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1 300 Math Formulas = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = fp_k= =VVQVNMTTQN= = ` = =OMMQ=^Kp K=^ =o =o K= = q = ~ = = ~ = = ~ K= i P reface = = = = q = ~ = = ~= = = = = = ~ = K= f = ~ = = = = = ~ = = ~ = =~ ~ = ~ ~ = = I= I= ~ = I=~ = ~ ~ K=q = = ~ = = = ~ I= ~ I= ~ = = = k = p I= ^ ~I= d I= q I= j~ = ~ = a ~ I= s I= ^ ~ = d I= `~ I= a ~ =b ~ I=p I=~ =m ~ =q K== q = = ~ = = I= I= ~ = ~ = ~ = = = ~ = ~ = = ~ = ~ I= = = ~ = = =~ =~ = ~ = = = K=== = = ii C ontents = = = = 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= iii 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= iv 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= v 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= vi 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= = = = = = vii = q = ~ = = ~ = = ~ K= = viii Chapter 1 N umber Sets = = = = 1.1 Set Identities = p W=^I=_I=`= r ~ = W=f= ` =W= ^ = m = W= ^ _ == b = W= = r = = W= ^ _ = f = = W= ^ _ = a = = W= ^ y _ = = = 1. = 2. = 3. 4. 5. ^ f= ^ ^= ^ = _ = = ^ _ =~ = _ ^ .= = b =p = ^= = r = =p == ` = ^ _ = { ^ _}= = 1 CHAPTER 1. NUMBER SETS = ===== = Figure 1. 6. = 7. = 8. = ` ~ = ^ _ = _ ^= ^ ~ = ^ (_ ` ) = (^ _ ) ` = f = =p = ` = ^ _ = { ^ ~ _} = = = ===== = Figure 2. 9. = 10. = = ` ~ = ^ _ = _ ^= ^ ~ = ^ (_ ` ) = (^ _ ) ` = = 2 CHAPTER 1. NUMBER SETS 11. = 12. = 13. = 14. a = ^ (_ ` ) = (^ _ ) (^ ` ) I= ^ (_ ` ) = (^ _ ) (^ ` ) K= f = ^ ^ = ^ I== ^ ^ = ^= a ~ = ^ = I= ^ f= f= f = ^ = ^ I== ^ f= ^ = 15. 16. 17. 18. ` = ^ = { f ^} = ` = =f =~ =r ^ ^ = f I== ^ ^ = = = a =j ~ =i~ (^ _ ) = ^ _ I== (^ _ ) = ^ _ = = a = =p ` = _ y ^ = { _ ~ ^} = = 3 CHAPTER 1. NUMBER SETS = ===== = Figure 3. = 19. _ y ^ = _ y (^ _ ) = 20. _ y ^ = _ ^ 21. ^y^= 22. ^ y _ = ^ = = ^ _ = . = = = ===== = Figure 4. = 23. (^ y _) ` = (^ `) y (_ `) 24. ^ = f y ^ 25. `~ ~ =m ` = ^ _ = {( I ) ^ ~ _} = = 4 = CHAPTER 1. NUMBER SETS 1.2 Sets of Numbers = 26. 27. = 28. = 29. = 30. k~ ~ = W=k= t = W= kM = f W=w= m = W= w + = k ~ = W= w = o~ ~ = W=n= o ~ = W=o== ` = W=`== = = k~ ~ =k ` = W k = {NI OI PI K} K= t =k ` = =~ = W= k M = {MI NI OI PI K} K= f t = =~ = = =~ = W= w + = k = {NI OI PI K}I= w = {KI PI OI N} I= w = w {M} w + = {KI PI OI NI MI NI OI PI K} K= o~ ~ =k o ~ = = ~ = ~ W== ~ n = = ~ ~ w ~ w ~ M K= f ~ ~ =k k ~ =~ = ~ = ~ K = 5 CHAPTER 1. NUMBER SETS 31. o ~ =k == r = = ~ ~ =~ = ~ ~ = W=oK= = 32. ` =k ` = { + o ~ o}I== = = = = ~ ~ = K = 33. k w n o `= = === = = Figure 5. = = = = = = 6 CHAPTER 1. NUMBER SETS 1.3 Basic Identities = o ~ = W=~I= I= = = = 34. ^ =f = ~+M=~ = = 35. ^ =f = ~ + ( ~ ) = M = = 36. ` ~ = =^ = ~ + = +~ = 37. ^ ~ = =^ = (~ + ) + = ~ + ( + ) = = = 38. a = =p ~ = ~ = ~ + ( ) = = 39. = 40. 41. 42. j ~ =f = ~ N = ~ = j ~ =f = N ~ = N I= ~ M ~ = j ~ =q =M ~ M = M = ` ~ = =j ~ = ~ = ~ = = 7 CHAPTER 1. NUMBER SETS 43. ^ ~ = =j ~ = (~ ) = ~ ( ) = a =i~ = ~ ( + ) = ~ + ~ = 44. = 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 = = = 46. = 47. = 48. N = = O = N = P = = Q = N= R = = S = N = T = = U = N = Q +N = = Q + O = N = Q + P = = Q = N = = ~ + = ` =m ~ = = 8 CHAPTER 1. NUMBER SETS = ===== = Figure 6. = 49. = 50. = 51. = (~ + ) ( + ) = (~ ) + ( ) = (~ + )( + ) = (~ ) + (~ + ) = ~ + ~ + ~ = + = + O + O O + O 52. = 53. (~ + ) + ( + ) = (~ + ) + ( + ) = ` ~ =` =k = ||||||| ~ + = ~ = = 54. ~ = I= = == = 9 CHAPTER 1. NUMBER SETS = = Figure 7. 55. = 56. = m ~ =m ~ = =` =k = ~ + = ( + ) = j =~ =^ = =~=` =k = f = ~ + = =~= = I= = = ~ O + O =E FI== = ~ ~ =E~ FK= ~ = 57. = 58. m = =m ~ =o ~ = N O = N ( N + N ) O ( O + O ) = = N O [ ( N + O ) + ( N + O )] = ` ~ =k = =m ~ =o ~ = ||||||||||||||||||||| ( + ) = [ ( ) + ( )] = = 59. f = =~=` =k = =m ~ =o ~ = N N = [ ( ) + ( )] = ( + ) 10 CHAPTER 1. NUMBER SETS 60. = 61. = 62. = 63. = 64. n = =m ~ =o ~ = N N ( N + N ) N = [ ( N O ) + ( N O )] = = O O ( O + O ) O m = =~=` =k = = [ ( + )] = [ ( ) + ( )] = c ~= a =j = ( + ) = ( ) + ( ) = k =o = =~=` =k = + O + O = ( + ) = + I== == = MI NI OI KI N K== b =c ~= = + = = = 11 Chapter 2 A lgebra = = = = 2.1 Factoring Formulas = o ~ = W=~I= I= == k~ ~ = W= = = = 65. = 66. = 67. = 68. = 69. = 70. = 71. = 72. ~ O O = (~ + )(~ ) = ~ P P = (~ )(~ O + ~ + O ) = ~ P + P = (~ + )(~ O ~ + O ) = ~ Q Q = (~ O O )(~ O + O ) = (~ )(~ + )(~ O + O ) = ~ R R = (~ )(~ Q + ~ P + ~ O O + ~ P + Q ) = ~ R + R = (~ + )(~ Q ~ P + ~ O O ~ P + Q ) = f = = = I= = ~ + = (~ + )(~ N ~ O + ~ P O K ~ O + N ) K== f = = = I= == ~ = (~ )(~ N + ~ O + ~ P O + K + ~ O + N ) I== 12 CHAPTER 2. ALGEBRA ~ + = (~ + )(~ N ~ O + ~ P O K + ~ O N ) K= = = = 2.2 Product Formulas 73. = 74. = 75. = 76. = 77. = 78. = 79. = 80. = 81. o ~ = W=~I= I= == t = W= I= = = = (~ )O = ~ O O~ + O = (~ + )O = ~ O + O~ + O = (~ )P = ~ P P~ O + P~ O P = (~ + )P = ~ P + P~ O + P~ O + P = (~ )Q = ~ Q Q~ P + S~ O O Q~ P + Q = (~ + )Q = ~ Q + Q~ P + S~ O O + Q~ P + Q = _ ~ =c ~= (~ + ) = ` M~ + `N~ N + ` O~ O O + K + ` N~ N + ` I > = ` = =~ = = ~ = K= > ( )> (~ + + )O = ~ O + O + O + O~ + O~ + O = (~ + + + K + + )O = ~ O + O + O + K + O + O + = + O(~ + ~ + K + ~ + ~ + + K + + + K + ) = 13 CHAPTER 2. ALGEBRA 2.3 Powers = _~ =E = ~ = FW=~I= == m =E ~ ~ = FW= I= = = = ~ ~ = ~ + = 82. = 83. ~ = ~ = ~ = 84. = (~ ) = ~ = 85. ~ ~ = = = 86. = 87. = 88. = (~ ) = ~ = ~ M = N I= ~ M = ~N = N = ~ = 89. N = ~ = ~ = ~ = 90. = = = = = 14 CHAPTER 2. ALGEBRA 2.4 Roots = 91. = _~ W=~I= == m =E ~ ~ = FW= I= = ~ I M = = = =E = O I= k F= = = ~ = ~ = 92. = ~ = ~ = 93. ~ ~ = I= M = = 94. = 95. = 96. = ~ ~ ~ I= M K= = = (~ ) ( ~) = ~ = =~= 97. = ~ = 98. = ~ =~ = 99. = 100. = ~ = ~ = ( ~) ~ = = ~ = 15 CHAPTER 2. ALGEBRA N ~ N = I= ~ M K= ~ ~ 101. = ~ = 102. ~ + ~O ~ ~O = O O = N ~m = = ~ ~ 103. = = = 2.5 Logarithms = 104. 105. 106. 107. 108. 109. m = ~ = W= I= I=~I= I= = k~ ~ = W= == = = a = =i ~ = = ~ = =~ = = = = ~ I= ~ > M I= ~ N K= = ~ N = M = = ~ ~ = N = = ~ > N ~ M = = + ~ < N = ~ ( ) = ~ + ~ = = ~ = ~ ~ = 16 CHAPTER 2. ALGEBRA 110. ~ ( ) = ~ = = N 111. ~ = ~ = = 112. ~ = = ~ I= > M I= N K= ~ = N 113. ~ = = ~ = 114. = ~ ~ = = 115. i ~ = =_~ =NM= NM = = = 116. k~ ~ =i ~ = = I== N = = N + = OKTNUOUNUOUK = = N 117. = = MKQPQOVQ = NM = N 118. = = OKPMORUR = = = = = = 17 CHAPTER 2. ALGEBRA 2.6 Equations = o ~ = W=~I= I= I= I= I= I= = p W= N I= O I= N I= O I= P = = = 119. i ~ =b ~ = =l =s~ ~ = ~ + = M I= = K== ~ = 120. n ~ ~ =b ~ = O Q~ ~ + + = M I= NI O = K= O~ = 121. a ~ = a = O Q~ = = 122. s =c ~ = f = O + + = M I= == N + O = K= N O = = 123. ~ O + = M I= N = M I= O = K= ~ = 124. ~ O + = M I= NI O = K= ~ = 125. ` =b ~ K=`~ ~ =c ~K== P + + = M I== O 18 CHAPTER 2. ALGEBRA N = + I= OI P = N ( + ) P ( + ) I== O O == O =P O O O + + I= = P + K== O O O P O P = = 2.7 Inequalities s~ ~ W= I= I= = ~ I I I o ~ = W= I= I= = ~N I ~ O I ~ P I KI ~ a ~ W=aI= a I= a I= a == = = 126. f ~ I=f ~ =k ~ =~ =d ~ == = f ~ = f ~ =k ~ = d ~ = [~I ]= ~ = ~ < = (~I ] = = ~ < = [~I ) = = ~ < < = (~I ) = = < I= = < < I= < = ~ < I= ~= ~ < < I= >~= ( I ] = = = ( I ) = = [~I ) = = (~I ) = = 19 CHAPTER 2. ALGEBRA 127. = 128. = 129. = 130. = 131. = 132. = 133. = f = ~ > I= = < ~ K= f = ~ > I= = ~ > M = = ~ < M K= f = ~ > I= = ~ + > + K= f = ~ > I= = ~ > K= f = ~ > =~ = > I= = ~ + > + K= f = ~ > =~ = > I= = ~ > K= f = ~ > =~ = > M I= = ~ > K= 134. f = ~ > =~ = > M I= = ~ > K= = 135. f = ~ > =~ = < M I= = ~ < K= = ~ 136. f = ~ > =~ = < M I= = < K= = 137. f = M < ~ < =~ = > M I= = ~ < K= = 138. f = M < ~ < =~ = < M I= = ~ > K= = 139. f = M < ~ < I= = ~ < K= = ~+ I== 140. ~ O = ~ > M =I= > M X=~ = ~ = = ~ = = = ~ = K== = N 141. ~ + O I= = ~ > M X=~ = ~ = ~ = ~ = =~ = ~ = N K= ~ 20 CHAPTER 2. ALGEBRA 142. ~N~ O K~ ~N + ~ O + K + ~ I= = ~N I ~ O I KI ~ > M K= = 143. f = ~ + > M =~ = ~ > M I= = > K= ~ = 144. f = ~ + > M =~ = ~ < M I= = < K== ~ = 145. ~ O + + > M = = = ~ > M= = = = = a>M= = = = a=M= = = = a<M= = < N I= > O = = N < I= > N = = = < < = = 21 ~ <M= = = N < < O = = = = = = = = = CHAPTER 2. ALGEBRA ~+ ~ + = 146. = 147. = 148. = 149. = 150. = f = < ~ I= = ~ < < ~ I= = ~ > M K= f = > ~ I= = < ~ =~ = > ~ I= = ~ > M K= f = O < ~ I= = < ~ I= = ~ > M K= f = O > ~ I= = > ~ I= = ~ > M K= 151. f = = ( ) ( ) > M ( ) > M I= = K= ( ) ( ) M ( ) ( ) < M ( ) < M I= = K= ( ) ( ) M 152. = = = 2.8 Compound Interest Formulas = c = ~ W=^= f ~ = W=`= ^ ~ = ~ = = W= = k = = ~ = W= = k = = = = = ~ W= = = = 153. d ~ =` =f =c ~= ^ = ` N + = = 22 CHAPTER 2. ALGEBRA 154. p =` =f =c ~= f = = = = = = ~ I= = = = ~= = W= ^ = `(N + ) K= = 155. ` =` =f = f = = = = ~ =E FI= == ^ = ` K= = = 23 Chapter 3 G eometry = = = = 3.1 Right Triangle = i = =~= = ~ W=~I= = e W= = ^ W= = j ~ W= ~ I= I= = ^ W= I = o~ = = = W=o= o~ = = = W= = ^ ~W=p= = = = = Figure 8. = 156. + = VM = = 24 CHAPTER 3. GEOMETRY 157. = ~ = = = 158. = = = = 159. ~ = ~ = = = 160. = = ~ = ~ = 161. = = = = 162. = = = ~ = 163. m ~ ~ =q = ~ O + O = O = = 164. ~ = I= = I== = = ~ = = ~ = = = = = ~= ~ = I= I= = = = K= = O = O = ===== Figure 9. = 25 CHAPTER 3. GEOMETRY 165. O = I=== = = = =~ = = = =~ K== = O O ~ 166. O = O I= O = ~ O I=== ~ Q Q = ~ =~ = =~ = = ~ = = = =~=~ = K== = = = Figure 10. = 167. = I== O = = = = ~ = = = = K= = 168. o = = = O = ~ + ~ = = 169. = O ~ + + = 170. ~ = = = = 26 CHAPTER 3. GEOMETRY 171. p = ~ = = O O = = = 3.2 Isosceles Triangle = _~ W=~= i W= = _~ =~ W= = s =~ W= = ^ = = = ~ W= = m W=i= ^ ~W=p= = = = = Figure 11. = 172. = VM = O = 173. O = O O ~ = Q 27 CHAPTER 3. GEOMETRY 174. i = ~ + O = = 175. p = O ~ = = O O = = = 3.3 Equilateral Triangle = p = =~= ~ ~ = ~ W=~= ^ W= = o~ = = = W=o= o~ = = = W= = m W=i= ^ ~W=p= = = = = Figure 12. = 176. = ~P = O = 28 CHAPTER 3. GEOMETRY O ~P = 177. o = = P P = N ~P o == 178. = = P S O = 179. i = P~ = = 180. p = O ~ ~ P = = O Q = = = 3.4 Scalene Triangle E^= ~ = = = = = ~ F= = = p = =~= ~ W=~I= I= = ~ + + == p W= = O ^ = =~= ~ W= I I = ^ = = = =~I= I= W= ~ I I = j ~ = = = =~I= I= W= ~ I I = _ = = =~ = I I W= ~ I I = o~ = = = W=o= o~ = = = W= = ^ ~W=p= = = 29 CHAPTER 3. GEOMETRY = ===== = Figure 13. = 181. + + = NUM = 182. ~ + > I== + > ~ I== ~ + > K= = 183. ~ < I== < ~ I== ~ < K= = = 184. j = ~ = I= ~ K= O = = = ===== Figure 14. = 30 CHAPTER 3. GEOMETRY 185. i~ = =` = ~ O = O + O O I= O = ~ O + O O~ I= O = ~ O + O O~ K= = 186. i~ = =p = ~ = = = Oo I== =o= = = ~ = = = = K== = ~ ~ ~ ~ = = = = = = 187. o = = O O O O ~ O O Qp = ( ~ )( )( ) I== 188. O = NN N N =+ + K= ~ = ( )( ) I= 189. = O ( ~ ) I= = O ~ = O ( )( ) K= ( ~ ) = O 190. ~ = ( ~ )( )( ) I= ~ O ( ~ )( )( ) I= = O = ( ~ )( )( ) K= 31 CHAPTER 3. GEOMETRY 191. ~ = = I= = ~ = I= = ~ = K= = + ~ I== O Q O O ~ + O O = I== O Q O O ~ + O O = K= O Q 192. O = ~ O O O = = = ===== Figure 15. = O O O 193. ^j = ~ I= _j = I= `j = =Ec KNRFK= P P P = Q ( ~ ) 194. O = I== ~ ( + )O Q~ ( ) O = I== (~ + )O Q~ ( ) O = K= (~ + )O = 32 CHAPTER 3. GEOMETRY ~ ~ = = I== O O O ~ ~ I== p= = = O O O p = ( ~ )( )( ) =Ee =c ~FI= p = I== ~ p= I= Qo p = Oo O I= p = O ~ ~ ~ K= O O O 195. p = = = = 3.5 Square p = =~= ~ W=~= a ~ ~ W= = o~ = = = W=o= o~ = = = W= = m W=i= ^ ~W=p= = = = Figure 16. 33 CHAPTER 3. GEOMETRY 196. = ~ O == = 197. o = ~O = = O O = ~ 198. = = O 199. i = Q~ = = = 200. p = ~ = = = = O 3.6 Rectangle = p = =~= ~ W=~I= = a ~ ~ W= = o~ = = = W=o= m W=i= ^ ~W=p= = = = = Figure 17. = 201. = ~ O + O == 34 CHAPTER 3. GEOMETRY 202. o = = O = 203. i = O(~ + ) = = 204. p = ~ = = = = 3.7 Parallelogram = p = =~= ~ ~ ~ W=~I= = a ~ ~ W= N I O = ` =~ W= I = ^ = = = ~ ~ W= = ^ W= == m W=i= ^ ~W=p= = = = ===== = Figure 18. = 205. + = NUM = 206. + = O(~ + ) = O N O O O = O = 35 CHAPTER 3. GEOMETRY 207. = = = 208. i = O(~ + ) = 209. p = ~ = ~ I== N p = N O K= O = = = = = 3.8 Rhombus = p = =~= W=~= a ~ ~ W= N I O = ` =~ W= I = ^ W=e= o~ = = = W= = m W=i= ^ ~W=p= = = = = ===== Figure 19. = 36 CHAPTER 3. GEOMETRY 210. + = NUM = = 211. + = Q~ = O N O O O = 212. = ~ = N O = O~ = ~ 213. = = N O = = O Q~ O = 214. i = Q~ = = 215. p = ~ = ~ I== N p = N O K= O = = = O 3.9 Trapezoid = _~ = =~= ~ W=~I= = j W= = ^ W= = ^ ~W=p= = = 37 CHAPTER 3. GEOMETRY = = Figure 20. = 216. = 217. p = ~+ = O ~+ = = O = = = = 3.10 Isosceles Trapezoid = _~ = =~= ~ W=~I= = i W= = j W= = ^ W= = a ~ ~ W= = o~ = = = W=o= ^ ~W=p= = = 38 CHAPTER 3. GEOMETRY = = Figure 21. = 218. = ~+ = O = 219. = ~ + = = N O 220. = O ( ~ ) = Q O = ~ + O = (O ~ + )(O + ~ ) = ~+ 222. p = = = O = = = = = = 221. o = 39 CHAPTER 3. GEOMETRY 3.11 Isosceles Trapezoid with Inscribed Circle = _~ = =~= ~ W=~I= = i W= = j W= = ^ W= = a ~ ~ W= = o~ = = = W=o= o~ = = = W= = m W=i= ^ ~W=p= = = = = Figure 22. = 223. ~ + = O = = ~+ 224. = = = O = 225. = + = O O O = 40 CHAPTER 3. GEOMETRY 226. = ~ = = O O = ~+ ~ N+ O + O = = = = +S+ = O Q O ~ O U ~ = 228. i = O(~ + ) = Q = = (~ + ) ~ = = = i == ~+ = 229. p = O O O = = = 227. o = O 3.12 Trapezoid with Inscribed Circle = _~ = =~= ~ W=~I= = i~ ~ = W= I= = j W= = ^ W= = a ~ ~ W= N I O = ^ = = = ~ ~ W= = o~ = = = W= = o~ = = = W=o= m W=i= ^ ~W=p= = 41 CHAPTER 3. GEOMETRY = = Figure 23. = 230. ~ + = + = ~+ + = = 231. = O O 232. i = O(~ + ) = O( + ) = = = = ~+ + = = I== O O N p = N O K= O 233. p = = = = 3.13 Kite = p = =~= W=~I= = a ~ ~ W= N I O = ^ W= I I = m W=i= ^ ~W=p= = = 42 CHAPTER 3. GEOMETRY = = Figure 24. = 234. + + O = PSM = 235. i = O(~ + ) = = = 236. p = N O = O = = = 3.14 Cyclic Quadrilateral p = =~= ~ ~ ~ W=~I= I= I= = a ~ ~ W= N I O = ^ = = = ~ ~ W= = f ~ =~ W= I I I = o~ = = = W=o= m W=i= p W= == ^ ~W=p= 43 CHAPTER 3. GEOMETRY = = Figure 25. = 237. + = + = NUM = = 238. m =q = ~ + = N O = 239. i = ~ + + + = = = N (~ + )(~ + )(~ + ) I== 240. o = Q ( ~ )( )( )( ) i = = K= O = N 241. p = N O I== O p = ( ~ )( )( )( ) I== i = = K= O = = = 44 CHAPTER 3. GEOMETRY 3.15 Tangential Quadrilateral = p = =~= ~ ~ ~ W=~I= I= I= = a ~ ~ W= N I O = ^ = = = ~ ~ W= = o~ = = = W= = m W=i= p W= == ^ ~W=p= = = = = Figure 26. = 242. ~ + = + = = 243. i = ~ + + + = O(~ + ) = O( + ) = = O N O (~ ) (~ + ) O I== O i = = K== O = O O 244. = 45 CHAPTER 3. GEOMETRY N 245. p = = N O = O = = = 3.16 General Quadrilateral = p = =~= ~ ~ ~ W=~I= I= I= = a ~ ~ W= N I O = ^ = = = ~ ~ W= = f ~ =~ W= I I I = m W=i= ^ ~W=p= = = = = ======= Figure 27. = 246. + + + = PSM = 247. i = ~ + + + = = = 46 CHAPTER 3. GEOMETRY N 248. p = N O = O = = = 3.17 Regular Hexagon = p W=~= f ~ =~ W= = p ~ = W= = o~ = = = W= = o~ = = = W=o= m W=i= p W= == ^ ~W=p= = = = = Figure 28. = 249. = NOM = = 250. = = ~P = O 47 CHAPTER 3. GEOMETRY 251. o = ~ = = 252. i = S~ = = O ~P P I== O i = = K= O = = = 253. p = = 3.18 Regular Polygon = p W=~= k = = W= = f ~ =~ W= = p ~ = W= = o~ = = = W= = o~ = = = W=o= m W=i= p W= == ^ ~W=p= = = 48 CHAPTER 3. GEOMETRY = = Figure 29. = 254. = 255. = O NUM = O = O NUM = O = 256. o = ~ O = = 257. = = ~ O ~ = oO ~O = Q = 258. i = ~ = = 259. p = o O I== O O p = = o O ~O I== Q 49 CHAPTER 3. GEOMETRY = = i K== O = = = 3.19 Circle = o~ W=o= a ~ W= = ` W=~= p ~ = W= I= = q~ = W= = ` ~ =~ W= = f =~ W= = m W=i= ^ ~W=p= = = 260. ~ = Oo = O = = = Figure 30. = 50 CHAPTER 3. GEOMETRY 261. ~N~ O = N O = = = = Figure 31. = 262. N = N = = = = ===== Figure 32. = 263. O = N = = 51 CHAPTER 3. GEOMETRY = ===== = Figure 33. = 264. = = O = = = Figure 34. = 265. i = O o = = = 266. p = o O = io = == Q O O = 52 CHAPTER 3. GEOMETRY 3.20 Sector of a Circle = o~ = =~= W=o= ^ = W= = ` ~ =~ =E = ~ ~ FW= = ` ~ =~ =E = FW= = m W=i= ^ ~W=p= = = = = Figure 35. = 267. = o = 268. = = o = NUM = 269. i = + Oo = = 270. p = o o o = = == O O PSM O O = = 53 CHAPTER 3. GEOMETRY 3.21 Segment of a Circle = o~ = =~= W=o= ^ = W= = ` W=~= ` ~ =~ =E = ~ ~ FW= = ` ~ =~ =E = FW= = e = = = W= = m W=i= ^ ~W=p= = = = = Figure 36. = 271. ~ = O O o O = = N 272. = o Qo O ~ O I= < o = O = 273. i = + ~ = = 54 CHAPTER 3. GEOMETRY O O N [ o ~(o )] = o = o ( ) I== O O NUM O O p ~ K= P 274. p = = = = 3.22 Cube = b W=~== a ~ ~ W= = o~ = = = W= = o~ = = = W= = p ~ =~ ~W=p= s W=s= = = = = === Figure 37. = 275. = ~ P = = ~ 276. = = O = 55 CHAPTER 3. GEOMETRY 277. o = ~P = O = 278. p = S~ = O = 279. s = ~ == = = = P 3.23 Rectangular Parallelepiped = b W=~I= I= == a ~ ~ W= = p ~ =~ ~W=p= s W=s= = = = = ===== Figure 38. = 280. = ~ O + O + O = 281. p = O(~ + ~ + ) = 282. s = ~ == = = 56 CHAPTER 3. GEOMETRY 3.24 Prism = i~ ~ = W= = e W= = i~ ~ =~ ~W= p i = ^ ~= = ~ W= p_ = q ~ = ~ =~ ~W=p= s W=s= = = = = ===== Figure 39. = 283. p = p i + Op_ K== = 284. i~ ~ =^ ~= =~=o =m = p i = (~ N + ~ O + ~ P + K + ~ ) = = 285. i~ ~ =^ ~= =~ =l =m = p i = I== = = = = = = = = K= = 57 CHAPTER 3. GEOMETRY 286. s = p_ = = 287. `~ ~ D =m == d = = = = = ~ ~ = ~ K=f = = ~ = = = ~ ~ = = = = ~ = ~ = = ~ = ~ ~= = = I= = = = = = =~ = ~ K= = = = 3.25 Regular Tetrahedron = q ~ = = W=~= e W= = ^ ~= = ~ W= p_ = p ~ =~ ~W=p= s W=s= = = = = Figure 40. = 288. = O ~= P = 58 CHAPTER 3. GEOMETRY 289. p_ = P~ O = Q = 290. p = P~ = = N ~P 291. s = p_ = K== P SO = = = O 3.26 Regular Pyramid = p = = ~ W=~= i~ ~ = W= = e W= = p ~ = W= == k = = W= == p = = ~ W= = o~ = = = = = ~ W= = ^ ~= = ~ W= p_ = i~ ~ = ~ =~ ~W= pi = q ~ = ~ =~ ~W=p= s W=s= = = 59 CHAPTER 3. GEOMETRY = = Figure 41. = 292. = O ~O = Q = 293. = O ~ = O Q O O = N N 294. p i = ~ = ~ Q O ~ O = = O Q = 295. p_ = = = 296. p = p_ + p i = = N N 297. s = p_ = == P P = = = 60 CHAPTER 3. GEOMETRY 3.27 Frustum of a Regular Pyramid = ~N I ~ O I ~ P IKI ~ = _~ =~ = = = W= N I O I P IKI e W= = p ~ = W= == ^ ~= = ~ W= pN I= pO = i~ ~ = ~ =~ ~W= p i = m = = ~ W= mN I= mO = p ~ = ~ W= = q ~ = ~ =~ ~W=p= s W=s= = = = = Figure 42. = 298. N O P = = =K= = = = ~N ~ O ~ P ~ ~ = 61 CHAPTER 3. GEOMETRY 299. pO = O = pN = (mN + mO ) = 300. p i = O = 301. p = p i + pN + pO = = 302. s = pN + pNpO + pO = P = O p p 303. s = N N + + = N N + + O = P ~ ~ P = = = ( ) [ ] 3.28 Rectangular Right Wedge = p = = ~ W=~I= = q = W= = e W= = i~ ~ = ~ =~ ~W= p i = ^ ~= = ~ W= p_ = q ~ = ~ =~ ~W=p= s W=s= = = 62 CHAPTER 3. GEOMETRY = = Figure 43. = N (~ + ) Q O + O + O + (~ )O = O = 305. p_ = ~ = = 306. p = p_ + p i = = (O~ + ) = 307. s = S = = = 304. p i = 3.29 Platonic Solids = b W=~= o~ = = = W= = o~ = = = W=o= p ~ =~ ~W=p= s W=s= = = 63 CHAPTER 3. GEOMETRY 308. c =m ~ =p = q = ~ = = ~ = = ~= = ~ = ~ = = = = = ~ = K== = p = k = k = k = p = =s =b = =c~ = 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 = 64 CHAPTER 3. GEOMETRY 311. p = O~ O P = = ~P O 312. s = = P = = Icosahedron = = = Figure 45. = 313. = ( = 314. o = ) ~ P P+ R = NO ( ) ~ O R+ R = Q = 315. p = R~ O P = = R~ P P + R 316. s = = NO = = ( ) 65 CHAPTER 3. GEOMETRY Dodecahedron = = = Figure 46. 317. = ( ~ NM OR + NN R = O = 318. o = ) = ( ) ~ P N+ R = Q = ( ) 319. p = P~ O R R + O R = = ~ P NR + T R 320. s = = Q = = = ( ) 3.30 Right Circular Cylinder = o~ = = ~ W=o= a ~ = = ~ W= = 66 CHAPTER 3. GEOMETRY e W=e= i~ ~ = ~ =~ ~W= p i = ^ ~= = ~ W= p_ = q ~ = ~ =~ ~W=p= s W=s= = = = ===== = Figure 47. = 321. p i = O oe = = 322. p = p i + Op_ = O o(e + o ) = e + = O = 323. s = p_ e = o O e = = = = 67 CHAPTER 3. GEOMETRY 3.31 Right Circular Cylinder with an Oblique Plane Face = o~ = = ~ W=o= q = ~ = = =~= W= N = q = = = =~= W= O = i~ ~ = ~ =~ ~W= p i = ^ ~= = ~ = = ~ W= p_ = q ~ = ~ =~ ~W=p= s W=s= = = = = Figure 48. = 324. p i = o( N + O ) = = O O 325. p_ = o + o o + N = O = O O 68 CHAPTER 3. GEOMETRY O N O O 326. p = p i + p_ = o N + O + o + o + = O = o O ( N + O ) = 327. s = O = = = 3.32 Right Circular Cone o~ = = ~ W=o= a ~ = = ~ W= = e W=e= p ~ = W= = i~ ~ = ~ =~ ~W= pi = ^ ~= = ~ W= p_ = q ~ = ~ =~ ~W=p= s W=s= = = = = Figure 49. 69 CHAPTER 3. GEOMETRY 328. e = O o O = = 329. p i = o = = O = 330. p_ = o O = = N 331. p = p i + p_ = o ( + o ) = + = O O = N N 332. s = p_ e = o O e = P P = = = 3.33 Frustum of a Right Circular Cone = o~ = = ~ W=oI= = e W=e= p ~ = W= = p ~ = ~ W= = ^ ~= = ~ W= pN I= pO = i~ ~ = ~ =~ ~W= p i = q ~ = ~ =~ ~W=p= s W=s= = = 70 CHAPTER 3. GEOMETRY = = Figure 50. = 333. e = O (o ) = = o 334. = = = p oO 335. O = O = O = pN = 336. p i = (o + ) = = 337. p = pN + pO + p i = o O + O + (o + ) = = 338. s = pN + pNpO + pO = P = O pN o o pN 339. s = N+ + O = N + + = P P = = = O [ ( ] ) [ 71 ] CHAPTER 3. GEOMETRY 3.34 Sphere = o~ W=o= a ~ W= = p ~ =~ ~W=p= s W=s= = = = Figure 51. = 340. p = Q o O = = Q N N 341. s = o P e = P = po = P S P = = = 3.35 Spherical Cap o~ = = W=o= o~ = = ~ W= = e W= = ^ ~= = ~ = ~ W= p_ = ^ ~= = ~ = ~ W= p` = q ~ = ~ =~ ~W=p= s W=s= 72 CHAPTER 3. GEOMETRY = = Figure 52. = 342. o = O + O = O = 343. p_ = O = = 344. p` = ( O + O )= = 345. p = p_ + p` = ( O + O O ) = (Oo + O ) = = 346. s = O (Po ) = (P O + O ) = S S = = = 3.36 Spherical Sector = o~ = = W=o= o~ = = ~ = = ~ = ~ W= = e W= = q ~ = ~ =~ ~W=p= s W=s= = 73 CHAPTER 3. GEOMETRY ====== = === = Figure 53. = 347. p = o(O + ) = = O 348. s = o O = P = k W= q = = ~ = ~ = = = = = ~ = = ~ = K= = = = 3.37 Spherical Segment = o~ = = W=o= o~ = = ~ W= N I= O = e W= = ^ ~= = ~ = ~ W= pp = ^ ~= = ~ = = ~ W= pN I= pO = q ~ = ~ =~ ~W=p= s W=s= = 74 CHAPTER 3. GEOMETRY = ===== = Figure 54. = 349. pp = O o = = 350. p = pp + pN + pO = (Oo + NO + OO ) = = N 351. s = (P NO + P OO + O )= S = = = 3.38 Spherical Wedge = o~ W=o= a ~ =~ = = W= = a ~ =~ = = ~ ~ W= = ^ ~= = ~ = W= p i = q ~ = ~ =~ ~W=p= s W=s= = = 75 CHAPTER 3. GEOMETRY = = Figure 55. = 352. p i = o O = Oo O = VM = 353. p = o O + o O = o O + Oo O = VM = 354. s = oP O = oP = OTM P = = = 3.39 Ellipsoid = p -~ W=~I= I= = s W=s= 76 CHAPTER 3. GEOMETRY = ======= = Figure 56. = Q 355. s = ~ = P = = = Prolate Spheroid = p -~ W=~I= I= =E ~ > F= p ~ =~ ~W=p= s W=s= = = ~ ~ 356. p = O + I== = = ~ O O K= ~ = Q 357. s = O~ = P = 77 CHAPTER 3. GEOMETRY Oblate Spheroid = p -~ W=~I= I= =E ~ < F= p ~ =~ ~W=p= s W=s= = = ~ ~ ~ I== 358. p = O + L ~ = = O ~ O K= = Q 359. s = O~ = P = = = 3.40 Circular Torus = j~ = ~ W=o= j = ~ W= = p ~ =~ ~W=p= s W=s= = 78 CHAPTER 3. GEOMETRY == Picture 57. = 360. p = Q Oo = = 361. s = O Oo O = = = 79 = Chapter 4 T rigonometry = = = = ^ W= I= = o ~ = =E ~ = =~= FW= I= == t = W= = = = 4.1 Radian and Degree Measures of Angles = 362. N ~ = = 363. N = = 364. N D = = 365. N ? = = 366. = = = = = NUM RT NT DQR? = ~ MKMNTQRP ~ = NUM ~ MKMMMOVN ~ = NUM SM ~ MKMMMMMR ~ = NUM PSMM ^ = E F= ^ = E ~ ~ F= M= PM= QR= SM= VM= NUM= OTM= PSM= M= = S = Q 80 = P = O = P = O O = CHAPTER 4. TRIGONOMETRY 4.2 Definitions and Graphs of Trigonometric Functions = = = = Figure 58. = 367. = = = 368. = = = 369. ~ = = = 370. = = = 81 CHAPTER 4. TRIGONOMETRY 371. = = = 372. = = = 373. p =c = = I= N N K= = = Figure 59. = 374. ` =c == = I= N N K= 82 CHAPTER 4. TRIGONOMETRY = = Figure 60. = 375. q~ =c = = ~ I= (O + N) I= ~ K = O = = = Figure 61. = 83 CHAPTER 4. TRIGONOMETRY 376. ` ~ =c == = I= I== K= = = = Figure 62. = 377. p ~ =c = = I= (O + N) K= O == 84 CHAPTER 4. TRIGONOMETRY = = Figure 63. = 378. ` ~ =c == = I= K= = Figure 64. 85 CHAPTER 4. TRIGONOMETRY 4.3. Signs of Trigonometric Functions 379. = = = = 380. = n ~ ~ = = f= ff= fff= fs= p = H= H= = = ` = H= = = H= q~ = H= = H= = ` = H= = H= = p = H= = = H= ` = = H= H= = = = = Figure 65. = = = = = = = = = = 86 CHAPTER 4. TRIGONOMETRY 4.4 Trigonometric Functions of Common Angles 381. = = ~ = M= M= = PM= S = QR= Q = SM= P = VM= O O = NOM= P NUM= = P = OTM= O PSM= O = = = = = = = = = = = = = = O = O P = O = N= P = O O = O N = O N= M= P = O M= N = O N= N= M= = M= N = O ~ = M= = N = P= P = N= O = P = = O= N= N= P= N = P O= O = P M= = N= = O= O= M= N P = N= O = P = M= = M= = N= N= M= = N= = P= 87 O= CHAPTER 4. TRIGONOMETRY 382. = = ~ = = NR= NO = = ~ = = S O = Q S+ O = Q O P = O+ P = R O R = R R+O R = NU= = NM R N = Q NM + O R Q PS= = R NM O R Q R +N = Q RQ= P = NM R +N = Q NM O R Q TO= O = R NM + O R Q R N = Q TR= R = NO S+ O = Q S O = Q = = = 4.5 Most Important Formulas = 383. O + O = N = = 384. O ~ O = N = = 385. O O = N = = = 386. ~ = 88 NM O R R +N R +N NM O R R +N NM O R = NM O R R +N = R+O R = R O R R = O+ P = O P = CHAPTER 4. TRIGONOMETRY 387. = = = 388. ~ = N = = N 389. = = = N 390. = = = = = 4.6 Reduction Formulas = 391. = = = = = = = = = VM = VM + = NUM NUM + OTM OTM + PSM = PSM + = = + = + = + = = = = = + = 89 = + = + = = = = = + = + = + = ~ = ~ = + = = ~ = + ~ = + = = ~ = + ~ = = = + ~ = ~ = = + = + ~ = ~ = = + = CHAPTER 4. TRIGONOMETRY 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. = N O = O= = N + ~ O O N (N O ) = O O N = O O Q O ~ = = 397. = N O = O= = N + ~ O O N (N + O ) = O O N = O O N ~ O = = 398. ~ = O N O = O N = = = N + O O 90 CHAPTER 4. TRIGONOMETRY N O O= = = N + O O N + ~ O O ~ = = N + O O = O N = = = O N O N ~ O N + O O= = = = N O O ~ O 399. = = N O= 400. = = N + ~ O = N ~ O O = N + ~ O N O= 401. = = N + O = O ~ O = = = N + ~ O 4.9 Addition and Subtraction Formulas = 402. ( + ) = + = = 403. ( ) = = = 404. ( + ) = = = 405. ( ) = + = 91 CHAPTER 4. TRIGONOMETRY 406. ~ ( + ) = = 407. ~ ( ) = = 408. ( + ) = = 409. ( ) = ~ + ~ = N ~ ~ ~ ~ = N + ~ ~ N ~ ~ = ~ + ~ N + ~ ~ = ~ ~ = = = 4.10 Double Angle Formulas = 410. O = O = = 411. O = O O = N O O = O O N = = O ~ O 412. ~ O = = = O N ~ ~ = O N ~ = = 413. O = O O = = = = = = 92 CHAPTER 4. TRIGONOMETRY 4.11 Multiple Angle Formulas = 414. P = P Q P = P O P = = 415. Q = Q U P = = 416. R = R OM P + NS R = = 417. P = Q P P = P P O = = 418. Q = U Q U O + N = = 419. R = NS R OM P + R = = P ~ ~ P 420. ~ P = = N P ~ O = Q ~ Q ~ P = 421. ~ Q = N S ~ O + ~ Q = ~ R NM ~ P + R ~ = 422. ~ R = N NM ~ O + R ~ Q = P P 423. P = = P O N = N S ~ O + ~ Q == 424. Q = Q ~ Q ~ P = 93 CHAPTER 4. TRIGONOMETRY 425. R = N NM ~ O + R ~ Q = ~ R NM ~ P + R ~ = = = 4.12 Half Angle Formulas = 426. N = = O O = 427. N + = = O O = 428. ~ N N = = = = = O N + N + = 429. N + N + = = = = + = O N N = = = 4.13 Half Angle Tangent Identities = O= 430. = N + ~ O O = O ~ 94 CHAPTER 4. TRIGONOMETRY O= 431. = O N + ~ O = O ~ O= 432. ~ = N ~ O O = N ~ O O= 433. = O ~ O = = = N ~ O 4.14 Transforming of Trigonometric Expressions to Product = 434. + = O = 435. = O + = O O + = O O = 436. + = O + = O O = 437. = O + = O O = 95 CHAPTER 4. TRIGONOMETRY 438. ~ + ~ = = 439. ~ ~ = = 440. + = = 441. = ( + ) = ( ) = ( + ) = ( ) = = 442. + = O = O + = Q Q = 443. = O = O + = Q Q = ( ) = 444. ~ + = = ( + ) = 445. ~ = = 446. N + = O O = O = 447. N = O O = O = 96 CHAPTER 4. TRIGONOMETRY 448. N + = O O = Q O = 449. N = O O = Q O = = = 4.15 Transforming of Trigonometric Expressions to Sum = 450. = ( ) ( + ) = O = 451. = = 452. = = 453. ~ ~ = = 454. = = 455. ~ = ( ) + ( + ) = O ( ) + ( + ) = O ~ + ~ = + + = ~ + ~ ~ + = + ~ = = = 97 CHAPTER 4. TRIGONOMETRY 4.16 Powers of Trigonometric Functions = 456. O = = 457. P = = 458. Q = = 459. R = = 460. S = = 461. O = = 462. P = = 463. Q = = 464. R = = 465. S = N O = O P P = Q Q Q O + P = U NM R P + R = NS NM NR O + S Q S = PO N + O = O P + P = Q Q + Q O + P = U NM + R P + R = NS NM + NR O + S Q + S = PO = 98 CHAPTER 4. TRIGONOMETRY 4.17 Graphs of Inverse Trigonometric Functions = 466. f =p =c == = ~ I= N N I= ~ K= O O = = = Figure 66. = 467. f =` =c == = ~ I= N N I= M ~ K= = 99 CHAPTER 4. TRIGONOMETRY = = Figure 67. = 468. f =q~ =c == = ~ ~ I= I= < ~ ~ < K= O O = = = ===== Figure 68. 100 CHAPTER 4. TRIGONOMETRY 469. f =` ~ =c == = ~ I= I= M < ~ < K= ===== = Figure 69. = 470. f =p ~ =c == = ~ = I ( I N] [NI )I ~ MI I K O O = Figure 70. 101 CHAPTER 4. TRIGONOMETRY 471. f =` ~ =c == = ~ I ( I N] [NI )I ~ I M MI K O O = = Figure 71. = = 4.18 Principal Values of Inverse Trigonometric Functions 472. = M= N = O PM = SM = O O ~ = M = ~ = VM N = O PM ~ = QR = NOM ~ = NPR = = O = O QR = QR = P O P O SM PM VM M = N= = VM = NUM NRM = = SM 102 N= = = CHAPTER 4. TRIGONOMETRY 473. = M= P P N= ~ ~ = M = PM QR SM ~ = VM SM QR PM P= P P 4.19 Relations between Inverse Trigonometric Functions = 474. ~ ( ) = ~ = = 475. ~ = ~ = O = 476. ~ = ~ N O I= M N K= = 477. ~ = ~ N O I= N M K= = O I= < N K= 478. ~ = ~ ~ O N = N O I= M < N K= = 480. ~ = ~ N O I= N < M K= = 481. ~ ( ) = ~ = 103 P= QR SM = = NPR NOM = NRM = = PM = = = 479. ~ = ~ N= CHAPTER 4. TRIGONOMETRY 482. ~ = ~ = O = 483. ~ = ~ N O I= M N K= = 484. ~ = ~ N O I= N M K= = 485. ~ = ~ ~ N O I= M < N K= = N O I= N < M K= 486. ~ = + ~ ~ = 487. ~ = ~ N O I= N N K= = 488. ~ ~ ( ) = ~ ~ = = 489. ~ ~ = ~ = O = = 490. ~ ~ = ~ N+ O = N I= M K= 491. ~ ~ = ~ N+ O = N I= M K= 492. ~ ~ = ~ N+ O = 104 CHAPTER 4. TRIGONOMETRY 493. ~ ~ = N ~ ~ I= > M K= O = N 494. ~ ~ = ~ ~ I= < M K= O = N 495. ~ ~ = ~ I= > M K= = N 496. ~ ~ = ~ I= < M K= = 497. ~ ( ) = ~ = = 498. ~ = ~ ~ = O = N I= > M K= 499. ~ = ~ N+ O = N I= < M K= 500. ~ = ~ N+ O = = 501. ~ = ~ N+ O = N 502. ~ = ~ ~ I= > M K= = N 503. ~ = + ~ ~ I= < M K= = = 105 CHAPTER 4. TRIGONOMETRY 4.20 Trigonometric Equations 504. 505. 506. 507. = t = W= = = = = ~ I= = ( N) ~ ~ + = = = ~ I= = ~ ~ + O = = ~ = ~ I= = ~ ~ ~ + = = = ~ I= = ~ ~ + = = = = 4.21 Relations to Hyperbolic Functions 508. 509. 510. 511. 512. = f ~ ~ = W= = = = ( ) = = = ~ ( ) = ~ = = ( ) = = = ( ) = = = ( ) = = = = = 106 Chapter 5 M atrices 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 ~ = ~ N ^ = N = ~ N O ~ O N = ~ O O = = = = = 107 CHAPTER 5. MATRICES AND DETERMINANTS 514. q =l =a ~ = ~NN ~NO ~NP ^ = ~ ON ~ OO ~ OP = ~NN~ OO~ PP + ~NO~ OP~ PN + ~ NP~ ON~ PO = ~ PN ~ PO ~ PP ~NN~ OP~ PO ~NO~ ON~ PP ~ NP~ OO~ PN = = 515. p~ =o =E^ =o F= = = Figure 72. = 516. k- =l =a ~ = ~NN ~NO K ~N ~ ON ~ OO K ~ O KKKK ^ = ~ N ~ O K ~ KKKK ~ N ~ O K ~ K ~N K ~ O KK K ~ = KK K ~ = 517. j = q = = j =~ ~ = = = = ~ = = - = = ~ = ^= = = ( N) - = = ~ = = = = ~ =^= = = = = - = =~ = - = K=== = 108 CHAPTER 5. MATRICES AND DETERMINANTS 518. ` ~ = + ` = ( N) j = = 519. i~ ~ =b ~ = = - =l =a ~ = i~ ~ = ~ = = = = = - = = ^ = ~ ` I= = NI OI KI K= =N i~ ~ = ~ = = = = = - = = ^ = ~ ` I= = NI OI KI K== =N = = = 5.2 Properties of Determinants = 520. q == ~ == =~= ~ = ~ == ~ = = =~ = ~ = = =~ = = = K= ~ ~ O ~N N = == =N N O ~ O O = 521. f = == ==E = = F=~ == ~ I= = = = = ~ = = ~ K= ~N N ~ O = O = ~ O O ~N N = 522. f = = ==E = = F=~ == ~ I= = ~ = = = ~ = = K= ~N ~N = M= ~O ~O = 109 CHAPTER 5. MATRICES AND DETERMINANTS 523. f == === == ==~ = ==E = F=~ = = ===== ~== == ~ I== == ~ == == == == ~ = ~ K= ~ N N ~ N = N = ~ O O ~ O O = 524. f == == == ==~ == ==E == F=~ = ~ =E = ~ F = ~ = = = = = = =~ = = ==E = FI== = ~ = = = ~ = = ~ K= ~N + N N ~N N = = ~ O + O O ~ O O = = = 5.3 Matrices = 525. a = ^ = = ~ =^= =~= ~ ~ =~ ~ = = =E = = F= = = =~ = = K== ~ NN ~ NO K ~ N ~ ~ OO K ~ O == ON ^ = ~ = M M M ~ N ~ O K ~ = 526. p ~ = ~ = =~= ~ = = = K== = 527. ^= ~ = ~ == ~ == == == == ~ = ~ I== K K== == = [] [] =~ = = ~ = ~ ~ K== = 528. ^= ~ = ~ = ~ = = - = = ~ = ~ K== = [] 110 CHAPTER 5. MATRICES AND DETERMINANTS 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 == ~NN ~NO K ~N ~ ~ OO K ~ O I== ^ = ~ = ON M M M ~ N ~ O K ~ NN NO K N OO K O I== _ = = ON M M M N O K = = = = = [] [] 111 CHAPTER 5. MATRICES AND DETERMINANTS == ~NO + NO K ~N + N ~NN + NN ~ + ~ OO + OO K ~ O + O K= ON ON ^+_= M M M ~ N + N ~ O + O K ~ + = 534. f = = =~= ~ ~ I=~ = ^ = ~ = =~= ~ I= = [] ~NN ~NO K ~N ~ ~ OO K ~ O K= ON ^ = ~ = M M M ~ N ~ O K ~ = 535. j ~ = =q =j~ = q = ~ = ~ = = = = = = = = = = = = = = ~ = = = = = = = = K== = f = ~NN ~NO K ~N ~ ~ OO K ~ O I== ^ = ~ = ON M M M ~ N ~ O K ~ NN NO K N OO K O I= _ = = ON M M M N O K = = = = = [] [] [] 112 CHAPTER 5. MATRICES AND DETERMINANTS == NN NO K N OO K O I== ON ^_ = ` = M M M N O K == = ~ N N + ~ O O + K + ~ = ~ = E = NI OI KI X = NI OI KI FK== = q = = [] ~ NN ^ = ~ = ~ ON ~ NO ~ OO =N N ~ NP I= _ = [ ] = O I== ~ OP P == ~ NN ~ NO ^_ = ~ ON ~ OO ~ NP N ~ NN N = ~ OP O ~ ON N P ~ NO O ~ OO O ~ NP P K== ~ OP P = 536. q ~ = =~=j~ = f = = =~ = = =~= ~ =~ = ~ I= = = = ~ = = ~ = = ~ = = = ~ = ~ K=== f = ^= = = ~ = ~ I= = ~ = = = ^ q = = ^ K== = 537. q = ~ =^= = ~ = = ^^ q = f K== = 538. f = = ~ = =^_= = I= == (^_ )q = _ q ^ q K= = = 113 CHAPTER 5. MATRICES AND DETERMINANTS 539. ^ = =j~ = f =^= =~= ~ = ~ I= =~ I= = = ~ ^ I= = = ~ = = = ~ = = ~ = ` = =^W= [] ~ ^ = ` K== = 540. q ~ = =~=j~ = f = ^= = ~= ~ = ~ I= = ~ I= = = ^ I= = = = == = = == = = = = ~ = ~ ~ W= ^ = ~NN + ~ OO + K + ~ K= = 541. f = =~=j~ = f =^= =~= ~ = ~ = =~= ~ = ~ = ^ I= = = = ^ N = = = = ~ ^ ^ N = K= ^ = 542. f = = ~ = =^_= = I= == (^_) N = _ N^ N K= = 543. f ==^== =~= ~ === == ~ I== == ==u=== ~ = = ~ = ^u = u I== = = ~ = = ~ = = ~ ~ = ~ = ^ f = M K=== = = = q 5.5 Systems of Linear Equations = = s~ ~ W= I= I= I= N I= O I K = o ~ = W= ~ N I ~ O I ~ P I N I ~ NN I ~ NO I K = 114 CHAPTER 5. MATRICES AND DETERMINANTS a ~ W=aI= a I= a I= a == j~ W=^I=_I=u= = = ~ + N = N I== 544. N ~ O + O = O a a =E` ~ = FI== = I= = a a == ~ N a= N = ~N O ~ O N I== ~ O O N a = N = N O O N I== O O ~ N a = N = ~N O ~ O N K== ~ O O = 545. f = a M I= = = = ~ =~= = W== a a K= = I= = a a f = a = M = ~ = a M E = a M FI= = = = ~ = = == K= f = a = a = a = M I= = = = ~ = = = = ~ == K= = ~N + N + N = N= 546. ~ O + O + O = O I== ~ + + = P P P P = a a a I= = I= = =E` ~ = FI== a a a = 115 CHAPTER 5. MATRICES AND DETERMINANTS == ~N N a = ~ O O ~ P P N N N N O I= a = O O O I= P P P P ~N N N ~N N N a = ~ O ~P O P O I= a = ~ O P ~P O P O K== P = 547. f = a M I= = = = ~ =~= = W== a a a I= = K= = I= = a a a f = a = M =~ = a M E = a M = = a M FI= = = = ~ = = K= f = a = a = a = a = M I= = = = ~ = = ~ = K= = 548. j~ =c = =~=p = = =i ~ =b ~ = ================= =r = q = = = ~ = ~ == ~NN N + ~ NO O + K + ~ N = N ~ + ~ + K + ~ = ON N OO O O O = KKKKKKKKKKKK ~ N N + ~ O O + K + ~ = ~ = = = = ~ = = ~ NN ~ NO K ~ N N N ~ ON ~ OO K ~ O O O I== = M M M M M ~ N ~ O K ~ K K== ^ u = _ I== 116 CHAPTER 5. MATRICES AND DETERMINANTS == ~ NN ~ ^ = ON M ~ N ~ NO K ~ N N N ~ OO K ~ O O I= u = I= _ = O K== M M M M ~ O K ~ = 549. p = =~=p = =i ~ =b ~ = = u = ^ N _ I== = ^ N = = = = =^K= = = 117 Chapter 6 V ectors = = = = rrrr s W= I= I= I= I= ^_ I= = rr s = W= I= I= = rrr r = W= I= I= = r k = W= M = r ` ~ = = = W= uN I vN I wN = r ` ~ = = = W= u O I vO I wO = p ~ ~ W= I = a = W= I= I= = ^ = = = W= = = = 6.1 Vector Coordinates = 550. r =s = r = (NI MI M) I= r = (MI NI M) I= r = (MI MI N) I= rrr = = = N K= = r r r r 551. = ^_ = ( N M ) + ( N M ) + ( N M ) = = 118 CHAPTER 6. VECTORS ======= = = Figure 73. = r = ^_ = 552. ( N M )O + ( N M )O + ( N M )O = = r r 553. f = ^_ = I= = _^ = K= = = = Figure 74. r 554. u = I= r v = I= r w = K= = 119 CHAPTER 6. VECTORS = ===== = Figure 75. = r r 555. f = (uI v I w ) = N (uN I vN I wN ) I= == u = uN I= v = vN I= w = wN K== == = 6.2 Vector Addition = rrr 556. = + = = = == = Figure 76. 120 CHAPTER 6. VECTORS = == = Figure 77. = r rrrr 557. = N + O + P + K + = = = = == Figure 78. = 558. ` ~ =i~ = rrrr + = + = = 559. ^ ~ =i~ = rr rrrr ( + ) + = + ( + ) = = rr 560. + = (uN + u O I vN + vO I wN + wO ) = = = = = = = 121 CHAPTER 6. VECTORS 6.3 Vector Subtraction = rrr rrr 561. = = = + = K= = = = Figure 79. = = == = Figure 80. = rrr r 562. = + ( ) = = rrr 563. = M = (MI MI M ) = = r 564. M = M = = rr 565. = (uN u O I vN vO I wN w O ) I== = = = 6.4 Scaling Vectors = r r 566. = = 122 CHAPTER 6. VECTORS = = Figure 81. = 567. r r = = = r 568. = ( uI v I w ) = = rr 569. = = = r r r 570. ( + ) = + = = r r r 571. ( ) = ( ) = ( ) = = rr r r 572. ( + ) = + = = = = 6.5 Scalar Product = r r 573. p ~ ~ =m = =s = =~ = rr r r = I== r r = = = =~ = = = =~ K==== = 123 CHAPTER 6. VECTORS = = = Figure 82. = 574. p ~ ~ =m = =` ~ =c = r r f = = (uN I vN I wN ) I= = (u O I vO I w O ) I= == rr = uNu O + vNvO + wNwO K= = 575. ^ =_ =q =s == r r f = = (uN I vN I wN ) I= = (u O I vO I w O ) I= == uNu O + vNvO + wNw O K= = O O O O O O uN + vN + wN u O + vO + w O = 576. ` ~ =m = rr rr = = = 577. ^ ~ =m = r r rr ( ) ( ) = = = 578. a =m = r r r rr rr ( + ) = + = = rr rr 579. = M = = I =~ = ~ =E = FK= O = rr 580. > M = = M < < K= O = 124 CHAPTER 6. VECTORS rr 581. < M = = < < K= O = rr r r 582. = = rr r r rr 583. = = = I =~ = ~ ~ =E = M FK= = r 584. f = = (uN I vN I wN ) I= == rr r rO O O O = O = = uN + vN + wN K= = rr rr rr 585. = = = N = = rr rr rr 586. = = = M = = = = 6.6 Vector Product = r r 587. s =m = =s = =~ = rrr = I= == r rr = I= = M X= O rr rr = ~ = X= rrr =s = I= I= = =~= - ~ = K= = 125 CHAPTER 6. VECTORS = ======= = Figure 83. = r rrr 588. = = u N uO r vN vO r wN = wO = uN wN uN vN r r r v wN = 589. = = N I I v w u O w O u O vO O O = rr rr 590. p = = =Ec KUPF= = 591. ^ =_ =q =s =Ec KUPF= rr = r r = = 592. k ~ =m = rr rr = ( ) == = 593. ^ ~ =m = r r rr ( ) ( ) = = = = 126 CHAPTER 6. VECTORS 594. a =m = r rr rrrr ( + ) = + = = rrr r r 595. = M = = =~ = =~ = ~ ~ =E = M FK= = rrrrrrr 596. = = = M = = rrrrrrrrr 597. = I= = I= = = = = = 6.7 Triple Product 598. 599. 600. 601. = p ~ ~ =q =m = rr r r r r r r r r r r [ ] = ( ) = ( ) = ( ) = = rr r r rr rr r rr r r rr rrr [ ] = [ ] = [ ] = [ ] = [ ] = [ ] = = rrr rr r ( ) = [ ] = = p ~ ~ =q =m = =` ~ =c = uN vN wN rrr ( ) = u O vO w O I== uP vP wP == r r r = (uN I vN I wN ) I= = (u O I vO I w O ) I= = (uP I vP I wP ) K== = 602. s = =m~ ~ = rrr s = ( ) = = 127 CHAPTER 6. VECTORS = ============ = Figure 84. = 603. s = =m ~ = Nr r r s = ( ) = S = = = Figure 85. = rrr rr r 604. f == ( ) = M I= = = == I= I=~ = =~ = ~ = r r r =I= = = + = = = ~ ~ = =~ = K== = rrr rr r 605. f == ( ) M I= = = == I= I=~ = =~ = ~ = K= = 128 CHAPTER 6. VECTORS 606. s =q =m = rrr r rr rrr ( ) = ( ) ( ) == = = = = = = = = 129 Chapter 7 A nalytic 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 = = ^_ = O N = N O = = = = Figure 86. = 608. a =~=i =p = = =o~ = = + O ^` I= = M = N I= N K= N+ `_ = = ======== Figure 87. 130 = CHAPTER 7. ANALYTIC GEOMETRY 609. j = =~=i =p = + O M = N I= = N K= O = = = 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 = = ^_ = = ( O N )O + ( O N )O = = = Figure 88. 131 CHAPTER 7. ANALYTIC GEOMETRY 611. a =~=i =p = = =o~ = = + O + O M = N I= M = N I== N+ N+ ^` = I= N K= `_ = ======= = = Figure 89. = = 132 CHAPTER 7. ANALYTIC GEOMETRY ======= = = Figure 90. = 612. j = =~=i =p = + O + O I= M = N I= = N K= M = N O O = 613. ` =Ef = =j ~ F= =~=q ~ = + O + P + O + P I= M = N M = N I== P P == ^( N I N ) I== _( O I O ) I==~ == `( P I P ) ==~ = = = = ~ = ^_` K= = = 133 CHAPTER 7. ANALYTIC GEOMETRY ========= = = Figure 91. = 614. f =Ef = =^ =_ F= =~=q ~ = ~ + O + P ~ + O + P I= M = N M = N I== ~ + + ~ + + = ~ = _` I= = `^ I= = ^_ K== = ======== = = Figure 92. 134 CHAPTER 7. ANALYTIC GEOMETRY 615. ` =Ef = = =p =m ~ ====================== _ F= =~=q ~ = O O O O N + N N N N N + N N O + O O N O O + O N O O O O O O O O P + P P N P P + P N M = I= M = = N N N N N N O O P O N P N O O P O N P N = = ======== == Figure 93. = = = = = = = 135 CHAPTER 7. ANALYTIC GEOMETRY 616. l =Ef = =^ F= =~=q ~ = O O N O P + N N N + O P N N O P N + O N O + P N O N O O O O P N O + P N P + N O P N I= M = = M = N N N N N N O O N O O N P P N P P N = = ====== = Figure 94. = 617. ^ ~= =~=q ~ = N N N N N O N p = ( ) O O N = ( ) O O P N P P N = = = 136 O N P N = CHAPTER 7. ANALYTIC GEOMETRY 618. ^ ~= =~=n ~ ~ ~ = N p = ( ) [( N O )( N + O ) + ( O P )( O + P ) + = O + ( P Q )( P + Q ) + ( Q N )( Q + N )] = = === = = Figure 95. = k W=f = ~ =SNTI=SNU= = = = =EHF= =E F= = ~ = = =~= =~ = =~ ~K== = 619. a ~ =_ =q =m = =m ~ =` ~ = = ^_ = NO + OO O N O ( O N ) = = 137 CHAPTER 7. ANALYTIC GEOMETRY = = Figure 96. = 620. ` =o ~ ~ =` ~ = =m ~ =` ~ = = I= = K= = = = Figure 97. = 621. ` =m ~ =` ~ = =o ~ ~ =` ~ = = O + O I= ~ = K= 138 CHAPTER 7. ANALYTIC GEOMETRY 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= = r k ~ = W= = rrr m = W= I= ~ I= = = = 622. d ~ =b ~ = =~=p ~ =i = ^ + _ + ` = M = = 623. k ~ =s = =~=p ~ =i = r q = = (^I _ ) = = ~ = = = = ^ + _ + ` = M K= = = = Figure 98. = 624. b =b ~ = =~=p ~ =i =Ep -f =c F= = + K== 139 CHAPTER 7. ANALYTIC GEOMETRY q = ~ = = = = = = ~ K= = = = Figure 99. = 625. d ~ = =~=i == N = ~ = O = O N = = = Figure 100. 140 CHAPTER 7. ANALYTIC GEOMETRY 626. b ~ = =~=i =d =~=m =~ = =d ~ = = M + ( M ) I== = = = = ~ I= m( M I M ) = =~= = = = K= = = = Figure 101. = 627. b ~ = =~=i =q ~ =m~ =q =q =m = N N = == O N O N = N N N N = M K= O O N = 141 CHAPTER 7. ANALYTIC GEOMETRY = = Figure 102. = 628. f =c = + =N= ~ = = = Figure 103. = = 142 CHAPTER 7. ANALYTIC GEOMETRY 629. k ~ =c = + = M = = = = Figure 104. = 630. m =a =c = N N = I== u v = (uI v ) = = = = = = = ~ = mN ( N I N ) = = = = K= = 143 CHAPTER 7. ANALYTIC GEOMETRY = = Figure 105. = 631. s ~ =i = =~= = 632. e ~ =i = = = = 633. s =b ~ = =~=p ~ =i = rr r = ~ + I== == l= = = = = = ~ I= u= =~ = ~ ~ = = = = I== r ~ = = = = = =~= = =^= = = =I= r = =~= = = = I= ~ ~ = = = I== = =~= ~ ~ I== r = lu = = = = = =~ = =u= = = K== = 144 CHAPTER 7. ANALYTIC GEOMETRY = = Figure 106. = 634. p ~ =i = =m~ ~ =c = = ~N + N I== = ~ O + O == ( I ) ~ = = ~ = =~ = = = = = I== (~N I ~ O ) =~ = = ~ = =~= = = = = I== ( N I O ) =~ = = ~ = =~= = ~ ~ = = = I== = =~= ~ ~ K= = 145 CHAPTER 7. ANALYTIC GEOMETRY = Figure 107. = 635. a ~ =c =~=m =q =~=i = q = ~ = = = = m(~ I ) = = = = ^ + _ + ` = M = == ^~ + _ + ` K= = ^ O + _O = = = Figure 108. 146 CHAPTER 7. ANALYTIC GEOMETRY 636. m~ ~ =i = q = = = N + N =~ = = O + O =~ = ~ ~ = == N = O K= q = = ^N + _N + `N = M = ~ = ^ O + _O + ` O = M = ~ = ~ ~ = = ^N _N = K= ^ O _O = = = Figure 109. = 637. m ~ =i = q = = = N + N =~ = = O + O =~ = ~ = == N O = = I= ~ I= N O = N K= N q = = ^N + _N + `N = M = ~ = ^ O + _ O + ` O = M = ~ = ~ = = ^N^ O + _N_ O = M K= = 147 CHAPTER 7. ANALYTIC GEOMETRY = = Figure 110. = 638. ^ =_ =q =i = N ~ = O I== N + N O ^N^ O + _N_ O = K= O O ^N + _N ^ O + _ O O O = 148 CHAPTER 7. ANALYTIC GEOMETRY = = Figure 111. = 639. f = =q =i = f = = = ^N + _N + `N = M =~ = ^ O + _ O + ` O = M = I= = = = ~ = ~ = `N_ O + ` O_N ^N` O + ^ O`N M = I= M = K= ^N_ O ^ O_N ^N_ O ^ O_N = = = 7.4 Circle = o~ W=o= ` = = W= (~ I ) = m = ~ W= I= I= N I= N I= = o ~ = W=^I=_I=`I=aI=bI=cI= = 149 CHAPTER 7. ANALYTIC GEOMETRY 640. b ~ = =~=` =` =~ = =l =Ep ~ ~ = c F= O + O = oO = ====== = = Figure 112. = 641. b ~ = =~=` =` =~ =^ =m = (~I ) ( ~ )O + ( )O = o O Figure 113. 150 CHAPTER 7. ANALYTIC GEOMETRY 642. q =m =c O + O N O O N + N N N N =M O + O O O N O O O O P + P P P N = = = Figure 114. = 643. m~ ~ =c = o I= M O K = o = 644. d ~ =c ^ O + ^ O + a + b + c = M =E^= I= aO + b O > Q ^c FK== q = = = = = ~ = ~ = (~ I ) I= == a b ~= I= = K= O^ O^ q = ~ = = = = 151 CHAPTER 7. ANALYTIC GEOMETRY o= aO + b O Q ^c K O^ = = = 7.5 Ellipse = p ~ =~ W=~= p =~ W= = c W= cN ( I M) I= cO ( I M) = a ~ = = = W=O = = b W= == o ~ = W=^I=_I=`I=aI=bI=cI= = m W=i= ^ ~W=p= = = 645. b ~ = =~ =b =Ep ~ ~ =c F O O + =N ~ O O = = Figure 115. 152 CHAPTER 7. ANALYTIC GEOMETRY 646. N + O = O~ I= == N I== O ==~ == ~ == ==~ == == m( I ) == = = = = = = K= = = = Figure 116. = 647. ~ O = O + O = 648. b = <N= ~ = 649. b ~ = =a ~ ~O = = = = 650. m~ ~ =c = ~ I= M O K = = = 153 CHAPTER 7. ANALYTIC GEOMETRY 651. d ~ =c ^ O + _ + ` O + a + b + c = M I== = _ O Q ^` < M K= = 652. d ~ =c = =^ =m~ ~ = = =` ~ =^ ^ O + ` O + a + b + c = M I== = ^` > M K = 653. ` i = Q~b( ) I== == == =b== == = == = ~ == = = = K== = 654. ^ ~ =c ~ = = =` i = NKR(~ + ) ~ I== ( i = O(~ O + O ) K= = 655. p = ~ = = = = ) 7.6 Hyperbola = q ~ =~ W=~= ` ~ =~ W= = c W= cN ( I M) I= cO ( I M) = a ~ = = = W=O = = b W= == ^ W= I= = o ~ = W=^I=_I=`I=aI=bI=cI= I= = = = = 154 CHAPTER 7. ANALYTIC GEOMETRY 656. b ~ = =~=e ~=Ep ~ ~ =c F= O O = N= ~ O O = = = Figure 117. = 657. N O = O~ I= == N I== O ==~ == ~ == ==~ = == m( I ) == = = ~= = = = K= = 155 CHAPTER 7. ANALYTIC GEOMETRY = = Figure 118. 658. 659. 660. 661. = b ~ = =^ = = = ~ = O = ~ O + O = = b = > N= ~ = b ~ = =a ~ ~O = = = = = = 156 CHAPTER 7. ANALYTIC GEOMETRY 662. m~ ~ =b ~ = = =o =_ ~ = =~=e ~= = ~ I= M O K = = 663. d ~ =c ^ O + _ + ` O + a + b + c = M I== = _ O Q ^` > M K= = 664. d ~ =c = =^ =m~ ~ = = =` ~ =^ ^ O + ` O + a + b + c = M I== = ^` < M K= 665. ^ =c = O = I== Q == O = I= = = K= Q f = = ~ = I= = ~ = ~ = ~ = = M = ~ = = M K== = 157 CHAPTER 7. ANALYTIC GEOMETRY = = Figure 119. = = = 7.7 Parabola = c ~ = ~ ~ W= = c W=c= s W= j( M I M ) = o ~ = W=^I=_I=`I=aI=bI=cI= I=~I= I= = = = 666. b ~ = =~=m~ ~ ~=Ep ~ ~ =c F O = O = 158 CHAPTER 7. ANALYTIC GEOMETRY = = Figure 120. = b ~ = = = = I= O ` ~ = = = = c I M I= O ` ~ = = = = j(MI M) K= = 667. d ~ =c ^ O + _ + ` O + a + b + c = M I== = _ O Q ^` = M K= = N 668. = ~ O I= = K= O~ b ~ = = = 159 CHAPTER 7. ANALYTIC GEOMETRY = I= O ` ~ = = = = c MI I= O ` ~ = = = = j(MI M) K= = = = Figure 121. = 669. d ~ =c I=^ =m~ ~ = = = -~ == ^ O + a + b + c = M =E^I=b= FI== N = ~ O + + I= = K== O~ b ~ = = = = M I= O ` ~ = = = = 160 CHAPTER 7. ANALYTIC GEOMETRY c M I M + I= O ` ~ = = = = Q~ O K= M = I= M = ~ O + M + = M O~ Q~ = = = 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= s W=s= = 161 CHAPTER 7. ANALYTIC GEOMETRY 670. a ~ =_ =q =m = = ^_ = = = ( O N )O + ( O N )O + ( O N )O = = === Figure 123. = 671. a =~=i =p = = =o~ = = + O + O + O M = N I= M = N I= M = N I== N+ N+ N+ = ^` = I= N K= `_ = 162 CHAPTER 7. ANALYTIC GEOMETRY ======== = = Figure 124. = = Figure 125. 163 CHAPTER 7. ANALYTIC GEOMETRY 672. j = =~=i =p = + O + O + M = N I= M = N I= M = N O I= = N K= O O O = 673. ^ ~= =~=q ~ = q =~ ~= =~= ~ = = = mN ( N I N I N ) I= mO ( O I O I O ) I=~ = mP ( P I P I P ) = = = == p= N O O O O N + O O N + O O N K= P P P P P N N N N P N N O N N N N O N N = 674. s = =~=q ~ = q = = =~= ~ = = = mN ( N I N I N ) I= mO ( O I O I O ) I= mP ( P I P I P ) I=~ = mQ ( Q I Q I Q ) = = = == N N N N N O O O N s= I== S P P P N Q Q Q N = N Q N Q N Q N s = O Q O Q O Q K= S P Q P Q P Q k W=t = = = =EHF= =E F= = ~ = = =~= = ~ = = K== = 164 CHAPTER 7. ANALYTIC GEOMETRY ==== = = 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= == rrr k ~ = W= I= N I= O = a = W= I= I= = a ~ = = = = ~ W= = = = 675. d ~ =b ~ = =~=m ~ = ^ + _ + ` + a = M = = = 165 CHAPTER 7. ANALYTIC GEOMETRY 676. k ~ =s = =~=m ~ = r q = = (^I _I ` ) = = ~ = = = ~ = ^ + _ + ` + a = M K= = = = === Figure 127. = 677. m~ ~ =`~ = = =b ~ = =~=m ~ = ^ + _ + ` + a = M = = f = ^ = M I= = ~ = = ~ ~ = = = -~ K= f = _ = M I= = ~ = = ~ ~ = = = -~ K= f = ` = M I= = ~ = = ~ ~ = = = -~ K= f = a = M I= = ~ = = = = K== = f = ^ = _ = M I= = ~ = = ~ ~ = = = - ~ K= f = _ = ` = M I= = ~ = = ~ ~ = = = - ~ K= f = ^ = ` = M I= = ~ = = ~ ~ = = = - ~ K= = 166 CHAPTER 7. ANALYTIC GEOMETRY 678. m =a =c = ^( M ) + _( M ) + `( M ) = M I== = = = m( M I M I M ) = = = = ~ I=~ = = = (^I _I ` ) = = ~ = = = ~ K=== = = = ==== Figure 128. = 679. f =c = + + = N= ~ = 167 CHAPTER 7. ANALYTIC GEOMETRY = ===== = Figure 129. = 680. q =m =c = P P P N P N P N P = M I== O P O P O P == N N N N N = M K= O O O N P P P N = 168 CHAPTER 7. ANALYTIC GEOMETRY = ===== = Figure 130. = 681. k ~ =c = + + = M I== == == == == ~ == ~ == == = = = = ~ =I=~ = I= I= =~ = = = = =~ = = ~ = = = ~ K== = 169 CHAPTER 7. ANALYTIC GEOMETRY = ====== = Figure 131. = 682. m~ ~ =c = = N + ~N + ~ O = N + N + O I== = + + N N O = ( I I ) =~ = = ~ = =~ = = = = = =I= = = m( N I N I N ) = = = = ~ I= = = (~N I N I N ) =~ = (~ O I O I O ) =~ = ~ ~ = = = ~ K= = 170 CHAPTER 7. ANALYTIC GEOMETRY = ===== = Figure 132. = 683. a ~ =^ =_ =q =m ~ = f = = ~ =~ = = == ^N + _N + `N + aN = M I== ^ O + _ O + ` O + aO = M I== = = ~ =~ = = = == rr N O ^N^ O + _N_ O + `N` O = r r = K= O O O N O ^N + _N + `N ^ O + _ O + ` O O O O = 171 CHAPTER 7. ANALYTIC GEOMETRY = ====== = Figure 133. = 684. m~ ~ =m ~ = q = ~ = ^N + _N + `N + aN = M =~ = ^ O + _ O + ` O + aO = M =~ = ~ ~ = == ^N _N `N = = K= ^ O _O `O = 685. m ~ =m ~ = q = ~ = ^N + _N + `N + aN = M =~ = ^ O + _ O + ` O + aO = M =~ = ~ = == ^N^ O + _N_ O + `N` O = M K= = 686. b ~ = =~=m ~ =q = m( N I N I N ) =~ =m~ ~ =q = =s = (~N I N I N ) =~ = (~ O I O I O ) =Ec KNPOF= 172 CHAPTER 7. ANALYTIC GEOMETRY N ~N ~O N N N N = M = O O = 687. b ~ = =~=m ~ =q = mN ( N I N I N ) =~ = mO ( O I O I O ) I= ~ =m~ ~ =q = =s = (~ I I ) = N N N O N ~ O N O N = M = = = Figure 134. = 688. a ~ =c =~=m =q =~=m ~ = q = ~ = = = = mN ( N I N I N ) = = = ~ = ^ + _ + ` + a = M = == 173 CHAPTER 7. ANALYTIC GEOMETRY = ^ N + _ N + ` N + a ^O + _O + `O K= = = = ====== Figure 135. = 689. f = =q =m ~ = f = = ~ = ^N + _N + `N + aN = M =~ = ^ O + _ O + ` O + aO = M = I= = = ~ = = = = = = N + ~ = N + I== = + N == N N N = = I== ~ == 174 CHAPTER 7. ANALYTIC GEOMETRY ~= _N _O `N ` I= = N `O `O aN aO aN aO ~ aN aO N = N = N = ^N ^ I= = N ^O ^O _N I== _O `N a _N N `O aO _ O I== O O ~ + + O ^N a `N ~ N ^O aO ` O I== O O ~ + + O _N a ^N N _O aO ^ O K== O O ~ + + O = = = 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= == rrr a = = =~= W= I= N I= O = r k ~ = = =~= ~ W= = ^ = = = W= = = 690. m =a =c = = =b ~ = =~=i == N N N = = I== ~ = = = mN ( N I N I N ) = = = = I=~ = (~ I I ) = = = = = = = K== = 175 CHAPTER 7. ANALYTIC GEOMETRY = ===== = Figure 136. = 691. q =m =c = N N N = = == O N O N O N = ===== Figure 137. 176 = CHAPTER 7. ANALYTIC GEOMETRY 692. m~ ~ =c == = N + = N + I== = + N = = = mN ( N I N I N ) = = = = ~ = I= I= I= =~ = = = = = = = = = = I= = ~ ~ = = =~ = ~ = K== = = ===== = Figure 138. = 693. ^ =_ =q =p ~ =i = rr ~N~ O + N O + N O = r N rO = = O O O N O ~N + N + N ~ O + O + O O O O = 177 CHAPTER 7. ANALYTIC GEOMETRY = ===== = Figure 139. = 694. m~ ~ =i = q = =~ = ~ ~ = == rr N O I== == ~ N N N = = K= ~ O O O = 695. m ~ =i = q = =~ = ~ ~ = == rr N O = M I== == ~N~ O + N O + N O = M K= = 696. f = =q =i = N N N = = =~ = q = = ~N N N 178 CHAPTER 7. ANALYTIC GEOMETRY O O O = = = = == ~O O O O N O N O N ~N N N = M K= ~O O O = 697. m~ ~ =i =~ =m ~ == N N N = = =~ = = ~ = q = ~ = = ~ ^ + _ + ` + a = M =~ = ~ ~ = = rr = M I== == ^~ + _ + ` = M K= = = ===== = Figure 140. = = 179 CHAPTER 7. ANALYTIC GEOMETRY 698. m ~ =i =~ =m ~ == N N N = = =~ = = ~ = q = ~ = = ~ ^ + _ + ` + a = M =~ = ~ = = rr I== == ^_` = = K= ~ = ==== = = Figure 141. = = 7.11 Quadric Surfaces = m = ~ = = = ~ = ~ W= I= I= = o ~ = W=^I=_I=`I=~I= I= I= N I O I P I= = 180 CHAPTER 7. ANALYTIC GEOMETRY 699. d ~ =n ~ ~ =b ~ = ^ O + _ O + ` O + Oc + Od + Oe + Om + On + Oo + a = M = 700. ` ~ ~ = =n ~ =p ~ = = `~ = o~ E F= o~ EbF= = = q = =p ~ = = N= P= Q= p~ = o ~ =b = <M= O= P= Q= p~ = f ~ ~ =b = > M= P= P= Q= e = =N=p = > M = a = Q= P= Q= e = =O=p = < M = a = R= S= T= U= V= NM= NN= NO= NP= NQ= NR= NS= NT= P= P= O= O= O= O= O= O= O= N= N= N= N= P= P= Q= Q= P= P= P= O= O= P= O= O= N= = = <M= > M= = = = = = = = = = a = p~ = p~ = a = p~ = p~ = a = a = p~ = = = = = o ~ =n ~ =` = f ~ ~ =n ~ =` = b =m~ ~ = e =m~ ~ = o ~ =b =` = f ~ ~ =b =` = e =` = o ~ =f =m ~ = f ~ ~ =f =m ~ = m~ ~ =` = o ~ =m~ ~ =m ~ = f ~ ~ =m~ ~ =m ~ = ` =m ~ = = e == ^ e n m ^ e d e _ c n = e _ c I= b = I= = (b ) I== d c ` o d c ` m n o a N I O I P =~ = = = = = ~ I== ^ e d e _ c = M K= d c ` = = 181 CHAPTER 7. ANALYTIC GEOMETRY 701. o ~ =b =E`~ =NF= O O O + + = N= ~ O O O = = ===== = Figure 142. = 702. f ~ ~ =b =E`~ =OF= O O O + + = N = ~ O O O = 703. e = =N=p =E`~ =PF= O O O + = N= ~ O O O = 182 CHAPTER 7. ANALYTIC GEOMETRY = ===== = Figure 143. = 704. e = =O=p =E`~ =QF= O O O + = N = ~ O O O = = = = Figure 144. 183 CHAPTER 7. ANALYTIC GEOMETRY 705. o ~ =n ~ =` =E`~ =RF= O O O + =M= ~ O O O = = == = Figure 145. = 706. f ~ ~ =n ~ =` =E`~ =SF= O O O + + =M= ~ O O O = 707. b =m~ ~ =E`~ =TF= O O + = M= ~ O O = 184 CHAPTER 7. ANALYTIC GEOMETRY ====== Figure 146. = 708. e =m~ ~ =E`~ =UF= O O = M= ~ O O ==== Figure 147. 185 = CHAPTER 7. ANALYTIC GEOMETRY 709. o ~ =b =` =E`~ =VF= O O + = N= ~ O O = = ===== = Figure 148. = 710. f ~ ~ =b =` =E`~ =NMF= O O + = N = ~ O O = 711. e =` =E`~ =NNF= O O = N= ~ O O = 186 CHAPTER 7. ANALYTIC GEOMETRY = ====== = Figure 149. = 712. o ~ =f =m ~ =E`~ =NOF= O O =M= ~ O O = 713. f ~ ~ =f =m ~ =E`~ =NPF= O O + = M= ~ O O = 714. m~ ~ =` =E`~ =NQF= O =M= ~O = 187 CHAPTER 7. ANALYTIC GEOMETRY = ===== = Figure 150. = 715. o ~ =m~ ~ =m ~ =E`~ =NRF= O = N= ~O = 716. f ~ ~ =m~ ~ =m ~ =E`~ =NSF= O = N = ~O = 717. ` =m ~ =E`~ =NTF= O = M = = = = = = 188 CHAPTER 7. ANALYTIC GEOMETRY 7.12 Sphere = o~ = =~= W=o= m = ~ W= I= I= I= N I= N I= N I= = ` = =~= W= (~ I I ) = o ~ = W=^I=aI=bI=cI=j= = = 718. b ~ = =~=p =` =~ = =l =Ep ~ ~ = c F= O + O + O = oO = = = ====== = Figure 151. = 719. b ~ = =~=` =` =~ =^ =m = (~ I I ) ( ~ )O + ( )O + ( )O = o O 720. a ~ =c ( N )( O ) + ( N )( O ) + ( N )( O ) = M I== 189 CHAPTER 7. ANALYTIC GEOMETRY == mN ( N I N I N ) I= mO ( O I O I O ) =~ = = = =~= ~ K== = 721. c =m =c = O + O + O N O O O N + N + N N N N N O + O + O O O O N = M = O O O O O O P + P + P P P P N O + O + O Q Q Q N Q Q Q = 722. d ~ =c ^ O + ^ O + ^ O + a + b + c + j = M =E^= = FK== q = = = = = ~ = ~ = (~ I I ) I= == a b c ~= I= = I= = K= O^ O^ O^ q = ~ = = = = aO + b O + c O Q ^ O j K o= O^ = = 190 Chapter 8 D ifferential Calculus = = = = c W= I= I= I= I= = ^ =E = ~ ~ FW= = o ~ = W=~I= I= I= = k~ ~ = W= = ^ W= = f = W= N = = = 8.1 Functions and Their Graphs 723. 724. 725. 726. = b =c = ( ) = ( ) = = l =c = ( ) = ( ) = = m =c = ( + q ) = ( ) = = f =c = = ( ) = =~ = I= = ( ) = = = N ( ) = = = = K== = 191 CHAPTER 8. DIFFERENTIAL CALCULUS = ===== = Figure 152. = 727. ` =c = = ( ) I= = ( ) I= = ( ( )) = =~= = K= = 728. i ~ =c = = ~ + I== o I== ~ = ~ == = = = = = I== == = = - K= = 192 CHAPTER 8. DIFFERENTIAL CALCULUS = ====== = Figure 153. = 729. n ~ ~ =c == = O I= o K= = = ====== = Figure 154. = 193 CHAPTER 8. DIFFERENTIAL CALCULUS 730. = ~ O + + I= o K= = = = === Figure 155. = 731. ` =c == = P I= o K= = 194 CHAPTER 8. DIFFERENTIAL CALCULUS = = = Figure 156. = 732. = ~ P + O + + I= o K= = 195 CHAPTER 8. DIFFERENTIAL CALCULUS = == = Figure 157. = 733. m =c == = I= k K= = 196 CHAPTER 8. DIFFERENTIAL CALCULUS ====== Figure 158. = = Figure 159. = 197 = CHAPTER 8. DIFFERENTIAL CALCULUS 734. p ~ =o =c == = I= [MI ) K= = ======= Figure 160. = = 735. b ~ =c = = ~ I= ~ > M I= ~ N I= = = = ~ = I= = OKTNUOUNUOUQSK = = = = = Figure 161. 198 CHAPTER 8. DIFFERENTIAL CALCULUS 736. i ~ =c = = ~ I= (MI ) I= ~ > M I= ~ N I= = = = ~ = I= > M K= = = Figure 162. = 737. e =p =c == = I= = I= o K= O = 199 = CHAPTER 8. DIFFERENTIAL CALCULUS = ==== = Figure 163. = 738. e =` =c == + = I= = I= o K= O = = ====== Figure 164. 200 CHAPTER 8. DIFFERENTIAL CALCULUS 739. e =q~ =c == = ~ I= = ~ = = I= o K= + = = ====== = Figure 165. = 740. e =` ~ =c == + = I= = = = I= o I= M K= = 201 CHAPTER 8. DIFFERENTIAL CALCULUS = ====== = Figure 166. = 741. e =p ~ =c == N O = I= = = = I= o K= + Figure 167. 202 = CHAPTER 8. DIFFERENTIAL CALCULUS 742. e =` ~ =c == N O = I= o I= M K= = I= = = = = ====== = Figure 168. = 743. f =e =p =c == = ~ I= o K= = 203 CHAPTER 8. DIFFERENTIAL CALCULUS = ===== = Figure 169. = 744. f =e =` =c == = ~ I= [NI ) K= = = ===== = Figure 170. = 745. f =e =q~ =c == = ~ ~ I= ( NI N) K= = 204 CHAPTER 8. DIFFERENTIAL CALCULUS = ===== = Figure 171. = 746. f =e =` ~ =c == = ~ I= ( I N) (NI ) K== = 205 CHAPTER 8. DIFFERENTIAL CALCULUS = ===== = Figure 172. = 747. f =e =p ~ =c == = ~ I= (MI N] K= = 206 CHAPTER 8. DIFFERENTIAL CALCULUS Figure 173. = = 748. f =e =` ~ =c == = ~ I= o I= M K== Figure 174. = 207 = CHAPTER 8. DIFFERENTIAL CALCULUS 8.2 Limits of Functions = c W= ( ) I= ( ) = ^ W= = o ~ = ~ W=~I= = = = 749. [ ( ) + ( )] = ( ) + ( ) = ~ ~ ~ = 750. [ ( ) ( )] = ( ) ( ) = ~ ~ ~ = 751. [ ( ) ( )] = ( ) ( ) = ~ = ~ ~ ( ) ( ) = ~ I= = ( ) M K= ~ ~ ( ) ( ) 752. ~ = 753. [ ( )] = ( ) = ~ ~ ( = ) 754. ( ( )) = ( ) = ~ ~ = 755. ( ) = (~ ) I= = = = ( ) = = =~ = = ~ K= ~ = = N= M 756. = 757. M ~ =N= = N = N= M 758. 208 CHAPTER 8. DIFFERENTIAL CALCULUS ~ N = N= M 759. = (N + ) = N= M 760. = N 761. N + = = = 762. N + = = = 763. ~ = N = M = = = 8.3 Definition and Properties of the Derivative = c W= I= I= I= I= = f = ~ ~ W= = o ~ = ~ W= = ^ W= = = = ( + ) ( ) 764. ( ) = == = = M M = 209 CHAPTER 8. DIFFERENTIAL CALCULUS = == = Figure 175. = = ~ == 765. = 766. = 767. = 768. ( + ) = + = ( ) = = ( ) = = = 769. m =o = ( ) = + == = = 210 CHAPTER 8. DIFFERENTIAL CALCULUS 770. n =o = = = O = 771. ` ~ =o = = ( ( )) I= = ( ) I== = K= = 772. a ~ = =f =c = N = I== = ( ) = = = = = ( ) K== = 773. o ~ =o = N = = O = 774. i ~ =a ~ = = ( ) I= = ( ) I== = ( ) [ ( )] K= = = 8.4 Table of Derivatives = f = ~ ~ W= = o ~ = ~ W=`I=~I= I= = k~ ~ = W= = 211 CHAPTER 8. DIFFERENTIAL CALCULUS (` ) = M = 775. = ( ) = N = 776. = (~ + ) = ~ = 777. = 778. = 779. = 780. (~ O + + ) = ~ + = ( ) = N = ( ) = +N = = N N = O = 781. = ( ) = O N = 782. ( )= = 783. N N = = ( ) = N = 784. = ( ~ ) = N I= ~ > M I= ~ N K= ~ 785. = 212 CHAPTER 8. DIFFERENTIAL CALCULUS (~ ) = ~ ~ I= ~ > M I= ~ N K= 786. = 787. ( ) = = = ( ) = = 788. = ( ) = = 789. = ( ~ ) = NO = O = 790. = ( ) = NO = O = 791. = ( ) = ~ = 792. = ( ) = = 793. = (~ ) = N O = N 794. = (~ ) = N O = N 795. = (~ ~ ) = N O = N+ 796. = 213 CHAPTER 8. DIFFERENTIAL CALCULUS (~ ) = N O = N+ 797. = (~ ) = NO = N 798. = (~ ) = NO = N 799. = ( ) = = 800. = ( ) = = 801. = ( ~ ) = N O = O = 802. = ( ) = N O = O = 803. = ( ) = ~ = 804. = ( ) = = 805. = (~ = ) = N = O +N 806. = 807. N (~ = ) = O = N 214 CHAPTER 8. DIFFERENTIAL CALCULUS (~ ~ = ) = N O I= < N K= N 808. = (~ = ) = ON I= > N K= N 809. = 810. ( ) = N + = = = = 8.5 Higher Order Derivatives 811. 812. 813. 814. 815. = c W= I= I= I= = f = ~ ~ W= = k~ ~ = W= = = = p = ~ = O = = = = = ( ) O = e -l = ~ = ( ) = = ( ) = ( ( N) ) = = ( + )( ) = ( ) + ( ) = = ( )( ) = ( ) ( ) = = i =c ~ = ( ) = + O + = 215 CHAPTER 8. DIFFERENTIAL CALCULUS ( ) = + P + P + = ( )( ) = ( ) + ( N) + ( N) ( O) + K + ( ) = N O = ( )( ) = ( > )> 816. = = = ( )( ) = > = 817. ( N) N ( N)> = ( ~ ) = ( ) 818. ~ = 819. N ( )( ) = ( N) ( N)> = = (~ )( ) = ~ ~ = ( )( ) = = 820. = 821. = (~ )( ) = ~ 822. ~ = = ( )( ) = + = 823. O = ( )( ) = + = 824. O = = = 216 CHAPTER 8. DIFFERENTIAL CALCULUS 8.6 Applications of Derivative = c W= I= I= = m = =~ = W= == s W= = ^ ~ W= = f = ~ ~ W= = q W= = k~ ~ = W= = = = 825. s =~ =^ ~ = = ( ) = = = = =~ = = ~ = =~= = ~ = =~ =~= = I== = = ( ) = = = ~ ~ = = = = I= = = = ( ) = = = ~ ~ =~ ~ = = = K== = 826. q~ =i = M = ( M )( M ) = = 217 CHAPTER 8. DIFFERENTIAL CALCULUS = = Figure 176. = 827. k ~ =i = N ( M ) =Ec =NTSF= M = ( M ) = 828. f ~ =~ =a ~ =c K== f = ( M ) > M I= = E F= = ~ =~ = M K=Ec =NTTI= < N I= O < FI= f = ( M ) < M I= = E F= = ~ =~ = M K=Ec =NTTI= N < < O FI= f = ( M ) = = = = = = I= = = = ~ K== = 218 CHAPTER 8. DIFFERENTIAL CALCULUS = = Figure 177. = 829. i ~ = ~= ^= = E F= ~ =~= ~ = ~ =~ = N = =~ = = = = = = ~ = ~ = N = = ~ = ( N ) ( ) = =~ = = = = ~ =Ec KNTTFK== = ^= = E F= ~ =~= ~ = =~ = O = =~ = = = = = = ~ = ~ = O = = ~ = ( O ) ( ) = =~ = = = = ~ =Ec KNTTFK= = 830. ` ~ =m = ^= ~ = = = E F= =~ = M = =~ = = = = ( M ) = = = = = ~ = = K= = 831. c =a ~ =q = =i ~ =b ~K= f = E F= == ~ ==E ( ) > M F= ==~ == == == == ~ = (~I N ] ==~ == E F== == ~ ==E ( ) < M F== =~ == = = = ~ = = [ N I ) I= = = E F= ~ = ~= = ~ = ~ = = ~ = = N = Ec KNTTFK== 219 CHAPTER 8. DIFFERENTIAL CALCULUS 832. f = E F= = ~ =E ( ) < M F= =~ = = = = ~ = (~I O ]=~ = E F= = ~ =E ( ) > M F= =~ = = = = ~ = [ O I ) I= = E F= ~ =~= ~ = =~ = O K== Ec KNTTFK= = 833. p =a ~ =q = =i ~ =b ~K= f = ( N ) = M =~ = ( N ) < M I= = E F= ~ =~= ~ = ~ = ~ == N K= f = ( O ) = M =~ = ( O ) > M I= = E F= ~ =~= ~ = = ~ = O K=Ec KNTTF= = 834. ` ~ K== E F= = = ~ = ~ = ~ = = M = = = = ~ = = = = = = ( ) = ============ ~ =~ = M =Ec KNTTI= P < FK=== E F= == ~ == ~ =~ == M == =~ = = == ( ) == =============== ~ =~ = M K=Ec KNTTI= < P FK=== = 835. p =a ~ =q = =` ~ K== f = ( M ) > M I= = E F= = ~ = ~ =~ = M K== f = ( M ) < M I= = E F= = ~ = ~ =~ = M K= f = ( ) = = = = = = I= = = = ~ K= = 836. f =m = f == ( P ) == ==~ == ( ) == ~ = =~ = = P I== = = = ( P I ( P )) = = ~ = = = = = ~ = = ( ) K=f = ( P ) = =~ = = = I= = ( P ) = M = Ec KNTTFK= = 837. i e ~ =o = M ( ) ( ) = = = ( ) = ( ) = K== ( ) ( ) = 220 CHAPTER 8. DIFFERENTIAL CALCULUS 8.7 Differential = c W= I= I= = f = ~ ~ W= = a ~ = =~= W= ( ) I= ( ) = o ~ = ~ W=`= a ~ = = = = ( ) W= = a ~ = = W= = p ~ = ~ = = W= = p ~ = ~ = = W= = = = 838. = = = 839. ( + ) = ( ) + ( ) = = = = Figure 178. 221 CHAPTER 8. DIFFERENTIAL CALCULUS 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= = = 222 CHAPTER 8. DIFFERENTIAL CALCULUS 847. p =l =m~ ~ =a ~ = O = I== = O O = = I== O O = I== = O = = K== f = = ~ =~ = I= == O O = K== = 848. ` ~ =o == f = ( I ) = ( ( I )) =E = =~= = = = ~ ~ = FI= == = ( ( I )) I= = ( ( I )) K== = + K== f = ( ) = ( ( )I ( )) I= = ( ) = = f = = ( ( I )I ( I )) I= == + K== + = I= = = 849. p ~ =` ~ = + = = = 223 CHAPTER 8. DIFFERENTIAL CALCULUS 850. i ~ =j~ ~=~ =j ~= ( I ) = ~ =~= ~ = ~ =~ = ( M I M ) = = ( I ) ( M I M ) = =~ = ( I ) = = = = ( M I M ) K== = ( I ) = ~ =~= ~ = =~ = ( M I M ) = = ( I ) ( M I M ) = =~ = ( I ) = = = = ( M I M ) K= = 851. p ~ ~ =m = = = M K= i ~ = ~ ~=~ = ~ = ~= =~ = ~ ~ = K= == 852. p~ =m = ^= ~ ~ == == == == ==~== ~ == ~ = =~= ~ = = = 853. p =a ~ =q = =p ~ ~ =m = = = M FK== i = ( M I M ) = =~= ~ ~ = =E ( M I M ) ( M I M ) K== a= ( M I M ) ( M I M ) = f = a > M I= ( M I M ) > M I== ( M I M ) == =~= = = ~ = ~K= f = a > M I= ( M I M ) < M I== ( M I M ) == =~= = = ~ = ~ ~K= f = a < M I= ( M I M ) = =~= ~ = K= f = a = M I= = = ~ K= = 854. q~ =m ~ = q = ~ = = = ~ = ~ = = = ~ = = ( I ) = ~ = ( M I M I M ) = == M = ( M I M )( M ) + ( M I M )( M ) K= = 224 CHAPTER 8. DIFFERENTIAL CALCULUS 855. k ~ = =p ~ = q = ~ = = = ~ = = = ~ = = ( I ) =~ = ( M I M I M ) = == M M M = = K= ( M I M ) ( M I M ) N = = = 8.9 Differential Operators = rrr r = =~ = = ~ =~ W= I= I= = p ~ ~ = =E ~ ~ = FW= ( I I ) I= ( N I O I KI ) = d ~ = =~= ~ ~ = W= ~ I= = a ~ = ~ W= = r s = =E = FW= c (mI nI o ) = r r a = =~= = W= c I= c = r r ` = =~= = W= c I= c = i~ ~ ~ = ~ W= O = = = 856. d ~ = =~=p ~ ~ =c = ~ = = I I I== K= ~ = = I IKI O N = 857. a ~ =a ~ = = + + I== 225 CHAPTER 8. DIFFERENTIAL CALCULUS = = = = = = = = r ( I I ) I= O + O + O = N K== = 858. a = =~=s =c = r r m n o + + = c = c = = 859. ` = =~=s =c = r r r r r = c = c = mno o n r m o r n m r = + + = = 860. i~ ~ ~ =l ~ = O O O O = O + O + O = = r r 861. c = c M = = 862. ( ~ ) = ( ) M = = 863. ( ~ ) = ( ) = O = = r r r r r 864. c = ~ c Oc = c Oc = = = ( ) ( ) ( ) ( ) 226 () Chapter 9 I ntegral Calculus = = = = c W= 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. = 866. = 867. = 868. = 869. ( ) = c( ) + ` = = c ( ) = ( ) K= ( ( ) ) = ( ) = ( ) = ( ) = [ ( ) + ( )] = ( ) + ( ) = [ ( ) ( )] = ( ) ( ) = = 870. N (~ ) = ~ c(~ ) + ` = 227 CHAPTER 9. INTEGRAL CALCULUS N (~ + ) = ~ c(~ + ) + ` = 871. = N ( ) ( ) = O ( ) + ` = O 872. = ( ) ( ) = ( ) + ` = 873. = 874. j = =p = ( ) = ( ( )) ( ) = = = ( ) K= = 875. f ~ = =m~ = = I== = ( ) I= ( ) =~ = ~ = K== = = = 9.2 Integrals of Rational Functions = ~ = ~ + ` = 876. = = 877. O +`= O = O = P +`= P = 878. +N + ` I= N K= +N = 879. = 228 CHAPTER 9. INTEGRAL CALCULUS (~ + ) +N + ` I= N K= (~ + ) = ~( + N) 880. = 881. = + ` = = N ~ + = ~ ~ + + ` = 882. = ~ + ~ + = + 883. ~ + + ` = O = + N ( + ~ )( + ) = ~ + ~ + ` I= ~ K= 884. = N ~ + = (~ + ~ ~ + ) + ` = 885. O = O N N O O ~ + = P O (~ + ) O~(~ + ) + ~ ~ + + ` = 886. = N (~ + ) = ~ 887. ~ + +`= = 888. N (~ + ) = ~ + ~ O O ~ + +`= = 889. (~ + ) O = N ~ ~ + + +`= O ~ + = 229 CHAPTER 9. INTEGRAL CALCULUS O N ~O +`= = P ~ + O~ ~ + (~ + )O ~ + 890. = (~ + ) 891. O = N N ~ + + O +`= ~ (~ + ) ~ = 892. N N +`= = O N O +N = N 893. O N N+ = +`= O N = N ~+ +`= = O O~ ~ ~ O 894. O = 895. ~ N +`= = O O~ + ~ ~ = N+ 896. O = ~ N + ` = = ~ O 897. O = 898. N = ~ N + ` = O ~ ~ + N = ( O + ~ O ) + ` = O +~ O = ~ + 899. O = N ~ ~ ~ + ` I= ~ > M K= ~ = 230 CHAPTER 9. INTEGRAL CALCULUS ~ + 900. O = ~ N O + + ` = O = N O = (~ + O ) O~ ~ + O + ` = 901. = ~ 902. O N ~ + = +`= OO O~ ~ = N O~ + O Q~ 903. O = + ` I= ~ + + O Q~ O~ + + O Q~ O Q~ > M K= = O O~ + ~ ~ + ` I= = + + Q~ O Q~ O O Q~ < M K= ~ 904. O = = = 9.3 Integrals of Irrational Functions = O ~ + + ` = = ~ + ~ ~ + = 905. O(~ O ) = ~ + + ` = P~ O ~ + = 906. = 907. P O (~ + ) O + ` = P~ = 231 CHAPTER 9. INTEGRAL CALCULUS 908. ~ + = P O(P~ O ) (~ + ) O + ` = O NR~ = ~ + ~ N = + ` I== ~ + + ~ ~ + ~ ~ > M K= ( + ) 909. = ~ + N = ~ ~ + ` I== ~ ~ + ~ ~ < M K= ( + ) 910. = N ~ + (~ + )( + ) = = + ~ ~ ( + ) + (~ + ) + ` I= ~ > M K= ~ 911. ~ + N = + = 912. ~ ( + ) ~ ~ ~ + ` I=E ~ < M I= > M FK== (~ + ) ~ = O ~ + = 913. = 914. (~ + )( + ) = O(U~ O NO~ + NR O O ) NMR P (~ + )P + ` = O O(U~ O Q~ + P O O ) = ~ + + ` = NR P ~ + = 915. N ~ + ~ = + ` I= ~ > M K= ~ + ~ ~ + + ~ = 232 CHAPTER 9. INTEGRAL CALCULUS 916. O ~ + = ~ ~ + ` I= ~ < M K= ~ ~ + ~ = ~ = + ~+ = (~ + )( ) (~ + )~ +`= ~+ 917. N+ = N O + ~ + ` = N (~ )( + ) + (~ + )~ + +`= ~+ = 918. = 919. = ( ~ )( ~ ) = O ~ 920. = 921. O ~ + O + = Q O Q~ O + ~ +`= U P O + Q~ ~ + O = = 922. ~ O + + ~ > M K= = N O~ + + O ~ (~ O + + ) + ` I== ~ = 923. ~ +` = ~ ~ + + O = N O~ + O ~ Q~ + ` I= ~ < M K= Q~ ~ = 924. O + ~ O = O O ~O + ~ + + O + ~ O + ` = O O = 233 CHAPTER 9. INTEGRAL CALCULUS 925. O + ~ O = N O O PO ( + ~ ) + ` = P = 926. O + ~ O = O (O O + ~ O ) O + ~ O = U ~Q + O + ~ O + ` = U = 927. O + ~O O + ~O = + + O + ~ O + ` = O = 928. +~ O O = + O + ~ O + ` = = 929. O + ~O = O + ~ O + ~ +`= ~ + O + ~O = 930. +~ O O = O + ~O + ` = = 931. O O + ~O = O O ~O + ~ + O + ~ O + ` = O O = 932. N = +`= O + ~O ~ ~ + O + ~O = 933. O ~ O = O O ~O ~ + O ~ O + ` = O O = 934. O ~ O = N O O PO ( ~ ) + ` = P 234 CHAPTER 9. INTEGRAL CALCULUS O ~O ~ = O ~ O + ~ ~ + ` = 935. O ~O O ~O = + + O ~ O + ` = O = 936. = 937. ~ O = + O ~ O + ` = O = 938. ~ O = O ~O + ` = O = 939. O O O ~O = ~ + + O ~ O + ` = O O O O ~ = N ~ = ~ + ` = ~ ~ 940. O O = ( + ~ ) 941. O ~O = N ~ +`= ~ +~ = ( ~ ) 942. ~ O O = N +~ +`= ~ ~ = 943. O O ~O = O ~O +`= ~O = ( 944. O ~O ) O P = ~O O ~O +`= = 235 CHAPTER 9. INTEGRAL CALCULUS ( 945. O ~ O ) O = P (O O R~ O ) O ~ O + = U P~ Q + + O ~ O + ` = U = 946. ~ O O = ~O O ~ O + ~ + ` = ~ O O = O O ~ = 947. NO (~ O )P O + ` = P = O O O ~ = 948. Q (O O ~ O ) ~ O O + ~U ~ + ` = U ~ = 949. ~O O = ~ O O + ~ +`= ~ + ~O O ~O O ~O O = ~ + ` = O ~ = 950. = 951. = ~ + ` = N O = 952. ~O O = +`= ~ = 953. ~ O O = ~O O + ` = = 954. O ~O O = O ~O ~ O + ~ + ` = O O ~ 236 CHAPTER 9. INTEGRAL CALCULUS ( + ~ ) 955. ~O O = N ~ +`= O ~+ = N ~+ +`= O ~ = ( ~ ) 956. ~O O = ( + ) 957. ~O O N = O ~ O ~ + ~ O + ` I= > ~ K= ~ ( + ) = ( + ) 958. ~ O O N = ~ O O + ~ O O ~ O O + ~ O + < ~ K= = 959. O ~O O = ~O O +`= ~O = (~ 960. O O ) O = P Q O (R~ O O ) ~ O O + P~ ~ + ` = U U ~ = (~ 961. O O ) P = O ~O ~O O +`= = = = 9.4 Integrals of Trigonometric Functions = = + ` = 962. = 963. = + ` = 237 + `I = CHAPTER 9. INTEGRAL CALCULUS 964. O = N O + ` = OQ = N + O + ` = OQ = 965. O = 966. P N N P = P + ` = P + ` = P NO Q = 967. P N N P = P + ` = P + + ` = P NO Q = = = ~ O + ` = 968. = = = ~ O + Q + ` = 969. = 970. O = O = + ` = = O = ~ + ` = = P = = P = = 971. O = 972. P N + ~ + ` = O O O O = 973. P N + ~ + + ` = O O O O Q = 974. N = Q O + ` = 238 CHAPTER 9. INTEGRAL CALCULUS 975. O N = P + ` = P = 976. O N = P + ` = P = 977. O O = N Q + ` = U PO = ~ = + ` = 978. = N = + ` = + ` = O 979. = O = ~ O + Q + ` = 980. = ~ 981. O = ~ + ` = = = + ` = 982. = N = + ` = + ` = O 983. = O = ~ + + ` = 984. O = 985. O = + ` = = 986. = ~ + ` = 239 CHAPTER 9. INTEGRAL CALCULUS 987. O N = + ~ + + ` = O Q = 988. O = N + ~ + ` = O = 989. O = ~ + ` = O = ( + ) = O( + ) 990. + ( ) + ` I= O( ) ( ) + ` I= O( ) O O K= = ( + ) = O( + ) 991. O O K= = ( + ) = O( + ) 992. O O K= = ~ = + ` = 993. = = + ` = 994. = = 995. +N +`= +N = +N 996. = +`= +N = 240 + ( ) + ` I= O( ) CHAPTER 9. INTEGRAL CALCULUS ~ = ~ + N O + ` = ~ = ~ 997. N O + ` = = 998. = ~ ~ = ~ ~ O ( N 999. O + N) + ` = = N 1000. ~ = ~ + ( O + N) + ` = O = = = 9.5 Integrals of Hyperbolic Functions = 1001. = + ` = = 1002. = + ` = = 1003. ~ = + ` = = 1004. = + ` = = 1005. O = ~ + ` = = 1006. O = + ` = = 1007. ~ = + ` = = 241 CHAPTER 9. INTEGRAL CALCULUS 1008. = + ` = = = = 9.6 Integrals of Exponential and Logarithmic Functions = 1009. = + ` = = ~ 1010. ~ = +`= ~ = ~ 1011. ~ = +`= ~ = ~ 1012. ~ = O (~ N) + ` = ~ = 1013. = + ` = = = + ` = 1014. = N 1015. = +N +`= O + N ( + N) = ~ ~ 1016. ~ = +`= ~ O + O = 242 CHAPTER 9. INTEGRAL CALCULUS 1017. ~ = ~ + ~ +`= ~ O + O = = = 9.7 Reduction Formulas = 1018. = N N = = = + I= N K= ( N) N N N 1019. = 1020. = = 1021. = = 1022. = = 1023. = N N O N = O N O I= N K= ( N) N N N O N + = O + N O I= N K= ( N) N = +N N 1024. = = + N O + = + = N +N 1025. = = + 243 CHAPTER 9. INTEGRAL CALCULUS N O = + = 1026. ~ = N ~ N + ~ O I= N K= N = 1027. = N N + O I= N K= N = 1028. = O ~ O O + I= N K= N N = N N O 1029. = N + = = O 1030. = + N O I= N K= ( N) N = N N O 1031. = N + = = O 1032. = + N O I= N K= ( N) N = +N N 1033. = = + N + O = + = N +N 1034. = = + 244 CHAPTER 9. INTEGRAL CALCULUS + N O = + = 1035. ~ = N ~ N ~ O I= N K= N = 1036. = N N O I= N K= N = 1037. = O ~ O O + I= N K= N N = 1038. = O O O + I= N K= N N = 1039. = +N N = +N +N = N = + I= N K= ( N) N N 1040. = 1041. = N = = 1042. = N = = 1043. = N = = 1044. = + N = = 1045. = N = 245 CHAPTER 9. INTEGRAL CALCULUS 1046. N = N +N +N = N + N N O +N = 1047. N = +N N +N N + = +N + N N O = 1048. ~ N = +N N +N ~ N = +N + N N+ O = 1049. = = ~ + ~ ~ ~ + = O~ 1050. = = O O (~ + + ) ( N)( Q~ )(~ O + + ) N O(O P)~ O (~ O + + ) N I= N K= ( N)( Q~ ) = ( 1051. +~ N K= O ) O = O( N)~ O ( O + ~ ) O N + O P O( N)~ O = ( 1052. O = = N O( N)~ O ( O ~ O ) O P I= N K= O O( N)~ ( O ~ O ) N ~ ) O = = = = = 246 ( O + ~O ) N I CHAPTER 9. INTEGRAL CALCULUS 9.8 Definite Integral = ~ ~ a = ~ = =~= W= ( ) I= ( ) I= = o ~ = W= ( ) == =N p ~ = ~ W= == ^ ~ W= c( ) I= d( ) == i = = ~ W=~I= I= I= = = = 1053. ( ) = ~ ( ) ~ M =N I== == = N I== N K== = = = Figure 179. = 247 CHAPTER 9. INTEGRAL CALCULUS 1054. N = ~ = ~ = ~ ~ 1055. ( ) = ( ) = = ~ ~ ~ ~ ~ ~ [ ( ) + ( )] = ( ) + ( ) = 1056. = [ ( ) ( )] = ( ) ( ) = 1057. = ~ 1058. ( ) = M = ~ = ~ ~ 1059. ( ) = ( ) = = ~ ~ 1060. ( ) = ( ) + ( ) = = ~ < < K= = 1061. ( ) M = = ( ) M = = [~ I ] K= ~ = 1062. ( ) M = = ( ) M = = [~ I ] K= ~ = = = = 248 CHAPTER 9. INTEGRAL CALCULUS 1063. c ~ ~ =q = =`~ = ( ) = c( ) ~ = c( ) c(~ ) = = c ( ) = ( ) K= ~ = 1064. j = =p == f = = ( ) I= == ~ ( ) = ( ( )) ( ) I== = = N (~ ) I= = N ( ) K= = 1065. f ~ = =m~ = = ( ) ~ = ~ ~ = 1066. q ~ ~ =o = N ~ ( ) = ( M ) + ( ) + O ( ) = O =N ~ = 249 CHAPTER 9. INTEGRAL CALCULUS = = Figure 180. = 1067. p =o == ~ ( ) = P [ ( M ) + Q ( N ) + O ( O ) + Q ( P ) + = ~ + O ( Q ) + K + Q ( N ) + ( )] I== == ~ = ~ + I= = MI NI OI KI K== = 250 CHAPTER 9. INTEGRAL CALCULUS = = Figure 181. = 1068. ^ ~=r =~=` = p = ( ) = c( ) c(~ ) I== ~ = c ( ) = ( ) K= = 251 CHAPTER 9. INTEGRAL CALCULUS = = Figure 182. = 1069. ^ ~=_ =q =` = p = [ ( ) ( )] = c( ) d( ) c(~ ) + d(~ ) I== ~ = c ( ) = ( ) I= d ( ) = ( ) K= = 252 CHAPTER 9. INTEGRAL CALCULUS = = Figure 183. = = = 9.9 Improper Integral = 1070. q = = ~ == ( ) = = ~ =~ = = ~ = ~ == ~= = = = I= ( ) == ~ == == == = = == = ===== = = ~ = [~ I ] K= = 1071. f = ( ) = =~= = = = [~ I ) I= == ( ) = ( ) K= ~ ~ = 253 CHAPTER 9. INTEGRAL CALCULUS = = Figure 184. = 1072. f = ( ) = =~= = = = ( I ] I= == ( ) = ( ) K= = = = Figure 185. 254 CHAPTER 9. INTEGRAL CALCULUS k =W=q = = ~ = =NMTNI=NMTO=~ = = = = = =~ =~ = X= = = ~ =~ = K= = ( ) = ( ) + ( ) = 1073. = = = Figure 186. = f = = = ~ = = I= = = = ~ = = = = = ~ = I= = = ~ = ( ) = = ~ ===== X= = = = K= = 1074. ` ~ =q = i == ( ) =~ == ( ) == == == == = = = ~ = [~ I ) K= p = ~ = M ( ) ( ) = = ~ = = = [~I ) K= 255 CHAPTER 9. INTEGRAL CALCULUS ~ ~ f = ( ) = = I= = ( ) = =~ = ===== I= ~ ~ f = ( ) = = I= = ( ) = =~ = K= = 1075. ^ =` = = ~ ~ f = ( ) = = I= = = ~ ( ) = =~ = K=== = 1076. a =f ~ = i = ( ) = =~= = = = = = = ~ == [~I ) = = = =~ = = K=q == ( ) = ~ M + ( ) = ~ = = = Figure 187. 256 CHAPTER 9. INTEGRAL CALCULUS 1077. i = ( ) = =~= = = =~ = ~ = == == = = ~ == [~ I ] == == == = == == = (~ I ) K=q = ( ) = ( ) + ( ) K= ~ M + M + ~ + = = = Figure 188. = = = 9.10 Double Integral = c = = = ~ ~ W= ( I ) I= ( I ) I= = a = ~ W= ( I ) I= ( I ) I= = o o ( ) o ~ = W= I = =N =N p ~ = ~ W= I= = o = = ~ W=oI=p== m ~ = ~ W= I= = 257 CHAPTER 9. INTEGRAL CALCULUS ^ ~W=^= p ~ =~ ~W=p= s = =~= W=s= j~ = =~= ~ ~W= = a W= ( I ) = c = W= j I= j = j = = ~W= f I= f I= fM = ` ~ = =~= ~ W=n= ` ~ = W= ( I ) = ` ~ = = = = ~ W= I= = ^ ~ = =~= W= = = 1078. a = =a =f ~ = q = = ~ = =~= ~ = [~I ] [ I ] = = = = == ( I ) ^ = [~ I ] [ I ] ( ) ( I ) ~ M ~ M =N =N I== = I = = = = = = ~ = ( N I ) N I I=~ = = N I= = N K= = === ( ) Figure 189. 258 = CHAPTER 9. INTEGRAL CALCULUS q = = ~ = =~= ~ = =o= == ( I ) ^ = ( I ) ^ I== o [~ I ] [ I ] = ~ = [~I ] [ I ] = ~ =oI== ( I ) = ( I ) = = ( I ) = = =o=~ = ( I ) = M = K= = = = Figure 190. = 1079. [ ( I ) + ( I )] ^ = ( I ) ^ + ( I ) ^ = o = 1080. o [ ( I ) ( I )] ^ = ( I ) ^ ( I ) ^ = o = 1081. o o o ( I ) ^ = ( I ) ^ I== o o = = =~= ~ K= = 1082. f = ( I ) ( I ) = =oI= = ( I ) ^ ( I ) ^ K= o = 1083. f = ( I ) M = =o=~ = p o I= = 259 o CHAPTER 9. INTEGRAL CALCULUS = ( I ) ^ ( I ) ^ K= p o = = = Figure 191. = 1084. f = ( I ) M = =o=~ =o=~ =p=~ = - ~ = I= = ( I ) ^ = ( I ) ^ + ( I ) ^ K== o p o p e = o p = = = = = = =o=~ =pK= = = = Figure 192. = = 260 CHAPTER 9. INTEGRAL CALCULUS 1085. f ~ =f ~ =~ =c =q = ( ) ( I ) ^ = ) ( I ) == ( o ~ =~= = = =fI== o = {( I ) ~ I ( ) ( )} K= = = = Figure 193. = ( ) ( I ) ^ = ( I ) == o ( ) =~= = = =ffI= o = {( I ) ( ) ( )I } K= = 261 CHAPTER 9. INTEGRAL CALCULUS = = Figure 194. = 1086. a =f ~ = =o ~ ~ =o = = f =o= = = ~ ~ = = [~ I ] [ I ] I= == ( I ) = ( I ) = ( I ) K== o ~ ~ = f = = ~ = ~ = = = ~ = ( I ) = ~ = = =~ = ( ) ( ) = = ~ == ( ) ( ) K== ( I ) = ( ) ( ) = o o ~ = 1087. ` ~ = =s~ ~ = ( I ) ( I ) = [ ( I )I ( I )] ( I ) I== o p ( I ) = = M = = ~ ~ = = = ~ ( I ) ~ = ( I ) ( I ) I=~ =p= = = ~ = =o= = 262 CHAPTER 9. INTEGRAL CALCULUS ~ = = = = = ( I ) I= = ( I ) = = = = =oK== = 1088. m ~ =` ~ = = I= = K== = = = Figure 195. = 1089. a =f ~ = =m ~ =` ~ = = q =a ~ = = =m ~ =` ~ = == ( I ) = = K== ( I ) = i = = =o= = =~ = W= M ( ) ( ) I= I= = O K== q == ( ) ( I ) = ( I ) K= o ( ) = 263 CHAPTER 9. INTEGRAL CALCULUS = = Figure 196. = f = = =o= = = ~ = ~ = = == M ~ I= I= = O I=== == ( I ) = ( I ) K== o ~ = = = Figure 197. = = 264 CHAPTER 9. INTEGRAL CALCULUS 1090. ^ ~= =~=o = ( ) ^ = =E =~= =f= FK= ~ ( ) = = = Figure 198. = ( ) ^ = =E =~= =ff= FK= ( ) = = = Figure 199. = = 265 CHAPTER 9. INTEGRAL CALCULUS 1091. s = =~=p = s = ( I ) ^ K== o = = = Figure 200. = f =o= =~= =f= = = = = ~ I= = I= = ( ) I= = ( ) I= == ( ) s = ( I ) ^ = ( I ) K== o ~ ( ) = f =o= =~= =ff= = = = = I= = I= = ( ) I= = ( ) I= = ( ) s = ( I ) ^ = ( I ) K== o ( ) = 266 CHAPTER 9. INTEGRAL CALCULUS f == ( I ) ( I ) == ==~== ==oI== == == == = = = = N = ( I ) = ~ = O = ( I ) = = o= = = = s = [ ( I ) ( I )] ^ K== o = 1092. ^ ~=~ =s = =m ~ =` ~ = f =p= =~= = = = - ~ = = = = I= = I= = ( ) I= = ( ) I== == ( ) ^ = ^ = I== p ( ) s = ( I ) K== p = = = Figure 201. = 1093. p ~ =^ ~= p = o O N + + = O = 267 CHAPTER 9. INTEGRAL CALCULUS 1094. j~ = =~=i~ ~= = ( I ) ^ I== o == == ~ ~== ==~== ==o=~ == == ==~ = ~= =E I F= = ( I ) K=== = 1095. j = q = = = = ~ ~=~ = == -~ == = = = ~= j = ( I ) ^ K== o = q = = = = ~ ~=~ = = -~ = = j = ( I ) ^ K= o = q = = = ~=~ = = -~ = = f = O ( I ) ^ K== o = q = = = ~=~ = = -~ = = f = O ( I ) ^ K== o = q = ~ = = = ~= = fM = ( O + O ) ( I ) ^ K== o = 1096. ` = =j~ = j N = = ( I ) ^ = o ( I ) ^ o ( I ) ^ o 268 I== CHAPTER 9. INTEGRAL CALCULUS j N = = ( I ) ^ = o ( I ) ^ o ( I ) ^ K== o = 1097. ` ~ = =~=m ~ = n = ( I ) ^ I== o = ~ = ~ = = = =~= =o=~ = = ~ = =~ =~= =E I F= = ( I ) K=== = 1098. ^ ~ = =~=c = N = ( I ) ^ I== po = p = ^ K== o = = = 9.11 Triple Integral = c = = = ~ ~ W= ( I I ) I= ( I I ) I= = q = ~ W= ( I I ) s I= ( I I ) s I= = d d ( ) o ~ = W= I I = =N =N =N p ~ = ~ W= I= I= = i = = ~ W=~I= I= I= I= I= = o = = ~ W=dI=qI=p== ` ~ = ~ W= I= I= = p ~ = ~ W= I= I= = s = =~= W=s= 269 CHAPTER 9. INTEGRAL CALCULUS j~ = =~= W= == a W= ( I I ) = ` ~ = = = = ~ W= I= I= = c = W= j I= j I= j = j = = ~W= f I= f I= f I= f I= f I= f I= fM = = = 1099. a = =q =f ~ = q = = ~ = =~= ~ ~ = [~I ] [ I ] [ I ] = = = = == ( I I ) s = [~ I ] [ I ] [ I ] ( ) ) ( I I ) ~ M ~ M =N =N =N ~ M I = I I = = = = = = ~ ~ = ( N I ) N I ( N I ) I=~ = = N I= = N I= = N K= = 1100. [ ( I I ) + ( I I )] s = ( I I ) s + ( I I ) s ( d d d = 1101. [ ( I I ) ( I I )] s = ( I I ) s ( I I ) s d d d = 1102. ( I I ) s = ( I I ) s I== d d = = =~= ~ K= = 1103. f == ( I I ) M =~ =d=~ =q=~ = ~ = ~ = I= == ( I I ) s = ( I I ) s + ( I I ) s K== d q d q e = d q = = = = = = =d=~ =qK= = 270 CHAPTER 9. INTEGRAL CALCULUS 1104. b ~ ~ = =q =f ~ = =o ~ =f ~ = f = = =d= = = = = = ( I I ) = = ~ = = ( I ) oI N ( I ) O ( I ) I= == O ( I ) = ( I I ) = ( I I ) I== d o N ( I ) =o= = = =d= = = - ~ K= = f = = =d= = = = = = ( I I ) = = ~ = ~ I N ( ) O ( )I N ( I ) O ( I ) I= == d O ( ) O ( I ) ( I I ) = ( I I ) == ~ N ( ) N ( I ) = 1105. q =f ~ = =m~ ~ = f =d= =~= ~ ~ = [~ I ] [ I ] [ I ] I= = ( I I ) = ( I I ) K== d ~ = f == = ~ = ~ == = = ~ == ( I I ) == ~ = = =~ = ( ) ( ) ( ) = = ~ == ( I I ) = ( ) ( ) ( ) K== d ~ = 1106. ` ~ = =s~ ~ = ( I I ) = = d = [ ( I I )I ( I I )I ( I I )] p 271 ( I I ) I = ( I I ) CHAPTER 9. INTEGRAL CALCULUS ( I I ) = M == == == ~ ~ == = == ( I I ) = ~ ~ = ( I I ) ( I I ) I= ~ = p= = = ~ = = d= = ~ = = = = = ( I I ) I= = ( I I ) = = ( I I ) = = = = =dK= = = 1107. q =f ~ = =` ~ =` ~ = q = ~ = = = ~ = ~ = == ( I I ) = = K== ( I I ) = i = = =d= = =~ = W= ( I ) oI N ( I ) O ( I ) I= =o= = = =d= = = - ~ K=q == ( I I ) = ( I I ) = d p O ( I ) ( I I ) K= o ( I ) N ( I ) e =p= = = ~ = =d= = ~ = ~ K= = 1108. q =f ~ = =p ~ =` ~ = q =a ~ = = =p ~ =` ~ = == ( I I ) = = O == ( I I ) = ( I I ) = = = d 272 CHAPTER 9. INTEGRAL CALCULUS ========= = ( I I ) O I== p = = =p= = = ~ = =d= = ~ = ~ K=q =~ == == ~ == ==M= == O I== ==~ == = ~ = =M= = K== = = ======= = Figure 202. = 1109. s = =~=p = s = = d = 1110. s = =` ~ =` ~ = s = = p ( I I ) = 1111. s = =p ~ =` ~ = s = O = p ( I I ) = 273 CHAPTER 9. INTEGRAL CALCULUS 1112. j~ = =~=p = = ( I I ) s I== d = = = = ~= = d= ~ = = = ~ ============ ~= = ( I I ) = = ( I I ) K=== = 1113. ` = =j~ = =~=p = j j j = I= = I= = I== == j = ( I I ) s I== d j = ( I I ) s I== d j = ( I I ) s = d ~ == == == ==~ == == ~ = ~ = = M I= = M I= = M I= I== ( I I ) = = = = K== = 1114. j = =f ~=~ == == - ~ =E = = M FI= - ~ == E = M FI=~ = - ~ =E = M F= f = O ( I I ) s I== d f = O ( I I ) s I== d f = O ( I I ) s K== d = 1115. j = =f ~=~ = = -~ I= -~ I=~ = -~ = f = f + f = ( O + O ) ( I I ) s I== d f = f + f = ( O + O ) ( I I ) s I== d 274 CHAPTER 9. INTEGRAL CALCULUS f = f + f = ( O + O ) ( I I ) s K== d = 1116. m ~ =j = =f ~= fM = f + f + f = ( O + O + O ) ( I I ) s = d = = 9.12 Line Integral = p ~ ~ = W= c( I I ) I= c( I ) I= ( ) = p ~ ~ = ~ W= ( I I ) = ` W=`I= `N I= ` O = i = = ~ W=~I= I= I= = m~ ~ W= I= = m ~ = ~ W= I= = r s = W= c (mI nI o ) = r m = W= ( ) = rrrr r = W= I= I= I= = ^ ~= = W=p= i = =~= W=i= j~ = =~= W= = a W= ( I I ) I= ( I ) = ` ~ = = = = ~ W= I= I= = c = W= j I= j I= j = j = = ~W= f I= f I= f = s = =~= W=s= t W=t= r j~ = W= _ = ` W=f= b = W= = j~ = W= = 275 CHAPTER 9. INTEGRAL CALCULUS 1117. i =f ~ = =~=p ~ ~ =c = rr i =~= =`= = = = = = = = ( ) I= M p I=~ =~= ~ ~ = =c= = = = = =`K== q == p r c( ( )) = c( I I ) = c I== M ` ` = = = =~ = = ~ K== = 1118. c = c + c = `N ` O `N `O = = = Figure 203. = rr 1119. f = = = =`= = ~ ~ = = = ( ) I= I= == c( I I ) = c( ( )I ( )I ( )) ( ( )) + ( ( )) + ( ( )) K= O O O ` = 1120. f =`= =~= = = = = - ~ = = = = ~ = = ( ) I= ~ I= == c( I ) = c( I ( )) N + ( ( )) K== O ` ~ = 1121. i =f ~ = =p ~ ~ =c = =m ~ =` ~ = 276 CHAPTER 9. INTEGRAL CALCULUS O c( I ) = c( I ) + I== ` = = =`= = = = = ~ = = E F K= = 1122. i =f ~ = =s =c = rr i =~= =`= = = = = = = = ( ) I= M p K=q == r r = = ( I I ) == = = = = = = ~ = = = = K== = O = = Figure 204. = r i = ~= = = c (mI nI o ) = = = = = = `K= r q = = = ~ = = = = = c =~ = = = `= == p m + n + o = (m + n + o ) K= ` M = = 277 CHAPTER 9. INTEGRAL CALCULUS 1123. m = =i =f ~ = =s =c = rr rr c = c I== ( ) ` ( ) ` =-`= == = = = = = ~ K= = rr rr rr rr c = c = c + c I== ( ) ( )( `N ` O ` )( `N ) `O =`= = = = = = = `N =~ = ` O K== = r 1124. f = = =`= = ~ ~ = = ( ) = ( )I ( )I ( ) I= I= == m + n + o = = ` = m( ( )I ( )I ( )) + n( ( )I ( )I ( )) + o( ( )I ( )I ( )) = 1125. f =`= = = = - ~ =~ = = = = ~ = = ( ) I= == m + n = m( I ( )) + n( I ( )) K== ` ~ = 1126. d =q = n m = m + n I== o ` r r r = c = m( I ) + n( I ) = = ~= = = m n = == == = ~ ~ == ~ = I= = =~= = ~ = oI= = = = = ~= I= = = =`K== = = 278 CHAPTER 9. INTEGRAL CALCULUS 1127. ^ ~= =~=o =o=_ = = =` =`= N p = = = O` o = 1128. m~ =f = =i =f ~ = r r r r q = = ~ = = ~= = = c = m + n + o = = ~ = = = ~ = I= =~ = = =mI=nI=~ =o=~ = = = ~= ~ = aI= ~ = = = = = ~ ~ = == = ( I I ) ==E~== ~ ~ == ~ F== ==a== = ~ = r = n I= c = ~ I= = = o K== = m I= q == rr r c( ) = m + n + o = (_) (^ ) K== ` ` = 1129. q = =~=` ~ =c = r ^= = = = = = c = ~ = = ~ =~= ~ = r r r r K=q = = ~ = =~= = = c = m + n + o = = ~ = = =~ = = == r r r r r c = = M K== mno = f = = = ~ = = ~ = = - ~ = = ~ == m + n = (_) (^ ) I== ` = = = = = =~= = = = ~ = ~ = = = = = == m n = K== = 279 CHAPTER 9. INTEGRAL CALCULUS 1130. i = =~=` = r O O O ( ) = + + I= i = = ` =`= ~=~= = = = = = = r = = ( ) I= K= = f = = =`= = - ~ I= = r O O ( ) = + K== i = = ` = f = = =`= = = ~ = =~= = = ( ) = = = ~ = (~ ) I= == i= ~ O N + K== = 1131. i = =~=` = =m ~ =` ~ = O i = + O I== = = =`= = = = = ~ = = ( ) I= = = ~ = ~ K=== = 1132. j~ = =~=t = = ( I I ) I== ` = ( I I ) = = = ~ = = = = = = K= = f =`= =~= = ~ ~ = = = = r ( ) = ( )I ( )I ( ) I== == == ~ == ~ == == = = = ~= 280 CHAPTER 9. INTEGRAL CALCULUS O O O = ( ( )I ( )I ( )) + + K== = f =`= =~= = = - ~ I= = = ~ = = = = = = == = ( I ) I= ` = = ( ( )I ( )) O O + =E = ~ ~ = FK= = 1133. ` = =j~ = =~=t = j j j = I= = I= = I= == j = ( I I ) I== ` j = ( I I ) I== ` j = ( I I ) K= ` = 1134. j = =f ~= q = = = ~=~ = = -~ I= -~ I=~ = -~ = ~ = = = = ~ = f = ( O + O ) ( I I ) I== ` f = ( O + O ) ( I I ) I== ` f = ( O + O ) ( I I ) K== ` = 281 CHAPTER 9. INTEGRAL CALCULUS 1135. ^ ~= =~=o =_ = =~=` =` = N p = = = K== O` ` ` = = = Figure 205. = f = = = =`= = = = ~ ~ = = r ( ) = ( )I ( ) I= = =~ ~= ~ = = ~ ~ = = = ~= N p = ( ) = ( ) = ( ) ( ) K== O = 1136. s = =~=p =c = =o ~ =~=` =` = ~ = = -~ = s = O = O = O + O == O` ` ` = 282 CHAPTER 9. INTEGRAL CALCULUS = = Figure 206. = 1137. t = r t = = =~= = c = =~ = = =~ =~= = `= = = = = = ~ = rr t = c I= ` r r = c = = = = = =~ = = = I= = = = = ~ = K== = = Figure 207. 283 CHAPTER 9. INTEGRAL CALCULUS f = = = = =~ =~= =`= = = - ~ I= = rr t = c = m + n I== ` ` = f = ~= ~ = `= = = = ~= ~ ~ = = E = = ~ = FI= = ~= = ~ ~ = = = t = m( ( )I ( )I ( )) + n( ( )I ( )I ( )) + o( ( )I ( )I ( )) I = = = = = = K== = r f = ~= = = c = = ~ = ~ = ( I I ) = = ~= ~ ~ = ~ = = = I= = = = = ~ = = = =^= =_= ~ = = = = = ~= t = (_ ) (^ ) K== = 1138. ^ =i~ = rr _ = M f K== ` r q = = ~ = =~= ~ = = _ =~ =~= = ~ = `= = ~ = = = ~ = = f= = = = ~ ~= = = = ~ K== = Figure 208. 284 = CHAPTER 9. INTEGRAL CALCULUS 1139. c~ ~ ~ =i~ = rr = = b = ` = q = = = E F= = = ~ = ~= = =`= = ~ = = = ~ = = = ~ = = ~ = = = ~ = = = K=== = = = Figure 209. = = = 9.13 Surface Integral = p ~ ~ = W= ( I I ) I= ( I ) = r r m = W= ( I ) I= ( I I ) = rrr r = W= I= I= = p ~ W=p= r s = W= c (mI nI o ) = r r a = =~= = W= c = c = 285 CHAPTER 9. INTEGRAL CALCULUS r r ` = =~= = W= c = c == r s = = =~= ~ W= p = r k ~ = = ~ W= = p ~ =~ ~W=^= j~ = =~= ~ W= = a W= ( I I ) = ` ~ = = = = ~ W= I= I= = c = W= j I= j I= j = j = = ~W= f I= f I= f I= f I= f I= f = s = =~= W=s= r c W= c = d ~ ~ ~ = ~ W=d= rr c = W= ( ) = c = W= = r m W= ( ) = j~ = I= = W= = p ~ = ~ W=n= ` ~ = W= ( I ) = r j~ = = = = W= b = = = 1140. p ~ =f ~ = =~=p ~ ~ =c = i =~= ~ =p= = = = = = = r r r r ( I ) = ( I ) + ( I ) + ( I ) I== = ( I ) = ~ = = = ~ = a( I ) = = = ~ K= q == ~ == ~ == ==~== ~ ~ = == ( I I ) = = = ~ =p= = =~ == r r ( I I ) p = ( ( I )I ( I )I ( I )) I== p a( I ) r r = = ~ ~ = ~ = =~ = =~ = = == 286 CHAPTER 9. INTEGRAL CALCULUS r r r r = ( I ) + ( I ) + ( I ) I== r r r r = ( I ) + ( I ) + ( I ) = r r ~ = = = = = K== = 1141. f == == ~ ==p== == = == = ~ = = ( I ) = = ( I ) == ==~== ~ == == = = ~ = a( I ) I= == ( I I ) p = ( ) ( I I ( I )) p a I O N + + K== O = r 1142. p ~ =f ~ = = =s =c = c = = =p ~ =p= f =p= = = ~ I= == r r r r ===== c( I I ) p = c( I I ) p = p ====== p r r r = c( ( I )I ( I )I ( I )) K== a( I ) = f =p= = = ~ I= == r r r r ===== c( I I ) p = c( I I ) p = p ====== = p r r r = c( ( I )I ( I )I ( I )) K== a( I ) rr p = p == == ~ == == = = = = ~ K==a = ~ = = = = ~ ~ = = = = = = = = ~ ~ = = K= r r q = ~ ~ = ~ = =~ == =~ = = == 287 CHAPTER 9. INTEGRAL CALCULUS r r r r = ( I ) + ( I ) + ( I ) I== r r r r = ( I ) + ( I ) + ( I ) K== = 1143. f == == ~ ==p== == = = = ~ = = ( I ) I= = ( I ) == ==~== ~ == == == = ~ = a( I ) I= == f = p= = = ~ I= K K= = - = = = = ~ = = = I= === r r r r ===== c( I I ) p = c( I I ) p = p p = ====== r r r r c( I I ) + I== a( I ) = f =p= = = ~ I= K K= = - = = = = ~ = = = ~ I= === r r r r ===== c( I I ) p = c( I I ) p p p r r r r = c( I I ) + K== a( I ) ====== = 1144. (c ) p = m + n + o = rr p p = (m + n + o ) p I== p = m( I I ) I= n( I I ) I= o( I I ) =~ = = = = r = = = c K== I= I= ==~ = = ~ = = = = = r ~ = = =~ = = -~ I= -~ I=~ = -~ I= K= = 288 CHAPTER 9. INTEGRAL CALCULUS 1145. f = = ~ =p= = = = ~ ~ = = = = = r ( ( I )I ( I )I ( I )) I== == == ~ = ~= ~ = = =~ == mno I ) I p p a( = ( I ) = ~ = = = ~ = a( I ) = = = ~ K= = 1146. a =q = rr r c p = c s I== () rr c p = m + n + o = ( p ) d == r c( I I ) = m( I I )I n( I I )I o( I I ) === ==~== == == == ==mI==nI==~ ==o== ~ == = ~ ~ = ~ I== r m n o c = + + == r r == == == == c I==~ == == c K==q == = = ~ == ~ = = ~ = ~ = = ~ = =~= = ~ K== = 1147. a =q = =` ~ =c = m n o m + n + o = + + K== p d = 1148. p =q = rr rr c = c p I== ( ` ) p 289 CHAPTER 9. INTEGRAL CALCULUS == r c( I I ) = m( I I )I n( I I )I o( I I ) === ==~= == == == ==mI==nI==~ =o== ~ = = ~ ~ = ~ I== r r r r o n r m o r n m r + c= = + mno r r = = = = c I=~ = = c K== q = == = ~ = ~ = = = ~ = = ~ = = ~= = K== = 1149. p =q = =` ~ =c = m + n + o = ` o n n m m o = + + p == 1150. p ~ =^ ~= ^ = p = p = 1151. f = = ~ =p= = ~ ~ = = = == r r r r ( I ) = ( I ) + ( I ) + ( I ) I== = = ~ =~ ~= == r r ^ = I== a( I ) r == a( I ) == == == ~ == == = ~ = ( I ) = = K== = 290 CHAPTER 9. INTEGRAL CALCULUS 1152. f =p= = = = = = = ( I ) I== == == ~ =~ ~= == O ^ = N + + I== a( I ) = a( I ) = = = = = = ~ =p= = = ~ K== = 1153. j~ = =~=p ~ = = ( I I ) p I== O p = ( I I ) = = = ~ = = = ~ ~= = E = FK= = 1154. ` = =j~ = =~=p = j j j = I= = I= = I== == j = ( I I ) p I== p j = ( I I ) p I== p j = ( I I ) p = p ~ == == == ==~ === = ~ = ~ = = M I= = M I= = M I== K== ( I I ) = = = = K= = 1155. j = =f ~=~ = = - ~ =E = = M FI== - ~ == E = M FI=~ = - ~ =E = M F= f = O ( I I ) p I== p f = O ( I I ) p I== p 291 CHAPTER 9. INTEGRAL CALCULUS f = O ( I I ) p K= p = 1156. j = =f ~=~ = = -~ I= -~ I=~ = -~ = f = ( O + O ) ( I I ) p I== p f = ( O + O ) ( I I ) p I== p f = ( O + O ) ( I I ) p K== p = 1157. s = =~=p =_ = =~=` =p ~ = s= N + + == P p = 1158. d ~ ~ ~ =c = r r c = d ( I I ) P p I= p = = =~= ~ =~ =~= = M I M I M = = = ~ I== r = M I M I M I== ( I I ) = = = = I== ~ =d= = ~ ~ ~ = ~ K= = 1159. m =c = r rr c = ( ) p I== p r = = == ( ) ==~ == == == ~ ==p== == = r = = = K= = 1160. c =c =E~ = = ~ =pF= rr r = ( ) p I== p 292 CHAPTER 9. INTEGRAL CALCULUS rr = ( ) = = = = K= == 1161. j~ =c =E~ = = ~ =pF= rr r = ( ) p I== p r r = c = = = = = I= = = = = K= = 1162. p ~ =` ~ = n = ( I ) p I== p = ( I ) = = = ~ = ~ = K= = 1163. d~ =i~ = q = = = =~ = = ~ = = ~ = = = ~ =n= = = = ~ = rrn = b p = I== M p == = = = = I== r b = = = ~ = = = = = I= c M = UIUR NM NO = = = = = ~ K== = = 293 Chapter 10 D ifferential Equations = = = = c = = = ~ ~ W= I= I= I= I= I= I=dI=eI= I= == ^ =E = ~ ~ FW= I= = c = = = ~ ~ W= ( I ) I= j( I ) I= k( I ) = & c = = ~ W= I= I= I= I= = Of p = = ~ W= I= && I= O I= = O m~ ~ = ~ W= I= O I= = k~ ~ = W= = m~ ~ = W= N I= = o ~ = W= I= I=`I= `N I= ` O I= I= I= I= = o = = = ~ ~ = ~ W= N I= O = q W= = q ~ W=qI=p= m ~ = W= m( ) == j~ = =~ = W= = p = =~= W= = a ~ = = = ~ = = W= = ^ = = = ~ W=^= c W= = a~ = W= = m ~ =~ = = = ~ W= = ^ ~ = ~ W= = m = W=i= 294 CHAPTER 10. DIFFERENTIAL EQUATIONS ^ ~ = = ~ W= = ` W=f= o ~ W=o= f ~ W=i= `~ ~ ~ W=`= = = 10.1 First Order Ordinary Differential Equations = 1164. i ~ =b ~ = + ( ) = ( ) K== = q = ~ = = == ( ) ( ) + ` I== = ( ) == ( ) = ( ) K= ( ) = 1165. p ~ ~ =b ~ = = ( I ) = ( ) ( ) = = q = ~ = = = = = ( ) = ( ) + ` I== = e( ) = d( ) + ` K= = = = 295 CHAPTER 10. DIFFERENTIAL EQUATIONS 1166. e =b ~ = = ( I ) = = I= = = = ( I ) = = I= ~ = == ( I ) = ( I ) K== = q = = = =E = = F= ~ = = = ~ ~ = ~ = + = (NI ) K== = 1167. _ =b ~ = + ( ) = ( ) K== = q = = = N = ~ = = = ~ = ~ == + (N ) ( ) = (N ) ( ) K== = 1168. o ~ =b ~ = = ( ) + ( ) + ( ) O = = f =~= ~ ~ = = N = = I= = = ~ = = ~ = = ~ = = = = = = N = I= = ~ = = = = = ~ = ~ == N = [ ( ) + O N ( )] ( ) K== = = = q = ~ = ~ = 296 CHAPTER 10. DIFFERENTIAL EQUATIONS 1169. b ~ =~ =k ~ =b ~ = q = ~ == j( I ) + k( I ) = M == = ~ = ~ = == j k = I== ~ = ~ = K= = q = ~ = = == j( I ) + k( I ) = ` K== = 1170. o~ ~ =a ~ = = I== = ( ) = = =~ = = ~ ~ = =~ = = I= = = = ~ = = ~ K== = q = = == ( ) = M I= = M = (M) = = = ~ =~ K= = 1171. k =i~ = =` = q = (q p ) I== = q( ) = = = ~ = =~ = =~ = = I=p= = = ~ = = = = I= = = ~= = ~ K== = q = = = q( ) = p + (qM p) I== = qM = q(M) = = = ~ = ~ = = = = ~ = = = M K== = = 297 CHAPTER 10. DIFFERENTIAL EQUATIONS 1172. m ~ =a ~ =Ei =j F= m m = m N I== j = m( ) = = ~ =~ = = I= = =~= = ~ I= j= =~= = = = = ~ K== = q = = = = ~ = ~ = == jmM m( ) = I= = mM = m(M) = = = ~ = mM + (j mM ) ~ =~ = = = M K=== = = = 10.2 Second Order Ordinary Differential Equations = 1173. e =i ~ =b ~ = =` ~ =` == + + = M K== q = ~ ~ = ~ = == O + + = M K== = f = N = ~ = O = ~ = = ~ = = = = ~ ~ = ~ I= = = ~ = = == = `N N + ` O O I= == `N =~ = ` O =~ = ~ = ~ K== == f = N = O = I= = = ~ = = == O = (`N + ` O ) O K== = f = N =~ = O =~ = = W= 298 CHAPTER 10. DIFFERENTIAL EQUATIONS N = + I= O = I= == Q O = I= = I== O O = = ~ = = == = (`N + ` O ) K== = 1174. f =i ~ =b ~ = =` ~ ======================= ` == + + = ( ) K== = q = ~ = = = = == = + I= == = ==~= ~ ~ == = == = = ~ = ~ = = = = ~ = = = =~ ~ = = ~ =E = = = =NNTPFK= = f = = = = ~ = = == ( ) = (mN ( ) + mN ( ) ) I== = == ~ ~ = = = = = == = (oN ( ) + o O ( ) ) I== = = ~ = oN ( ) = ~ = o O ( ) = ~ = = = = = = = = = = K== f = + = = =~= = = = ~ ~ = ~ I= = = = = M I= f = + = =~= = I= = = N I= f = + = =~= = I= = = O K== = 1175. a ~ =b ~ = = =j = = ( I ) K== p = = K=q = = = ~ = ~ = = = == = ( I ) I== = =~= = = ~ = ~ K= 299 CHAPTER 10. DIFFERENTIAL EQUATIONS 1176. a ~ =b ~ = = =j = = ( I ) K== p = = K=p == = = = I== = ~ = = ( I ) I== = =~= = = ~ = ~ K== = 1177. c =r ~ =s ~ = q = = =~=j~ = =~=p = = = = = ~ == && + = M I== == = = = ~ = = = I= = = = = = = I= = = ~ = = = ~ = = K= = q = ~ = = == = ^ ( M ) I== == ^= = =~ = = = ~ I= O I= M = = = ~ ~ = I= = = = q = M = = ~ =~ = = = ~ K= q = =~ = ~ = = = ~ = K== = 1178. c =a~ =s ~ = && + + = M I= == & = = = ~ = K== q =~ =P= ~ = = = ~ = W= = 300 CHAPTER 10. DIFFERENTIAL EQUATIONS `~ =NK O > Q =E ~ F= ( ) = ^ N + _ O I== == N = O Q + O Q I= O = K== O O = `~ =OK= O = Q E ~ = ~ F= ( ) = (^ + _ ) I== == = K= O = `~ =PK= O < Q =E ~ F== ( ) = O ^ ( ) I= == = Q O K== = 1179. p =m = O + = M I= O i = = = = ~ ~ = ~ I= i= = = = I= = = =~ ~ = = ~ K= = q = ~ = = = ~ =~ = = == i ( ) = ~ I= = = = q = O K== i = 1180. oi`=` = Of f N i O + o + f = s ( ) = b M ( ) I= ` 301 CHAPTER 10. DIFFERENTIAL EQUATIONS =f= = = = =~ =oi`= = =~ =~ = ~ = = s( ) = b M ( ) K=== = q = ~ = = == f( ) = `N N + ` O O + ^ ( ) I= == Qi o oO ` I== NI O = Oi b M ^= I== O N O OO i + o ` N i = ~ ~ I== o o` `N I= ` O =~ = ~ = = = ~ = K= = = = 10.3. Some Partial Differential Equations = 1181. q =i~ ~ =b ~ = O O + = M= O O ~ = = ~ = = = ( I ) == ==~== ~ = = = = = - ~ K= m~ ~ = ~ = ~ = = = =~ = ~ = K== = 1182. q =e ~ =b ~ = O O + = = O O 302 CHAPTER 10. DIFFERENTIAL EQUATIONS ~ = = = ~ = = ( I ) = = = ~ = = ~ = =~ = = = = ~ =~ ~ = = = K=q = ~ = = = =~ = ~ = ~ ~ K== = 1183. q =t~ =b ~ = O O O + = = O O O ~ = = = ~ = ( I ) = = ~ = ~ = ~ = = ~ = K= q = ~ = = = = ~ = ~ = K== = = 303 Chapter 11 S eries = = = = 11.1 Arithmetic Series = f ~ = W= ~N = k = W= ~ = a = = = W= = k = = = = = W= = p = = = = = W= p = = = 1184. ~ = ~ N + = ~ O + O = K = ~N + ( N) = = 1185. ~N + ~ = ~ O + ~ N = K = ~ + ~ +N = = ~ +~ 1186. ~ = N +N = O = ~ +~ O~ + ( N) 1187. p = N = N = O O = = = = = 304 CHAPTER 11. SERIES 11.2 Geometric Series = f ~ = W= ~N = k = W= ~ = ` = ~ W= = k = = = = = W= = p = = = = = W= p = p = = W=p= = = 1188. ~ = ~ N = ~N N = = 1189. ~N ~ = ~ O ~ N = K = ~ ~ +N = = 1190. ~ = ~ N ~ +N = = ~ ~N ~N ( N) 1191. p = = = N N = ~ 1192. p = p = N = N c = < N I= = =p= =~ = K= = = = 11.3 Some Finite Series = k = = = = = W= = = = 305 CHAPTER 11. SERIES 1193. N + O + P + K + = ( + N) = O = 1194. O + Q + S + K + O = ( + N) = = 1195. N + P + R + K + (O N) = O = = 1196. + ( + N) + ( + O) + K + ( + N) = = 1197. NO + OO + PO + K + O = (O + N) = O ( + N)(O + N) = S = ( + N) 1198. NP + OP + PP + K + P = = O = (Q O N) O O O O 1199. N + P + R + K + (O N) = = P = P 1200. NP + PP + RP + K + (O N) = O (O O N) = = NNN N 1201. N + + + + K + + K = O = OQU O = N N N N 1202. + + +K+ +K = N= N O O P P Q ( + N) = NNN N 1203. N + + + + K + +K = = ( N)> N> O> P> = = = O 306 CHAPTER 11. SERIES 11.4 Infinite Series = p W= {~ }= c = W= ~N = k = W= ~ = = = 1204. f =p = ~ =N = ~N + ~ O + K + ~ + K = = 1205. k =m~ ~ =p = p = ~ = ~N + ~ O + K + ~ = =N = 1206. ` = =f =p = ~ =N = i I= = p = i = = 1207. k =q =q = f = = = ~ = I= = ~ = M K== =N f = ~ M I= = = = = K= = = = 11.5 Properties of Convergent Series = =N =N ` =p W= ~ = ^ I= = _ = o ~ = W= = 307 CHAPTER 11. SERIES =N =N =N 1208. (~ + ) = ~ + = ^ + _ = = ~ 1209. =N = ~ = ^ K== =N = = = 11.6 Convergence Tests = 1210. q =` ~ =q = =N =N i = ~ =~ = = = = = ~ = M < ~ = =~ = K== f = = = = ~ =~ = K== =N =N =N =N f = ~ = = = =~ = K= = 1211. q =i =` ~ =q = =N =N i = ~ =~ = = = = = ~ = ~ =~ = =~ = = =~ = K== ~ f = M < < = = ~ = ~ = ~ = = = =N =N = = = K= ~ f = = M = = = = ~ = ~ = = =N =N ~ = K= 308 CHAPTER 11. SERIES ~ = = = = = ~ = ~ = = =N =N ~ = K= f = = 1212. - = N = = = > N =~ = = = =N M < N K= = 1213. q =f ~ =q = i = ( ) = = ~= = = = I= I= ~ = ~ = =~ = N K=q = == - = ( ) = (N) + (O) + (P) + K + ( ) + K = =N = = ( ) I=~ = = = N ( ) =~ = K= N = 1214. q =o~ =q = i = ~ = =~= = = = K= =N ~ +N < N = = ~ = K= ~ =N f = f = ~ +N > N = = ~ = = K= ~ =N ~ f = +N = N = = ~ = ~ = = = = ~ = ~ =N = ~ = = = X= = = = = = K== = 309 CHAPTER 11. SERIES 1215. q =o =q = i = ~ = =~= = = = K= =N f = ~ < N = = ~ = = K= =N f = ~ > N = = ~ = = K= =N f = ~ = N = = ~ = ~ = = = I= = =N = = ~ = = ~ = = = K= = = = 11.7 Alternating Series = 1216. q =^ ~ =p =q =Ei =q F= = i = {~ }= =~= = = = = = ~ = ~ +N < ~ = =~ = K= ~ = M K== q = = ~ ~ = = ( N) ~ =N = ~ = ( N) =N N ~ = = K==== = 1217. ^ =` = ^= = ~ =N = = ~ = = = = = ~ =N = = K== 310 CHAPTER 11. SERIES f = = = ~ =~ = = = = = - =N K= = 1218. ` ~ =` = ^= = ~ =N = ~ = = = = = = = = = =~ = K= = = = 11.8 Power Series = o ~ = W= I= M = =M =M m = W= ~ I= ~ ( M ) = t = W= = o~ = =` W=o= = 1219. m =p = = = ~ =M = ~ M + ~N + ~ O O + K + ~ + K = = 1220. m =p = = ( M ) = ~ ( ) =M M = ~ M + ~ N ( M ) + ~ O ( M ) + K + ~ ( M ) + K O = 1221. f ~ = =` === q = = = = ~ = = = = = = = ( ) = ~ ( M ) = = = = ~ == == ~ = = =M K= 311 CHAPTER 11. SERIES 1222. o~ = =` = f = = ~ = = = == ( M oI M + o ) == == = o M I= =o= = ~ == = ~ = = K==f = = = ~ = ~ N = = o = K== o = ~ ~ +N = = = 11.9 Differentiation and Integration of Power Series = ` = W= ( ) = m = W= ~ = =M t = W= = o~ = =` W=o= = = 1223. a ~ = =m =p = i = ( ) = ~ = ~ M + ~ N + ~ O O + K = = < o K== =M q I= = = < o I= ( ) = = I= = ~ = ( ) = =~ = ( ) = ~ M + ~ N + ~ O O + K = = ~N + O~ O + P~ P O + K = ~ N K= =N = = = 312 CHAPTER 11. SERIES 1224. f ~ = =m =p = i = ( ) = ~ = ~ M + ~ N + ~ O O + K = = < o K== =M q I== = < o I= = = ~ = ( ) =~ == ( ) = ~ + ~ + ~ + K = O M = ~ M + ~N N O O P +N + ~O + K = ~ + ` K= O P +N =M = = = 11.10 Taylor and Maclaurin Series = t = W= = a ~ = W= ( ) = o ~ = W= o = = = 1225. q~ =p = ( ~ ) = (~ ) + (~ )( ~ ) + (~ )( ~ )O + K ( ) = ( ) (~ ) > O> =M == + ( ) (~ )( ~ ) + o K== > = 1226. q =o ~ =^ = HN=q = = = == +N ( +N) ( )( ~ ) I=== ~ < < K= o = ( + N)> = 1227. j~ ~ =p = 313 CHAPTER 11. SERIES ( ) = ( ) (M ) =M (M) O ( ) (M ) = (M) + (M) + +K+ + o > O> > = = = = 11.11 Power Series Expansions for Some Functions = t = W= = o ~ = W= = = = O P 1228. = N + + + +K+ + K= O> P> > = O P ( ~ ) + K = ~ ( ~ ) ( ~ ) 1229. ~ = N + + + +K+ N> O> P> > = ( N) +N K I= N < N K= O P Q 1230. (N + ) = + + K + OP Q +N = N+ P R T = O + + + + K I= < N K= 1231. N P R T = N N N P N N R 1232. = O + K I= > M K= + + N P + N R + N = ( N) O K = O Q S 1233. = N + + K + (O )> O> Q> S> = 314 CHAPTER 11. SERIES ( N) O +N K = P R T 1234. = + + K + (O + N)> P> R> T> = P O R NT T SO V 1235. ~ = + + + + + K I= < K= O P NR PNR OUPR = N P O R O T 1236. = + + P QR VQR + QTOR + K I= < K= = P N P R N P RK (O N) O +N 1237. ~ = + + +K+ + K I= O P O Q R O Q SK (O )(O + N) < N K= = P N P R N P RK (O N) O +N + +K+ + K I 1238. ~ = + O O P O Q R O Q SK (O )(O + N) < N K= = 1239. ~ ~ = ( N) O +N K I= N K= P R T + +K+ P RT O + N = 1240. = N + O Q S O + + +K+ + K= (O )> O> Q> S> = 1241. = + P R T O +N + + +K+ + K= (O + N)> P> R> T> = = = = = 315 CHAPTER 11. SERIES 11.12 Binomial Series = t = W= I= = o ~ = W= = ` ~ W= ` = = = 1242. (N + ) = N + `N + ` O O + K + ` + K + = = ( N)K[ ( N)] 1243. ` = I= < N K= > = N 1244. = N + O P + K I= < N K= N+ = N 1245. = N + + O + P + K I= < N K= N = O N P P N P R Q 1246. N + = N + + + K I= N K= O O Q O Q S O Q S U = N O O N O R P N O R U Q 1247. P N + = N + + + K I= N K= P P S P S V P S V NO = = = 11.13 Fourier Series = f ~ = W= ( ) = c = W= ~ M I= ~ I= = t = W= == 316 CHAPTER 11. SERIES ~ 1248. ( ) = M + (~ + ) = O =N = N 1249. ~ = ( ) == = N = ( ) == 1250. = = 317 Chapter 12 P robability = = = = 12.1 Permutations and Combinations = m ~ W= m = ` ~ W= ` = t = W= I= = = = 1251. c~ ~ = > = N O PK( O)( N) = M> = N = = 1252. m = > = = > 1253. m = = ( )> = 1254. _ ~ =` = > ` = = > ( )> = = 1255. ` = ` = = 1256. ` + ` +N = +N ` +N = = 318 CHAPTER 12. PROBABILITY 1257. ` M + `N + ` O + K + ` = O = = 1258. m~ ~ =q ~ = = o =M= = = = = = = o =N= = = = = = N= o =O= = = = = N= = o =P= = = = N= = P= o =Q= = = N= = Q= = o =R= = N= = R= = NM= o =S= N= = S= = NR= = = = = N= = = = N= = O= = N= = P= = S= = Q= = NM= = OM= = NR= 12.2 Probability Formulas = b W=^I=_= m ~ W=m= o~ = ~ ~ W=uI=vI=w= s~ = = ~ = ~ ~ W= I= I= = b = ~ = =uW= = ^ = = ~ = W= == p ~ ~ = ~ W= = s~ ~ W= O = a = W= ( ) I= ( ) = = = 1259. m ~ = =~ =b = m(^ ) = I== == = = = = = = = I== = = = ~ = = = = K= = 319 = = = N= = R= = = = = = N= = S= = = = = = N= = = = = = = = N= CHAPTER 12. PROBABILITY 1260. o~ = =m ~ =s~ = M m(^ ) N = = 1261. ` ~ =b = m( ^ ) = N = = 1262. f =b = m( ^ ) = M = = 1263. ` = m(^ ) = N m(^ ) = = 1264. f =b = m(^ L _ ) = m(^ ) I== m(_ L ^ ) = m(_ ) = = 1265. ^ =o = =f =b = m(^ _ ) = m(^ ) + m(_ ) = = 1266. j ~ =o = =f =b = m(^ _ ) = m(^ ) m(_ ) = = 1267. d ~ =^ =o = m(^ _ ) = m(^ ) + m(_ ) m(^ _ ) I== == ^ _ = = = = = =^=~ =_I== ^ _ = = = = = =^=~ =_K= = 1268. ` ~ =m ~ = m(^ _ ) m( ^ L _ ) = = m(_ ) = 1269. m(^ _ ) = m(_ ) m(^ L _ ) = m(^ ) m(_ L ^ ) = 320 CHAPTER 12. PROBABILITY 1270. i~ = =q ~ =m ~ = m(^ ) = m(_ )m(^ L _ ) I== =N = _ = =~= = = ~ = = K== = 1271. _~ =q = m(^ L _ ) m(_ ) m(_ L ^ ) = = m(^ ) = 1272. _~ =c ~= m(_ ) m(^ L _ ) m(_ L ^ ) = I== m(_ ) m(^ L _ ) =N == _ = =~= = = ~ = = =E FI= ^ = = = ~ = I== m(_ ) =~ = = = ~ I= m(_ L ^ ) =~ = = = ~ K= = 1273. i~ = =i~ =k = p m M =~ = I== p m < N =~ = I== == p = = = = = ~ = ~ ~ I= = = = = = = K= = 1274. ` =f ~ = s(u ) m( u ) O I== = s(u ) = = = ~ ~ = =uK= 321 CHAPTER 12. PROBABILITY 1275. k ~ =a =c = ( ) O N O ( ) = O I== O = = =~= ~ ~ = K= = 1276. p ~ ~ =k ~ =a =c = O N O = ( ) = O ^ ~ = ~ = = M I= ~ = = N K= = = ===== = Figure 210. = 1277. p ~ ~ =w=s~ = u w= = = 1278. ` ~ =k ~ =a =c = N c( ) = O ( ) O O O I== 322 CHAPTER 12. PROBABILITY == = =~= ~ ~ = I== = =~= ~ ~ = = ~ K= = 1279. m( < u < ) = c I= c = u= = ~ = = ~ = ~ ~ I= c= = ~ = ~ = = I== m( < u < ) = = ~ = ~ K= = 1280. m( u < ) = Oc I== == u= = ~ = = ~ = ~ ~ I= c= = ~ = ~ = = K= = 1281. ` ~ =a =c = c( ) = m(u < ) = ( ) I== = = =~= ~ ~ = = ~ K= = 1282. _ =q ~ =m = = = I= O = I== == = =~= = = I== = = = ~ = = = = ~ = I= = = = ~ = = ~ I= = N K= = 1283. _ ~ =a =c ======= ( I I ) = I== 323 CHAPTER 12. PROBABILITY = I= O = I= ( ) = ( + ) I== == = = = = = ~ = = I= = = = ~ = = I= = = = ~ = = ~ I= = N K= = 1284. d =a = m(q = ) = N I== N = I= O = O I== == q= = = = = = = = I= = = = = I= = = = ~ = ~ =~ = = = = I== = = = ~ = = ~ I= = N K= = 1285. m =a = m(u = ) I= = I== > O = I= = I== == = = = ~ = = I= = = = = = = K= = 1286. a =c == m(~ u ) = ( ) == ~ = 1287. ` =r =a = ~+ N I== I= = = ~ O 324 CHAPTER 12. PROBABILITY = = = = = K= = 1288. b ~ =a =c = ( ) = I= = I= O = O = = = = I= = = = ~ = ~ K= = 1289. b ~ =a =c = c( ) = N I== = = = I= = = = ~ = ~ K= = 1290. b =s~ = =a =o~ =s~ ~ = = b(u ) = I== =N = = =~= ~ ~ = I= = = = ~ K= = 1291. b =s~ = =` =o~ =s~ ~ = = b(u ) = ( ) == = 1292. m = =b ~ = b(u + v ) = b(u ) + b(v ) I== b(u v ) = b(u ) b(v ) I= b( u ) = b(u ) I== b(uv ) = b(u ) b(v ) I== = = =~= ~ K= = 1293. b(u O ) = s(u ) + O I== == = b(u ) = = = = ~ I=== s(u ) = = = ~ ~ K= = = = 325 CHAPTER 12. PROBABILITY 1294. j~ =f ~ = b(u ) m(u > ) I== = = = = ~ K= = 1295. s~ ~ = =a =o~ =s~ ~ = [ ] O = s(u ) = b (u ) = ( ) I== O O =N == = =~= ~ ~ = I= = = = ~ K= = 1296. s~ ~ = =` =o~ =s~ ~ = [ ] ( ) ( ) == = s(u ) = b (u ) = O O O = 1297. m = =s~ ~ = s(u + v ) = s(u ) + s(v ) I== s(u v ) = s(u ) + s(v ) I= s(u + ) = s(u ) I= s( u ) = O s(u ) I= = = =~= ~ K= = 1298. p ~ ~ =a ~ = [ ] a(u ) = s(u ) = b (u ) = = 1299. ` ~ ~ = (uI v ) = b[(u (u ))(v (v ))] = b(uv ) (u ) (v ) I== == u = = ~ = ~ ~ I== s(u ) = = = ~ ~ = =uI== = = = = ~ = =u= =vK= O 326 CHAPTER 12. PROBABILITY = 1300. ` ~ = (uI v ) I== (uI v ) = s(u )s(v ) == s(u ) = = = ~ ~ = =uI== s(v ) = = = ~ ~ = =vK= = = = = = = = = = = = = = = = = = = = = = = = = = = = = 327 = i = = = ~ =~ = = = =~ = K ~ - K K= = = = =

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