1300 Math Formulas

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1300 Math Formulas - Alex Svirin

Transcript of 1300 Math Formulas

  • 1300 Math Formulas ==============================

    fp_k= =VVQVNMTTQN==

    `==OMMQ=^KpK=^=o=oK=

  • i

    =q=~==~==~K=

  • ii

    Preface ====q= ~= = ~= = = = = -= ~= K= f= ~= = = = =~==~==~~=~~==I=I=~=I=~=~~K=q==~= = = ~I= ~I= ~= = =k= pI= ^~I= dI= qI= j~=~= a~I= sI= ^~= dI= `~I=a~=b~I=pI=~=m~=qK==q= = ~= = I= I= ~= ~= ~== = ~= ~= = ~= ~I= = =~===~=~=~===K===

    ==

  • iii

    Contents ====

    1 krj_bo=pbqp= NKN= p=f==1= NKO= p==k==5= NKP= _~=f==7= NKQ= `=k==8= =2 ^idb_o^= OKN= c~=c~==12= OKO= m=c~==13= OKP= m==14= OKQ= o==15= OKR= i~==16= OKS= b~==18= OKT= f~==19= OKU= `=f=c~==22= =3 dbljbqov= PKN= o=q~==24= PKO= f=q~==27= PKP= b~~=q~==28= PKQ= p~=q~==29= PKR= p~==33= PKS= o~==34= PKT= m~~~==35= PKU= o==36= PKV= q~==37= PKNM= f=q~==38= PKNN= f=q~==f=`==40= PKNO= q~==f=`==41=

  • iv

    PKNP= h==42= PKNQ= `=n~~~==43= PKNR= q~~=n~~~==45= PKNS= d~=n~~~==46= PKNT= o~=e~==47= PKNU= o~=m==48= PKNV= `==50= PKOM= p==~=`==53= PKON= p==~=`==54= PKOO= `==55= PKOP= o~~=m~~==56= PKOQ= m==57= PKOR= o~=q~==58= PKOS= o~=m~==59= PKOT= c==~=o~=m~==61= PKOU= o~~=o=t==62= PKOV= m~=p==63= PKPM= o=`~=`==66= PKPN= o=`~=`==~=l=m~=c~==68= PKPO= o=`~=`==69= PKPP= c==~=o=`~=`==70= PKPQ= p==72= PKPR= p~=`~==72= PKPS= p~=p==73= PKPT= p~=p==74= PKPU= p~=t==75= PKPV= b==76= PKQM= `~=q==78= = =4 qofdlkljbqov= QKN= o~~=~=a=j~==^==80= QKO= a=~=d~==q=c==81= QKP= p==q=c==86= QKQ= q=c==`=^==87= QKR= j=f~=c~==88=

  • v

    QKS= o=c~==89= QKT= m==q=c==90= QKU= o~==q=c==90= QKV= ^=~=p~=c~==91= QKNM= a=^=c~==92= QKNN= j=^=c~==93= QKNO= e~=^=c~==94= QKNP= e~=^=q~=f==94= QKNQ= q~==q=b==m==95= QKNR= q~==q=b==p==97=== QKNS= m==q=c==98= QKNT= d~==f=q=c==99= QKNU= m~=s~==f=q=c==102= QKNV= o~==f=q=c==103= QKOM= q=b~==106= QKON= o~==e=c==106= = =5 j^qof`bp=^ka=abqbojfk^kqp= RKN= a~==107= RKO= m==a~==109= RKP= j~==110= RKQ= l~==j~==111= RKR= p==i~=b~==114= = =6 sb`qlop= SKN= s=`~==118= SKO= s=^==120= SKP= s=p~==122= SKQ= p~=s==122= SKR= p~~=m==123= SKS= s=m==125= SKT= q=m=127= = =7 ^k^ivqf`=dbljbqov= TKN= l=-a~=`~=p==130=

  • vi

    TKO= q=-a~=`~=p==131= TKP= p~=i==m~==139= TKQ= `==149= TKR= b==152= TKS= e~==154= TKT= m~~~==158= TKU= q=-a~=`~=p==161= TKV= m~==165= TKNM= p~=i==p~==175= TKNN= n~=p~==180= TKNO= p==189= = =8 afccbobkqf^i=`^i`rirp= UKN= c=~=q=d~==191= UKO= i==c==208= UKP= a=~=m===a~==209= UKQ= q~==a~==211= UKR= e=l=a~==215= UKS= ^~==a~==217= UKT= a~==221= UKU= j~~=c==222= UKV= a~=l~==225= = =9 fkqbdo^i=`^i`rirp= VKN= f=f~==227= VKO= f~==o~~=c==228= VKP= f~==f~~=c==231= VKQ= f~==q=c==237= VKR= f~==e=c==241= VKS= f~==b~=~=i~=c==242= VKT= o=c~==243= VKU= a=f~==247= VKV= f=f~==253= VKNM= a=f~==257= VKNN= q=f~==269=

  • vii

    VKNO= i=f~==275= VKNP= p~=f~==285= = =10 afccbobkqf^i=bnr^qflkp= NMKN= c=l=l~=a~=b~==295= NMKO= p=l=l~=a~=b~==298= NMKP= p=m~~=a~=b~==302= = =11 pbofbp= NNKN= ^=p==304= NNKO= d=p==305= NNKP= p=c=p==305= NNKQ= f=p==307= NNKR= m==`=p==307= NNKS= `=q==308= NNKT= ^~=p==310= NNKU= m=p==311= NNKV= a~=~=f~==m=p==312= NNKNM= q~=~=j~~=p==313= NNKNN= m=p=b~==p=c==314= NNKNO= _~=p==316= NNKNP= c=p==316= = =12 mol_^_fifqv= NOKN= m~=~=`~==318= NOKO= m~=c~==319= = = =

    ==

  • viii

    =q=~==~==~K=

    =

  • 1

    Chapter 1

    Number Sets ====

    1.1 Set Identities =pW=^I=_I=`=r~=W=f=`=W= ^ =m=W= _^ ==b=W= =r==W= _^ =f==W= _^ =a==W= _y^ ==

    =1. f^ =

    =2. ^^ =

    =3. _^ = == _^ =~= ^_ .=

    =4. b=p=

    ^ ==

    5. r==p=={ }_^_^` == =

    =

  • CHAPTER 1. NUMBER SETS

    2

    ===== ==

    Figure 1. =

    6. `~=^__^ = =

    =7. ^~=

    ( ) ( ) `_^`_^ = ==8. f==p=

    { }_~^_^` == = ==

    ===== ==

    Figure 2. =

    9. `~=^__^ = =

    =10. ^~=

    ( ) ( ) `_^`_^ = ==

  • CHAPTER 1. NUMBER SETS

    3

    11. a=( ) ( ) ( )`^_^`_^ = I=( ) ( ) ( )`^_^`_^ = K=

    =12. f=

    ^^^ = I==^^^ = =

    =13. a~=

    =^ I=ff^ = =

    =14. f=

    ^^ = I==^f^ =

    =15. `=

    { }^f^ = =

    16. `==f=~=r f^^ = I===^^ =

    =17. a=j~=i~

    ( ) _^_^ = I==( ) _^_^ = ==

    18. a==p { }^~_^y_` == =

    =

  • CHAPTER 1. NUMBER SETS

    4

    ===== ==

    Figure 3. =

    19. ( )_^y_^y_ = =

    20. ^_^y_ = =

    21. =^y^ =

    22. ^_y^ = == =_^ . =

    ===== ==

    Figure 4. =

    23. ( ) ( ) ( )`_y`^`_y^ = 24. ^yf^ = 25. `~~=m

    ( ){ }_~^I_^` == ==

  • CHAPTER 1. NUMBER SETS

    5

    1.2 Sets of Numbers =k~~=W=k=t=W= Mk =

    fW=w=m=W= +w =k~=W= w =o~~=W=n=o~=W=o==`=W=`====

    26. k~~=k `=W { }KIPIOINk = K=

    27. t=k

    `==~=W= { }KIPIOINIMkM = K==28. f

    t==~===~=W={ }KIPIOINkw ==+ I=

    { }NIOIPIw = K I={ } { }KK IPIOINIMINIOIPIwMww == + K=

    =29. o~~=k

    o~==~=~W==

    == M~w~w~~

    ~n K=

    =30. f~~=k

    k~=~=~=~K =

  • CHAPTER 1. NUMBER SETS

    6

    31. o~=k==r==~~=~=~~=W=oK=

    =32. `=k

    { }o~o` += I======~~=K

    =33. `onwk =

    =

    === ==

    Figure 5. ======

  • CHAPTER 1. NUMBER SETS

    7

    1.3 Basic Identities =o~=W=~I=I==

    ==

    34. ^=f=~M~ =+ =

    =35. ^=f=

    ( ) M~~ =+ ==

    36. `~==^=~~ +=+ =

    =37. ^~==^=

    ( ) ( )~~ ++=++ ==38. a==p~=

    ( )~~ += ==

    39. j~=f=~N~ = =

    =40. j~=f=

    N~

    N~ = I= M~

    =41. j~=q=M

    MM~ = =

    42. `~==j~=~~ =

    ==

  • CHAPTER 1. NUMBER SETS

    8

    43. ^~==j~=( ) ( )~~ =

    =44. a=i~=

    ( ) ~~~ +=+ ==45. a==a=

    N~

    ~= =

    ===

    1.4 Complex Numbers =k~~=W==f~~=W==`=W==o~=~W=~I==f~~=~W=I==j==~==W=I= N I= O =^==~==W= I= N I= O =

    ==

    N = = R = = NQ =+ =NO = = NS = = N OQ =+ =P = = T = = PQ =+ =

    46.

    NQ = = NU = = N Q = ==47. ~ += ==48. `=m~=

    =

  • CHAPTER 1. NUMBER SETS

    9

    ===== ==

    Figure 6. =

    49. ( ) ( ) ( ) ( )~~ +++=+++ ==50. ( ) ( ) ( ) ( )~~ +=++ ==51. ( )( ) ( ) ( )~~~ ++=++ ==

    52.

    ~

    ~

    ~OOOO

    +

    +++

    =++

    =

    =53. `~=`=k=

    ~~|||||||

    =+ ==54. = ~ I= = ==

    =

  • CHAPTER 1. NUMBER SETS

    10

    ==

    Figure 7. =

    55. m~=m~==`=k=( )+=+ ~ =

    =56. j=~=^==~=`=k=

    f= ~ + ==~==I==OO ~ += =EFI==

    ~

    ~~= =E~FK=

    =57. m==m~=o~=

    ( ) ( )OOONNNON ++= =( ) ( )[ ]ONONON +++= =

    =58. `~=k==m~=o~=

    ( ) ( ) ( )[ ]+=+ |||||||||||||||||||||

    ==59. f==~=`=k==m~=o~=

    ( ) ( ) ( )[ ]+=+ N

    N=

  • CHAPTER 1. NUMBER SETS

    11

    60. n==m~=o~=( )( ) ( ) ( )[ ]ONONO

    N

    OOO

    NNN

    O

    N

    +=

    ++

    = =

    =61. m==~=`=k=

    ( )[ ] ( ) ( )[ ]+=+= ==62. c~=a=j=

    ( ) ( ) ( )+=+ ==63. k=o==~=`=k=

    ( )

    ++

    +=+=

    O

    O I==

    ==NIIOINIM = K K==

    =64. b=c~=

    += ===

  • 12

    Chapter 2

    Algebra ====

    2.1 Factoring Formulas =o~=W=~I=I===k~~=W==

    ==

    65. ( )( )~~~ OO += ==

    66. ( )( )OOPP ~~~~ ++= ==67. ( )( )OOPP ~~~~ ++=+ ==68. ( )( ) ( )( )( )OOOOOOQQ ~~~~~~ ++=+= ==

    69. ( )( )QPOOPQRR ~~~~~~ ++++= ==70. ( )( )QPOOPQRR ~~~~~~ +++=+ ==71. f===I==

    ( )( )NOOPON ~~~~~~ +++=+ K K===72. f===I===

    ( )( )NOOPON ~~~~~~ +++++= K I==

  • CHAPTER 2. ALGEBRA

    13

    ( )( )NOOPON ~~~~~~ +++=+ K K====

    2.2 Product Formulas o~=W=~I=I===t=W=I====

    73. ( ) OOO ~O~~ += ==

    74. ( ) OOO ~O~~ ++=+ ==

    75. ( ) POOPP ~P~P~~ += ==

    76. ( ) POOPP ~P~P~~ +++=+ ==

    77. ( ) QPOOPQQ ~Q~S~Q~~ ++= ==

    78. ( ) QPOOPQQ ~Q~S~Q~~ ++++=+ ==79. _~=c~=

    ( ) I`~`~`~`~`~ NNOOONNM +++++=+ K

    = ( )>>>

    `

    = =~==~=K=

    =

    80. ( ) O~O~O~~ OOOO +++++=++ ==

    81. ( ) ++++++=+++++ OOOOOO ~~ KK =( )~~~~O +++++++++++ KKK =

  • CHAPTER 2. ALGEBRA

    14

    2.3 Powers =_~=E=~=FW=~I===m=E~~=FW=I==

    ==

    82. ~~~ += ==

    83.

    ~~

    ~ = =

    =

    84. ( ) ~~ = ==

    85.

    ~

    ~=

    =

    =

    86. ( ) ~~ = ==87. N~M = I= M~ ==88. N~N = ==

    89.

    ~

    N~ = =

    =

    90.

    ~~ = ======

  • CHAPTER 2. ALGEBRA

    15

    2.4 Roots =_~W=~I===m=E~~=FW=I==

    MI~ ====E O = I= k F===

    91. ~~ = ==

    92. ~~ = ==

    93.

    ~

    ~= I= M =

    =

    94.

    ~

    ~

    ~== I= M K=

    =

    95. ( ) ~~ = ==

    96. ( ) ~~ = ==

    97. ~~ = ==

    98.

    ~~ = ==

    99. ~~ = ==

    100. ( ) ~~ = ==

  • CHAPTER 2. ALGEBRA

    16

    101. ~

    ~

    ~

    N N

    = I= M~ K=

    =

    102. O

    ~~

    O

    ~~~

    OO

    += =

    =

    103. ~

    ~

    ~

    N

    =

    m

    =

    ===

    2.5 Logarithms =m=~=W=I=I=~I=I==k~~=W=====

    104. a==i~= ~= ==~===

    ~ = I= M~ > I= N~ K==

    105. MN~ = ==

    106. N~~ = ==

    107.

    =N~

    N~M~ =

    =108. ( ) ~~~ += =

    =

    109.

    ~~~ = =

  • CHAPTER 2. ALGEBRA

    17

    110. ( ) ~~ = ==

    111.

    N ~

    ~ = =

    =

    112. ~

    ~

    ~ == I= M > I= N K=

    =

    113. ~

    N

    ~ = =

    =114. ~~ = =

    =115. i~==_~=NM=

    NM = ==

    116. k~~=i~= = I==

    = KTNUOUNUOUKO

    NN

    =

    +=

    =

    =

    117. QPQOVQKMNM

    N == =

    =

    118. PMORURKO

    N == =

    =====

  • CHAPTER 2. ALGEBRA

    18

    2.6 Equations =o~=W=~I=I=I=I=I=I==pW= N I= O I= N I= O I= P =

    ==

    119. i~=b~==l=s~~=

    M~ =+ I=~

    = K==

    =120. n~~=b~=

    M~O =++ I=~O

    ~Q

    O

    OIN

    = K=

    =121. a~=

    ~Qa O = ==

    122. s=c~=f= MO =++ I===

    ==+

    ON

    ON K=

    =

    123. M~O =+ I= MN = I= ~

    O = K=

    =

    124. M~ O =+ I=~

    OIN = K=

    =125. `=b~K=`~~=c~K==

    MP =++ I==

  • CHAPTER 2. ALGEBRA

    19

    N += I= ( ) ( ) OP

    O

    N PIO ++= I==

    ==

    P

    OO

    P

    O

    O

    +

    += I= P

    OO

    P

    O

    O

    +

    = K==

    ==

    2.7 Inequalities s~~W=I=I==

    o~=W=

    PON ~II~I~I~

    III~

    KI=I==

    a~W=aI= a I= a I= a ==

    ==

    126. f~I=f~=k~=~=d~===

    f~= f~=k~= d~=~ = [ ]I~ =

    =~ < = ( ]I~ =

    =~

  • CHAPTER 2. ALGEBRA

    20

    127. f= ~ > I== ~ < K==128. f= ~ > I== M~ > == M~ I== ~ +>+ K==130. f= ~ > I== ~ > K==131. f= ~ > =~= > I== ~ +>+ K==132. f= ~ > =~= > I== ~ > K==133. f= ~ > =~= M > I== ~ > K==

    134. f= ~ > =~= M > I==

    ~> K=

    =135. f= ~ > =~= M < I== ~ < K==

    136. f= ~ > =~= M < I==

    ~< K=

    =137. f= ~M I== ~ < K==138. f= ~M K==

    139. f= ~M =I= M > X=~=~==~=== ~ = K===

    141. O~

    N~ + I== M~ > X=~=~=~=~==~= N~ = K=

  • CHAPTER 2. ALGEBRA

    21

    142.

    ~~~~~~ ON ON

    +++

    KK I== M~II~I~ ON >K K=

    =

    143. f= M~ >+ =~= M~ > I==~

    > K=

    =

    144. f= M~ >+ =~= M~ < I==~

    < K==

    =145. M~ O >++ ==

    = M~ > = M~ < ====

    Ma> =

    =

    =

    N < I= O > =

    =

    ==

    ON =

    == =

    ===

    Ma< =

    =

    =

  • CHAPTER 2. ALGEBRA

    22

    146. ~~ ++ ==147. f= ~ < I== ~~ I== ~ < =~= ~ > I== M~ > K==

    149. f= ~O < I== ~ < I== M~ > K==

    150. f= ~O > I== ~ > I== M~ > K==

    151. f= ( )( ) M

    > I==( ) ( )( )

    >

    M

    MK=

    =

    152. ( )( ) M

    < I==( ) ( )( )

  • CHAPTER 2. ALGEBRA

    23

    154. p=`=f=c~=f======~I====~==W=

    ( )N`^ += K==155. `=`=f=

    f====~=E FI===`^ = K=

    ==

  • 24

    Chapter 3

    Geometry ====

    3.1 Right Triangle =i==~==~W=~I==eW==^W==j~W= ~ I= I= =

    ^W= I =o~===W=o=o~===W==^~W=p===

    ==

    Figure 8. =

    156. =+ VM ==

  • CHAPTER 3. GEOMETRY

    25

    157. ==

    ~ =

    =

    158. ==

    =

    =

    159. ==

    ~~ =

    =

    160. == ~~

    =

    =

    161. ==

    =

    =

    162. == ~

    =

    =163. m~~=q=

    OOO ~ =+ ==

    164. ~O = I= O = I=== = ~= = ~= = = = = ~= ~= I= -I====K==

    ===== ==

    Figure 9. =

  • CHAPTER 3. GEOMETRY

    26

    165. O = I=======~====~K==

    =

    166. Q

    ~

    OOO

    ~ = I= Q

    ~O

    OO = I===

    = ~ =~= =~==~====~=~=K==

    =

    ==

    Figure 10. =

    167. O

    = I==

    = ===~====K=

    =

    168. O

    o == =

    =

    169. ~

    ~

    O

    ~

    ++=

    += =

    =170. ~ = =

    ==

  • CHAPTER 3. GEOMETRY

    27

    171. O

    O

    ~p == =

    ===

    3.2 Isosceles Triangle =_~W=~=iW==_~=~W= =s=~W= =^===~W==mW=i=^~W=p=

    ==

    ==

    Figure 11. =

    172. O

    VM

    = =

    =

    173. Q

    ~

    OOO = =

  • CHAPTER 3. GEOMETRY

    28

    174. O~i += ==

    175. == O

    O

    ~p

    O

    =

    ===

    3.3 Equilateral Triangle =p==~=~~=~W=~=^W==o~===W=o=o~===W==mW=i=^~W=p===

    ==

    Figure 12. =

    176. O

    P~ = =

    =

  • CHAPTER 3. GEOMETRY

    29

    177. P

    P~

    P

    Oo == =

    =

    178. O

    o

    S

    P~

    P

    N === =

    =179. ~Pi = =

    =

    180. Q

    P~

    O

    ~p

    O

    == =

    ===

    3.4 Scalene Triangle E^=~=====~F=

    ==p==~=~W=~I=I==

    pW=O

    ~

    ++= ==

    ^==~=~W= II =^====~I=I=W= ~ II =

    j~====~I=I=W= ~ II =

    _===~= II W= ~ II =o~===W=o=o~===W==^~W=p===

  • CHAPTER 3. GEOMETRY

    30

    ===== ==

    Figure 13. =

    181. =++ NUM ==

    182. ~ >+ I==~ >+ I==~ >+ K=

    =183. ~

  • CHAPTER 3. GEOMETRY

    31

    185. i~==`=+= O~ OOO I=+= ~O~ OOO I=+= ~O~ OOO K=

    =186. i~==p=

    oO

    ~=

    =

    =

    I==

    =o===~====K===

    187. pQ

    ~

    O

    ~

    O

    ~

    O

    O

    O

    O

    ~o

    ~

    ====

    =

    =

    = =

    =

    188. ( )( )( )

    ~O

    = I==

    ~

    N

    N

    N

    N++= K=

    =

    189. ( )( )

    O

    =

    I=

    ( )

    ~

    O

    =

    I=

    ( )( )( )~

    O~

    =

    K=

    =

    190. ( )( )( )~~

    O~ = I=

    ( )( )( )~

    O = I=

    ( )( )( )~

    O = K=

  • CHAPTER 3. GEOMETRY

    32

    191. == ~ I=== ~ I=== ~ K=

    =

    192. Q

    ~

    O

    OOOO~

    += I==

    Q

    O

    ~

    OOOO

    += I==

    Q

    O

    ~

    OOOO

    += K=

    =

    ===== ==

    Figure 15. =

    193. ~PO

    ^j = I= PO

    _j = I= PO

    `j = =EcKNRFK=

    =

    194. ( )( )O

    O~

    ~Q

    +

    = I==

    ( )( )O

    O

    ~

    ~Q

    +

    = I==

    ( )( )O

    O

    ~

    ~Q

    +

    = K=

    =

  • CHAPTER 3. GEOMETRY

    33

    195. O

    O

    O

    ~p ~ === I==

    O

    O

    ~

    O

    ~p

    =

    =

    = I==

    ( )( )( )~p = =Ee=c~FI=p = I==

    oQ

    ~p = I=

    = oOp O I=

    O~

    O~

    O~p O

    = K=

    ===

    3.5 Square p==~=~W=~=a~~W==o~===W=o=o~===W==mW=i=^~W=p==

    ==

    Figure 16.

  • CHAPTER 3. GEOMETRY

    34

    196. O~ = ===

    197. O

    O~

    O

    o == =

    =

    198. O

    ~ = =

    =199. ~Qi = =

    =

    200. O~p = ====

    3.6 Rectangle =p==~=~W=~I==a~~W==o~===W=o=mW=i=^~W=p===

    ==

    Figure 17. =

    201. OO ~ += ==

  • CHAPTER 3. GEOMETRY

    35

    202. O

    o = =

    =203. ( )~Oi += =

    =204. ~p = =

    ===

    3.7 Parallelogram =p==~=~~~W=~I==a~~W= ON I =`=~W= I =^===~~W= =^W===mW=i=^~W=p===

    ===== ==

    Figure 18. =

    205. =+ NUM ==

    206. ( )OOOOON ~O +=+ ==

  • CHAPTER 3. GEOMETRY

    36

    207. == ==

    208. ( )~Oi += ==

    209. == ~~p I==

    = O

    Np ON K=

    ===

    3.8 Rhombus =p==~=W=~=a~~W= ON I =`=~W= I =^W=e=o~===W==mW=i=^~W=p===

    ===== ==

    Figure 19. =

  • CHAPTER 3. GEOMETRY

    37

    210. =+ NUM ==

    211. OOOON ~Q =+ ==

    212. ~O

    ~ ON== =

    =

    213. O

    ~

    ~Q

    O

    ON

    === =

    =214. ~Qi = =

    =

    215. == ~~p O I==

    ONO

    Np = K=

    ===

    3.9 Trapezoid =_~==~=~W=~I==jW==^W==^~W=p===

  • CHAPTER 3. GEOMETRY

    38

    ==

    Figure 20. =

    216. O

    ~

    += =

    =

    217. O

    ~p =

    += =

    ===

    3.10 Isosceles Trapezoid =_~==~=~W=~I==iW==jW==^W==a~~W==o~===W=o=^~W=p===

  • CHAPTER 3. GEOMETRY

    39

    ==

    Figure 21. =

    218. O

    ~

    += =

    =

    219. O~ += ==

    220. ( )OO ~Q

    N = =

    =

    221. ( )( )~O~O

    ~o

    O

    +++

    = =

    =

    222. O

    ~p =

    += =

    ======

  • CHAPTER 3. GEOMETRY

    40

    3.11 Isosceles Trapezoid with Inscribed Circle

    =_~==~=~W=~I==iW==jW==^W==a~~W==o~===W=o=o~===W==mW=i=^~W=p===

    ==

    Figure 22. =

    223. O~ =+ ==

    224. O

    ~ =

    += =

    =225. OOO += =

    =

  • CHAPTER 3. GEOMETRY

    41

    226. O

    ~

    O

    == =

    =

    227. ~

    S

    ~

    U

    ~

    O

    ~

    N

    O

    Q

    O

    o OO

    O

    +++

    =+=+=== =

    =228. ( ) Q~Oi =+= =

    =

    229. ( )O

    i

    O

    ~~

    O

    ~p ===

    +=

    += ==

    ===

    3.12 Trapezoid with Inscribed Circle =_~==~=~W=~I==i~~=W=I==jW==^W==a~~W= ON I =^===~~W= =o~===W==o~===W=o=mW=i=^~W=p==

  • CHAPTER 3. GEOMETRY

    42

    ==

    Figure 23. =

    230. ~ +=+ ==

    231. O

    O

    ~

    +=

    += =

    =232. ( ) ( )O~Oi +=+= =

    =

    233. O

    O

    ~p =

    +=

    += I==

    = O

    Np ON K=

    ===

    3.13 Kite =p==~=W=~I==a~~W= ON I =^W= II =mW=i=^~W=p===

  • CHAPTER 3. GEOMETRY

    43

    ==

    Figure 24. =

    234. =++ PSMO ==

    235. ( )~Oi += ==

    236. O

    p ON= =

    ===

    3.14 Cyclic Quadrilateral p==~=~~~W=~I=I=I==a~~W= ON I =^===~~W= =f~=~W= III =o~===W=o=mW=i=pW===^~W=p=

  • CHAPTER 3. GEOMETRY

    44

    ==

    Figure 25. =

    237. =+=+ NUM ==

    238. m=q=ON~ =+ =

    =239. ~i +++= =

    =

    240. ( )( )( )( )( )( )( )~~~~

    Q

    No

    +++

    = I==

    =O

    i = K=

    =

    241. = O

    Np ON I==

    ( )( )( )( )~p = I==

    =O

    i = K=

    ===

  • CHAPTER 3. GEOMETRY

    45

    3.15 Tangential Quadrilateral =p==~=~~~W=~I=I=I==a~~W= ON I =^===~~W= =o~===W==mW=i=pW===^~W=p===

    ==

    Figure 26. =

    242. ~ +=+ ==

    243. ( ) ( )O~O~i +=+=+++= ==

    244. ( ) ( )

    O

    ~~

    OOOO

    ON += I==

    =O

    i = K==

    =

  • CHAPTER 3. GEOMETRY

    46

    245. == O

    Np ON =

    ===

    3.16 General Quadrilateral =p==~=~~~W=~I=I=I==a~~W= ON I =^===~~W= =f~=~W= III =mW=i=^~W=p===

    ======= ==

    Figure 27. =

    246. =+++ PSM ==

    247. ~i +++= ==

  • CHAPTER 3. GEOMETRY

    47

    248. = O

    Np ON =

    ===

    3.17 Regular Hexagon =pW=~=f~=~W= =p~=W==o~===W==o~===W=o=mW=i=pW===^~W=p===

    ==

    Figure 28. =

    249. = NOM ==

    250. O

    P~ == =

  • CHAPTER 3. GEOMETRY

    48

    251. ~o = ==

    252. ~Si = ==

    253. O

    PP~p

    O

    == I==

    =O

    i = K=

    ===

    3.18 Regular Polygon =pW=~=k==W==f~=~W= =p~=W==o~===W==o~===W=o=mW=i=pW===^~W=p===

  • CHAPTER 3. GEOMETRY

    49

    ==

    Figure 29. =

    254. = NUMO

    O=

    =

    255. = NUMO

    O=

    =

    256.

    O

    ~o

    = =

    =

    257. Q

    ~o

    ~O

    ~

    OO =

    == =

    =258. ~i = =

    =

    259.

    O

    O

    op

    O = I==

    Q

    ~op

    OO == I==

  • CHAPTER 3. GEOMETRY

    50

    =O

    i = K==

    ===

    3.19 Circle =o~W=o=a~W==`W=~=p~=W=I==q~=W==`~=~W= =f=~W= =mW=i=^~W=p===

    260. O

    oO~

    = =

    =

    ==

    Figure 30. =

  • CHAPTER 3. GEOMETRY

    51

    261. ONON ~~ = ==

    ==

    Figure 31. =

    262. NN = ==

    ===== ==

    Figure 32. =

    263. NO = ==

  • CHAPTER 3. GEOMETRY

    52

    ===== ==

    Figure 33. =

    264. O

    = =

    =

    ==

    Figure 34. =

    265. oOi == ==

    266. O

    io

    Q

    op

    OO =

    == ==

    =

  • CHAPTER 3. GEOMETRY

    53

    3.20 Sector of a Circle =o~==~=W=o=^=W==`~=~=E=~~FW==`~=~=E=FW= =mW=i=^~W=p===

    ==

    Figure 35. =

    267. o = ==

    268.

    =NUM

    o =

    =269. oOi += =

    =

    270.

    ===PSM

    o

    O

    o

    O

    op

    OO

    ==

    ==

  • CHAPTER 3. GEOMETRY

    54

    3.21 Segment of a Circle =o~==~=W=o=^=W==`W=~=`~=~=E=~~FW==`~=~=E=FW= =e===W==mW=i=^~W=p===

    ==

    Figure 36. =

    271. OoOO~ = ==

    272. OO ~oQO

    No = I= o < =

    =273. ~i += =

    =

  • CHAPTER 3. GEOMETRY

    55

    274. ( )[ ] ( )O

    o

    NUMO

    oo~o

    O

    Np

    OO

    =

    == I==

    ~P

    Op K=

    ===

    3.22 Cube =bW=~==a~~W==o~===W==o~===W==p~=~~W=p=sW=s===

    === ==

    Figure 37. =

    275. P~ = ==

    276. O

    ~ = =

    =

  • CHAPTER 3. GEOMETRY

    56

    277. O

    P~o = =

    =278. O~Sp = =

    =279. P~s = ==

    ===

    3.23 Rectangular Parallelepiped =bW=~I=I===a~~W==p~=~~W=p=sW=s===

    ===== ==

    Figure 38. =

    280. OOO ~ ++= ==

    281. ( )~~Op ++= ==

    282. ~s = ==

  • CHAPTER 3. GEOMETRY

    57

    3.24 Prism =i~~=W==eW==i~~=~~W= ip =^~==~W= _p =q~=~=~~W=p=sW=s===

    ===== ==

    Figure 39. =

    283. _i pOpp += K===

    284. i~~=^~==~=o=m=( )~~~~p PONi ++++= K =

    =285. i~~=^~==~=l=m=

    pi = I==========K=

    =

  • CHAPTER 3. GEOMETRY

    58

    286. ps _= ==

    287. `~~D=m==d=== ==~~=~K=f==~===~~====~=~==~=~~===I=======~=~K====

    3.25 Regular Tetrahedron =q~==W=~=eW==^~==~W= _p =p~=~~W=p=sW=s===

    ==

    Figure 40. =

    288. ~P

    O = =

    =

  • CHAPTER 3. GEOMETRY

    59

    289. Q

    ~Pp

    O

    _ = =

    =

    290. O~Pp = ==

    291. OS

    ~p

    P

    Ns

    P

    _ == K==

    ===

    3.26 Regular Pyramid =p==~W=~=i~~=W==eW==p~=W===k==W===p==~W==o~=====~W==^~==~W= _p =i~~=~=~~W= ip =q~=~=~~W=p=sW=s===

  • CHAPTER 3. GEOMETRY

    60

    ==

    Figure 41. =

    292. Q

    ~

    OO = =

    =

    293.

    O

    ~

    Q

    OOO

    = =

    =

    294. ~Q~Q

    N~

    O

    Np OOi === =

    =295. p_ = =

    =296. i_ ppp += =

    =

    297. P

    Np

    P

    Ns _ == ==

    ===

  • CHAPTER 3. GEOMETRY

    61

    3.27 Frustum of a Regular Pyramid =

    _~=~===W=

    PON

    PON

    IIII

    ~II~I~I~

    K

    K=

    eW==p~=W===^~==~W= Np I= Op =i~~=~=~~W= ip =m==~W= Nm I= Om =p~=~W==q~=~=~~W=p=sW=s===

    ==

    Figure 42. =

    298. ~

    ~

    ~

    ~

    ~

    P

    P

    O

    O

    N

    N ====== K =

    =

  • CHAPTER 3. GEOMETRY

    62

    299. ON

    O p

    p= =

    =

    300. ( )O

    mmp ONi

    += =

    =301. ONi pppp ++= =

    =

    302. ( )OONN ppppP

    s ++= =

    =

    303. [ ]ONO

    N NP

    p

    ~

    ~

    N

    P

    ps ++=

    ++= =

    ===

    3.28 Rectangular Right Wedge =p==~W=~I==q=W==eW==i~~=~=~~W= ip =^~==~W= _p =q~=~=~~W=p=sW=s===

  • CHAPTER 3. GEOMETRY

    63

    ==

    Figure 43. =

    304. ( ) ( )OOOOi ~Q~ON

    p ++++= =

    =305. ~p_ = =

    =306. i_ ppp += =

    =

    307. ( )~OS

    s += =

    ===

    3.29 Platonic Solids =bW=~=o~===W==o~===W=o=p~=~~W=p=sW=s===

  • CHAPTER 3. GEOMETRY

    64

    308. c=m~=p=q= ~= = ~= = ~= = ~=~=====~=K==

    =p= k=

    =sk==b=

    k==c~=

    p=

    q~== Q= S= Q= PKOR=`= U= NO= S= PKOO=

    l~= S= NO= U= PKOT=f~= NO= PM= OM= PKOT=

    a~= OM= PM= NO= PKOT===

    Octahedron =

    ==

    Figure 44. =

    309. S

    S~ = =

    =

    310. O

    O~o = =

    =

  • CHAPTER 3. GEOMETRY

    65

    311. P~Op O= ==

    312. P

    O~s

    P

    = =

    ==

    Icosahedron =

    ==

    Figure 45. =

    313. ( )NO

    RPP~

    += =

    =

    314. ( )RROQ

    ~o += =

    =

    315. P~Rp O= ==

    316. ( )NO

    RP~Rs

    P += =

    ==

  • CHAPTER 3. GEOMETRY

    66

    Dodecahedron =

    ==

    Figure 46. =

    317. ( )O

    RNNORNM~

    += =

    =

    318. ( )Q

    RNP~o

    += =

    =

    319. ( )RORR~Pp O += ==

    320. ( )Q

    RTNR~s

    P += =

    ===

    3.30 Right Circular Cylinder =o~==~W=o=a~==~W==

  • CHAPTER 3. GEOMETRY

    67

    eW=e=i~~=~=~~W= ip =^~==~W= _p =q~=~=~~W=p=sW=s===

    ===== ==

    Figure 47. =

    321. oeOpi = ==

    322. ( )

    +=+=+=

    O

    eoeoOpOpp _i =

    =

    323. eoeps O_ == ====

  • CHAPTER 3. GEOMETRY

    68

    3.31 Right Circular Cylinder with an Oblique Plane Face

    =o~==~W=o=q=~===~=W= N =q====~=W= O =i~~=~=~~W= ip =^~==~==~W= _p =q~=~=~~W=p=sW=s===

    ==

    Figure 48. =

    324. ( )ONi op += ==

    325. O

    ONOO_ O

    ooop

    ++= =

    =

  • CHAPTER 3. GEOMETRY

    69

    326.

    ++++=+=

    O

    ONOON_i O

    oooppp =

    =

    327. ( )ONO

    O

    os +

    = =

    ===

    3.32 Right Circular Cone o~==~W=o=a~==~W==eW=e=p~=W==i~~=~=~~W= ip =^~==~W= _p =q~=~=~~W=p=sW=s===

    ==

    Figure 49.

  • CHAPTER 3. GEOMETRY

    70

    328. OO oe = ==

    329. O

    opi

    == =

    =

    330. O_ op = ==

    331. ( )

    +=+=+=

    O

    O

    Nooppp _i =

    =

    332. eoP

    Nep

    P

    Ns O_ == =

    ===

    3.33 Frustum of a Right Circular Cone =o~==~W=oI==eW=e=p~=W==p~=~W==^~==~W= Np I= Op =i~~=~=~~W= ip =q~=~=~~W=p=sW=s===

  • CHAPTER 3. GEOMETRY

    71

    ==

    Figure 50. =

    333. ( )OO oe = ==

    334.

    o= =

    =

    335. OO

    O

    N

    O

    o

    p

    p== =

    =336. ( )opi += ==337. ( )[ ]oopppp OOiON +++=++= ==

    338. ( )OONN ppppP

    s ++= =

    =

    339. [ ]ONO

    N NP

    p

    o

    oN

    P

    ps ++=

    ++= =

    ===

  • CHAPTER 3. GEOMETRY

    72

    3.34 Sphere =o~W=o=a~W==p~=~~W=p=sW=s==

    ==

    Figure 51. =

    340. OoQp = ==

    341. poP

    N

    S

    Neo

    P

    Qs PP === =

    ===

    3.35 Spherical Cap o~==W=o=o~==~W==eW==^~==~=~W= _p =^~==~=~W= `p =

    q~=~=~~W=p=sW=s=

  • CHAPTER 3. GEOMETRY

    73

    ==

    Figure 52. =

    342. O

    o

    OO += =

    =343. O_ p = ==344. ( )OO` p += ==345. ( ) ( )OOO`_ oOOppp +=+=+= ==

    346. ( ) ( )OOO PS

    oPS

    s +

    =

    = =

    ===

    3.36 Spherical Sector =o~==W=o=o~==~==~=~W==eW==q~=~=~~W=p=sW=s==

  • CHAPTER 3. GEOMETRY

    74

    ====== === ==

    Figure 53. =

    347. ( )Oop += ==

    348. oP

    Os O= =

    =kW=q== ~=~=== == ~==~=K====

    3.37 Spherical Segment =o~==W=o=o~==~W= N I= O =eW==^~==~=~W= pp =

    ^~==~==~W= Np I= Op =q~=~=~~W=p=sW=s==

  • CHAPTER 3. GEOMETRY

    75

    ===== ==

    Figure 54. =

    349. oOpp = ==350. ( )OOONONp oOpppp ++=++= ==

    351. ( )OOOON PPSN

    s ++= =

    ===

    3.38 Spherical Wedge =o~W=o=a~=~==W==a~=~==~~W= =^~==~=W= ip =q~=~=~~W=p=sW=s===

  • CHAPTER 3. GEOMETRY

    76

    ==

    Figure 55. =

    352. oOVM

    op O

    O

    i =

    = =

    =

    353. oOoVM

    oop OO

    OO +=

    += =

    =

    354. oP

    O

    OTM

    os P

    P

    =

    = =

    ===

    3.39 Ellipsoid =p-~W=~I=I==sW=s=

  • CHAPTER 3. GEOMETRY

    77

    ======= ==

    Figure 56. =

    355. ~P

    Qs = =

    ===

    Prolate Spheroid =p-~W=~I=I==E ~ > F=p~=~~W=p=sW=s===

    356.

    +=

    ~~Op I==

    =~

    ~

    OO = K=

    =

    357. ~P

    Qs O= =

    =

  • CHAPTER 3. GEOMETRY

    78

    Oblate Spheroid =p-~W=~I=I==E ~ < F=p~=~~W=p=sW=s===

    358.

    +=~L

    ~

    ~~

    Op I==

    =

    ~

    OO = K=

    =

    359. ~P

    Qs O= =

    ===

    3.40 Circular Torus =j~=~W=o=j=~W==p~=~~W=p=sW=s==

  • CHAPTER 3. GEOMETRY

    79

    == =Picture 57.

    =360. oQp O= ==361. OOoOs = =

    ==

  • 80

    Chapter 4

    Trigonometry ====^W= I= =o~==E~==~=FW=I===t=W====

    4.1 Radian and Degree Measures of Angles =

    362. ?QRDNTRTNUM~N

    = =

    =

    363. ~MNTQRPKM~NUM

    N

    = =

    =

    364. ~MMMOVNKM~SMNUM

    DN

    = =

    =

    365. ~MMMMMRKM~PSMMNUM

    ?N

    = =

    =366. ==

    ^=EF=

    M= PM= QR= SM= VM= NUM= OTM= PSM=

    ^=E~~F= M= S

    =

    Q

    =

    P

    =

    O

    = =

    O

    P= O =

    ===

  • CHAPTER 4. TRIGONOMETRY

    81

    4.2 Definitions and Graphs of Trigonometric Functions

    =

    = ==

    Figure 58. =

    367.

    = =

    =

    368.

    = =

    =

    369.

    ~ = =

    =

    370.

    = =

    =

  • CHAPTER 4. TRIGONOMETRY

    82

    371.

    = =

    =

    372.

    = =

    =373. p=c=

    = I= NN K==

    =

    Figure 59. =

    374. `=c== = I= NN K=

  • CHAPTER 4. TRIGONOMETRY

    83

    ==

    Figure 60. =

    375. q~=c=

    ~ = I= ( )O

    NO

    + I= K~ =

    =

    ==

    Figure 61. =

  • CHAPTER 4. TRIGONOMETRY

    84

    376. `~=c== = I= I== K=

    =

    ==

    Figure 62. =

    377. p~=c=

    = I= ( )O

    NO

    + K=

    ==

  • CHAPTER 4. TRIGONOMETRY

    85

    ==

    Figure 63. =

    378. `~=c== = I= K=

    =

    Figure 64.

  • CHAPTER 4. TRIGONOMETRY

    86

    4.3. Signs of Trigonometric Functions 379. ==

    n~~=p =

    ` =

    q~ =

    ` =

    p =

    `= =

    f= H= H= H= H= H= H=ff= H= = = = = H=fff= = = H= H= = =fs= = H= = = H= ==

    ==

    380. ==

    =

    Figure 65.

    ==========

  • CHAPTER 4. TRIGONOMETRY

    87

    4.4 Trigonometric Functions of Common Angles 381. = = ~ = = = ~ = = =

    M= M= M= N= M= = N= =

    PM=S

    =

    O

    N=

    O

    P=

    P

    N= P =

    P

    O= O=

    QR=Q

    =

    O

    O=

    O

    O= N= N= O = O =

    SM=P

    =

    O

    P=

    O

    N= P =

    P

    N= O=

    P

    O=

    VM=O

    = N= M= = M= = N=

    NOM=P

    O=

    O

    P=

    O

    N = P =

    P

    N O =

    P

    O=

    NUM= = M= N = M= = N = =

    OTM=O

    P= N = M= = M= = N =

    PSM= O = M= N= M= = N= ==============

  • CHAPTER 4. TRIGONOMETRY

    88

    382. = = ~ = = = ~ = =

    NR=NO

    =

    Q

    OS =

    Q

    OS += PO = PO+ =

    NU=NM

    =

    Q

    NR =

    Q

    RONM+R

    ROR= ROR+ =

    PS=R

    =

    Q

    RONMQ

    NR +=

    NR

    RONM

    +

    RONM

    NR

    +

    =

    RQ=NM

    P=

    Q

    NR +=

    Q

    RONMRONM

    NR

    +NR

    RONM

    +

    =

    TO=R

    O=

    Q

    RONM+Q

    NR = ROR+ = R

    ROR

    =

    TR=NO

    R=

    Q

    OS +=

    Q

    OS = PO+ = PO =

    ===

    4.5 Most Important Formulas =

    383. N OO =+ ==

    384. N~ OO = ==

    385. N OO = ==

    386.

    =

    ~ =

  • CHAPTER 4. TRIGONOMETRY

    89

    387.

    =

    =

    =388. N~ = =

    =

    389.

    =

    N =

    =

    390.

    =

    N =

    ===

    4.6 Reduction Formulas =

    391. ==

    = = = ~ = = = = + = ~ = =VM = + = + = + = + ~ =+VM = + = = = ~ =NUM + = = ~ = =+NUM = = + ~ = + =OTM = = + = + ~ =+OTM = + = = ~ =PSM = + = ~ = =+PSM + = + = + ~ = + ==

    =====

  • CHAPTER 4. TRIGONOMETRY

    90

    4.7 Periodicity of Trigonometric Functions =

    392. ( ) = O I== O == PSM K==

    393. ( ) = O I== O == PSM K==

    394. ( ) = ~~ I== == NUM K==

    395. ( ) = I== == NUM K====

    4.8 Relations between Trigonometric Functions

    =

    396. ( ) NQO

    OONO

    NN OO

    === =

    =

    O~N

    O~O

    O +

    = =

    =

    397. ( ) NO

    OONO

    NN OO

    =+== =

    =

    O~N

    O~N

    O

    O

    +

    = =

    =

    398.

    =+

    ==

    =O

    ON

    ON

    ON

    ~ O =

  • CHAPTER 4. TRIGONOMETRY

    91

    =

    O~N

    O~O

    ON

    ON

    O +

    =+

    = =

    =

    399.

    =

    +

    ==

    =ON

    O

    O

    ONN

    O =

    =

    O~O

    O~N

    ON

    ONO

    =+

    = =

    =

    400.

    O~N

    O~N

    ~N

    N

    O

    O

    O

    +

    =+=

    = =

    =

    401.

    O~O

    O~N

    N

    N

    O

    O

    +

    =+=

    = =

    ===

    4.9 Addition and Subtraction Formulas =

    402. ( ) +=+ ==

    403. ( ) = ==

    404. ( ) =+ ==

    405. ( ) += =

  • CHAPTER 4. TRIGONOMETRY

    92

    406. ( )+

    =+~~N

    ~~~ =

    =

    407. ( )+

    =~~N

    ~~~ =

    =

    408. ( )+

    =+~~

    ~~N =

    =

    409. ( )+

    =~~

    ~~N =

    ===

    4.10 Double Angle Formulas =

    410. = OO ==

    411. NOONO OOOO === ==

    412.

    =

    =~

    O

    ~N

    ~OO~

    O=

    =

    413. O

    ~

    O

    NO

    O =

    = =

    ======

  • CHAPTER 4. TRIGONOMETRY

    93

    4.11 Multiple Angle Formulas =

    414. == POP PQPP ==

    415. = UQQ P ==

    416. += RP NSOMRR ==

    417. == OPP PPQP ==

    418. NUUQ OQ += ==

    419. += ROMNSR PR ==

    420.

    =O

    P

    ~PN

    ~~PP~ =

    =

    421. +

    =QO

    P

    ~~SN

    ~Q~QQ~ =

    =

    422. ++

    =QO

    PR

    ~R~NMN

    ~R~NM~R~ =

    =

    423. NP

    PP

    O

    P

    = =

    =

    424. +

    =P

    QO

    ~Q~Q

    ~~SNQ ==

    =

  • CHAPTER 4. TRIGONOMETRY

    94

    425. +

    +=

    ~R~NM~

    ~R~NMNR

    PR

    QO

    =

    ===

    4.12 Half Angle Formulas =

    426. O

    N

    O

    =

    =

    =

    427. O

    N

    O

    +=

    =

    =

    428. =

    =+

    =

    +

    =

    N

    N

    N

    N

    O~ =

    =

    429. +=+

    =

    =

    +

    =

    N

    N

    N

    N

    O =

    ===

    4.13 Half Angle Tangent Identities =

    430.

    O~N

    O~O

    O +

    = =

    =

  • CHAPTER 4. TRIGONOMETRY

    95

    431.

    O~N

    O~N

    O

    O

    +

    = =

    =

    432.

    O~N

    O~O

    ~O

    = =

    =

    433.

    O~O

    O~N

    O

    = =

    ===

    4.14 Transforming of Trigonometric Expressions to Product

    =

    434. O

    O

    O+

    =+ =

    =

    435. O

    O

    O+

    = =

    =

    436. O

    O

    O+

    =+ =

    =

    437. O

    O

    O+

    = =

    =

  • CHAPTER 4. TRIGONOMETRY

    96

    438. ( )

    +=+

    ~~ =

    =

    439. ( )

    =

    ~~ =

    =

    440. ( )

    +=+

    =

    =

    441. ( )

    =

    =

    =

    442.

    +

    =

    =+Q

    OQ

    O =

    =

    443.

    +

    =

    =Q

    OQ

    O =

    =

    444. ( )

    =+

    ~ =

    =

    445. ( )

    +=

    ~ =

    =

    446. O

    ON O

    =+ =

    =

    447. O

    ON O

    = =

    =

  • CHAPTER 4. TRIGONOMETRY

    97

    448.

    =+OQ

    ON O =

    =

    449.

    =OQ

    ON O =

    ===

    4.15 Transforming of Trigonometric Expressions to Sum

    =

    450. ( ) ( )O

    += =

    =

    451. ( ) ( )O

    ++= =

    =

    452. ( ) ( )O

    ++= =

    =

    453. ++

    =

    ~~~~ =

    =

    454. ++

    =~~

    =

    =

    455. ++

    =~

    ~~ =

    ===

  • CHAPTER 4. TRIGONOMETRY

    98

    4.16 Powers of Trigonometric Functions =

    456. O

    ONO

    = =

    =

    457. Q

    PPP

    = =

    =

    458. U

    POQQQ

    += =

    =

    459. NS

    RPRNMR

    += =

    =

    460. PO

    SQSONRNMS

    += =

    =

    461. O

    ONO

    += =

    =

    462. Q

    PPP

    += =

    =

    463. U

    POQQQ

    ++= =

    =

    464. NS

    RPRNMR

    ++= =

    =

    465. PO

    SQSONRNMS

    +++= =

    =

  • CHAPTER 4. TRIGONOMETRY

    99

    4.17 Graphs of Inverse Trigonometric Functions

    =466. f=p=c==

    ~ = I= NN I=O

    ~O

    K=

    =

    ==

    Figure 66. =

    467. f=`=c==~ = I= NN I= ~M K=

    =

  • CHAPTER 4. TRIGONOMETRY

    100

    ==

    Figure 67. =

    468. f=q~=c==

    ~~ = I= I=O

    ~~O

  • CHAPTER 4. TRIGONOMETRY

    101

    469. f=`~=c==~ = I= I=

  • CHAPTER 4. TRIGONOMETRY

    102

    471. f=`~=c==

    ( ] [ ) KO

    IMMIO

    ~IINNII~

    =

    ==

    Figure 71. ==

    4.18 Principal Values of Inverse Trigonometric Functions 472. = M= O

    N=

    O

    O=

    O

    PN=

    ~ = M = PM = QR = SM VM~ = VM SM = QR = PM M =

    = O

    N

    O

    O

    O

    P N = =

    ~ =PM

    = QR SM VM

    ==

    ~ =NOM

    =NPR = NRM = NUM

    ==

  • CHAPTER 4. TRIGONOMETRY

    103

    473. = M=

    P

    PN= P =

    P

    P N = P =

    ~~ = M = PM QR SM PM QR=

    SM =

    ~ = VM SM QR PM NOM = NPR=

    NRM = ===

    4.19 Relations between Inverse Trigonometric Functions

    =474. ( ) ~~ = =

    =

    475. ~O

    ~

    = =

    =

    476. ON~~ = I= NM K==

    477. ON~~ = I= MN K==

    478. ON

    ~~~

    = I= N

    O < K=

    =

    479.

    N~~

    O= I= NM < K=

    =

    480. =

    N~~

    O

    I= MN

  • CHAPTER 4. TRIGONOMETRY

    104

    482. ~O

    ~

    = =

    =

    483. ON~~ = I= NM K==

    484. ON~~ = I= MN K==

    485.

    N~~~

    O= I= NM < K=

    =

    486.

    N~~~

    O+= I= MN

  • CHAPTER 4. TRIGONOMETRY

    105

    493.

    N~~

    O~~

    = I= M > K=

    =

    494.

    N~~

    O~~

    = I= M < K=

    =

    495.

    N~~~ = I= M > K=

    =

    496. =

    N~~~ I= M < K=

    =497. ( ) ~~ = =

    =

    498. ~~O

    ~

    = =

    =

    499. ON

    N~~

    += I= M > K=

    =

    500. ON

    N~~

    += I= M < K=

    =

    501. ON

    ~~

    += =

    =

    502.

    N~~~ = I= M > K=

    =

    503.

    N~~~ += I= M < K=

    ==

  • CHAPTER 4. TRIGONOMETRY

    106

    4.20 Trigonometric Equations =t=W====

    504. ~ = I= ( ) ~~N += ==

    505. ~ = I= O~~ += ==

    506. ~~ = I= ~~~ += ==

    507. ~ = I= ~~ += ====

    4.21 Relations to Hyperbolic Functions =f~~=W====

    508. ( ) = ==

    509. ( ) ~~ = ==

    510. ( ) = ==

    511. ( ) = ==

    512. ( ) = ====

  • 107

    Chapter 5

    Matrices and Determinants ====j~W=^I=_I=`=b==~=~W= ~ I= I= ~ I= I= =

    a~==~=~W= ^ =j==~== ~ W= j =

    `~==~== ~ W= ` =

    q~==~=~W= q^ I= ^

    =^==~=~W= ^~ =q~==~=~W= ^ =

    f==~=~W= N^ =o~=W==o~=~~W= =k~~=W=I======

    5.1 Determinants =

    513. p=l=a~=

    NOONOO

    NN ~~~

    ~^ == =

    =====

  • CHAPTER 5. MATRICES AND DETERMINANTS

    108

    514. q=l=a~=

    ++== POONNPPNOPNOPPOONNPPPOPN

    OPOOON

    NPNONN

    ~~~~~~~~~

    ~~~

    ~~~

    ~~~

    ^ =

    PNOONPPPONNOPOOPNN ~~~~~~~~~ ==

    515. p~=o=E^=oF=

    ==

    Figure 72. =

    516. k-=l=a~=

    ON

    ON

    OOOOON

    NNNONN

    ~~~~

    ~~~~

    ~~~~

    ~~~~

    ^

    KK

    KKKKKK

    KK

    KKKKKK

    KK

    KK

    = =

    =517. j=

    q== j =~~==== ~ ==-==~= ^= = = ( )N -= = ~= = ==~=^=====-==~=-=K====

  • CHAPTER 5. MATRICES AND DETERMINANTS

    109

    518. `~=( ) jN` += =

    =519. i~~=b~==-=l=a~=

    i~~=~=====-==

    =

    =

    N`~^ I= IIOIN K= K=

    i~~=~=====-==

    =

    =

    N`~^ I= IIOIN K= K==

    ===

    5.2 Properties of Determinants =

    520. q==~===~=~=~==~===~=~===~===K=

    =OO

    NN

    ON

    ON

    ~

    ~

    ~~= ==

    =521. f=====E==F=~==~I====

    =~==~K=

    NN

    OO

    OO

    NN

    ~

    ~

    ~

    ~= =

    =522. f====E==F=~==~I==~===

    ~==K=

    M~~

    ~~

    OO

    NN = =

    =

  • CHAPTER 5. MATRICES AND DETERMINANTS

    110

    523. f=========~===E=F=~=======~====~I====~========~=~K=

    OO

    NN

    OO

    NN

    ~

    ~

    ~

    ~= =

    =524. f========~====E==F=~=~=E=

    ~F=~=======~====E=FI===~===~==~K=

    OO

    NN

    OOO

    NNN

    ~

    ~

    ~

    ~=

    ++

    =

    ===

    5.3 Matrices =

    525. a=^= =~=^==~=~~=~~===E-==F====~==K==

    [ ]

    ==

    ON

    OOOON

    NNONN

    ~~~

    ~~~

    ~~~

    ~^

    K

    MMM

    K

    K

    ==

    =526. p~=~==~=~=== K==

    =527. ^=~=~== [ ]~ ======== ~~ = I==KK=====

    =~==~=~~K===

    528. ^=~=~= [ ]~ ==-== ~~ = K===

  • CHAPTER 5. MATRICES AND DETERMINANTS

    111

    529. a~~=~====~=~==~==~=========~=~~K===

    530. r=~====~=~~==~========~=~~=~=~=K=q==~==============fK===

    531. ^==~=====~=~=K====

    5.4 Operations with Matrices =

    532. q=~=^=~=_=~=~=I=~==I==~======~==~== ==~===~=~K==

    533. q=~==^=~=_==~==~=E=~F=I=~==I==~==~=~= K=f==

    [ ]

    ==

    ON

    OOOON

    NNONN

    ~~~

    ~~~

    ~~~

    ~^

    K

    MMM

    K

    K

    I==

    [ ]

    ==

    ON

    OOOON

    NNONN

    _

    K

    MMM

    K

    K

    I==

    =====

  • CHAPTER 5. MATRICES AND DETERMINANTS

    112

    ==

    +++

    ++++++

    =+

    OONN

    OOOOOOONON

    NNNONONNNN

    ~~~

    ~~~

    ~~~

    _^

    K

    MMM

    K

    K

    K=

    =534. f===~=~~I=~= [ ]~^ = ==~=~I==

    [ ]

    ==

    ON

    OOOON

    NNONN

    ~~~

    ~~~

    ~~~

    ~^

    K

    MMM

    K

    K

    K=

    =535. j~==q=j~=

    q= ~= ~= = = = = = == = = = = = = ~= = = = ====K===f=

    [ ]

    ==

    ON

    OOOON

    NNONN

    ~~~

    ~~~

    ~~~

    ~^

    K

    MMM

    K

    K

    I==

    [ ]

    ==

    ON

    OOOON

    NNONN

    _

    K

    MMM

    K

    K

    I=

    =====

  • CHAPTER 5. MATRICES AND DETERMINANTS

    113

    ==

    ==

    ON

    OOOON

    NNONN

    `^_

    K

    MMM

    K

    K

    I==

    ==

    =

    =+++=

    NOONN ~~~~ K =

    E IIOIN K= X IIOIN K= FK===q==

    [ ]

    ==

    OPOOON

    NPNONN ~~~

    ~~~~^ I= [ ]

    ==

    P

    O

    N

    _ I==

    ==

    =

    =

    POPOOONON

    PNPONONNN

    P

    O

    N

    OPOOON

    NPNONN

    ~~~

    ~~~

    ~~~

    ~~~^_ K==

    =536. q~==~=j~=

    f===~===~=~=~=~I====~==~==~===~=~K===f= ^= = = ~= ~I= = ~= = = q^ = =

    ^

    K===

    537. q=~=^==~== f^^q = K===

    538. f==~==^_==I===( ) qqq ^_^_ = K===

  • CHAPTER 5. MATRICES AND DETERMINANTS

    114

    539. ^==j~=f=^==~=~= ~I==~I=== ^~ I===~===~==~= ` ==^W=

    [ ]q`^~ = K===

    540. q~==~=j~=f= ^= = ~= ~= ~I= = ~I= = = ^ I= =============~=~~W=

    OONN ~~~^ +++= K K==541. f==~=j~=

    f=^==~=~= ~==~=~=~=^ I==== N^ ====

    ^

    ^~^ N = K=

    =542. f==~==^_==I===

    ( ) NNN ^_^_ = K==

    543. f==^===~=~=== ==~I======u===~==~=

    u^u = I====~= =~==~~=~=

    Mf^ = K======

    5.5 Systems of Linear Equations ==s~~W=I=I=I= N I= KIO =o~=W= KI~I~II~I~I~ NONNNPON =

  • CHAPTER 5. MATRICES AND DETERMINANTS

    115

    a~W=aI= a I= a I= a ==

    j~W=^I=_I=u===

    544.

    =+=+

    OOO

    NNN

    ~

    ~I==

    a

    a = I=

    a

    a = =E`~=FI==

    ==

    NOONOO

    NN ~~~

    ~a == I==

    NOONOO

    NN

    a == I==

    NOONOO

    NN ~~~

    ~a == K==

    =545. f= Ma I====~=~==W==

    a

    a = I=

    a

    a = K=

    f= Ma= = ~= Ma E= Ma FI= = = = ~= = ==K=f= Maaa === I= = = = ~= = = = ~==K==

    546.

    =++=++=++

    PPPP

    OOOO

    NNNN

    ~

    ~

    =~

    I==

    a

    a = I=

    a

    a = I=

    a

    a = =E`~=FI==

    =

  • CHAPTER 5. MATRICES AND DETERMINANTS

    116

    ==

    PPP

    OOO

    NNN

    ~

    ~

    ~

    a= I=

    PPP

    OOO

    NNN

    a = I=

    PPP

    OOO

    NNN

    ~

    ~

    ~

    a = I=

    PPP

    OOO

    NNN

    ~

    ~

    ~

    a = K==

    =547. f= Ma I====~=~==W==

    a

    a = I=

    a

    a = I=

    a

    a = K=

    f= Ma= =~= Ma E= Ma == Ma FI====~==K=f= Maaaa ==== I= = = = ~= =~=K==

    548. j~=c==~=p===i~=b~===================r=q===~=~==

    =+++

    =+++=+++

    OONN

    OOOOONON

    NNONONNN

    ~~~

    ~~~

    ~~~

    K

    KKKKKKKKKKKK

    K

    K

    =

    ~====~==

    =

    O

    N

    O

    N

    ON

    OOOON

    NNONN

    ~~~

    ~~~

    ~~~

    MM

    K

    MMM

    K

    K

    I==

    KK==_u^ = I==

  • CHAPTER 5. MATRICES AND DETERMINANTS

    117

    ==

    =

    ON

    OOOON

    NNONN

    ~~~

    ~~~

    ~~~

    ^

    K

    MMM

    K

    K

    I=

    =

    O

    N

    uM

    I=

    =

    O

    N

    _M

    K==

    =549. p==~=p==i~=b~= =

    _^u N = I=== N^ =====^K===

  • 118

    Chapter 6

    Vectors ====

    sW= r

    I= r

    I= r

    I= r

    I=

    ^_ I==s=W=

    rI= r

    I==

    r=W= r

    I= r

    I= r

    =

    k=W= Mr

    =`~===

    rW= NNN wIvIu =

    `~=== r

    W= OOO wIvIu =p~~W= I =a=W= I= I= =^===W= ===

    6.1 Vector Coordinates =

    550. r=s=( )MIMIN =

    rI=

    ( )MINIM =r

    I=

    ( )NIMIM =r

    I=

    N ===rrr

    K=

    =

    551. ( ) ( ) ( )^_ MNMNMNrrrr

    ++==

    =

    =

  • CHAPTER 6. VECTORS

    119

    ======= ==

    Figure 73. =

    552. ( ) ( ) ( )OMNOMNOMN ^_ ++==r

    =

    =

    553. f= ^_ r=

    I== _^r

    =

    K==

    ==

    Figure 74. =

    554. = u r I== v

    rI=

    = wr

    K=

  • CHAPTER 6. VECTORS

    120

    ===== ==

    Figure 75. =

    555. f= ( ) ( )NNNN wIvIuwIvIurr

    = I===

    Nuu = I= Nvv = I= Nww = K=====

    6.2 Vector Addition =

    556. rrr += ==

    == ==

    Figure 76.

  • CHAPTER 6. VECTORS

    121

    == ==

    Figure 77. =

    557. PON r

    Krrrr

    ++++= ==

    == ==

    Figure 78. =

    558. `~=i~=rrrr

    +=+ ==

    559. ^~=i~=( ) ( ) rrrrrr ++=++ ==

    560. ( )ONONON wwIvvIuu +++=+rr

    =======

  • CHAPTER 6. VECTORS

    122

    6.3 Vector Subtraction =

    561. rrr = == rrr =+ K==

    ==

    Figure 79. =

    == ==

    Figure 80. =

    562. ( ) rrrr += ==

    563. ( )MIMIMM ==rrr

    ==

    564. MM =r

    =

    =565. ( )ONONON wwIvvIuu =

    rrI==

    ===

    6.4 Scaling Vectors =

    566. rr = =

  • CHAPTER 6. VECTORS

    123

    ==

    Figure 81. =

    567. rr = ==

    568. ( )wIvIu =r ==

    569. = rr ==

    570. ( ) rrr +=+ ==

    571. ( ) ( ) ( ) rrr == ==

    572. ( ) rrrr +=+ ====

    6.5 Scalar Product =

    573. p~~=m==s= r =~ r ==

    rrrrI==

    = ===~=== r

    =~ r

    K=====

  • CHAPTER 6. VECTORS

    124

    = ==

    Figure 82. =

    574. p~~=m==`~=c=f= ( )NNN wIvIu =r

    I= ( )OOO wIvIu =r

    I===

    ONONON wwvvuu ++=rr

    K==

    575. ^=_=q=s==f= ( )NNN wIvIu =r

    I= ( )OOO wIvIu =r

    I===

    OO

    OO

    OO

    ON

    ON

    ON

    ONONON

    wvuwvu

    wwvvuu

    ++++

    ++= K=

    =576. `~=m=

    rrrr= =

    =577. ^~=m=

    ( ) ( ) rrrr = ==

    578. a=m=( ) rrrrrrr +=+ =

    =

    579. M = rr == r I r =~=~=EO

    = FK=

    =

    580. M > rr ==O

    M

  • CHAPTER 6. VECTORS

    125

    581. M

  • CHAPTER 6. VECTORS

    126

    ======= ==

    Figure 83. =

    588. OOO

    NNN

    wvu

    wvu

    rrr

    rrr== =

    =

    589.

    ==

    OO

    NN

    OO

    NN

    OO

    NN

    vu

    vuI

    wu

    wuI

    wv

    wvrrr

    =

    =590. == p rrrr =EcKUPF=

    =591. ^=_=q=s=EcKUPF=

    rr

    rr

    = =

    =592. k~=m=

    ( ) rrrr = ===

    593. ^~=m=( ) ( ) rrrr = ===

  • CHAPTER 6. VECTORS

    127

    594. a=m=( ) rrrrrrr +=+ =

    =595. M

    rrr= ==

    r=~=

    r=~=~~=E M= FK=

    =

    596. Mrrrrrrr

    === ==

    597. rrr

    = I= rrr

    = I= rrr

    = ====

    6.7 Triple Product =

    598. p~~=q=m=[ ] ( ) ( ) ( ) rrrrrrrrrrrr === ==

    599. [ ] [ ] [ ] [ ] [ ] [ ] rrrrrrrrrrrrrrrrrr ===== ==

    600. ( ) [ ] rrrrrr = ==

    601. p~~=q=m==`~=c=

    ( )PPP

    OOO

    NNN

    wvu

    wvu

    wvu

    =rrr

    I==

    ==( )NNN wIvIu =

    rI= ( )OOO wIvIu =r

    I= ( )PPP wIvIu =r

    K==

    =602. s==m~~=

    ( )s rrr = ==

  • CHAPTER 6. VECTORS

    128

    ============ ==

    Figure 84. =

    603. s==m~=

    ( )S

    Ns

    rrr= =

    =

    ==

    Figure 85. =

    604. f== ( ) M = rrr I===== r I= r I=~= r =~=~==I==

    rrr+= ===~~= =~= K==

    =605. f== ( ) M rrr I===== r I= r I=~= r =~=~=

    K==

  • CHAPTER 6. VECTORS

    129

    606. s=q=m=( ) ( ) ( ) rrrrrrrrr = ==

    ========

  • 130

    Chapter 7

    Analytic Geometry ====

    7.1 One-Dimensional Coordinate System =m=~W= M I= N I= O I= M I= N I= O =

    o~=W= ==a~===W====

    607. a~=_=q=m=ONNO ^_ === =

    =

    ==

    Figure 86. =

    608. a=~=i=p===o~= =

    ++

    =N

    ONM I= `_

    ^`= I= N K=

    =

    ======== ==

    Figure 87.

  • CHAPTER 7. ANALYTIC GEOMETRY

    131

    609. j==~=i=p=

    O

    ONM

    += I= N= K=

    ===

    7.2 Two-Dimensional Coordinate System =m=~W= M I= N I= O I= M I= N I= O =

    m~=~W= I =o~=W= ==m=~=W=~I=I=I==a~===W==^~W=p===

    610. a~=_=q=m=( ) ( )ONOONO ^_ +== =

    =

    ==

    Figure 88.

  • CHAPTER 7. ANALYTIC GEOMETRY

    132

    611. a=~=i=p===o~= =

    ++

    =N

    ONM I= +

    +=

    N

    ONM I==

    `_

    ^`= I= N K=

    =

    ======= ==

    Figure 89. ==

  • CHAPTER 7. ANALYTIC GEOMETRY

    133

    ======= ==

    Figure 90. =

    612. j==~=i=p=

    O

    ONM

    += I=

    O

    ONM

    += I= N= K=

    =613. `=Ef==j~F==~=q~=

    P

    PONM

    ++= I=

    P

    PONM

    ++= I==

    == ( )NN I^ I== ( )OO I_ I==~== ( )PP I` ==~====~= ^_` K= ==

  • CHAPTER 7. ANALYTIC GEOMETRY

    134

    ========= ==

    Figure 91. =

    614. f=Ef==^=_F==~=q~=

    ~

    ~ PONM ++

    ++= I=

    ~

    ~ PONM ++

    ++= I==

    = _`~ = I= `^ = I= ^_ = K===

    ======== ==

    Figure 92.

  • CHAPTER 7. ANALYTIC GEOMETRY

    135

    615. `=Ef===p=m~======================_F==~=q~=

    N

    N

    N

    O

    N

    N

    N

    PP

    OO

    NN

    POP

    OP

    OOO

    OO

    NON

    ON

    M

    +++

    = I=

    N

    N

    N

    O

    N

    N

    N

    PP

    OO

    NN

    OP

    OPP

    OO

    OOO

    ON

    ONN

    M

    +++

    = =

    =

    ======== ===

    Figure 93. =======

  • CHAPTER 7. ANALYTIC GEOMETRY

    136

    616. l=Ef==^F==~=q~=

    N

    N

    N

    N

    N

    N

    PP

    OO

    NN

    OPONP

    OONPO

    ONPON

    M

    +++

    = I=

    N

    N

    N

    N

    N

    N

    PP

    OO

    NN

    PONOP

    ONPOO

    NPOON

    M

    +++

    = =

    =

    ====== ==

    Figure 94. =

    617. ^~==~=q~=

    ( ) ( )NPNP

    NONO

    PP

    OO

    NN

    O

    N

    N

    N

    N

    O

    Np

    == =

    ===

  • CHAPTER 7. ANALYTIC GEOMETRY

    137

    618. ^~==~=n~~~=

    ( ) ( )( ) ( )( )[ ++++= POPOONON ON

    p =

    ( )( ) ( )( )]NQNQQPQP ++++ ==

    === ==

    Figure 95. =kW=f=~=SNTI=SNU=====EHF==EF==~===~==~==~~K==

    =619. a~=_=q=m==m~=`~=

    ( )NOONOOON O^_ +== ==

  • CHAPTER 7. ANALYTIC GEOMETRY

    138

    ==

    Figure 96. =

    620. `=o~~=`~==m~=`~== I= = K=

    =

    ==

    Figure 97. =

    621. `=m~=`~==o~~=`~=OO += I=

    ~ = K=

  • CHAPTER 7. ANALYTIC GEOMETRY

    139

    7.3 Straight Line in Plane =m=~W=uI=vI=I= M I= N I== M I= N I= N~ I= O~ I===

    o~=W=I=~I=I=I=I=^I=_I=`I= N^ I= O^ I==^W= I= =^===W= =k~=W=

    r=

    m=W= r

    I= ~r

    I= r

    ===

    622. d~=b~==~=p~=i=M`_^ =++ =

    =623. k~=s==~=p~=i=

    q== ( )_I^r ==~==== M`_^ =++ K==

    ==

    Figure 98. =

    624. b=b~==~=p~=i=Ep-f=cF= += K==

  • CHAPTER 7. ANALYTIC GEOMETRY

    140

    q=~===== = ~ K==

    ==

    Figure 99. =

    625. d~==~=i==

    NO

    NO

    ~

    == =

    =

    ==

    Figure 100.

  • CHAPTER 7. ANALYTIC GEOMETRY

    141

    626. b~==~=i=d=~=m=~==d~=( )MM += I==

    ====~I= ( )MM Im ==~====K==

    ==

    Figure 101. =

    627. b~==~=i=q~=m~=q=q=m=

    NO

    N

    NO

    N

    =

    ==

    =

    M

    N

    N

    N

    OO

    NN = K=

    =

  • CHAPTER 7. ANALYTIC GEOMETRY

    142

    ==

    Figure 102. =

    628. f=c=

    N

    ~

    =+ =

    =

    ==

    Figure 103. ==

  • CHAPTER 7. ANALYTIC GEOMETRY

    143

    629. k~=c=M =+ =

    =

    ==

    Figure 104. =

    630. m=a=c=

    v

    u

    NN =

    I==

    = ( )vIu = = = = = = = ~= ( )NNN Im = ===K==

  • CHAPTER 7. ANALYTIC GEOMETRY

    144

    ==

    Figure 105. =

    631. s~=i=~ = =

    =632. e~=i=

    = ==

    633. s=b~==~=p~=i=~rrr

    += I====l======~I=u==~=~~====I==~r

    ======~===^====I=

    r

    ==~====I=~~===I====~=~~I==

    =lur

    ======~==u===K===

  • CHAPTER 7. ANALYTIC GEOMETRY

    145

    ==

    Figure 106. =

    634. p~=i==m~~=c=

    +=+=

    OO

    NN

    ~

    ~I==

    ==( )I ~==~==~=====I==( )ON ~I~ =~==~==~=====I==( )ON I =~==~==~==~~===I====~=~~K==

  • CHAPTER 7. ANALYTIC GEOMETRY

    146

    =Figure 107.

    =635. a~=c=~=m=q=~=i=

    q=~==== ( )I~m ====M`_^ =++ ===

    OO _^

    `_^~

    +

    ++= K=

    =

    ==

    Figure 108.

  • CHAPTER 7. ANALYTIC GEOMETRY

    147

    636. m~~=i=q== NN += =~= OO += =~=~~===

    ON = K=q= = M`_^ NNN =++ = ~= M`_^ OOO =++ = ~=~~==

    O

    N

    O

    N

    _

    _

    ^

    ^= K=

    =

    ==

    Figure 109. =

    637. m~=i=q== NN += =~= OO += =~=~===

    NO

    N = =I=~I= N ON = K=

    q= = M`_^ NNN =++ = ~= M`_^ OOO =++ = ~=~==

    M__^^ ONON =+ K==

  • CHAPTER 7. ANALYTIC GEOMETRY

    148

    ==

    Figure 110. =

    638. ^=_=q=i=

    ON

    NO

    N

    ~

    +

    = I==

    OO

    OO

    ON

    ON

    ONON

    _^_^

    __^^

    ++

    += K=

    =

  • CHAPTER 7. ANALYTIC GEOMETRY

    149

    ==

    Figure 111. =

    639. f==q=i=f=== M`_^ NNN =++ =~= M`_^ OOO =++ =-I====~=~=

    NOON

    NOONM _^_^

    _`_`

    +

    = I=NOON

    NOONM _^_^

    `^`^

    +

    = K=

    ===

    7.4 Circle =o~W=o=`==W= ( )I~ =m=~W=I=I= N I= N I==o~=W=^I=_I=`I=aI=bI=cI==

  • CHAPTER 7. ANALYTIC GEOMETRY

    150

    640. b~==~=`=`=~==l=Ep~~=cF=

    OOO o =+ =

    ====== ==

    Figure 112. =

    641. b~==~=`=`=~=^=m= ( )I~ ( ) ( ) OOO o~ =+

    Figure 113.

  • CHAPTER 7. ANALYTIC GEOMETRY

    151

    642. q=m=c

    M

    N

    N

    N

    N

    PPOP

    OP

    OOOO

    OO

    NNON

    ON

    OO

    =

    ++++

    =

    ==

    Figure 114. =

    643. m~~=c

    ==

    o

    oI= OM K

    =644. d~=c

    Mcba^^ OO =++++ =E^=I= ^cQba OO >+ FK==q=====~=~= ( )I~ I===

    ^O

    a~ = I=

    ^O

    b = K=

    q=~====

  • CHAPTER 7. ANALYTIC GEOMETRY

    152

    ^O

    ^cQbao

    OO += K

    ===

    7.5 Ellipse =p~=~W=~=p=~W==cW= ( )MIcN I= ( )MIcO =a~===W=O= =bW===o~=W=^I=_I=`I=aI=bI=cI==mW=i=^~W=p===

    645. b~==~=b=Ep~~=cF

    N

    ~

    O

    O

    O

    O

    =+

    ==

    Figure 115.

  • CHAPTER 7. ANALYTIC GEOMETRY

    153

    646. ~O ON =+ I= == N I== O ==~==~====~==== ( )Im ========K==

    ==

    Figure 116. =

    647. OOO ~ += =

    648. b

    N~

  • CHAPTER 7. ANALYTIC GEOMETRY

    154

    651. d~=c Mcba`_^ OO =+++++ I==

    = M^`Q_O K =

    653. ` ( )~bQi = I==

    =====b========~=====K===

    654. ^~=c~===` ( )( )~~RKNi += I==( )OO ~Oi += K=

    =655. ~p = =

    ===

    7.6 Hyperbola =q~=~W=~=`~=~W==cW= ( )MIcN I= ( )MIcO =a~===W=O= =bW===^W=I==o~=W=^I=_I=`I=aI=bI=cI=I=====

  • CHAPTER 7. ANALYTIC GEOMETRY

    155

    656. b~==~=e~=Ep~~=cF=

    N

    ~

    O

    O

    O

    O

    = =

    =

    ==

    Figure 117. =

    657. ~O ON = I= = = N I= = O = =~= =~= ===~== = ( )Im ====~====K==

  • CHAPTER 7. ANALYTIC GEOMETRY

    156

    ==

    Figure 118. =

    658. b~==^=

    ~

    = =

    =659. OOO ~ += =

    =660. b

    N~

    >= =

    =661. b~==a

    ~

    ~

    O

    == =

    ===

  • CHAPTER 7. ANALYTIC GEOMETRY

    157

    662. m~~=b~===o=_~==~=e~=

    ==

    ~I= OM K

    =663. d~=c

    Mcba`_^ OO =+++++ I=== M^`Q_O > K==

    664. d~=c==^=m~~===`~=^ Mcba`^ OO =++++ I==

    = M^` < K=

    665. ^=c=

    Q

    O

    = I==

    ==

    = I==

    Q

    O

    = K=

    f= = ~= I= = ~= ~= ~= M = = ~=M = K==

    =

  • CHAPTER 7. ANALYTIC GEOMETRY

    158

    ==

    Figure 119. ===

    7.7 Parabola =c~=~~W==cW=c=sW= ( )MM Ij =o~=W=^I=_I=`I=aI=bI=cI=I=~I=I====

    666. b~==~=m~~~=Ep~~=cF OO =

    =

  • CHAPTER 7. ANALYTIC GEOMETRY

    159

    ==

    Figure 120. =b~===

    O

    = I=

    `~====

    MIO

    c I=

    `~====( )MIMj K=

    =667. d~=c

    Mcba`_^ OO =+++++ I=== M^`Q_O = K==

    668. O~ = I=~O

    N = K=

    b~===

  • CHAPTER 7. ANALYTIC GEOMETRY

    160

    O

    = I=

    `~====

    O

    IMc I=

    `~====( )MIMj K=

    =

    ==

    Figure 121. =

    669. d~=cI=^=m~~===-~==Mcba^ O =+++ =E^I=b=FI==

    ~ O ++= I=~O

    N = K==

    b~===

    O

    M = I=

    `~====

  • CHAPTER 7. ANALYTIC GEOMETRY

    161

    +

    O

    Ic MM I=

    `~====

    ~O

    M = I= ~Q

    ~Q~

    O

    MOMM

    =++= K=

    =

    ==

    Figure 122. ===

    7.8 Three-Dimensional Coordinate System =m=~W= M I= M I= M I= N I= N I= N I==

    o~=W= ==a~===W==^~W=p=sW=s==

  • CHAPTER 7. ANALYTIC GEOMETRY

    162

    670. a~=_=q=m=( ) ( ) ( )ONOONOONO ^_ ++== =

    =

    === ==

    Figure 123. =

    671. a=~=i=p===o~= =

    ++

    =N

    ONM I= +

    +=

    N

    ONM I= +

    +=

    N

    ONM I==

    =

    `_

    ^`= I= N K=

    =

  • CHAPTER 7. ANALYTIC GEOMETRY

    163

    ======== ==

    Figure 124. =

    =

    Figure 125.

  • CHAPTER 7. ANALYTIC GEOMETRY

    164

    672. j==~=i=p=

    O

    ONM

    += I=

    O

    ONM

    += I=

    O

    ONM

    += I= N= K=

    =673. ^~==~=q~=

    q=~~==~=~=== ( )NNNN IIm I=( )OOOO IIm I=~= ( )PPPP IIm =====

    O

    PP

    OO

    NN

    O

    PP

    OO

    NN

    O

    PP

    OO

    NN

    N

    N

    N

    N

    N

    N

    N

    N

    N

    O

    Np ++= K=

    =674. s==~=q~=

    q===~=~=== ( )NNNN IIm I=( )OOOO IIm I= ( )PPPP IIm I=~= ( )QQQQ IIm =====

    N

    N

    N

    N

    S

    Ns

    QQQ

    PPP

    OOO

    NNN

    = I==

    =

    QPQPQP

    QOQOQO

    QNQNQN

    S

    Ns

    = K=

    kW=t====EHF==EF==~===~==~==K===

  • CHAPTER 7. ANALYTIC GEOMETRY

    165

    ==== ==

    Figure 126. ===

    7.9 Plane =m=~W=I=I=I= M I= M I= M I= N I= N I= N I==

    o~=W=^I=_I=`I=aI= N^ I= O^ I=~I=I=I= N~ I= O~ I= I=I=I===k~=W=

    rI= Nr

    I= Or

    =a=W= I= I= =a~====~W====

    675. d~=b~==~=m~=Ma`_^ =+++ =

    ==

  • CHAPTER 7. ANALYTIC GEOMETRY

    166

    676. k~=s==~=m~=q== ( )`I_I^r ==~===~=

    Ma`_^ =+++ K==

    === ==

    Figure 127. =

    677. m~~=`~===b~==~=m~=Ma`_^ =+++ =

    =f= M^ = I==~==~~===-~K=f= M_ = I==~==~~===-~K=f= M` = I==~==~~===-~K=f= Ma= I==~====K===f= M_^ == I==~==~~===-~K=f= M`_ == I==~==~~===-~K=f= M`^ == I==~==~~===-~K==

  • CHAPTER 7. ANALYTIC GEOMETRY

    167

    678. m=a=c=( ) ( ) ( ) M`_^ MMM =++ I==

    === ( )MMM IIm == ==~I=~==-= ( )`I_I^ ==~===~K====

    ==== ==

    Figure 128. =

    679. f=c=

    N

    ~

    =++ =

    =

  • CHAPTER 7. ANALYTIC GEOMETRY

    168

    ===== ==

    Figure 129. =

    680. q=m=c=

    M

    POPOPO

    PNPNPN

    PPP

    =

    I==

    ==

    M

    N

    N

    N

    N

    PPP

    OOO

    NNN = K=

    =

  • CHAPTER 7. ANALYTIC GEOMETRY

    169

    ===== ==

    Figure 130. =

    681. k~=c=M =++ I==

    ========~==~========~=I=~= I= I= =~=====~==~===~K===

  • CHAPTER 7. ANALYTIC GEOMETRY

    170

    ====== ==

    Figure 131. =

    682. m~~=c=

    ++=++=++=

    ~~

    ONN

    ONN

    ONN

    I==

    = ( )II =~==~==~======I=== ( )NNN IIm ====~I===( )NNN II~ =~= ( )OOO II~ =~=~~===~K==

  • CHAPTER 7. ANALYTIC GEOMETRY

    171

    ===== ==

    Figure 132. =

    683. a~=^=_=q=m~=f==~=~====

    Ma`_^ NNNN =+++ I==Ma`_^ OOOO =+++ I==

    ==~=~=====

    OO

    OO

    OO

    ON

    ON

    ON

    ONONON

    ON

    ON

    `_^`_^

    ``__^^

    ++++

    ++=

    = rrrr

    K=

    =

  • CHAPTER 7. ANALYTIC GEOMETRY

    172

    ====== ==

    Figure 133. =

    684. m~~=m~=q=~= Ma`_^ NNNN =+++ =~=

    Ma`_^ OOOO =+++ =~=~~===

    O

    N

    O

    N

    O

    N

    `

    `

    _

    _

    ^

    ^== K=

    =685. m~=m~=

    q=~= Ma`_^ NNNN =+++ =~=Ma`_^ OOOO =+++ =~=~===

    M``__^^ ONONON =++ K==

    686. b~==~=m~=q= ( )NNN IIm =~=m~~=q==s= ( )NNN II~ =~= ( )OOO II~ =EcKNPOF=

  • CHAPTER 7. ANALYTIC GEOMETRY

    173

    M

    ~

    ~

    OOO

    NNN

    NNN

    =

    =

    =687. b~==~=m~=q= ( )NNNN IIm =~= ( )OOOO IIm I=

    ~=m~~=q==s= ( )II~ =

    M

    ~

    NONONO

    NNN

    =

    =

    =

    =

    Figure 134. =

    688. a~=c=~=m=q=~=m~=q=~==== ( )NNNN IIm ===~=

    Ma`_^ =+++ ===

  • CHAPTER 7. ANALYTIC GEOMETRY

    174

    OOO

    NNN

    `_^

    a`_^

    ++

    +++= K=

    =

    ====== ==

    Figure 135. =

    689. f==q=m~=f==~= Ma`_^ NNNN =+++ =~=

    Ma`_^ OOOO =+++ = I= = = ~=====

    +=+=+=

    ~

    N

    N

    N

    I==

    ==

    ~

    NNN =

    =

    I==

    ==

  • CHAPTER 7. ANALYTIC GEOMETRY

    175

    OO

    NN

    `_

    `_~ = I=

    OO

    NN

    ^`

    ^` = I=

    OO

    NN

    _^

    _^ = I==

    OOO

    OO

    NN

    OO

    NN

    N ~

    _a

    _a

    `a

    `a

    ++

    = I==

    OOO

    OO

    NN

    OO

    NN

    N ~

    `a

    `a~

    ^a

    ^a

    ++

    = I==

    OOO

    OO

    NN

    OO

    NN

    N ~

    ^a

    ^a

    _a

    _a~

    ++

    = K==

    ===

    7.10 Straight Line in Space =m=~W=I=I=I= N I= N I= N I==a=W= I= I= =o~=W=^I=_I=`I=aI=~I=I=I= N~ I= O~ I=I===a===~=W=

    rI= Nr

    I= Or

    =k~===~=~W=

    r=

    ^===W= ==

    690. m=a=c===b~==~=i==

    ~

    NNN =

    =

    I==

    === ( )NNNN IIm ====I=~= ( )II~ =======K===

  • CHAPTER 7. ANALYTIC GEOMETRY

    176

    ===== ==

    Figure 136. =

    691. q=m=c=

    NO

    N

    NO

    N

    NO

    N

    =

    =

    ==

    =

    ===== =Figure 137.

  • CHAPTER 7. ANALYTIC GEOMETRY

    177

    692. m~~=c==

    +=+=+=

    N

    N

    N

    I==

    === ( )NNNN IIm ====~=I= I= I= =~=======

    ===I==~~===~=~=K===

    ===== ==

    Figure 138. =

    693. ^=_=q=p~=i=

    OO

    OO

    OO

    ON

    ON

    ON

    ONONON

    ON

    ON

    ~~

    ~~

    ++++

    ++=

    = rrrr

    =

    =

  • CHAPTER 7. ANALYTIC GEOMETRY

    178

    ===== ==

    Figure 139. =

    694. m~~=i=q==~=~~===

    ON rr

    I====

    O

    N

    O

    N

    O

    N

    ~

    ~== K=

    =695. m~=i=

    q==~=~~===M ON =

    rrI==

    ==M~~ ONONON =++ K=

    =696. f==q=i=

    q==N

    N

    N

    N

    N

    N

    ~

    =

    =

    =~=

  • CHAPTER 7. ANALYTIC GEOMETRY

    179

    O

    O

    O

    O

    O

    O

    ~

    =

    =

    ====

    M

    ~

    ~

    OOO

    NNN

    NONONO

    =

    K=

    =697. m~~=i=~=m~==

    q=~==

    ~

    NNN =

    =

    =~==~=

    Ma`_^ =+++ =~=~~==M =

    rrI==

    ==M`_^~ =++ K=

    =

    ===== ==

    Figure 140. ==

  • CHAPTER 7. ANALYTIC GEOMETRY

    180

    698. m~=i=~=m~==

    q=~==

    ~

    NNN =

    =

    =~==~=

    Ma`_^ =+++ =~=~==rr

    I====

    `

    _

    ~

    ^== K=

    =

    ==== ==

    Figure 141. ==

    7.11 Quadric Surfaces =m=~===~=~W=I=I==o~=W=^I=_I=`I=~I=I=I= PON II I==

  • CHAPTER 7. ANALYTIC GEOMETRY

    181

    699. d~=n~~=b~=MaoOnOmOeOdOcO`_^ OOO =+++++++++

    =700. `~~==n~=p~=

    =`~= o~EF= o~EbF= = == q==p~=

    N= P= Q= M< = p~= o~=b=O= P= Q= M> = p~= f~~=b=P= P= Q= M> = a= e==N=p=Q= P= Q= M< = a= e==O=p=R= P= P= = a= o~=n~=`=S= P= P= = p~= f~~=n~=`=T= O= Q= M< = p~= b=m~~=U= O= Q= M> = a= e=m~~=V= O= P= = p~= o~=b=`=

    NM= O= P= = p~= f~~=b=`=NN= O= P= = a= e=`=NO= O= O= = a= o~=f=m~=NP= O= O= = p~= f~~=f=m~=NQ= N= P= = = m~~=`=NR= N= O= = = o~=m~~=m~=NS= N= O= = = f~~=m~~=m~=NT= N= N= = = `=m~=

    =e==

    =

    `cd

    c_e

    de^

    I=

    =

    aonm

    o`cd

    nc_e

    mne^

    b I= ( )b= I==

    PON II =~=====~I==

    M

    `cd

    c_e

    de^

    =

    K=

    ==

  • CHAPTER 7. ANALYTIC GEOMETRY

    182

    701. o~=b=E`~=NF=

    N

    ~

    O

    O

    O

    O

    O

    O

    =++ =

    =

    ===== ==

    Figure 142. =

    702. f~~=b=E`~=OF=

    N

    ~

    O

    O

    O

    O

    O

    O

    =++ =

    =703. e==N=p=E`~=PF=

    N

    ~

    O

    O

    O

    O

    O

    O

    =+ =

    =

  • CHAPTER 7. ANALYTIC GEOMETRY

    183

    ===== ==

    Figure 143. =

    704. e==O=p=E`~=QF=

    N

    ~

    O

    O

    O

    O

    O

    O

    =+ =

    =

    = ==

    Figure 144.

  • CHAPTER 7. ANALYTIC GEOMETRY

    184

    705. o~=n~=`=E`~=RF=

    M

    ~

    O

    O

    O

    O

    O

    O

    =+ =

    =

    == ==

    Figure 145. =

    706. f~~=n~=`=E`~=SF=

    M

    ~

    O

    O

    O

    O

    O

    O

    =++ =

    =707. b=m~~=E`~=TF=

    M

    ~

    O

    O

    O

    O

    =+ =

    =

  • CHAPTER 7. ANALYTIC GEOMETRY

    185

    ====== =Figure 146.

    708. e=m~~=E`~=UF=

    M

    ~

    O

    O

    O

    O

    = =

    ==== =Figure 147.

  • CHAPTER 7. ANALYTIC GEOMETRY

    186

    709. o~=b=`=E`~=VF=

    N

    ~

    O

    O

    O

    O

    =+ =

    =

    ===== ==

    Figure 148. =

    710. f~~=b=`=E`~=NMF=

    N

    ~

    O

    O

    O

    O

    =+ =

    =711. e=`=E`~=NNF=

    N

    ~

    O

    O

    O

    O

    = =

    =

  • CHAPTER 7. ANALYTIC GEOMETRY

    187

    ====== ==

    Figure 149. =

    712. o~=f=m~=E`~=NOF=

    M

    ~

    O

    O

    O

    O

    = =

    =713. f~~=f=m~=E`~=NPF=

    M

    ~

    O

    O

    O

    O

    =+ =

    =714. m~~=`=E`~=NQF=

    M~

    O

    O

    = =

    =

  • CHAPTER 7. ANALYTIC GEOMETRY

    188

    ===== ==

    Figure 150. =

    715. o~=m~~=m~=E`~=NRF=

    N~

    O

    O

    = =

    =716. f~~=m~~=m~=E`~=NSF=

    N~

    O

    O

    = =

    =717. `=m~=E`~=NTF=

    MO = ======

  • CHAPTER 7. ANALYTIC GEOMETRY

    189

    7.12 Sphere =o~==~=W=o=m=~W=I=I=I= N I= N I= N I==`==~=W= ( )II~ =o~=W=^I=aI=bI=cI=j===

    718. b~==~=p=`=~==l=Ep~~=cF=

    OOOO o =++ ==

    ====== ==

    Figure 151. =

    719. b~==~=`=`=~=^=m= ( )II~ ( ) ( ) ( ) OOOO o~ =++

    720. a~=c

    ( )( ) ( )( ) ( )( ) M ONONON =++ I==

  • CHAPTER 7. ANALYTIC GEOMETRY

    190

    ==( )NNNN IIm I= ( )OOOO IIm =~====~=~K==

    =721. c=m=c=

    M

    N

    N

    N

    N

    N

    QQQOQ

    OQ

    OQ

    PPPOP

    OP

    OP

    OOOOO

    OO

    OO

    NNNON

    ON

    ON

    OOO

    =

    ++++++++++

    =

    =722. d~=c

    Mjcba^^^ OOO =++++++ =E^==FK==q=====~=~= ( )II~ I===

    ^O

    a~ = I=

    ^O

    b = I=

    ^O

    c = K=

    q=~====

    ^O

    j^Qcbao

    OOOO ++= K

    ==

  • 191

    Chapter 8

    Differential Calculus ====cW=I=I=I=I==^=E=~~FW==o~=W=~I=I=I==k~~=W==^W= =f=W= N ===

    8.1 Functions and Their Graphs =

    723. b=c=( ) ( ) = =

    =724. l=c=

    ( ) ( ) = ==

    725. m=c=( ) ( )q =+ =

    =726. f=c=

    ( ) = ==~=I= ( ) = == ( ) N= ====K===

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    192

    ===== ==

    Figure 152. =

    727. `=c=( ) = I= ( ) = I= ( )( ) = ==~==K=

    =728. i~=c=

    ~ += I= = o I= = = ~~ =======I======-K==

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    193

    ====== ==

    Figure 153. =

    729. n~~=c==O = I= o K=

    =

    ====== ==

    Figure 154. =

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    194

    730. ~ O ++= I= o K==

    === ==

    Figure 155. =

    731. `=c==P = I= o K=

    =

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    195

    = ==

    Figure 156. =

    732. ~ OP +++= I= o K==

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    196

    == ==

    Figure 157. =

    733. m=c== = I= k K=

    =

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    197

    ====== =Figure 158.

    =

    =Figure 159.

    =

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    198

    734. p~=o=c== = I= [ ) IM K=

    =

    ======= =Figure 160.

    =735. b~=c=

    ~ = I= M~ > I= N~ I= = == ~ = I= KSTNUOUNUOUQKO = =

    =

    = ==

    Figure 161.

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    199

    736. i~=c= ~= I= ( ) IM I= M~ > I= N~ I=

    = == ~ = I= M > K==

    ==

    Figure 162. =

    737. e=p=c==

    = I=O

    = I= o K=

    =

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    200

    ==== ==

    Figure 163. =

    738. e=`=c==

    = I=O

    += I= o K=

    ====== ==

    Figure 164.

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    201

    739. e=q~=c==

    ~ = I=

    ~

    +

    === I= o K=

    =

    ====== ==

    Figure 165. =

    740. e=`~=c==

    = I=

    +

    === I= o I= M K=

    =

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    202

    ====== ==

    Figure 166. =

    741. e=p~=c==

    = I=

    O

    N +

    === I= o K=

    =Figure 167.

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    203

    742. e=`~=c==

    = I=

    O

    N

    === I= o I= M K=

    =

    ====== ==

    Figure 168. =

    743. f=e=p=c==~ = I= o K=

    =

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    204

    ===== ==

    Figure 169. =

    744. f=e=`=c==~ = I= [ ) IN K=

    =

    ===== ==

    Figure 170. =

    745. f=e=q~=c==~~ = I= ( )NIN K=

    =

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    205

    ===== ==

    Figure 171. =

    746. f=e=`~=c==~ = I= ( ) ( ) INNI K==

    =

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    206

    ===== ==

    Figure 172. =

    747. f=e=p~=c==~ = I= ( ]NIM K=

    =

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    207

    =Figure 173.

    =748. f=e=`~=c==

    ~ = I= o I= M K==

    = Figure 174.

    =

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    208

    8.2 Limits of Functions =cW= ( ) I= ( ) =^W==o~=~W=~I====

    749. ( ) ( )[ ] ( ) ( )~~~

    +=+ =

    =750. ( ) ( )[ ] ( ) ( )

    ~~~ = =

    =751. ( ) ( )[ ] ( ) ( )

    ~~~ = =

    =

    752. ( )( )( )( )

    ~

    ~

    ~

    = I== ( ) M

    ~

    K=

    =753. ( )[ ] ( )

    ~~ = =

    =754. ( )( ) ( )( )

    ~~ = =

    =755. ( ) ( )~

    ~=

    I==== ( ) ===~= ~ = K=

    =

    756. N

    M=

    =

    =

    757. N

    ~

    M=

    =

    =

    758. N

    N

    M=

    =

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    209

    759. N

    ~

    N

    M=

    =

    =

    760. ( ) N

    N

    M=

    +

    =

    =

    761.

    NN

    =

    +

    =

    =

    762.

    N =

    +

    =

    =763. N~

    M=

    =

    ===

    8.3 Definition and Properties of the Derivative =cW=I=I=I=I==f=~~W==o~=~W==^W= ===

    764. ( ) ( ) ( )

    MM=

    =

    +=

    ==

    =

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    210

    == ==

    Figure 175. =

    765. = ~

    ==

    =

    766. ( )

    +=

    +=

    =

    767. ( )

    =

    =

    =

    768. ( )

    = =

    =769. m=o=

    ( )

    +=

    ==

    ==

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    211

    770. n=o=

    O

    =

    =

    =771. `~=o=

    ( )( ) = I= ( ) = I==

    = K=

    =772. a~==f=c=

    N

    = I==

    = ( ) ===== ( ) K===

    773. o~=o=

    O

    N

    =

    =

    =774. i~=a~=

    ( ) = I= ( ) = I==

    ( ) ( )[ ]

    = K=

    ==

    8.4 Table of Derivatives =f=~~W==o~=~W=`I=~I=I==k~~=W==

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    212

    775. ( ) M`

    = =

    =

    776. ( ) N

    = =

    =

    777. ( ) ~~

    =+ =

    =

    778. ( ) ~~

    O +=++ =

    =

    779. ( ) N

    = =

    =

    780. ( )N

    +

    = =

    =

    781. O

    N

    N

    =

    =

    =

    782. ( )O

    N

    = =

    =

    783. ( ) N

    N

    = =

    =

    784. ( )

    N

    = =

    =

    785. ( )~

    N

    ~ = I= M~ > I= N~ K=

    =

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    213

    786. ( ) ~~~

    = I= M~ > I= N~ K=

    =

    787. ( )

    = =

    =

    788. ( )

    = =

    =

    789. ( )

    = =

    =

    790. ( )

    N~

    OO

    == =

    =

    791. ( )

    N

    OO

    == =

    =

    792. ( ) ~

    = =

    =

    793. ( )

    = =

    =

    794. ( )ON

    N~

    = =

    =

    795. ( )ON

    N~

    = =

    =

    796. ( )ON

    N~~

    += =

    =

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    214

    797. ( )ON

    N~

    += =

    =

    798. ( )N

    N~

    O

    = =

    =

    799. ( )N

    N~

    O

    = =

    =

    800. ( )

    = =

    =

    801. ( )

    = =

    =

    802. ( )

    N~

    OO

    == =

    =

    803. ( )

    N

    OO

    == =

    =

    804. ( ) ~

    = =

    =

    805. ( )

    = =

    =

    806. ( )N

    N=~

    O +

    = =

    =

    807. ( )N

    N=~

    O

    = =

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    215

    808. ( )ON

    N=~~

    = I= N < K=

    =

    809. ( )N

    N=~

    O

    = I= N > K=

    =

    810. ( )

    N += =

    ===

    8.5 Higher Order Derivatives =cW=I=I=I==f=~~W==k~~=W====

    811. p=~=

    ( )O

    O

    =

    =

    == =

    =812. e-l=~=

    ( ) ( ) ( )( ) === N

    =

    =

    813. ( )( ) ( ) ( ) +=+ ==

    814. ( )( ) ( ) ( ) = ==

    815. i=c~=( ) O ++= =

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    216

    ( ) PP +++= =

    ( )( ) ( ) ( ) ( ) ( ) ( )ON ON

    N ++

    ++= K =

    =

    816. ( )( ) ( )

    >

    >

    = =

    =

    817. ( )( ) > = ==

    818. ( )( ) ( ) ( )~

    >NN

    N

    ~

    =

    =

    =

    819. ( )( ) ( ) ( )

    N

    >NN

    =

    =

    =

    820. ( )( ) ~~~ = ==

    821. ( )( ) = ==

    822. ( )( ) ~~~ = ==

    823. ( )( )

    +=

    O

    =

    =

    824. ( )( )

    +=

    O

    =

    ===

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    217

    8.6 Applications of Derivative =cW=I=I==m==~=W===sW==^~W==f=~~W==qW==k~~=W====

    825. s=~=^~=( ) = =====~==~==~==

    ~==~=~==I==( ) == ===~~====I=

    ( ) === ===~~=~~===K===

    826. q~=i=( )( )MMM = =

    =

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    218

    ==

    Figure 176. =

    827. k~=i=

    ( )( )MMM

    N

    = =Ec=NTSF=

    =828. f~=~=a~=cK==

    f= ( ) M M > I==EF==~=~= M K=Ec=NTTI= N < I=O < FI=

    f= ( ) M M

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    219

    ==

    Figure 177. =

    829. i~=~=^==EF=~=~=~=~=~= N ==~======~=~= N ==~=( ) ( ) N ==~====~=EcKNTTFK==

    =^==EF=~=~=~==~= O ==~======~=~= O ==~=( ) ( ) O ==~====~=EcKNTTFK=

    =830. `~=m=

    ^=~===EF==~= M ==~====

    ( )M =====~==K==

    831. c=a~=q==i~=b~K=f=EF===~==E ( ) M > F===~========~=( ]NI~ ==~==EF====~==E ( ) M

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    220

    832. f=EF==~=E ( ) M F==~====~= [ )I O I==EF=~=~=~==~= O K==EcKNTTFK==

    833. p=a~=q==i~=b~K=f= ( ) M N = =~= ( ) M N I==EF=~=~=~==~= O K=EcKNTTF==

    834. `~K==EF= = = ~= ~= ~= = M = = = = ~= = = = = = ( ) = ============~=~= M =EcKNTTI= P < FK===EF= = =~= =~=~= = M = = =~== = = ( ) = = ===============~=~= M K=EcKNTTI= P < FK====

    835. p=a~=q==`~K==f= ( ) M M > I==EF==~=~=~= M K==f= ( ) M M

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    221

    8.7 Differential =cW=I=I==f=~~W==a~==~=W= ( ) I= ( ) =o~=~W=`=a~=== ( ) = W==a~==W==p~=~==W= =p~=~==W= ===

    838. = ==

    839. ( ) ( ) ( ) +=+ ==

    ==

    Figure 178.

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    222

    840. p~=`~===( ) ( ) += =

    =841. ( ) +=+ =

    =842. ( ) = =

    =843. ( ) `` = =

    =844. ( ) += =

    =

    845. O

    =

    =

    ===

    8.8 Multivariable Functions =c===~~W= ( )I I= ( )I I= ( )I I= ( )I ==^W=I=I==p~=~==I=I=I=W= I= I= K===

    846. c=l=m~~=a~=q=~~=~=====

    =

    =E~=

    =

    FI=

    q=~~=~=====

    =

    =E~=

    =

    FK=

    ==

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    223

    847. p=l=m~~=a~=

    O

    O

    =

    =

    I==

    O

    O

    =

    =

    I==

    O

    =

    =

    I==

    O

    =

    =

    K==

    f==~=~=I===

    OO

    =

    K==

    =848. `~=o==

    f= ( ) ( )( )II = =E==~====~~=FI===

    ( )( )

    I

    =

    I= ( )( )

    I

    =

    K==

    =

    f= ( ) ( ) ( )( )I = I== ( )

    +

    = K==

    =f= ( ) ( )( )III = I===

    +

    =

    I=

    +

    =

    K==

    =849. p~=`~=

    +

    =

    ==

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    224

    850. i~=j~~=~=j~=( )I =~=~=~=~=~= ( )MM I == ( ) ( )MM II =

    =~= ( )I ==== ( )MM I K===( )I =~=~=~==~= ( )MM I == ( ) ( )MM II =

    =~= ( )I ==== ( )MM I K==

    851. p~~=m=

    M

    =

    =

    K=

    i~=~~=~=~=~==~=~~=K===

    852. p~=m=^=~~= == == = = == =~= = ~= =~==~=~===

    853. p=a~=q==p~~=m=

    i= ( )MM I ==~=~~==E M

    =

    =

    FK==

    ( ) ( )( ) ( )MMMM

    MMMM

    II

    IIa = K==

    =f= Ma> I= ( ) MI MM > I== ( )MM I ===~===~=~K=f= Ma> I= ( ) MI MM < I== ( )MM I ===~===~=~~K=f= Ma< I= ( )MM I ==~=~=K=f= Ma= I===~K==

    854. q~=m~=q=~===~=~===~= ( )I = =~= ( )MMM II ===

    ( )( ) ( )( )MMMMMMM II += K==

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    225

    855. k~==p~=q=~===~===~= ( )I = =~=( )MMM II ===

    ( ) ( ) N

    I

    I

    M

    MM

    M

    MM

    M

    =

    =

    K=

    ===

    8.9 Differential Operators =

    r==~==~=~W= r

    I= r

    I= r

    =p~~==E~~=FW= ( )II I= ( )ON III K =d~==~=~~=W= ~ I= =

    a~=~W=

    =

    s==E=FW= ( )oInImcr

    =

    a==~==W= cr

    I= cr =

    `==~==W= cr

    I= cr

    =i~~~=~W= O ===

    856. d~==~=p~~=c=

    ==

    I

    I

    ~ I==

    ==ON

    II

    I

    ~ K K=

    =857. a~=a~=

    +

    +

    =

    I==

  • CHAPTER 8. DIFFERENTIAL CALCULUS

    226

    ========

    ( ) IIr

    I= N OOO =++ K===

    858. a==~=s=c=

    o

    n

    mcc

    +

    +

    ==rr

    =

    =859. `==~=s=c=

    onm

    cc

    ==

    rrr

    rr=

    m

    n

    o

    m

    n

    o rrr

    +

    +

    = =

    =860. i~~~=l~=

    O

    O

    O

    O

    O

    OO

    +

    +

    = =

    =

    861. ( ) ( ) Mcc = rr ==

    862. ( ) ( ) M~ = ==

    863. ( ) ( ) ~ O== ==

    864. ( ) ( ) ( ) cccc~c OO rrrrr == ===

  • 227

    Chapter 9

    Integral Calculus ====cW=I=I=I==f=~~W=I=I= =

    f=~==~=W= ( ) I= ( ) I==a~==~=W= ( ) I= ( ) I= ( )c I==o~=~W=`I=~I=I=I=I==k~~=W=I=I=I====

    9.1 Indefinite Integral =

    865. ( ) ( ) `c += == ( ) ( )c = K==

    866. ( )( ) ( ) = ==

    867. ( ) ( ) = ==

    868. ( ) ( )[ ] ( ) ( ) +=+ ==

    869. ( ) ( )[ ] ( ) ( ) = ==

    870. ( ) ( ) `~c~

    N~ += =

  • CHAPTER 9. INTEGRAL CALCULUS

    228

    871. ( ) ( ) `~c~

    N~ ++=+ =

    =

    872. ( ) ( ) ( ) `O

    N O += =

    =

    873. ( )( ) ( ) `

    +=

    ==

    874. j==p=( ) ( )( ) ( ) = == ( ) = K=

    =875. f~==m~=

    = I=== ( ) I= ( ) =~=~=K=====

    9.2 Integrals of Rational Functions =

    876. `~~ += ==

    877. `O

    O

    += ==

    878. `P

    PO += =

    =

    879. `N

    N +

    +=

    +

    I= N K==

  • CHAPTER 9. INTEGRAL CALCULUS

    229

    880. ( ) ( )( ) `N~~

    ~N

    ++

    +=+

    +

    I= N K==

    881. `

    += =

    =

    882. `~~

    N

    ~

    ++=

    + ==

    883. `

    ~

    ~

    ~O

    ++

    +=++

    ==

    884. ( )( ) `~

    ~

    N

    ~

    +

    ++

    =

    ++ I= ~ K==

    885. ( ) `~~~

    N

    ~

    O

    +++=+ =

    =

    886. ( ) ( ) `~~~~O~O

    N

    N

    ~

    OOP

    O

    +

    ++++=

    + ==

    887. ( ) `~

    ~

    N

    ~

    +

    +=

    + ==

    888. ( ) `~

    ~

    ~

    N

    ~

    OO

    ++

    +=+ =

    =

    889. ( )

    `~

    ~~

    N

    ~

    OO

    +

    +++=

    + ==

  • CHAPTER 9. INTEGRAL CALCULUS

    230

    890. ( )

    `~

    ~~~O~

    N

    ~

    O

    PO

    O

    +

    +

    ++=+ =

    =

    891. ( ) ( )

    `

    ~

    ~

    N

    ~~

    N

    ~

    OO

    ++

    ++

    =+ =

    =

    892. `N

    N

    O

    N

    N

    O

    ++

    = =

    =

    893. `N

    N

    O

    N

    N

    O

    ++

    = =

    =

    894. `~

    ~

    ~O

    N

    ~

    OO

    ++

    = =

    =

    895. `~

    ~

    ~O

    N

    ~

    OO

    ++

    = =

    =

    896. `~N

    NO

    +=+

    ==

    897. `~

    ~

    ~

    N

    ~

    NOO

    +=+

    ==

    898. ( ) `~O

    N

    ~

    OOOO

    ++=+ =

    =

    899. `~

    ~~

    ~

    N

    ~

    O

    +

    =

    + I= M~ > K=

    =

  • CHAPTER 9. INTEGRAL CALCULUS

    231

    900. `

    ~

    O

    N

    ~

    OO

    ++=+ =

    =

    901. ( ) `~

    ~O

    N

    ~

    O

    O

    O+

    +=

    + ==

    902. `~

    ~

    ~O

    N

    ~

    OOO

    ++

    = =

    =

    903. `~Q~O

    ~Q~O

    ~Q

    N

    ~

    O

    O

    OO+

    ++

    +

    =

    ++ I=

    M~QO > K==

    904. `~Q

    ~O~~

    ~Q

    O

    ~

    OOO+

    +

    =

    ++ I=

    M~QO

  • CHAPTER 9. INTEGRAL CALCULUS

    232

    908. ( )( ) `~~NR

    O~PO~ O

    P

    O++

    =+ =

    =

    909. ( )

    `~~

    ~~

    ~

    N

    ~

    +

    +++

    =

    ++ I==

    M~ > K==

    910. ( )

    `~

    ~~~

    ~

    N

    ~

    +

    +

    =

    ++ I==M~ K=

    =

    912. ( )( ) ++=++

    ~N

    ~=

    ( )( ) `~

    ~~~

    ~

    ~+

    ++

    I=E M~ < I= M > FK==

    =

    913. ( ) ( ) `~NMR

    NR~NO~UO~ P

    P

    OOOO ++

    +=+ =

    =

    914. ( ) `~NR

    P~Q~UO

    ~

    P

    OOOO

    +++

    =+ =

    =

    915. `~~

    ~~

    ~

    N

    ~

    +

    +++

    =+ I= M~ > K=

    =

  • CHAPTER 9. INTEGRAL CALCULUS

    233

    916. `~

    ~~~

    ~

    O

    ~

    +