. `~ññäå~ñ
Çñ OO
OO+−+=
−∫ =
=
938. `~ñ~ñ
ñÇñ OO
OO+−=
−∫ =
=
939. `~ññäåO
~~ñ
O
ñ
~ñ
Çññ OOO
OO
OO
O
+−++−=−∫ =
=
940. `ñ
~~êÅëáå
~
N
~ññ
ÇñOO
+−=−∫ =
=
941. ( )
`~ñ
~ñ
~
N
~ñ~ñ
ÇñOO
++−
=−+∫ =
=
942. ( )
`~ñ
~ñ
~
N
~ñ~ñ
ÇñOO
+−+
−=−−∫ =
=
943. `ñ~
~ñ
~ññ
ÇñO
OO
OOO+
−=
−∫ =
=
944. ( )
`~ñ~
ñ
~ñ
ÇñOOOO
POO
+−
−=−
∫ =
=
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CHAPTER 9. INTEGRAL CALCULUS
236
945. ( ) ( ) +−−−=−∫ OOOOOP
OO ~ñ~RñOU
ñÇñ~ñ =
`~ññäåU
~P OOQ
+−++ =
=
946. `~
ñ~êÅëáå
O
~ñ~
O
ñÇññ~
OOOOO ++−=−∫ =
=
947. ( ) `ñ~P
NÇññ~ñ O
POOOO +−−=−∫ =
=
948. ( ) `~
ñ~êÅëáå
U
~ñ~~ñO
U
ñÇññ~ñ
QOOOOOOO ++−−=−∫ =
=
949. `ñ~~
ñäå~ñ~Çñ
ñ
ñ~OO
OOOO
+−+
+−=−
∫ =
=
950. `~
ñ~êÅëáå
ñ
ñ~Çñ
ñ
ñ~ OO
O
OO
+−−
−=−
∫ =
=
951. `ñ~êÅëáåñN
ÇñO
+=−
∫ =
=
952. `~
ñëáå
ñ~
ÇñOO
+=−
∫ =
=
953. `ñ~ñ~
ñÇñ OO
OO+−−=
−∫ =
=
954. `~
ñ~êÅëáå
O
~ñ~
O
ñ
ñ~
Çññ OOO
OO
O
++−−=−∫ =
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CHAPTER 9. INTEGRAL CALCULUS
237
955. ( )
`ñ~
ñ~
O
N
ñ~~ñ
ÇñOO
++−
−=−+∫ =
=
956. ( )
`ñ~
ñ~
O
N
ñ~~ñ
ÇñOO
+−+
−=−−∫ =
=
957. ( ) ( ) `
Äñ~
~Äñ~êÅëáå
~Ä
N
ñ~Äñ
Çñ O
OOOO+
++
−=
−+∫ I= ~Ä > K=
=
958. ( )
I`Äñ~ñ~Ä~
Äñäå
Ä~
N
ñ~Äñ
ÇñOOOOOOOOO
+++−−
+
−=
−+∫ =
~Ä < K==
959. `ñ~
ñ~
ñ~ñ
ÇñO
OO
OOO+
−−=
−∫ =
=
960. ( ) ( ) `~
ñ~êÅëáå
U
~Pñ~ñO~R
U
ñÇññ~
QOOOOO
POO ++−−=−∫ =
=
961. ( )
`ñ~~
ñ
ñ~
ÇñOOOO
POO
+−
=−
∫ =
===
9.4 Integrals of Trigonometric Functions =
962. `ñÅçëñÇñëáå +−=∫ =
=
963. `ñëáåñÇñÅçë +=∫ =
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CHAPTER 9. INTEGRAL CALCULUS
238
964. `ñOëáåQ
N
O
ñÇññëáåO +−=∫ =
=
965. `ñOëáåQ
N
O
ñÇññÅçëO ++=∫ =
=
966. `ñÅçëQ
PñPÅçë
NO
N`ñÅçëñÅçë
P
NÇññëáå PP +−=+−=∫ =
=
967. `ñëáåQ
PñPëáå
NO
N`ñëáå
P
NñëáåÇññÅçë PP ++=+−=∫ =
=
968. `O
ñí~åäåÇññÅëÅ
ñëáå
Çñ+== ∫∫ =
=
969. `QO
ñí~åäåÇññëÉÅ
ñÅçë
Çñ+
π
+== ∫∫ =
=
970. `ñÅçíÇññÅëÅñëáå
Çñ O
O+−== ∫∫ =
=
971. `ñí~åÇññëÉÅñÅçë
Çñ O
O+== ∫∫ =
=
972. `O
ñí~åäå
O
N
ñëáåO
ñÅçëÇññÅëÅ
ñëáå
ÇñO
P
P++−== ∫∫ =
=
973. `QO
ñí~åäå
O
N
ñÅçëO
ñëáåÇññëÉÅ
ñÅçë
ÇñO
P
P+
π
++== ∫∫ =
=
974. `ñOÅçëQ
NÇññÅçëñëáå +−=∫ =
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CHAPTER 9. INTEGRAL CALCULUS
239
975. `ñëáåP
NÇññÅçëñëáå PO +=∫ =
=
976. `ñÅçëP
NÇññÅçëñëáå PO +−=∫ =
=
977. `ñQëáåPO
N
U
ñÇññÅçëñëáå OO +−=∫ =
=
978. `ñÅçëäåñÇñí~å +−=∫ =
=
979. `ñëÉÅ`ñÅçë
NÇñ
ñÅçë
ñëáåO
+=+=∫ =
=
980. `ñëáåQO
ñí~åäåÇñ
ñÅçë
ñëáåO
+−
π
+=∫ =
=
981. `ññí~åÇññí~åO +−=∫ =
=
982. `ñëáåäåñÇñÅçí +=∫ =
=
983. `ñÅëÅ`ñëáå
NÇñ
ñëáå
ñÅçëO
+−=+−=∫ =
=
984. `ñÅçëO
ñí~åäåÇñ
ñëáå
ñÅçëO
++=∫ =
=
985. `ññÅçíÇññÅçíO +−−=∫ =
=
986. `ñí~åäåñëáåñÅçë
Çñ+=∫ =
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CHAPTER 9. INTEGRAL CALCULUS
240
987. `QO
ñí~åäå
ñëáå
N
ñÅçëñëáå
ÇñO
+
π
++−=∫ =
=
988. `O
ñí~åäå
ñÅçë
N
ñÅçëñëáå
ÇñO
++=∫ =
=
989. `ñÅçíñí~åñÅçëñëáå
ÇñOO
+−=∫ =
=
990. ( )( )
( )( ) `
åãO
ñåãëáå
åãO
ñåãëáåÇñåñëáåãñëáå +
−−
+++
−=∫ I=
OO åã ≠ K==
991. ( )( )
( )( ) `
åãO
ñåãÅçë
åãO
ñåãÅçëÇñåñÅçëãñëáå +
−−
−++
−=∫ I=
OO åã ≠ K==
992. ( )( )
( )( ) `
åãO
ñåãëáå
åãO
ñåãëáåÇñåñÅçëãñÅçë +
−−
+++
=∫ I=
OO åã ≠ K==
993. `ñëÉÅñÇñí~åñëÉÅ +=∫ =
=
994. `ñÅëÅñÇñÅçíñÅëÅ +−=∫ =
=
995. `Nå
ñÅçëÇññÅçëñëáå
Nåå +
+−=
+
∫ =
=
996. `Nå
ñëáåÇññÅçëñëáå
Nåå +
+=
+
∫ =
=
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CHAPTER 9. INTEGRAL CALCULUS
241
997. `ñNñ~êÅëáåñÇññ~êÅëáå O +−+=∫ =
=
998. `ñNñ~êÅÅçëñÇññ~êÅÅçë O +−−=∫ =
=
999. ( ) `NñäåO
Nñ~êÅí~åñÇññ~êÅí~å O ++−=∫ =
=
1000. ( ) `NñäåO
NñÅçí~êÅñÇññÅçí~êÅ O +++=∫ =
===
9.5 Integrals of Hyperbolic Functions =
1001. `ñÅçëÜñÇñëáåÜ +=∫ =
=
1002. `ñëáåÜñÇñÅçëÜ +=∫ =
=
1003. `ñÅçëÜäåÇññí~åÜ +=∫ =
=
1004. `ñëáåÜäåÇññÅçíÜ +=∫ =
=
1005. `ñí~åÜñÇñëÉÅÜO +=∫ =
=
1006. `ñÅçíÜñÇñÅëÅÜO +−=∫ =
=
1007. `ñëÉÅÜñÇñí~åÜñëÉÅÜ +−=∫ =
=
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CHAPTER 9. INTEGRAL CALCULUS
242
1008. `ñÅëÅÜñÇñÅçíÜñÅëÅÜ +−=∫ =
===
9.6 Integrals of Exponential and Logarithmic Functions
=
1009. `ÉÇñÉ ññ +=∫ =
=
1010. `~äå
~Çñ~
ññ +=∫ =
=
1011. `~
ÉÇñÉ
~ñ~ñ +=∫ =
=
1012. ( ) `N~ñ~
ÉÇññÉ
O
~ñ~ñ +−=∫ =
=
1013. `ññäåñÇññäå +−=∫ =
=
1014. `ñäåäåñäåñ
Çñ+=∫ =
=
1015. ( )
`Nå
N
Nå
ñäåñÇññäåñ O
Nåå +
+
−+
= +∫ =
=
1016. `ÉÄ~
ÄñÅçëÄÄñëáå~ÇñÄñëáåÉ ~ñ
OO
~ñ ++−
=∫ =
=
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CHAPTER 9. INTEGRAL CALCULUS
243
1017. `ÉÄ~
ÄñëáåÄÄñÅçë~ÇñÄñÅçëÉ ~ñ
OO
~ñ +++
=∫ =
===
9.7 Reduction Formulas =
1018. ∫∫ −−= ÇñÉñã
åÉñ
ã
NÇñÉñ ãñNåãñåãñå =
=
1019. ( ) ∫∫ −− −
+−
−= Çññ
É
Nå
ã
ñNå
ÉÇñ
ñ
ÉNå
ãñ
Nå
ãñ
å
ãñ
I= Nå ≠ K=
=
1020. ∫∫ −− −−= ñÇñëáåÜ
å
NåñÅçëÜñëáåÜ
å
NñÇñëáåÜ OåNåå =
=
1021. ( ) ∫∫ −− −−
−−
−=ñëáåÜ
Çñ
Nå
Oå
ñëáåÜNå
ñÅçëÜ
ñëáåÜ
ÇñOåNåå
I= Nå ≠ K=
=
1022. ∫∫ −− −+= ñÇñÅçëÜ
å
NåñÅçëÜñÅçëÜñëáåÜ
å
NñÇñÅçëÜ OåNåå =
=
1023. ( ) ∫∫ −− −−
+−
−=ñÅçëÜ
Çñ
Nå
Oå
ñÅçëÜNå
ñëáåÜ
ñÅçëÜ
ÇñOåNåå
I= Nå ≠ K=
=
1024. ãå
ñÅçëÜñëáåÜñÇñÅçëÜñëáåÜ
NãNåãå
+=
−+
∫ =
∫ −
+−
+ ñÇñÅçëÜñëáåÜãå
Nã Oãå =
=
1025. ãå
ñÅçëÜñëáåÜñÇñÅçëÜñëáåÜ
NãNåãå
+=
+−
∫ =
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CHAPTER 9. INTEGRAL CALCULUS
244
∫ −
+−
− ñÇñÅçëÜñëáåÜãå
Nå ãOå =
=
1026. ∫∫ −− +−
−= ñÇñí~åÜñí~åÜNå
NñÇñí~åÜ OåNåå I= Nå ≠ K=
=
1027. ∫∫ −− +−
−= ñÇñÅçíÜñÅçíÜNå
NñÇñÅçíÜ OåNåå I= Nå ≠ K=
=
1028. ∫∫ −−
−−
+−
= ñÇñëÉÅÜNå
Oå
Nå
ñí~åÜñëÉÅÜñÇñëÉÅÜ Oå
Oåå I= Nå ≠ K=
=
1029. ∫∫ −− −+−= ñÇñëáå
å
NåñÅçëñëáå
å
NñÇñëáå OåNåå =
=
1030. ( ) ∫∫ −− −−
+−
−=ñëáå
Çñ
Nå
Oå
ñëáåNå
ñÅçë
ñëáå
ÇñOåNåå
I= Nå ≠ K=
=
1031. ∫∫ −− −+= ñÇñÅçë
å
NåñÅçëñëáå
å
NñÇñÅçë OåNåå =
=
1032. ( ) ∫∫ −− −−
+−
=ñÅçë
Çñ
Nå
Oå
ñÅçëNå
ñëáå
ñÅçë
ÇñOåNåå
I= Nå ≠ K=
=
1033. ãå
ñÅçëñëáåñÇñÅçëñëáå
NãNåãå
+=
−+
∫ =
∫ −
+−
+ ñÇñÅçëñëáåãå
Nã Oãå =
=
1034. ãå
ñÅçëñëáåñÇñÅçëñëáå
NãNåãå
+−=
+−
∫ =
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CHAPTER 9. INTEGRAL CALCULUS
245
∫ −
+−
+ ñÇñÅçëñëáåãå
Nå ãOå =
=
1035. ∫∫ −− −−
= ñÇñí~åñí~åNå
NñÇñí~å OåNåå I= Nå ≠ K=
=
1036. ∫∫ −− −−
−= ñÇñÅçíñÅçíNå
NñÇñÅçí OåNåå I= Nå ≠ K=
=
1037. ∫∫ −−
−−
+−
= ñÇñëÉÅNå
Oå
Nå
ñí~åñëÉÅñÇñëÉÅ Oå
Oåå I= Nå ≠ K=
=
1038. ∫∫ −−
−−
+−
−= ñÇñÅëÅNå
Oå
Nå
ñÅçíñÅëÅñÇñÅëÅ Oå
Oåå I= Nå ≠ K=
=
1039. ∫∫ −+
+−
+= ñÇñäåñ
Nå
ã
Nå
ñäåññÇñäåñ Nãå
ãNåãå =
=
1040. ( ) ∫∫
−
− −+
−−= Çñ
ñ
ñäå
Nå
ã
ñNå
ñäåÇñ
ñ
ñäåå
Nã
Nå
ã
å
ã
I= Nå ≠ K=
=
1041. ∫∫ −−= ñÇñäååñäåññÇñäå Nååå =
=
1042. ∫∫ −−= ñÇñÅçëÜñåñÅçëÜññÇñëáåÜñ Nååå =
=
1043. ∫∫ −−= ñÇñëáåÜñåñëáåÜññÇñÅçëÜñ Nååå =
=
1044. ∫∫ −+−= ñÇñÅçëñåñÅçëññÇñëáåñ Nååå =
=
1045. ∫∫ −−= ñÇñëáåñåñëáåññÇñÅçëñ Nååå =
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CHAPTER 9. INTEGRAL CALCULUS
246
1046. ∫∫ −+−
+=
+−
+− Çñ
ñN
ñ
Nå
Nñëáå
Nå
ññÇñëáåñ
O
NåN
NåNå =
=
1047. ∫∫ −++
+=
+−
+− Çñ
ñN
ñ
Nå
NñÅçë
Nå
ññÇñÅçëñ
O
NåN
NåNå =
=
1048. ∫∫ ++−
+=
+−
+− Çñ
ñN
ñ
Nå
Nñí~å
Nå
ññÇñí~åñ
O
NåN
NåNå =
=
1049. ∫∫ +−=
+ Ä~ñ
Çñ
~
Ä
~
ñ
Ä~ñ
Çññåå
å
=
=
1050. ( ) ( )( )( ) NåOOåO ÅÄñ~ñ~ÅQÄNå
Ä~ñO
ÅÄñ~ñ
Çñ−
++−−
−−=
++∫ =
( )( )( ) ( )∫ −
++−−−
−NåOO
ÅÄñ~ñ
Çñ
~ÅQÄNå
~PåOOI= Nå ≠ K=
=
1051. ( ) ( ) ( ) ( ) ( ) I~ñ
Çñ
~NåO
PåO
~ñ~NåO
ñ
~ñ
ÇñNåOOONåOOOåOO ∫∫ −−
+−−
++−
=+
Nå ≠ K==
1052. ( ) ( ) ( ) NåOOOåOO ~ñ~NåO
ñ
~ñ
Çñ−
−−−=
−∫ =
( ) ( )∫ −−−
−−
NåOOO~ñ
Çñ
~NåO
PåOI= Nå ≠ K=
=====
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CHAPTER 9. INTEGRAL CALCULUS
247
9.8 Definite Integral =
aÉÑáåáíÉ=áåíÉÖê~ä=çÑ=~=ÑìåÅíáçåW= ( )∫Ä
~
ÇññÑ I= ( )∫Ä
~
ÇññÖ I=£=
oáÉã~åå=ëìãW= ( )∑=
∆ξå
Nááá ñÑ ==
pã~ää=ÅÜ~åÖÉëW= áñ∆ ==^åíáÇÉêáî~íáîÉëW= ( )ñc I= ( )ñd ==iáãáíë=çÑ=áåíÉÖê~íáçåëW=~I=ÄI=ÅI=Ç==
=
1053. ( ) ( )∑∫=→∆
∞→∆ξ=
å
Nááá
Mñã~ñå
Ä
~
ñÑäáãÇññÑá
I==
ïÜÉêÉ== Nááá ñññ −−=∆ I== ááNá ññ ≤ξ≤− K===
==
Figure 179. =
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CHAPTER 9. INTEGRAL CALCULUS
248
1054. ~ÄÇñNÄ
~
−=∫ =
=
1055. ( ) ( )∫∫ =Ä
~
Ä
~
ÇññÑâÇññâÑ =
=
1056. ( ) ( )[ ] ( ) ( )∫∫∫ +=+Ä
~
Ä
~
Ä
~
ÇññÖÇññÑÇññÖñÑ =
=
1057. ( ) ( )[ ] ( ) ( )∫∫∫ −=−Ä
~
Ä
~
Ä
~
ÇññÖÇññÑÇññÖñÑ =
=
1058. ( ) MÇññÑ~
~
=∫ =
=
1059. ( ) ( )∫∫ −=~
Ä
Ä
~
ÇññÑÇññÑ =
=
1060. ( ) ( ) ( )∫∫∫ +=Ä
Å
Å
~
Ä
~
ÇññÑÇññÑÇññÑ =Ñçê= ÄÅ~ << K=
=
1061. ( ) MÇññÑÄ
~
≥∫ =áÑ= ( ) MñÑ ≥ =çå= [ ]ÄI~ K=
=
1062. ( ) MÇññÑÄ
~
≤∫ =áÑ= ( ) MñÑ ≤ =çå= [ ]ÄI~ K=
====
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CHAPTER 9. INTEGRAL CALCULUS
249
1063. cìåÇ~ãÉåí~ä=qÜÉçêÉã=çÑ=`~äÅìäìë=
( ) ( ) ( ) ( )~cÄcñcÇññÑÄ
~
Ä
~
−==∫ =áÑ= ( ) ( )ñÑñc =′ K=
=1064. jÉíÜçÇ=çÑ=pìÄëíáíìíáçå==
fÑ= ( )íÖñ = I=íÜÉå==
( ) ( )( ) ( )∫∫ ′=Ç
Å
Ä
~
ÇííÖíÖÑÇññÑ I==
ïÜÉêÉ=( )~ÖÅ N−= I= ( )ÄÖÇ N−= K=
=1065. fåíÉÖê~íáçå=Äó=m~êíë=
( ) ∫∫ −=Ä
~
Ä
~
Ä
~
îÇììîìÇî =
=1066. qê~éÉòçáÇ~ä=oìäÉ=
( ) ( ) ( ) ( )
++
−= ∑∫
−
=
Nå
NááåM
Ä
~
ñÑOñÑñÑåO
~ÄÇññÑ =
=
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CHAPTER 9. INTEGRAL CALCULUS
250
==
Figure 180. =
1067. páãéëçå∞ë=oìäÉ==
( ) ( ) ( ) ( ) ( )[ ++++−
=∫ PONM
Ä
~
ñÑQñÑOñÑQñÑåP
~ÄÇññÑ =
( ) ( ) ( )]åNåQ ñÑñÑQñÑO ++++ −K I==ïÜÉêÉ==
áå
~Ä~ñ á
−+= I= åIIOINIMá K= K==
=
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CHAPTER 9. INTEGRAL CALCULUS
251
==
Figure 181. =
1068. ̂ êÉ~=råÇÉê=~=`ìêîÉ=
( ) ( ) ( )~cÄcÇññÑpÄ
~
−== ∫ I==
ïÜÉêÉ= ( ) ( )ñÑñc =′ K==
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CHAPTER 9. INTEGRAL CALCULUS
252
==
Figure 182. =
1069. ̂ êÉ~=_ÉíïÉÉå=qïç=`ìêîÉë=
( ) ( )[ ] ( ) ( ) ( ) ( )~d~cÄdÄcÇññÖñÑpÄ
~
+−−=−= ∫ I==
ïÜÉêÉ= ( ) ( )ñÑñc =′ I= ( ) ( )ñÖñd =′ K==
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CHAPTER 9. INTEGRAL CALCULUS
253
==
Figure 183. ===
9.9 Improper Integral =
1070. qÜÉ=ÇÉÑáåáíÉ=áåíÉÖê~ä== ( )∫Ä
~
ÇññÑ =áë=Å~ääÉÇ=~å=áãéêçéÉê=áåíÉÖê~ä=
áÑ==• ~=çê=Ä=áë=áåÑáåáíÉI=• ( )ñÑ ==Ü~ë==çåÉ==çê==ãçêÉ=éçáåíë=çÑ==ÇáëÅçåíáåìáíó======áå=íÜÉ=áåíÉêî~ä= [ ]ÄI~ K==
1071. fÑ= ( )ñÑ =áë=~=Åçåíáåìçìë=ÑìåÅíáçå=çå= [ )∞I~ I=íÜÉå==
( ) ( )∫∫ ∞→
∞
=å
~å
~
ÇññÑäáãÇññÑ K=
=
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CHAPTER 9. INTEGRAL CALCULUS
254
==
Figure 184. =
1072. fÑ= ( )ñÑ =áë=~=Åçåíáåìçìë=ÑìåÅíáçå=çå= ( ]ÄI∞− I=íÜÉå==
( ) ( )∫∫ ∞−→∞−
=Ä
åå
Ä
ÇññÑäáãÇññÑ K=
=
==
Figure 185.
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CHAPTER 9. INTEGRAL CALCULUS
255
kçíÉ=W=qÜÉ=áãéêçéÉê=áåíÉÖê~äë=áå=NMTNI=NMTO=~êÉ=ÅçåîÉêÖÉåí=áÑ= íÜÉ= äáãáíë=Éñáëí=~åÇ=~êÉ=ÑáåáíÉX=çíÜÉêïáëÉ=íÜÉ= áåíÉÖê~äë=~êÉ=ÇáîÉêÖÉåíK==
1073. ( ) ( ) ( )∫∫∫∞
∞−
∞
∞−
+=Å
Å
ÇññÑÇññÑÇññÑ =
=
==
Figure 186. =fÑ=Ñçê=ëçãÉ=êÉ~ä=åìãÄÉê=ÅI=ÄçíÜ=çÑ=íÜÉ=áåíÉÖê~äë=áå=íÜÉ=êáÖÜí=
ëáÇÉ= ~êÉ= ÅçåîÉêÖÉåíI= íÜÉå= íÜÉ= áåíÉÖê~ä= ( )∫∞
∞−
ÇññÑ = áë= ~äëç=====
ÅçåîÉêÖÉåíX=çíÜÉêïáëÉ=áí=áë=ÇáîÉêÖÉåíK==
1074. ̀ çãé~êáëçå=qÜÉçêÉãë=iÉí== ( )ñÑ =~åÇ== ( )ñÖ ==ÄÉ==Åçåíáåìçìë==ÑìåÅíáçåë==çå=íÜÉ=ÅäçëÉÇ=áåíÉêî~ä= [ )∞I~ K= pìééçëÉ= íÜ~í= ( ) ( )ñÑñÖM ≤≤ = Ñçê= ~ää= ñ= áå=[ )∞I~ K=
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CHAPTER 9. INTEGRAL CALCULUS
256
• fÑ= ( )∫∞
~
ÇññÑ =áë=ÅçåîÉêÖÉåíI=íÜÉå= ( )∫∞
~
ÇññÖ =áë=~äëç=
=====ÅçåîÉêÖÉåíI=
• fÑ= ( )∫∞
~
ÇññÖ =áë=ÇáîÉêÖÉåíI=íÜÉå= ( )∫∞
~
ÇññÑ =áë=~äëç=ÇáîÉêÖÉåíK=
=1075. ̂ ÄëçäìíÉ=`çåîÉêÖÉåÅÉ=
=
fÑ= ( )∫∞
~
ÇññÑ =áë=ÅçåîÉêÖÉåíI=íÜÉå=íÜÉ=áåíÉÖê~ä ( )∫∞
~
ÇññÑ =áë=~Äëç-
äìíÉäó=ÅçåîÉêÖÉåíK====
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( ) ( )∫∫ε−
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Ä
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Ä
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ÇññÑäáãÇññÑ =
=
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CHAPTER 9. INTEGRAL CALCULUS
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1077. iÉí= ( )ñÑ =ÄÉ=~=Åçåíáåìçìë=ÑìåÅíáçå=Ñçê=~ää=êÉ~ä=åìãÄÉêë==ñ==áå=íÜÉ=áåíÉêî~ä== [ ]ÄI~ ==ÉñÅÉéí==Ñçê==ëçãÉ=éçáåí==Å==áå= ( )ÄI~ K=qÜÉå=
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Ä
ÅM
Å
~M
Ä
~
ÇññÑäáãÇññÑäáãÇññÑ K=
=
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açìÄäÉ=áåíÉÖê~äëW= ( )∫∫o
ÇñÇóóIñÑ I= ( )∫∫o
ÇñÇóóIñÖ I=£=
oáÉã~åå=ëìãW= ( )∑∑= =
∆∆ã
Ná
å
Nààáàá óñîIìÑ =
pã~ää=ÅÜ~åÖÉëW= áñ∆ I= àó∆ =
oÉÖáçåë=çÑ=áåíÉÖê~íáçåW=oI=p==mçä~ê=ÅççêÇáå~íÉëW= ê I=θ =
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^êÉ~W=^=pìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉ=çÑ=~=ëçäáÇW=s=j~ëë=çÑ=~=ä~ãáå~W=ã=aÉåëáíóW= ( )óIñρ =cáêëí=ãçãÉåíëW= ñj I= ój =
jçãÉåíë=çÑ=áåÉêíá~W= ñf I= óf I= Mf =
`Ü~êÖÉ=çÑ=~=éä~íÉW=n=`Ü~êÖÉ=ÇÉåëáíóW= ( )óIñσ =`ççêÇáå~íÉë=çÑ=ÅÉåíÉê=çÑ=ã~ëëW= ñ I= ó =^îÉê~ÖÉ=çÑ=~=ÑìåÅíáçåW=µ ==
1078. aÉÑáåáíáçå=çÑ=açìÄäÉ=fåíÉÖê~ä=qÜÉ=ÇçìÄäÉ=áåíÉÖê~ä=çîÉê=~=êÉÅí~åÖäÉ= [ ] [ ]ÇIÅÄI~ × =áë=ÇÉÑáåÉÇ=íç=ÄÉ==
( )[ ] [ ]
( )∑∑∫∫= =→∆
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∆∆=ã
Ná
å
Nààáàá
Móã~ñMñã~ñ
ÇIÅÄI~
óñîIìÑäáãÇ^óIñÑà
á
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( ) ( )àNàáNá óIóñIñ −− × I=~åÇ= Nááá ñññ −−=∆ I= Nààà óóó −−=∆ K=
=
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CHAPTER 9. INTEGRAL CALCULUS
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qÜÉ=ÇçìÄäÉ=áåíÉÖê~ä=çîÉê=~=ÖÉåÉê~ä=êÉÖáçå=o=áë==
( ) ( )[ ] [ ]∫∫∫∫×
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Ç^óIñÖÇ^óIñÑ I==
ïÜÉêÉ=êÉÅí~åÖäÉ= [ ] [ ]ÇIÅÄI~ × =Åçåí~áåë=oI==( ) ( )óIñÑóIñÖ = =áÑ= ( )óIñÑ =áë=áå=o=~åÇ= ( ) MóIñÖ = =çíÜÉêïáëÉK=
=
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1079. ( ) ( )[ ] ( ) ( )∫∫∫∫∫∫ +=+ooo
Ç^óIñÖÇ^óIñÑÇ^óIñÖóIñÑ =
=
1080. ( ) ( )[ ] ( ) ( )∫∫∫∫∫∫ −=−ooo
Ç^óIñÖÇ^óIñÑÇ^óIñÖóIñÑ =
=
1081. ( ) ( )∫∫∫∫ =oo
Ç^óIñÑâÇ^óIñâÑ I==
ïÜÉêÉ=â=áë=~=Åçåëí~åíK==
1082. fÑ= ( ) ( )óIñÖóIñÑ ≤ =çå=oI=íÜÉå= ( ) ( )∫∫∫∫ ≤oo
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CHAPTER 9. INTEGRAL CALCULUS
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= ( ) ( )∫∫∫∫ ≤op
Ç^óIñÑÇ^óIñÑ K=
=
==
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1084. fÑ= ( ) MóIñÑ ≥ =çå=o=~åÇ=o=~åÇ=p=~êÉ=åçå-çîÉêä~ééáåÖ=
êÉÖáçåëI=íÜÉå= ( ) ( ) ( )∫∫∫∫∫∫ +=∪ popo
Ç^óIñÑÇ^óIñÑÇ^óIñÑ K==
eÉêÉ= po∪ =áë=íÜÉ=ìåáçå=çÑ=íÜÉ=êÉÖáçåë=o=~åÇ=pK==
==
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CHAPTER 9. INTEGRAL CALCULUS
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1085. fíÉê~íÉÇ=fåíÉÖê~äë=~åÇ=cìÄáåá∞ë=qÜÉçêÉã=
( ) ( )( )
( )
∫ ∫∫∫ =Ä
~
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ñéo
ÇóÇñóIñÑÇ^óIñÑ ==
Ñçê=~=êÉÖáçå=çÑ=íóéÉ=fI==( ) ( ) ( ){ }ñèóñéIÄñ~öóIño ≤≤≤≤= K=
=
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( )
∫ ∫∫∫ =Ç
Å
óî
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ÇñÇóóIñÑÇ^óIñÑ ==
Ñçê=~=êÉÖáçå=çÑ=íóéÉ=ffI=( ) ( ) ( ){ }ÇóÅIóîñóìöóIño ≤≤≤≤= K=
=
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CHAPTER 9. INTEGRAL CALCULUS
262
==
Figure 194. =
1086. açìÄäÉ=fåíÉÖê~äë=çîÉê=oÉÅí~åÖìä~ê=oÉÖáçåë==fÑ=o=áë=íÜÉ=êÉÅí~åÖìä~ê=êÉÖáçå= [ ] [ ]ÇIÅÄI~ × I=íÜÉå==
( ) ( ) ( )∫ ∫∫ ∫∫∫
=
=
Ç
Å
Ä
~
Ä
~
Ç
Åo
ÇóÇñóIñÑÇñÇóóIñÑÇñÇóóIñÑ K==
=få=íÜÉ=ëéÉÅá~ä=Å~ëÉ=ïÜÉêÉ=íÜÉ=áåíÉÖê~åÇ= ( )óIñÑ =Å~å=ÄÉ=ïêáí-íÉå=~ë= ( ) ( )óÜñÖ =ïÉ=Ü~îÉ==
( ) ( ) ( ) ( ) ( )
== ∫∫∫∫∫∫
Ç
Å
Ä
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ÇóóÜÇññÖÇñÇóóÜñÖÇñÇóóIñÑ K==
=1087. ̀ Ü~åÖÉ=çÑ=s~êá~ÄäÉë=
( ) ( ) ( )[ ] ( )( )ÇìÇî
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po ∂∂
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ïÜÉêÉ=( )( ) M
î
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ì
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ñ
ì
ñ
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∂∂
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áë= íÜÉ= à~ÅçÄá~å= çÑ= íÜÉ= íê~åë-
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CHAPTER 9. INTEGRAL CALCULUS
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Å~å=ÄÉ=ÅçãéìíÉÇ=Äó= ( )îIìññ = I= ( )îIìóó = =áåíç=íÜÉ=ÇÉÑáåá-íáçå=çÑ=oK===
1088. mçä~ê=`ççêÇáå~íÉë=θ= Åçëêñ I= θ= ëáåêó K==
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( )( ) θ=θθ∂
∂= êÇêÇÇêÇ
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=iÉí=íÜÉ=êÉÖáçå=o=áë=ÇÉíÉêãáåÉÇ=~ë=ÑçääçïëW=
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α
θ
θ
θθθ=Ü
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CHAPTER 9. INTEGRAL CALCULUS
264
==
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Äê~M ≤≤≤ I= β≤θ≤α I=ïÜÉêÉ= π≤α−β O I===íÜÉå==
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Figure 197. ==
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CHAPTER 9. INTEGRAL CALCULUS
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1090. ̂ êÉ~=çÑ=~=oÉÖáçå=
( )
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Figure 199. ==
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CHAPTER 9. INTEGRAL CALCULUS
266
1091. sçäìãÉ=çÑ=~=pçäáÇ=
( )∫∫=o
Ç^óIñÑs K==
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( )
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ÇóÇñóIñÑÇ^óIñÑs K==
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∫ ∫∫∫ ==Ç
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=
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CHAPTER 9. INTEGRAL CALCULUS
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fÑ== ( ) ( )óIñÖóIñÑ ≥ ==çîÉê==~==êÉÖáçå==oI==íÜÉå==íÜÉ==îçäìãÉ==çÑ=íÜÉ= ëçäáÇ= ÄÉíïÉÉå= ( )óIñÑòN = = ~åÇ= ( )óIñÖòO = = çîÉê= o= áë=ÖáîÉå=Äó=
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268
1094. j~ëë=çÑ=~=i~ãáå~=
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ñ Ç^óIñój K==
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o
o
o
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N
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CHAPTER 9. INTEGRAL CALCULUS
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( )( )
( )∫∫
∫∫∫∫ ρ
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o
o
o
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( )∫∫=µo
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NI==
ïÜÉêÉ= ∫∫=o
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===
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qêáéäÉ=áåíÉÖê~äëW= ( )∫∫∫d
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CHAPTER 9. INTEGRAL CALCULUS
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j~ëë=çÑ=~=ëçäáÇW=ã==aÉåëáíóW= ( )òIóIñµ =`ççêÇáå~íÉë=çÑ=ÅÉåíÉê=çÑ=ã~ëëW= ñ I= ó I= ò =cáêëí=ãçãÉåíëW= ñój I= óòj I= ñòj =
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ïÜÉêÉ= =( )( ) M
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áN β+α=λ I= áO β−α=λ I=ïÜÉêÉ==
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1176. aáÑÑÉêÉåíá~ä=bèì~íáçåë=ïáíÜ=ñ=jáëëáåÖ=( )óIóÑó ′=′′ K==
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~ééäáÉë= íç= íÜÉ= íÉãéÉê~íìêÉ= ÇáëíêáÄìíáçå= ( )óIñì = áå= íÜÉ= ñó-éä~åÉ=ïÜÉå=ÜÉ~í=áë=~ääçïÉÇ=íç=Ñäçï=Ñêçã=ï~êã=~êÉ~ë=íç=Åççä=çåÉëK=qÜÉ=Éèì~íáçåë=çÑ=íÜáë=íóéÉ=~êÉ=Å~ääÉÇ=é~ê~ÄçäáÅK===
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Chapter 11
Series ====
11.1 Arithmetic Series =fåáíá~ä=íÉêãW= N~ =kíÜ=íÉêãW= å~ =aáÑÑÉêÉåÅÉ=ÄÉíïÉÉå=ëìÅÅÉëëáîÉ=íÉêãëW=Ç=kìãÄÉê=çÑ=íÉêãë=áå=íÜÉ=ëÉêáÉëW=å=pìã=çÑ=íÜÉ=Ñáêëí=å=íÉêãëW= åp ===
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11.2 Geometric Series =fåáíá~ä=íÉêãW= N~ =kíÜ=íÉêãW= å~ =`çããçå=ê~íáçW=è=kìãÄÉê=çÑ=íÉêãë=áå=íÜÉ=ëÉêáÉëW=å=pìã=çÑ=íÜÉ=Ñáêëí=å=íÉêãëW= åp =pìã=íç=áåÑáåáíóW=p===
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1193. ( )O
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11.4 Infinite Series =pÉèìÉåÅÉW={ }å~ =cáêëí=íÉêãW= N~ =kíÜ=íÉêãW= å~ ===
1204. fåÑáåáíÉ=pÉêáÉë=
KK ++++=∑∞
=åON
Nåå ~~~~ =
=1205. kíÜ=m~êíá~ä=pìã=
åON
å
Nååå ~~~~p +++==∑
=
K =
=1206. ̀ çåîÉêÖÉåÅÉ=çÑ=fåÑáåáíÉ=pÉêáÉë=
i~Nå
å =∑∞
=
I=áÑ= ipäáã åå
=∞→
=
=1207. kíÜ=qÉêã=qÉëí=
• fÑ=íÜÉ=ëÉêáÉë=∑∞
=Nåå~ áë=ÅçåîÉêÖÉåíI=íÜÉå= M~äáã å
å=
∞→K==
• fÑ= M~äáã åå
≠∞→
I=íÜÉå=íÜÉ=ëÉêáÉë=áë=ÇáîÉêÖÉåíK=
===
11.5 Properties of Convergent Series =
`çåîÉêÖÉåí=pÉêáÉëW= ^~Nå
å =∑∞
=
I= _ÄNå
å =∑∞
=
=
oÉ~ä=åìãÄÉêW=Å=
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CHAPTER 11. SERIES
308
1208. ( ) _^Ä~Ä~Nå
åNå
åNå
åå +=+=+ ∑∑∑∞
=
∞
=
∞
=
=
=
1209. Å^~ÅÅ~Nå
åNå
å == ∑∑∞
=
∞
=
K==
===
11.6 Convergence Tests =
1210. qÜÉ=`çãé~êáëçå=qÉëí=
iÉí=∑∞
=Nåå~ =~åÇ=∑
∞
=NååÄ =ÄÉ=ëÉêáÉë=ëìÅÜ=íÜ~í= åå Ä~M ≤< =Ñçê=~ää=åK==
• fÑ=∑∞
=NååÄ áë=ÅçåîÉêÖÉåí=íÜÉå=∑
∞
=Nåå~ áë=~äëç=ÅçåîÉêÖÉåíK==
• fÑ=∑∞
=Nåå~ áë=ÇáîÉêÖÉåí=íÜÉå=∑
∞
=NååÄ áë=~äëç=ÇáîÉêÖÉåíK=
=1211. qÜÉ=iáãáí=`çãé~êáëçå=qÉëí=
iÉí=∑∞
=Nåå~ =~åÇ=∑
∞
=NååÄ =ÄÉ=ëÉêáÉë=ëìÅÜ=íÜ~í= å~ =~åÇ= åÄ =~êÉ=éçëá-
íáîÉ=Ñçê=~ää=åK==
• fÑ= ∞<<∞→
å
å
å Ä
~äáãM = íÜÉå= ∑
∞
=Nåå~ = ~åÇ= ∑
∞
=NååÄ ~êÉ= ÉáíÜÉê= ÄçíÜ=
ÅçåîÉêÖÉåí=çê=ÄçíÜ=ÇáîÉêÖÉåíK=
• fÑ= MÄ
~äáã
å
å
å=
∞→=íÜÉå=∑
∞
=NååÄ ÅçåîÉêÖÉåí=áãéäáÉë=íÜ~í=∑
∞
=Nåå~ =áë=
~äëç=ÅçåîÉêÖÉåíK=
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CHAPTER 11. SERIES
309
• fÑ= ∞=∞→
å
å
å Ä
~äáã =íÜÉå=∑
∞
=NååÄ ÇáîÉêÖÉåí=áãéäáÉë=íÜ~í=∑
∞
=Nåå~ =áë=
~äëç=ÇáîÉêÖÉåíK==
1212. é-ëÉêáÉë=
é-ëÉêáÉë=∑∞
=Nåéå
N=ÅçåîÉêÖÉë=Ñçê= Né > =~åÇ=ÇáîÉêÖÉë=Ñçê=
NéM ≤< K==
1213. qÜÉ=fåíÉÖê~ä=qÉëí=iÉí= ( )ñÑ = ÄÉ= ~= ÑìåÅíáçå= ïÜáÅÜ= áë= ÅçåíáåìçìëI= éçëáíáîÉI= ~åÇ=ÇÉÅêÉ~ëáåÖ=Ñçê=~ää= Nñ ≥ K=qÜÉ=ëÉêáÉë==
( ) ( ) ( ) ( ) ( ) KK +++++=∑∞
=
åÑPÑOÑNÑåÑNå
=
ÅçåîÉêÖÉë=áÑ= ( )∫∞
N
ÇññÑ ÅçåîÉêÖÉëI=~åÇ=ÇáîÉêÖÉë=áÑ=
( ) ∞→∫å
N
ÇññÑ =~ë= ∞→å K=
=1214. qÜÉ=o~íáç=qÉëí=
iÉí=∑∞
=Nåå~ =ÄÉ=~=ëÉêáÉë=ïáíÜ=éçëáíáîÉ=íÉêãëK=
• fÑ= N~
~äáã
å
Nå
å<+
∞→=íÜÉå=∑
∞
=Nåå~ áë=ÅçåîÉêÖÉåíK=
• fÑ= N~
~äáã
å
Nå
å>+
∞→=íÜÉå=∑
∞
=Nåå~ =áë=ÇáîÉêÖÉåíK=
• fÑ= N~
~äáã
å
Nå
å=+
∞→= íÜÉå=∑
∞
=Nåå~ =ã~ó=ÅçåîÉêÖÉ=çê=ÇáîÉêÖÉ=~åÇ=
íÜÉ= ê~íáç= íÉëí= áë= áåÅçåÅäìëáîÉX= ëçãÉ= çíÜÉê= íÉëíë= ãìëí= ÄÉ=ìëÉÇK==
=
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310
1215. qÜÉ=oççí=qÉëí=
iÉí=∑∞
=Nåå~ =ÄÉ=~=ëÉêáÉë=ïáíÜ=éçëáíáîÉ=íÉêãëK=
• fÑ= N~äáã åå
å<
∞→=íÜÉå=∑
∞
=Nåå~ =áë=ÅçåîÉêÖÉåíK=
• fÑ= N~äáã åå
å>
∞→=íÜÉå=∑
∞
=Nåå~ =áë=ÇáîÉêÖÉåíK=
• fÑ= N~äáã åå
å=
∞→= íÜÉå=∑
∞
=Nåå~ =ã~ó=ÅçåîÉêÖÉ=çê=ÇáîÉêÖÉI=Äìí=
åç=ÅçåÅäìëáçå=Å~å=ÄÉ=Çê~ïå=Ñêçã=íÜáë=íÉëíK====
11.7 Alternating Series =
1216. qÜÉ=^äíÉêå~íáåÖ=pÉêáÉë=qÉëí=EiÉáÄåáò∞ë=qÜÉçêÉãF==iÉí={ }å~ =ÄÉ=~=ëÉèìÉåÅÉ=çÑ=éçëáíáîÉ=åìãÄÉêë=ëìÅÜ=íÜ~í=
åNå ~~ <+ =Ñçê=~ää=åK=M~äáã å
å=
∞→K==
qÜÉå= íÜÉ= ~äíÉêå~íáåÖ= ëÉêáÉë= ( )∑∞
=
−Nå
åå ~N = ~åÇ= ( )∑
∞
=
−−Nå
åNå ~N =
ÄçíÜ=ÅçåîÉêÖÉK=====
1217. ̂ ÄëçäìíÉ=`çåîÉêÖÉåÅÉ=
• ^= ëÉêáÉë= ∑∞
=Nåå~ = áë= ~ÄëçäìíÉäó= ÅçåîÉêÖÉåí= áÑ= íÜÉ= ëÉêáÉë=
∑∞
=Nåå~ =áë=ÅçåîÉêÖÉåíK==
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CHAPTER 11. SERIES
311
• fÑ=íÜÉ=ëÉêáÉë=∑∞
=Nåå~ áë=~ÄëçäìíÉäó=ÅçåîÉêÖÉåí=íÜÉå=áí=áë=Åçå-
îÉêÖÉåíK==
1218. ̀ çåÇáíáçå~ä=`çåîÉêÖÉåÅÉ=
^= ëÉêáÉë= ∑∞
=Nåå~ áë= ÅçåÇáíáçå~ääó= ÅçåîÉêÖÉåí= áÑ= íÜÉ= ëÉêáÉë= áë=
ÅçåîÉêÖÉåí=Äìí=áë=åçí=~ÄëçäìíÉäó=ÅçåîÉêÖÉåíK====
11.8 Power Series =oÉ~ä=åìãÄÉêëW=ñI= Mñ =
mçïÉê=ëÉêáÉëW=∑∞
=Må
ååñ~ I= ( )∑
∞
=
−Må
åMå ññ~ =
tÜçäÉ=åìãÄÉêW=å=o~Çáìë=çÑ=`çåîÉêÖÉåÅÉW=o==
1219. mçïÉê=pÉêáÉë=áå=ñ=
KK +++++=∑∞
=
åå
OONM
Må
åå ñ~ñ~ñ~~ñ~ =
=1220. mçïÉê=pÉêáÉë=áå= ( )Mññ − =
( ) ( ) ( ) ( ) KK +−++−+−+=−∑∞
=
åMå
OMOMNM
Må
åMå ññ~ññ~ññ~~ññ~
=1221. fåíÉêî~ä=çÑ=`çåîÉêÖÉåÅÉ===
qÜÉ=ëÉí=çÑ=íÜçëÉ=î~äìÉë=çÑ=ñ=Ñçê=ïÜáÅÜ=íÜÉ=ÑìåÅíáçå=
( ) ( )∑∞
=
−=Må
åMå ññ~ñÑ =áë=ÅçåîÉêÖÉåí=áë=Å~ääÉÇ==íÜÉ==áåíÉêî~ä=çÑ=
ÅçåîÉêÖÉåÅÉK=
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CHAPTER 11. SERIES
312
1222. o~Çáìë=çÑ=`çåîÉêÖÉåÅÉ=fÑ=íÜÉ=áåíÉêî~ä=çÑ=ÅçåîÉêÖÉåÅÉ=áë== ( )oñIoñ MM +− ==Ñçê==ëçãÉ=
Mo ≥ I=íÜÉ=o=áë=Å~ääÉÇ==íÜÉ=ê~Çáìë=çÑ=ÅçåîÉêÖÉåÅÉK==fí=áë=ÖáîÉå=~ë=
åå
å ~
Näáão
∞→= =çê=
Nå
å
å ~
~äáão
+∞→
= K==
===
11.9 Differentiation and Integration of Power Series
=`çåíáåìçìë=ÑìåÅíáçåW= ( )ñÑ =
mçïÉê=ëÉêáÉëW=∑∞
=Må
ååñ~ =
tÜçäÉ=åìãÄÉêW=å=o~Çáìë=çÑ=`çåîÉêÖÉåÅÉW=o===
1223. aáÑÑÉêÉåíá~íáçå=çÑ=mçïÉê=pÉêáÉë=
iÉí= ( ) K+++==∑∞
=
OONM
Må
åå ñ~ñ~~ñ~ñÑ =Ñçê= oñ < K==
qÜÉåI= = Ñçê= oñ < I= ( )ñÑ = áë= ÅçåíáåìçìëI= íÜÉ= ÇÉêáî~íáîÉ= ( )ñÑ ′ =
Éñáëíë=~åÇ=
( ) K+++=′ OONM ñ~
Çñ
Çñ~
Çñ
Ç~
Çñ
ÇñÑ =
∑∞
=
−=+++=Nå
Nåå
OPON ñå~ñ~Pñ~O~ K K=
===
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CHAPTER 11. SERIES
313
1224. fåíÉÖê~íáçå=çÑ=mçïÉê=pÉêáÉë=
iÉí= ( ) K+++==∑∞
=
OONM
Må
åå ñ~ñ~~ñ~ñÑ =Ñçê= oñ < K==
qÜÉåI==Ñçê= oñ < I=íÜÉ=áåÇÉÑáåáíÉ=áåíÉÖê~ä= ( )∫ ÇññÑ Éñáëíë=~åÇ==
( ) K+++= ∫∫∫∫ Çññ~ñÇñ~Çñ~ÇññÑ OONM =
`Nå
ñ~
P
ñ~
O
ñ~ñ~
Må
Nå
å
P
O
O
NM ++
=+++= ∑∞
=
+
K K=
===
11.10 Taylor and Maclaurin Series =tÜçäÉ=åìãÄÉêW=å=aáÑÑÉêÉåíá~ÄäÉ=ÑìåÅíáçåW= ( )ñÑ =oÉã~áåÇÉê=íÉêãW= åo ===
1225. q~óäçê=pÉêáÉë=
( ) ( )( )( ) ( ) ( )( ) ( )( )K+
−′′+−′+=
−=∑
∞
= >O
~ñ~Ñ~ñ~Ñ~Ñ
>å
~ñ~ÑñÑ
O
Må
åå
= =( ) ( )( )
å
åå
o>å
~ñ~Ñ+
−+ K==
=1226. qÜÉ=oÉã~áåÇÉê=^ÑíÉê=åHN=qÉêãë=áë=ÖáîÉå=Äó==
( )( )( )( )>Nå
~ñÑo
NåNå
å +−ξ
=++
I=== ñ~ <ξ< K=
=1227. j~Åä~ìêáå=pÉêáÉë=
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314
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )å
ååO
Må
åå o
>å
ñMÑ
>O
ñMÑñMÑMÑ
>å
ñMÑñÑ +++
′′+′+==∑
∞
=
K
====
11.11 Power Series Expansions for Some Functions
=tÜçäÉ=åìãÄÉêW=å=oÉ~ä=åìãÄÉêW=ñ===
1228. KK ++++++=>å
ñ
>P
ñ
>O
ññNÉ
åPOñ =
=
1229. ( ) ( ) ( )KK ++++++=
>å
~äåñ
>P
~äåñ
>O
~äåñ
>N
~äåñN~
åPOñ =
=
1230. ( ) ( )KK ±
+−
++−+−=++
Nå
ñN
Q
ñ
P
ñ
O
ñññNäå
NååQPO
I= NñN ≤<− K=
=
1231.
++++=
−+
KT
ñ
R
ñ
P
ññO
ñN
ñNäå
TRP
I= Nñ < K=
=
1232.
+−
+
+−
++−
= KRP
Nñ
Nñ
R
N
Nñ
Nñ
P
N
Nñ
NñOñäå I= Mñ > K=
=
1233. ( )( ) KK ±−
++−+−=>åO
ñN
>S
ñ
>Q
ñ
>O
ñNñÅçë
åOåSQO
=
=
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315
1234. ( )( ) KK ±
+−
++−+−=+
>NåO
ñN
>T
ñ
>R
ñ
>P
ñññëáå
NåOåTRP
=
=
1235. K+++++=OUPR
ñSO
PNR
ñNT
NR
ñO
P
ñññí~å
VTRP
I=O
ñπ
< K=
=
1236.
++++−= K
QTOR
ñO
VQR
ñO
QR
ñ
P
ñ
ñ
NñÅçí
TRP
I= π<ñ K=
=
1237. ( )( )( ) K
K
KK +
+⋅⋅−⋅⋅
++⋅⋅
⋅+
⋅+=
+
NåOåOSQO
ñNåORPN
RQO
ñPN
PO
ñññ~êÅëáå
NåORP
I=
Nñ < K=
=
1238. ( )( )( ) I
NåOåOSQO
ñNåORPN
RQO
ñPN
PO
ññ
Oñ~êÅÅçë
NåORP
+
+⋅⋅−⋅⋅
++⋅⋅
⋅+
⋅+−
π=
+
KK
KK
Nñ < K=
=
1239. ( )KK ±
+−
++−+−=+
NåO
ñN
T
ñ
R
ñ
P
ñññ~êÅí~å
NåOåTRP
I= Nñ ≤ K=
=
1240. ( ) KK ++++++=>åO
ñ
>S
ñ
>Q
ñ
>O
ñNñÅçëÜ
åOSQO
=
=
1241. ( ) KK ++
+++++=+
>NåO
ñ
>T
ñ
>R
ñ
>P
ñññëáåÜ
NåOTRP
=
=====
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316
11.12 Binomial Series =tÜçäÉ=åìãÄÉêëW=åI=ã=oÉ~ä=åìãÄÉêW=ñ=`çãÄáå~íáçåëW= ã
å ` ===
1242. ( ) åãå
ãOO
åN
åå ññ`ñ`ñ`NñN ++++++=+ KK ==
1243. ( ) ( )[ ]>ã
NãåNåå`ã
å −−−=
KI= Nñ < K=
=
1244. K+−+−=+
PO ñññNñN
NI= Nñ < K=
=
1245. K++++=−
PO ñññNñN
NI= Nñ < K=
=
1246. K+⋅⋅⋅
⋅⋅−
⋅⋅⋅
+⋅
−+=+USQO
ñRPN
SQO
ñPN
QO
ñ
O
ñNñN
QPO
I= Nñ ≤ K=
=
1247. K+⋅⋅⋅⋅⋅⋅
−⋅⋅⋅⋅
+⋅⋅
−+=+NOVSP
ñURON
VSP
ñRON
SP
ñON
P
ñNñN
QPOP I= Nñ ≤ K=
===
11.13 Fourier Series =fåíÉÖê~ÄäÉ=ÑìåÅíáçåW= ( )ñÑ =cçìêáÉê=ÅçÉÑÑáÅáÉåíëW= M~ I= å~ I= åÄ =
tÜçäÉ=åìãÄÉêW=å==
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CHAPTER 11. SERIES
317
1248. ( ) ( )∑∞
=
++=Nå
ååM åñëáåÄåñÅçë~
O
~ñÑ =
=
1249. ( )∫π
π−π= ÇñåñÅçëñÑ
N~å ==
=
1250. ( )∫π
π−π= ÇñåñëáåñÑ
NÄå ==
==
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Chapter 12
Probability ====
12.1 Permutations and Combinations =
mÉêãìí~íáçåëW= ãå m =
`çãÄáå~íáçåëW= ãå ` =
tÜçäÉ=åìãÄÉêëW=åI=ã===
1251. c~Åíçêá~ä=( )( )åNåOåPON>å −−⋅⋅= K =
N>M = ==
1252. >åmåå = =
=
1253. ( )>ãå
>åmã
å
−= =
=1254. _áåçãá~ä=`çÉÑÑáÅáÉåí=
( )>ãå>ã
>å
ã
å`ã
å
−=
= =
=1255. ãå
åã
å `` −= ==
1256. NãNå
Nãå
ãå ``` +
++ =+ =
=
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CHAPTER 12. PROBABILITY
319
1257. åå
åO
åN
åM
å O```` =++++ K =
=1258. m~ëÅ~ä∞ë=qêá~åÖäÉ=
=oçï=M= = = = = = = N= = = = = = =oçï=N= = = = = = N= = N= = = = = =oçï=O= = = = = N= = O= = N= = = = =oçï=P= = = = N= = P= = P= = N= = = =oçï=Q= = = N= = Q= = S= = Q= = N= = =oçï=R= = N= = R= = NM= = NM= = R= = N= =oçï=S= N= = S= = NR= = OM= = NR= = S= = N=
===
12.2 Probability Formulas =bîÉåíëW=^I=_=mêçÄ~ÄáäáíóW=m=o~åÇçã=î~êá~ÄäÉëW=uI=vI=w=s~äìÉë=çÑ=ê~åÇçã=î~êá~ÄäÉëW=ñI=óI=ò=bñéÉÅíÉÇ=î~äìÉ=çÑ=uW=µ =^åó=éçëáíáîÉ=êÉ~ä=åìãÄÉêW= ε ==pí~åÇ~êÇ=ÇÉîá~íáçåW=σ =s~êá~åÅÉW= Oσ =aÉåëáíó=ÑìåÅíáçåëW= ( )ñÑ I= ( )íÑ ===
1259. mêçÄ~Äáäáíó=çÑ=~å=bîÉåí=
( )å
ã^m = I==
ïÜÉêÉ==ã=áë=íÜÉ=åìãÄÉê=çÑ=éçëëáÄäÉ=éçëáíáîÉ=çìíÅçãÉëI==å=áë=íÜÉ=íçí~ä=åìãÄÉê=çÑ=éçëëáÄäÉ=çìíÅçãÉëK==
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CHAPTER 12. PROBABILITY
320
1260. o~åÖÉ=çÑ=mêçÄ~Äáäáíó=s~äìÉë=( ) N^mM ≤≤ =
=1261. ̀ Éêí~áå=bîÉåí=
( ) N^m = ==
1262. fãéçëëáÄäÉ=bîÉåí=( ) M^m = =
=1263. ̀ çãéäÉãÉåí=
( ) ( )^mN^m −= ==
1264. fåÇÉéÉåÇÉåí=bîÉåíë=( ) ( )^m_L^m = I==( ) ( )_m^L_m = =
=1265. ̂ ÇÇáíáçå=oìäÉ=Ñçê=fåÇÉéÉåÇÉåí=bîÉåíë=
( ) ( ) ( )_m^m_^m +=∪ ==
1266. jìäíáéäáÅ~íáçå=oìäÉ=Ñçê=fåÇÉéÉåÇÉåí=bîÉåíë=( ) ( ) ( )_m^m_^m ⋅=∩ =
=1267. dÉåÉê~ä=^ÇÇáíáçå=oìäÉ=
( ) ( ) ( ) ( )_^m_m^m_^m ∩−+=∪ I==ïÜÉêÉ==
_^∪ =áë=íÜÉ=ìåáçå=çÑ=ÉîÉåíë=^=~åÇ=_I==_^∩ =áë=íÜÉ=áåíÉêëÉÅíáçå=çÑ=ÉîÉåíë=^=~åÇ=_K=
=1268. ̀ çåÇáíáçå~ä=mêçÄ~Äáäáíó=
( ) ( )( )_m
_^m_L^m
∩= =
=1269. ( ) ( ) ( ) ( ) ( )^L_m^m_L^m_m_^m ⋅=⋅=∩ =
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CHAPTER 12. PROBABILITY
321
1270. i~ï=çÑ=qçí~ä=mêçÄ~Äáäáíó=
( ) ( ) ( )∑=
=ã
Nááá _L^m_m^m I==
ïÜÉêÉ= á_ =áë=~=ëÉèìÉåÅÉ=çÑ=ãìíì~ääó=ÉñÅäìëáîÉ=ÉîÉåíëK===
1271. _~óÉë∞=qÜÉçêÉã=
( ) ( ) ( )( )^m
_m_L^m^L_m
⋅= =
=1272. _~óÉë∞=cçêãìä~=
( ) ( ) ( )
( ) ( )∑=
⋅
⋅=
ã
Nâáá
ááá
_L^m_m
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