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 £ln £n g}V”…vsgh eCˆeg}V˜‘’gd e¢‘’’‘’“‘’‘’’‘“’‘’‘’’’’’‘’‘“’‘’’‘’“‘’‘’’‘“’‘’‘’ † es ž † es d ž                                              ¡ {n ™vo“• vogh eŸ“g—xgi’E˜˜gjxv˜™’“‘’‘’’‘“’‘’‘’’’’’‘’‘“’‘’’‘’“‘’‘’’‘“’‘’‘’ j h ž ””d ‚ ed s e h               œ {n ¥¥“#—Pyv˜vhvy’‘“’‘’’‘’“‘’‘’’‘“’‘’‘’’’’’‘’‘“’‘’’‘’“‘’‘’’‘“’‘’‘’ i–od ”ff hu  i– ed › {n ¥¥go’’’’’’‘’‘“’‘’’‘’“‘’‘’’‘“’‘’‘’’’’’‘’‘“’‘’’‘’“‘’‘’’‘“’‘’‘’ f w• s ee h ˜ ok i  – {n x˜ “šg}Ye¥3„kg‚P¥qoy2• 3xy’‘’“‘’‘’’‘“’‘’‘’’’’’‘’‘“’‘’’‘’“‘’‘’’‘“’‘’‘’ sf” s ‚ eo ƒ {n ”P™˜˜—j“z•¥‘’‘“’‘’’‘’“‘’‘’’‘“’‘’‘’’’’’‘’‘“’‘’’‘’“‘’‘’’‘“’‘’‘’ “ ed ff ed r {n —vgj’tPx|˜gh’¥‘’‘“’‘’’‘’“‘’‘’’‘“’‘’‘’’’’’‘’‘“’‘’’‘’“‘’‘’’‘“’‘’‘’ ‰ {n “p¥¥’’’‘’‘“’‘’’‘’“‘’‘’’‘“’‘’‘’’’’’‘’‘“’‘’’‘’“‘’‘’’‘“’‘’‘’ dd i–o” q {n x˜ “|•”„…™’‘qxz˜Žxg}Vgs’’‘’“‘’‘’’‘“’‘’‘’’’’’‘’‘“’‘’’‘’“‘’‘’’‘“’‘’‘’ f w h “o s h e} † e ed                n {n 0”¥¥“#0”Žg}Ye3’‘’‘“’‘’’‘’“‘’‘’’‘“’‘’‘’’’’’‘’‘“’‘’’‘’“‘’‘’’‘“’‘’‘’ †–od † s n g}Vgs’EW˜p¥¥go’x’‘’‘“’‘’’‘“’‘’‘’’’’’‘’‘’“‘’’‘’‘“’‘’’‘“’‘’‘’’’’’‘’‘’“ † e ed s i– ed  ˆ „y ‡ wŒ „ „ ¥ˆ c‡P9ƒ™‘†‹3€Šx‡ ∼ ∗ qUn qUn nmn nmn nmn £ln œ – – – ƒ ƒ ƒ ‰ ƒ – ¡ › › r r r r ‰ h ‰ ‰ {n ml qh¥gˆe‡x˜ go„eq‚EW˜#Pyv˜vhv@“’‘’‘’’’’’‘’‘’“‘’’‘’“‘’‘’’‘“’‘’‘’’’’’‘ h † … h s ff hu q {n ml ™¥vgh(e3kvsv€ gw¥qw‘~—Px|˜vhvC’‘’’’’’‘’‘’“‘’’‘’“‘’‘’’‘“’‘’‘’’’’’‘ ”o ‚ h u e h u }ff h u           n {n ml z˜¥yv˜‘x’‘’“‘’‘’’‘“’‘’‘’’’’’‘’‘’“‘’’‘’“‘’‘’’‘“’‘’‘’’’’’‘ eo hu  n cl —x™™vov#——xv˜vhv™“’‘’’‘’“‘’‘’’‘“’‘’‘’’’’’‘’‘“’‘’’‘’“‘’‘’’‘“’‘’‘’ ”f iw h u ”ff h u                 ‰ cl ml ‘xqighP“txW˜Pgf’E’’‘“’‘’‘’’’’’‘’‘’“‘’’‘’“‘’‘’’‘“’‘’‘’’’’’‘ kj efd ” s ed  q cl ml ‘xqighPgf’#x¥¥p‘’’‘“’‘’‘’’’’’‘’‘’“‘’’‘’“‘’‘’’‘“’‘’‘’’’’’‘ kj e ed ” i–o  n cl ml ‘xqighPgf’#x™—•‘’’‘“’‘’‘’’’’’‘’‘’“‘’’‘’“‘’‘’’‘“’‘’‘’’’’’‘ k j e ed ” ˜–”               ˆ 9†…Pƒx23€y2¨xvstsr2qpi ‡ „y wu‚ rp wuu r fd ` R R gecbaYXWTURVUTSQ $ " ¢ ¢  ©¢ ¦ ¤¢ %#!   ¥¥¥¨§¥£¡   I1G D B 'B '18 '1 5 '1 P3HF7EC(1C§(A3@9(7366(4320)('& ∗ q hxq˜g‚ž £‰n vxvighCgf’d ¢ kj e e d ed e ‚ jo e d k j e e de h– ed f f ff f i h u ff h ueo j f w h u ff h f h h u kd s 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v…™go’š„}vgjg—P“’‘’‘’’’’’‘’‘’“‘’’‘’“‘’‘’’‘“’‘’‘’’’’’‘ j h ede h ed ‚ e–” q {†nn €q ¥—–‘“Ÿ‘x“• Pgfvu ’’‘“’‘’‘’’’’’‘’‘’“‘’’‘’“‘’‘’’‘“’‘’‘’’’’’‘ } hjd kji o e  n {†nn €q gz(eg‚’6gq–v” ’‘’’‘“’‘’‘’’’’’‘’‘’“‘’’‘’“‘’‘’’‘“’‘’‘’’’’’‘ d ed s} hj  } h j d ‚ j h” s if d e d “ e                                   ¥—–‘“¨gw„ev¥v—‡…™—Vgz„eg‚‘¤„”0‚„†™go’d ‘“’‘’‘’’’’’‘’‘“’‘’’‘’“‘’‘’’‘“’‘’‘’ †f ed s h e˜ e  e  r c€ln €q  –se—3gk’’x|˜—gfH~„}g‚Hi “d |•vhYhgs™o ’’’’’‘’‘’“‘’’‘’“‘’‘’’‘“’‘’‘’’’’’‘ ‰ c€ln €q g‚Hi “Ÿv•vhhgs¥Wo“‘’‘’’‘“’‘’‘’’’’’‘’‘’“‘’’‘’“‘’‘’’‘“’‘’‘’’’’’‘ e d e  q c€ln €q ” ™g2E(˜q†3¨‘’‘’’‘“’‘’‘’’’’’‘’‘’“‘’’‘’“‘’‘’’‘“’‘’‘’’’’’‘ …od hk”                n c€ln €q 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„ u „ k u h i – o ” } … s d f i w d dh i – oe d k o kh‚ž f d ‚ ie d s ke de ”† }e s k“h ” i – o ” ” f je oh d kh‚ ” f d ‚ i   "  5 # " ‘‘x˜ ¥¥6V’§x™™o “q¦¥¥g’šx˜ ~v„gš“g—xg’eCg‘g§0ŽgY~—q§xg™—‡—vg¥v“šq——€’gqx’d  k „k€•’E¥¥™W˜s ¦ “ s dd i–o”  h˜se — x’g”|h ̐¤    05  " vu k„q“¤xg™—” h ” i–o d jo dd i–o v™Ck “p¥¥”  d” i–od f iw h u s”ed h ‚ ‚” i–o ¥¥“#x™™vov¨g…YeP’q”vh‘“gd(eEg™—” 4 G E C 4 IHFDB  " vh ci£ ep2vw ‡‡ ‚y a < 1 n rn < k a > 1 an (n = 1, 2, 3 . . .) k>α k α a rn a nk → a a n α 1 ≤ n 1 < n2 < n3 . . . {ank }∞ = {an1 , an2 . . .} k=1 α an → a aα = ∞ 1 (a) lim an = lim bn ↓ b n , ↑ a n , an < b n n ∈ N a→∞ I = [an , bn ] an ≤ an+1 ≤ bn ≤ bn+1 {an } , {bn } bn − a n − − → 0 −− an ¤ {rn }n=1 a rn < a k 1 α ¨  k šCx˜ o qjvhvu  W˜gWs v”—” h s hh ‚—q‚h  ¦ —v”h Wxg”e  h s˜ |¦vu  @ s k k vu „kv“h  n>N f (1 + cx → c x n ε>0 cx n cx ¦ § ¤ ¥ s e”  h s” “o } ‚” ”}k k g”‘€™voV V„…™~“Y•„…#ŽC—” ¢ ‘u „kq“h  f gCgkž ¢  s ff ‡gWs Pyv˜h  ¦  ¡ ¢  ¢ gE¥¥” & s  s i – o xq‚h‚„ ‘™g}YeVY• —…‘—qj‚s E(˜g†—e” ¦„e„kg“™x˜vd™” ¦ „kq“2Yyh vŽ—g}˜† f Yegs V' ‘“V… ˜† ” hv  ˜ s “ ” h h ¦ ev“3h“ h o h ˜ e s j • hi h ‚ e h  h s j e” ˜ hk dd i–o x˜ vk„hxgi™§ ¡ ‘u ¦„kq“6vg}g—¤„”g“e ¦ q”3˜“Eg™—”  kh } o ” i – o ” f s } – f i o e wh ˜ kh oh u ‚ f  kh u } j • … ”“ ¡ ¢  Ws vŽ™vh x¥¥#Ž¥tx™w Egi¥vpWs vŽ} q‘’—¥3vv~‘“V#——”  & g”‘” ¢ v”h Wxg”e ‘x“g‡…˜“V… e h s˜ ‚id s s j•  k u s i – o ” ‚ d e ‚ x‘Žg}YevvjŸk ¥P—eh v¥– ¦ „kq“€Ws xg™—” h ˜ † s s h – h h ” i–o "    5 ¥ # " ‘’x¥¥v“g(” ”0¥¥” 7g}VvvjE˜E¥¥‘“gd(e” g”2‘Žg}YeE˜˜“Eg™—™W˜—”  †–o ¢ † es s h s i–o” ‚ ‚ ek † s s d d i–o” s ”o ‚ h u } s j … e h e sk e f h u i–o” ‚ e h ¦ s ž f iw ff h ek † s s i–od o ‚ h ™¥vgh(e3kvsv~gW˜“Y•™gi¥vwŸgk’3™gi¥vwh g” e—e‘™¥¥tqk—hxgi¥qw“d YegzšWs x™™o —x|˜#g”2vg}YepW˜™¥¥“#” ¥qgh(eCYkvsvu  an → a an (a − ε0 , a + ε0 ) a 2 3 3 4 3 5 2 5 ¢ £ ”o ‚ h u ™¥vgh(e3kvsv~} ci£ 2"9‡ xvstsr2p ‚y wuu r r= n m, 0 h˜ f •} xšxi ¨ Vq…h  ˜d h #2gs—e¨} ¤ ε Y˜Žg}¥goSgWs …ees hxv”f —¨h i ” —x|˜P‘™30”Žg}Ye¥W˜˜¥¥2’gd„e—” ff hh jok † s s i–o”‚ ‚ k s vu gs 0”¥¥“yg}VEW˜™¥¥” †–o d † es s i–o i–o” hk † u ‚ j h ˜f… h j” k d ” • h u sff h ¥¥#v”3‘‘§gw„ev¥v—” —V“qd‘—¦“¢“g} v…vhv6——xv˜vhvu Cy˜ o k q”3¤3qk™¢” W˜#x¥¥”  hk ” h}o s ” i–o i¥¥¦qk—hxgi¥qw“’3“˜Žg}gE3y¤¥0—š¥g™—” –o” ‚ e hd“kd j e” ˜}”h ˜ ” †” } eo ci£ 2"†yH‡ ‚ „   ¡ n>N N f →R ε>0 1 1 2 1 3 2 7 3 7 (a − ε, a + ε) a1 , a2 , ..an .. k [0, 1] 1 an an ) ¦ § x 1+ an an = δ>0 bx → 1∀x n x 1+ an an x 1 − δ < bn < 1 + δ x bn → 1 → ex (1 + {an } ε0 > 0 → e an → ∞ x (k = 1, 2, 3...)ak ∈ R f (k) = ak (k = 1, 2, 3...) a=l x an an ) ε>0 a1 , a2 , a3 ..., an |an − a| < ε →1 → ex cn → c x>0 ≤n≤m N  •} Vq…h ¦ h q”h ”e “z•¥o ¢ |an − a| ≤ |an − ank | + |ank − a| < ε + ε = 2ε fd h h “‘qjvhPvfvu ‚Ž‘™o “d  } k  k ™„kv“h ‘u ‚  fi hj† e ef } ™o x™vwvH¥ –Vs WgsP™o ¦  fd h E„kv“h ’‘qjvhvu ˆevu † † h „kvv“k §vjvhP“¥“sed pu ¥¥”s „kv“h ¦ h u h f d d‘ i – o ” ˜  “ ‚s i d † •g}dk —x“”0sŽg}Yes ¦ Ye9gg‚s z˜h|že vjvhš|u šWtx” g™¥—C”o˜– xVo v—v”h q”” e gfe “„d3‚ vkh ¥„kq“h f vh s i k ˜  h h h  }  |ank − a| < ε |ank N ε>0 an → a − a| < ε nk > N nk > N |an − am | < 1 N 1 − a m < an < 1 + a m a a nk → a nk n>N {ank }∞ k=1 {an } |ank − an | < ε N N |A™˜ “g—x“’0sŽg}Ye™gi™qwh ’v‚vj  @ s fd ‚id † s e  h „‚v‚h —P“vvjqh‘¤“„d3vk‡…s  efd hu ” ‚ h {an } N an → a an (e} ™o ‚  h 2v”h ¦ ku† e2v‘vu vjvu h ¦ „kv“h ˜… e e s Vg}¥goegs hh q”” ‚„q‚‘ ¦ „kv“h vjqh‘u Y W˜gWs q”” h h s hh g”‘” ¢ e fx˜ “• ¢ ¥¥” “g—x“x0sg}Vs gi™qwš‘g™v~‘egWs w i–o fd ‚ id † e e h k ueo j k u s ci£ ©y9% ©y£r©„ xe2 †‚x‡ C• ©tr r ww‡ „ tŠ „ ¤ ¥ ¦ ¢ £   ¡ ¦ § |an − am | < ε 0 < δ < min 1, N < n, m |am − a| < ε/2 < (1 − δ)x < bx < (1 + δ)x < (1 + δ)L = n < 1+δ 1 − ε < bx < 1 + ε n bx → 1 n 1 − ε < 1 − δ · 2L < 1 − Lδ < (1 − δ) < L = [x] + 1 k 0 |an − am | ≤ |an − a| + |am − a| < L K 2[x]+1 N 0 |an − a| < m=N +1 L k=0 L ε 2 ε ε + =ε 2 2 L k−1 δ <1−δ K n>N ε=1 N = max (N, N ) |ank − an | < ε L 1 L K {an } mn· = n+1 n+2 2n 2n 2 sn ↑ bn = sup{an, an+1 , an+2 ...} −∞ < ≤ ¯ < ∞ l l = liman ¯ = liman ¯ l {an } ¡ ¢ sžs gz”gWs hh v””  ≤ = {ajn }n=1 a kn |Sn − Sm | bn − cn → liman 1 n ≤ a kn ≤ b n = 1 1 1 1 (−1)n−1 − + − + .. m+1 m+2 m+3 m+4 n 1 1 1 1 − − − + .. m+1 m+2 m+3 m+4 1 <ε m+1 b1 b2 = sup{a1 , a2 ...an } = sup{a2 ...an } b cn bn → b cn = inf{an , an+1 + ...} {an } {an , an+1 ...} |Sn − Sm | < ε dn ≤ sup{ajn , ajn +1 , ..} = bjn dn = sup {aj1 , aj2 , ..ajn ...} l≤b ¯ bn → b = lim an l 0 a (a − ε0 , a + ε0 ) ∩ E = {a} b 3− an → = 1 2, 3 a liman ∞ 1 k , 0 k=1 E + 1 2 ∩E a⇔E E E E (k) = E k−1 an = a, n = 1, 2, 3, . . . ¯ lim an = liman = (E ) E = a a∈E a {1, 2, 3, . . .} a E ε0 > 0 a∈R ¯ liman limn→∞ cn = liman E E = 1 1 n, n E a a + E ∞ 1 n , 0 n=1 E E (b − ε1 , b + ε1 ) ⊂ (a − ε0 , a + ε0 ) E ∞ 1 m , 0 n,m=1 E E = {−1, 1} E = k a   E 1 (M − n , M ) M ∈E M ∈E M ∈E / 1 M − n (n = 1, 2, 3, . . .) M ε M = sup E E = {a} E = {an |an = a, n = 1, 2, 3, . . .} E= E ⊂ E E = [0, 1] E = (0, 1) an = a a E = a 1 n ε0 > 0 + E = {0} E = E = Φ E = {1, 2, 3, 4} (a − ε0 , a + ε0 ) 1 m |n, m ↑ cn ↓ bn ¯ limn→∞ bn = liman E =Φ 1 n E = {an }∞ n=1 E = {1, 2, 3, . . .} + 1 m E ⊂R 1 m M∈E / M {an } + (−1)n an → a ε a +1 l E a E 1 n ¡ ¡ M E = 1, 2, 3, . . . 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g…Yegs’¢yg} e€q}gh€gw(ev¥vgh’d ¢ ed ˜ ž ž ‚j e £i ¤h ‡ „ t “ {„ H‡ CŠ £{„ £rH‡ „ gw(ev¥v” —xv˜“’g“e ” “‘—EW˜¥Žg}g” ¦ 2v™3•d gy#0”¥¥E˜s ¦ k§g}g™2‘™3k z• ee #™gw(e‘¥q”  ‚jh ff hd …d j” s j e ˜d jok e u †–o” s j e” ˜d jo  ” ‚ j h ¤  d T  £V§f ¢ R d ¡ ‚ x h¥j s g}g e€e“dgd e t e 3k š o e €w e e –   e s j e e e e e k s k } ”   e  e f  ‰  k g“ggde ¨tsW˜sYe– qk„hxgii ™¨e qw™¥˜egih qw™W˜™g’g}ej gg‘™gˆi’g†e ™de go‘g”e 2’g”k 2™g†}ˆi g™e” o ¦ ‘2vkh“ggdg•vkghž gi¥vwh W˜Eg}gg’d 0ig†¥go’d g”2‡3—gVo p0ig†¥o ’CgCgkž f s e d w e d sk  q  q”C˜Žg}g€“gdtv—¥vt‘”gs x˜q}™W˜s —gfW“~}  hk j e” w e” kueoj ku s h˜ i h}” ˜ e sd n  vkh ggWxg”pW˜¥Žg}gg’~“gdg’#v”3˜g}g€“gd” tgCgkž ¢ v“gde  Wxg”™W˜s g}gg’Eˆig†™go‘¤q”3˜g}g™0ig†¥” e– s ˜ e s j e ed w e ed hk j e” w e  f h s ˜ e e ed e ed hk j e” eo |s  @ Yegzž s  g”‘” 1x—gf} ¡ “˜g}g’d 0”g}Yegs‘¢” vvjqh‘™W˜s §™voV g”|h   e ¢‚ i ž kd j e † ed s hu   h s e f i ‚ k i–o x‘„j” ¢ ‘u ¥¥” ¦ „kv“˜Žg}g“d 0”Žg}Ygs’¤‘Y‘qjvhvEWs v”3˜g}gp0ig†¥”  h j e † e e d ”s h u ˜ h k j e” eo  hq”h ¦ „kv“h f ‚ e e d ¦es g|hs vgjg –’E˜s Vgzž ¡ s s  ˜ qkghž } f  ‘vh vj g‚vueh W6„”v‚h ™PB¦k §i q}v“gj}‚h gde —E˜W˜s s gvh§v¦xi ¨ gi¥vw ¨˜x‘gjg –‘d ¨„kv“pq}yg“e g|h—” vgjg –’¨˜s yz˜„eg‚‘d ¦  y  ˜ –” f s oj f e h s‚ e e  h i h}” f s ‚ e ed ˜ e k ¦‘~Ws q}” gvhtvk„hxgie 1vgjg –’™W˜s ¦ —kq“‘vpW˜s ¦ Yegzž   ku i h} f s ‚ ‚ e ed h†u s g”‘” )v‘ e ¢‚j ¦ „kv“h ‘gjg‘¥“g™go’¥Ws ‚ e – d ”  • } e d ˜ t„kv“‡Ws Žv}—” ˜s gvYh qk—hxgi™§ vgjg –’˜W˜s   h i h} f ‚ e ‚ eed f w s s x˜ “• ¢ gWs ¦ Yegs fd hu “vvjqh‘p˜ ¢ ¦ Ž™™“• ¦ e}wo ¦ |h ¦  e —g†” 0i xqsW˜o ” ‘pW˜§Žg}gp0ig†¥v‘™3“§Žg}g” # ¦ ©¡  CxgiP„f“d gke “—gfŽg}ž ¦ Py˜vsv™W˜Eg}g~“gd” q”3§g}g—” f¥ h u s j e  ” e o ” ‚ j o k d j e  ¨   e f e ‚ e e ”f hu s j e” w e hk j e 0ig†¥€|h eo” gfWš} ¡  e s 0”¥¥” ¢ g}g” ‘™3k  £ (   vu  †–o j e d jo ¨ k " " # "  "  £ A( ¨ ¦ '   ¢!   £ ¨ $ A∪E E ⊆ (E ∪ E ) ⊆ E E ⊆ E ∪ E E A ⊆E ⊂E ∞ x∈E ⊂E E A∪E E c x A ∈ (x − ε0 , x + ε0 ) (A ∪ E ) ⊆ E x∈E A E ⊆E y = f (x) ⇔ x = f f (f g ) (x) = f (x)g(x) , ∀f (x) > 0 – ∞ (f + g)(x) = f (x) + g(x) – E E= f · g(x) = f (x) · g(x) – x ε0 (x − ε0 , x + ε0 ) ∞ {an }1 A⊆E x ∈ (A ∪ E ) f /g(x) = E = (E ) y = f (x) y = f (x) y = f (x) (y) A E y = x2 – ε0 > 0 f −1 f (x) g(x) E E f (x) A ⊆A ε – – – ε>0 E • • • x E •  ¦  cqn g…Ye~} s” ¦ ¦  ” ‚jh ™gw„ev¥v—” gf(eg‚’z•(ev‚“2gh¥¥‘d —xš} ¡ “z• e —™gw(e‘¥q” ™gw¤v”3k e hd eo– ff ˜ ” e e” ‚j h ”” e” h  } e “d gyu ¦ ™gw(ev¥v§g…Y” —xP |'… 2„k“ ” ‚ j h” es ff ˜e @ ¦e ¦  … eo ” ‚j h g}¥#™gw(e‘¥q” ¢ ¡ ‘ ¦ ¡ ¡ ¡ V”} Žg}¥¤qfghŽ0…e q}†h EVgq–ghH|˜gih 9g…e ™ne ¦ #” ™g—g“~¤3„”o vgj|˜g’—d–he P¥v™qo‘u ¦ |˜hg…… g}¥¥tv”‚ gjg—˜– ’”… ¤—f 0…†§—f gŽ™“’‘#™g—g“€™¥—–ghH¢q”ghg™2g“ eqe“d ‘‡q™o … e o” s ˜ o e o Pf o e ˜s } ‚ x ” ”e h h ee o“ e ”  s ” s ˜ e } o d … u f o e } o e i e  o h … u h e k o s o h hu e” ¥q–3„kv‚vpz˜ e—Ss  ¥——–”o f 30…h gk†tv”3k vf¥Pp˜Ÿgk¥V Pvfqh¥0“yg“ eqe“d eg†gY–t„”g“ e—e“#—™g—v““d W˜s f 0…†tg”2#v”h xšq}gh•gk¥V ‘x2d gfVs —”g“ e—e“‡™¥—–ghHi ž s h h – ” s e o ” h † d h  e s h d ” f o h s ek  i ž eo ” j ˜ ˜ e hd so e #¨& $¨ § '©¦ ¦ 2 2 3 3 2 ( ¨&¨ (¨ # £ ¢ ¦ #©¦ q‚“gd”g”|h 2x|˜™go’t—0”¥¥CP’gz„eg‚‘¨gw(e‘¥q“#” —v” šgf~zž ¦ h ž s e h h ed f” †–o”efd ed ‚j hd “j   e x loga y loga y logb y = logb y = x loga b = logb y · loga b loga y = loga b h q”h ¦ gf(eg‚’z•(ev‚“d Px”g}¥o e h ff˜ … e qex6‡gw(ev¥v#™g|“™v”3k h ˜ s” ‚ j h” ”w hd ˜ h  " 2 # 3 2 #¨&  ¦   §A $ ©§'  ¨ £ ( ©¤  ¦    ¦    gw(e‘¥qgh’Ÿ• qo0hg†(eg‚P“• vogh’“—™gwz’d  ‚j ed ef ede”” e e gw(e‘¥q“„‘— ¦¤ gw(ev¥v#”gsˆe™qo“‘u  ‚j hd “j” ‚j h” † hd gwYevs„hg‚yu ¦ gw(e‘¥q¨¥qgh(e‚vs“ „f™W˜§—˜v‚§gwYevs„hg‚Pqf‘Ìu¤ e ‚j h” o hd ‚” s h e h hvh £†„ £rH‡ 2“†u % €y£rH‡ i „ …Š „ ”™gg—” xgiP„f“~}  ”o} f e ‚d d h” ””o}” —v”#™gg—EW˜s e  d ‚”‚ e —v”vh„hv“gd(eg‚’d e k e ” †–o” e ” e  Px˜ o d gvu ˆ”¥¥EW˜s g”2k “z• eEW˜s Ÿ„k“   '( ¥ !  '  §A '©¦  £ #¨& $¨ f g z = g(y) y = f (x) f g g◦f ax • • •  " " # # 2 θ θ = arccos y z = g (y) = g (f (x)) loga y = x −1 ≤ y ≤ 1 θ = arcsin y y loga (m/p) = loga m − loga p m aloga (m/p) = p m loga m−loga p a = p loga (m, p) = loga m + loga p [−1, 1] −∞ < y < ∞ θ = arctan y = bx x [0, π] x≥0 cos θ −π, π 2 2 −π ≤ θ ≤ 2 y = sin θ y y = cos θ 180o π 2 y = tan θ = sin θ y = ax 1 = a > 0 sin θ cos θ – • • • z(x) x π €‰n f w• ” ‚ h x˜ “¤x„i¥ – eqeV' ¢ }i h}”‚ e}” ek xq}“vgjŽ—EW˜s 3vu „k“  g”‘” ¢ —x˜ “|•v“gz’„dv‚v”x¥¥go’d ¦ Yegz¤q”vh„h#xvivu  e ”f w hu‚ e hu s i– e s ž ‚” f h k vu ˆe‘u † „kv“h ¦ ¦ } ‚h “„d3vk™˜   h˜ f w hu x#y˜ “v•‘€(g† ež ¢ kh v”” h h ek 3vu ¦ „kv“(—kex˜ q}h ˜s ” ¦ “ h i }” ˜}  „kxx˜ “h Y C‘u ˜k   (  05 ) 2 vu ‚} Ž‘™o ¦ 2„k6gs e “ s hh v””  ggW}s |ghs  e ež x—˜v‚h ¢ „‚ v‚h  q”” hh ¦ e} e ef ¦ s !‘3gk™gwž  –C” £gWs h s ˜ … evu g f s w ”  ™voV —g”|h  ¢ vjqh‘u  gWs Vg}¥qje vhgopWxi ™o h se hs Y˜Žg}q”goeh … h¥ e ¦ Cvk™” x3—PvfPvfgh˜q””  ˜ hw f ˜f kse h ž d h 4 eb " 4 8 „q‘ d ‚‚h z˜ eqevu ~gsP™o ¦ d ‚ k h ”  h e f i w h j ˜ s w e ‚ j  h  y‘˜vjh v”¥—ghh Ho ¦† g}Ye¢„}‘gjgP”¤  g”2™‘vjqh—qf‘Ÿ0”g}Yegs‘¦” ¥– ¡ ¥q–vhvu } ¡ —vu h  ef s ˜ u ”¥ s ‚ e – e k ˜ h h u † e d   – h e ” e     05  2 „Cqtg o v3Px™q‘¨W™g(v¥v”  u h ¤  u ‚ k ˜ o  e|h goe †g s h f ž h ¡ e ” s f i ‚ h2 h‘ … s d ” h ¦ gke gfˆeg™’ŸxŽv}—xP¦—fq‚“vW˜gss s  vjqh‘¥3Pf VŸg}¥™0”g}WYes ™vfPv˜ f ghvvjqh} ‘u z˜ eP¤Px™w o (egv”gd—}” † – o d } i h } ” h h kd ¤ e 4 G E C 4 ¨ IHFD% 2 q” s qw#v”vh„h” ¢ gWs h ‚ s dh q””  qjIvh4 vu FDB3Q 2 q”h  W˜gWs gfˆeg™’¢xŽv}’d ¦ qh ” ”“˜gkYeqs€s ¡  ¤ ¨3—™ƒ˜ ¡ h HE C 4 d s † – o d } i h } h ™o • h  ‚ k f o w ¦ ‘~gWs g –Y¤ ¥g(e„‚g‚e ¦ k u s † es o G 4 xb " 4 8 v‘ŽgYEE¥¥” g23h v¥— 0Y¤PxV“¤q¥ e … x¥¥g’̤  d h † }e s ˜ s i – o ”e k h – }† s s ” f ˜ w • ”h – } i – oe d 5 VT B3 2 ŽgYE¨g(‘¥q~„‘ggP¤g2k h vg–  X 4 2 † }e s ˜ s we‚ j h ” }‚ je – ” ”e h eYgzž s ¦ „k“  ™qw¢q”vh„h̔¤ (eg“p“vd™”  s h ‚ ¦ e i ho v”qh—h—” ¦ gjHv˜h „‚v‚h  ’gjHv˜e šCk “vvjqh‘u ¢ „k“  vjqh‘u ‚ e ¢ ˜ e¦ h s f d h h ˜… e e s wo • k u h j Vg}¥gogWs ¦ } ™“§‘xv‚v‘ ¦ h qjvhvu sgWs j¦ e2„k˜ “ v”qh—h—” ¢ v”h  e ‚ d  @ ¨|s h  e } †–od e  h Cq”ghe —gf‡0”¥¥“ŸgjH|˜h t„k“ vjqh‘u  ˜ ¡ 4 xb " 4 8 gxvh d d q”h gfˆe¥¥“’xq}“d ™“• gkYeqsh s ¡ ¦¤ 3—™o †–od } i h} wo ‚kf k s i–o vu gWs ¥¥” hh q””  ”0¥¥™’gjYe—” ¦ ™vw¤v”qh„h” ¢ †–o” ˜ s s h ‚  ˜w s dq”h gf¦ˆe¥¥g– k e “d xq}” W˜s ¦ v¦’V‘hdefgk vs qjvhvu˜} gWs ¤ ¨3—™o vjqh|‘u z˜2k ’t—¦s x™og}¥go~CxsWk ˜ o z˜3k s ¡ (egv”gd—}” † o } i h } k u  h 3k ¢ s  ‚ k f hh s e w d f iV…™ e e  g e § ‚ h2 h‘ e s wh je ˜ 4 G E C 4 " IHFD¥# 2 ™v¥gH|h ¢ i Œ „ tŠ r h ‘†‹CŠ C¥£{„ ’w   ¡  ¦  0 < 0 < |xn − 0| < δ = limx→∞ f (xn ) 0 < δ(ε) 0 < x0 − x < δ x0 x0 x 0 < |x − x0 | < δ 0 < x − x0 < δ δ = m=0 1 n M (n = 1, 2, 3 . . .) δ > 0 N x0 xn → x 0 f (x) g(m) limx→x0 f (x) = +∞ f (x) < M → ε > 0 f (x) → δ (ε) > 0 ε>0 x m f (x) 0 < δ = δ(ε) f (x)g(x) = m f (x) + g(x) → + m x0 x0 δ δ limx→x0 f (x) = |f (x) − | < ε f (x0 ± 0) x > x 0 x0 ε > 0 ε, δ ε>0 f (x) < M x0 f (x) > M ε0 > 0 f (xn ) → x0 f (x) g→ h, f → x0 M > = limx→x0 f (x) M − > f (x) − > − M 0 < |x − x0 | < δ = f (x0 + 0) = f (x0 − 0) |f (x − | < ε 0 < x − x0 < δ |f (x) − | < ε limx→x+ f (x) = limx→x− f (x) x0 f (xn ) → +∞ x0 0 < |x − x0 | < δ ε0 < | − f (xn )| = limx→x+ f (x) xn →= x0 ⇐ | − f (xn )| < ε x0 g(x) → m 0 0 f (x) − − → −− f (xn ) → x→x0 x0 f |f (x) − | x n → x0 |xn − x0 | < f (x) → (x → x0 ) δ = min(δ, δ ) δ(M ) > 0 xn →= x0 N 0 |f (xm ) − f (xn )| < 2ε |xn − x0 | < ε/2 f x, x {f (xn )} δ>0 M x n − x0 → inf x∈A f (x) > −∞ supx∈A f (x) < ∞ |f (x) − | < ε |x0 − x | < δ ε>0 ε > |f (x) − | x n − − → x0 −− ε > 0 x0 |f (x1 ) − f (x2 )| < ε n→∞ m, n > N f (xn ) = ε0 > 0 |xm − x0 | < δ f (x) = lim f (x) δ x0 |x − x0 | < δ x0 x0 limx→∞ f (x) δ>0 ε>0 |f (xi )− | < ε/2 x→x0 0 < |x − x0 | < δ x0 f (x) |f (xn ) − | > ε0 x0 ∈ I E = {xn } N δ>0 f (xn ) − f (xm ) < ε ε>0 |xn − x0 | < δ I f x0 f (x) = limx→x0 f (x) 0 < |xi −x0 | < δ x0 x, x = x0 f (x) = limx→x0 f (x) A i = 1, 2 0 < |xi − x0 | < δ limx→x0 f (x) f (x) → |xn − x0 | < δ m, n > N |f (x) − f (x )| < 2ε f (xn ) → = inf E f (x) 1 n ε>0 x n → x0 |xn − xm | < δ f (x) f (x) 0 < δ(ε) x → x0 f (xn ) xn |f (x) − f (x )| < ε x1, x2 N x n → x0 x n → x0 ε>0 ε f (x) inf A f  nƒ ¢ d” e ¡ ¥v™~} ¡ ¦ „kq“Cx§vi—EW˜s ” o hw” hh˜ h}” ˜}  "    05  2 —vh  Ž™™“• ¦ ‘u g…YŸ„k“ e } w o 4 k ¥ " ek s ” gf„eg‚’z•(eq‚“š‘“V… „k“ gWyvgjg—¦– ” e ¡ ‘“V… 9vjqh—vfv¦ˆ”g}Yegs’¦‘g¥g¤– q– qh‘u e hd }j• } s‚ e }j• ¦ h hu † ed ”–  h    T %& 2 ‘u e —gs}W x˜ f “¦w o ~„”|‚s ¦ vu 3€ xgiP¢fs v…hs ¦ „kq“h gWs Vq…yPvgjg—” fC • @ k s k  e f ” s •} hh˜‚ e– Y ˜ ¡ ¦ 2„k“ gWs e s e} e ž ef Ž3gk¥gwš –P—” ¦    inf x n M |x − x0 | < δ yn → y0 f (b − 0) f y= xn < b f (x) > 0 f (a + 0) = inf f (x) x0 x a n limx→x0 f (x) xn < x f (b − 0) = sup f (x) M0 = lim f (x) x→b−δ a1 n [a, b] x0 f f (x) > 0 f f limx→b− = ±∞ f x0 M0 = supx f (x) f   f n– f ⇔ f f (x) f f j• “V…  2 d ”   05 ¥ ! 2 —vh ¦ Px˜ o kk 4 2  T B( 3 2 vu  ” 2    5  3 2 dvh k vu    T 4 2 %¨ ! 2 (e‚vgjg—” g—” e– ˜} ˜} ”   f (c) = γ x1 4  o hw” ¦ „kv“h ¥v™‘e ¡ ‚f h } e h h ˜ ‚ e– ˜} —¥3qk‘u “z• e  —kq“yyxvgjg—” g” 4 eb " 4 8 d k o¥qh™”¨vj“Y•… ¦ 2‘•z• ee  g|YhŸ“V… w } eku f s j• —kq“h gWs s h x˜ 2    T 4BQ 3 2 vu     T 4 2 B2 ! 2 k ‘u e¡ e ¡ e ¡ ¥q™gw’~‘“V… ohed }j• g f „kCx˜ ‘“VE” “h } j •… ˜} c ¤ 4 xb " 4 8 d d” o hw” } j ¥v™~v“Y•… CV“qd‘—¦“#v“Y•‡Y…‡¥—gf¦„kq“2y“‘gjg‘Eg’d ¦ ef… h j” k d ” j … s … e hh ˜ ‚ e– d ˜}    " 2  5 A3 2 vh ¦ Yegzž s  ̉¤ ¦ s ”sf” ˜s ‚ e ¦ hf ™vwh Y—™W˜„j“z•™o vjqh—vu ˜} g” ¦   q¤ zn¤ k §‘u Pf „og‚e 7 ¤ † ‚ f h ¦ s ‚ j s j s j h u ‚ ¦ “ ‚ d  –   h —x|˜V (e‚ v€ ” ˜k “d Y•6”… xq˜• q‚vhv(e‘Žxpvhv‚s}sj —˜ ¥v¥u–hi )(ed } “™Y•j ˜‘“ukhq…o ¦ ˜k ¥Ÿ” ¥3vkvu  y’u o ˜˜ k §vu ¦ —™„og‚Fe©¤¦ Y †f e 4 xb " 4 8 ‘¥—˜“¦v“Y… d  d ” ¥q™§v“V… (e„‚v‚h  o hw” } j•  h hfd‚ e– ¦„kv“˜vjqh—“‘‘gj—” ˜} g” 2   5  3 2 vh ¡ ¥v™Pe ‚ o hw”  ¦„kq“h qghŽv}’§xŽv}—EW˜s  e hd }i h}” 4 deb " 4 8 ‘j  k hh˜ i h}” ˜}  e˜ h žohed x“vdg”€¥v™gw’~} ¡ „kv“3x¥Žv}ŽEW˜s g” "    T 4% # 2 ‘u „kq“h gWs x•q}” g” s h ˜ i h} ˜} ¦ Ž™™“¤‘Cv‘ e}wo• k u‚ j —k“ 4 d eb " 48 f (x0 ) < f (x) f (x0 ) = x x0 0= f (c) = 0 f (c) = g(c) ε= f (a1 ) · f (b1 ) < 0 an , bn → c c ∈ [an , bn ] ∀n f (x0 ) 2 f (x) > g(x) a 0 x0 f (x) = 0 [0, 1] f (b) ≤ g(b) [a, b] δ f 2 (x) ≤ 0 0 > f (an )f (bn ) → f (x)2 h(x) = f (x) − g(x) g(x0 ) < f (x0 ) [a, b] g(a) ≤ f (a) d eb " 48 c ∈ [a, b] m = f (x1 ) m≤γ≤M x1 h(x1 ) h(x2 ) [m, M ] = f (x1 ) − γ = m − γ ≤ 0 = f (x2 ) − γ = M − γ ≥ 0 „kv“3x˜ hh gfˆeg™’§v“Y•… #ˆig†™o “gdtgk¥v¥ˆig†™o ¦ gi™qwtgktvk„hxgi¥vwÌh¤ Pgf(eg‚’z•(ev‚“d „kq“h ¥v™” z• e g”2k †–od } j  e ¦ w e e  h e e h e ‚ e e e h o hw e e o hw” } j• ¥v™Ÿ‘“V… ¦ „kq“h z• e g” 2k “Y•…  ™™“• vu e e j }wo ¦ k 0ig†¥o „kv“h z• e g”2k “Y•… 0ig†¥o e¦ e e j e gf(eg‚’z•„eq‚“~‘“V… ¦„kv“h ¥qh —€v“Vt‘P¥‘¦‘u z• e g”2˜“Y•…  e hd }j•  o w” } j •… k ueo j k e ek j o¥v™~} ¡ 2vu hw” ek e „kv“~W‡™V q}ghž e ¡ “vY}s  h s …o fdj ˜} g—” s gzž  f M f m x = [a, b] γ M = f (x2 ) m = f (x1 ) y f (x) m, M f (x) = γ [a, b] h h(c) = 0 [a, b] f f x f M = supx f (x) m = inf x f (x) [a, b] 0= h(x) = f (x) − g(x) h(x) = f (x) − γ M = f (x2 ) y = f (x) [an+1 , bn+1 ] ⊂ [an , bn ] ε>0 f (x) f (x) x 1 ≤ C ≤ x2 f (an ) · f (bn ) < 0 bn − a n = y = [m, M ] b−a 2n n› k o hw” } j• ¥v™€‘“V… ¦ „k“ ¥v™~—qf¥—pz˜ e—~‘“V…  o hw” } h–” e” } j•   " Q  5 A3 2 vu vjqhh—vfvu 4 vvjqh—vfvu Pgfˆe¥¥“d v“Y•…¦ e ¡ 3¢¥q™” —qf¥—”  z˜ e—” } ¡ ‘‘¥‘™vu vjvhvu hC ˜f h h ” †–o } j sk o hw } h– e kueoj k h X D% !Q 2 4 δ(ε) X ε X y = f (x) ε0 ≤ |f (x1 ) − f (x2 )| |x1 − x2 | < δ x 1 , x2 ¤   ¢ ˜… e e s Vg}¥gogWs „‚Žhkv‚ }˜¥zž ¦ |“@ k s ˜ o w” } ¥qh —¨—vf@ ˜ Y…s t„kv“h gWs qjvhvu ¦ Y¥W˜gWs  s h  s hh v”—”  o hw” ¥q™~} ¡ Ž}’C~Pvf¥—” z˜ e—” 7 k s sk } h– e † ‚  –se—” ™Yw 3Y–H’zC es ž qk¢ „hxgie § 7 ¦ † f ˜ ” ” s gk s g ‚ o w” ¥qh —¢} ¡ Yk3#—vfg—” z˜ eP” §v—‘™Žv}v§“d C™xi ¦ „kq“h  sk } h– e   ˜f jo h u k skw o h ed } h– e” eood o h ed s” } s‚ e– ¥q™gw’~—vfg—” z˜ e—yg†™™“§¥v™gw’‡ˆov†hšg’vgjg—”  ‚‘Igj4 gH –Ge ‘d D9 32 2 —v”h vvjvhPgfgf’ˆee ¥d† ¥“¢‘“V…  ¦ v”‘™”j 3~¥v™¢—vfg—ƒz˜ e—e ” v§gWs eFC 4 d ˜ f h – o d } j • h d  o k o h w ” } h – ” k u s h Y W˜gWxg“se† s s vjqh‘u h E i y r „ ‹ r „ h y P£H‡ ysu3 w {t{„‘w   Yegz¥vjqh—“v‘¤™gw(e‘¥q” ™gw” ¦ s ž hfd†u ” ‚j h ”” e e hk o w”e e h Pv”3¨¥qh —PCgf(eg‚’z•„eq‚“d  arccos x, . . . eŽ3gk¥gw~ –P” ¦ W¨gw(ev¥v•• vo0hg†(eg‚P“• qo“d ™gw™¦ } e ž ef s ‚j h” ef h ” e” ¦ “„d3vk™v”3tgfˆg™’d  } ‚ h ˜ h k e† – o  –ˆeP3gk’d ¢ †f e ¦ ¦ es ff ˜eo hw ¦es ž hf deo hw” es ” ‚ j h g…Y” —x—¥¥v™” Vgz¥vjvhP“P™¥q™Ce g…Y—” ¦ ™gw(e‘¥q” ¦  gf„eg‚’z•(eq‚“d CV“qd‘—¦“d Y…s ¦ —v”gh‘e ¡ “VPYe™vw#—x˜ “¤—”v‚As ¥v™”  e h ¦ef… h j” k e e d j •… s h ”f w • @ o hw 4 xb " 4 8 ™g¥E˜ ¡ d } oe o ¦ —q”C§¥v™CPgf(eg‚’z•(ev‚“d ¦ Pf “vdvŸ“d s  e h f k i o j ‚ h w ” k e e gf(eg‚’z•(ev‚e“¤…d —h xj ¨”˜ ‘k “V… ¦ ˆe… ‘”¥q™¨v“Y•… ¦ qeq”h z• ee  ¦ „kq“Ÿqjvh—’d ™gw(e‘¥q¢—” gw” eh h f f } j • † u o h w ” } j h h hf ” ‚j h” ” e 4 2    T B ! 2 x‘„” ¢ vu f sk g|Yh¦“d  }f jo” —v™ƒ˜ ¡ ¦ WgsCf o ¦ 3¥qh —¢v¥•‘“VŸx—CYkvsh ¦ „kv“h e sk o w” } j” ” j •… ” ˜f ¦ —k¢q–gh e# –C” f 3” 2z• e 3k “ ž ef ok ˜ e s ek j• v˜z• ee  g”3˜“V…  h s”dj s f˜f s h ˜ “v}¤x3kSs ¡  |˜‘u ¤ YgzŸz• e ™gw(e‘¥q” 3¤g|s “¤™go e es ž e ” ‚ j h s k f h k d ” e ¦ (eg“§“qdh —” e i o ‚vj  ‘¨z#x™™o k u ž f iw  … skh s s”dj s h ž skh s s”dj s s e ¦ skh e #” ™Yo C3gzYeS“v}¢vfPvfgh’s ¡ ¦ Ye3CgzYeS—“‘Ž}6v”gh’d Ye3CgzVs ¡  ‚ ghž s s } h s  e ” ¦  h j ¤ h¦ Pfy„‚˜q‚¦ ‘g…h”™v”Ye—zh • es ” vjvhvg”’2ŽpYeg}k†eu “Vvf• … —qf(egh„‚ž v‚h  3¥v™—¥„}f 3‘€gv}egjk‚ Ppq˜q‚¦ ”D™v…¦h yj „‚qhgv‚o ‘hh |v~Ÿg}—Vke x—W˜sf ¤” s“—‚—q˜ ‚ h C”sYkf v—gx—W˜–fesgj š„wv‚Pvf“€‘“V…  q”gh’™d ¦“e q”¡ gh’gfd e (e¡ g‚’z•(e3v‚s “d ¦¥q—‚h q‚—” s uh  u  f k s k o h w ¤ – ” e vs† s x ‚ ” h h d s j • e o e e h‡o wh ” ”fi ‚ d xb " 4 8 g“e —xv„j” 4 −1 ≤ y ≤ 1 x = arcsin y f f (x0 ) f f (x0 − 0), f (x0 ) f limx↑x0 f (x) = f (x − 0) f (x0 ) > f (x0 − 0) f −1 f (x) f (x) < f (a + 0) y x0 X x0 X X X x0 f f limx↓x0 f (x) = f (x0 + 0) f (x0 ) f (x0 − 0) < f (x0 + 0) y y γ f (x) ≤ f (x0 − 0) f (x0 − 0) < γ < f (x0 ) −π ≤ x ≤ 2 f f f −1 (y) f (a), f (a + 0) x0 π 2 δ > 0 x2 − x 2 < ε 1 2 x1 , x2 0 < δ = δ(ε) x2 − x2 = |(x1 − x2 )(x1 + x2 )| < δ · (x1 + x2 ) = δ(ε)(x1 + x2 ) 1 2 |x1 − x2 | < δ ε0 > 0 ε>0 y = ax , 0 < a = 1, 0 ≤ x ≤ ∞ ⇒ |f (x1 ) − f (x2 )| < ε y = ex , x = ln y (0, 1] [0, ∞) f (x) 0 n (nk <)f (xnk ) → f (c) = 1, 2, . . .) Σ [an , bn ] ⊂ σ [a, b] ⊆ ε0 ≤ 0 x nk → x a ≤ x ≤ b ε0 ≤ f (xnk ) − f (xnk ) → |f (x) − x(f )| = 0 σi Σ A = [a, b] σ1 , σ2 , . . . , σ n ∈ Σ xn k − x n k < (0, 1) ⊂ 1 1 Σ = {σn }∞ n=1 σn = ( n , 1 − n )n = 1, 2, 3 . . . A = (0, 1) σ∈Σ n=1 ∞ [a, b] ∞ σ [a, b] σn n 1 nk 1 1 ,1− n n →0 x∈σ ∞ {xnk }k=1 A {xnk }∞ k=1 1 n > |xn − xn | σ∈Σ (0, 1) f (c) = ∞ f (xnk ) → f (x) f f (x) xn x∈A xn ∈ [a, b] xn k a≤c≤b [an , bn ] f (xnk ) → ∞ xn ∞ k=1  s ž h †P v”3¦h k gzvix¥v”u –  i ho” s sk h 6(eg“e “qd™~g W˜3Hv‚’gdž [a, b] n £¡n ¦ sh hu‚i Vq…“vdv“x—gfž ‚iž e e xPgf•gk’gdš} ¡ „e3Pf p˜ ¡ ‚k o gke ¦ k‘š™go¥o ¦ …3¥q–h ¦ gk€…C¥q–h u }e sh e sh gk§|…@ h  e „†v“™~„}‘gjgP”  vj 3¤„”0‚„†“ ’™W˜s } ¡  ho” ‚ e– h ok od x0 f (x) y y (x0 ) n ∆x→0 k hu† vu vjvhvyg}Yes kv“q‚“gd§“—Ss ¡ u h ž …d” „kv“h x|˜™gw™… z˜2k § x|˜™gwe ¤—xv˜‘~„}‘gj—” ” h e e s h … ”f hj ‚ e– ‚ jok ” h …e e hd s ” hd ‘™3#yv˜™gwe C—g”2k „”g“ eqe“™W˜#—qf“™o ∆x = 0 x0 α y = f (x) ∆x → 0 k hu f hj “ vu vjvhv#x|˜v€—”  … u ho v€v™#” ¡ ” vghŽq}h  e x   ¡  d” ˜ †–od } i h}” —gfˆe¥¥“€xv}ŽE˜s qeq”h ’x—gfž h ‚i 4 G E C 4 ¨ IHF9D  2 vh i h cw cŒH‡ ! y eYgzž s  xgiv‡Y…Cgq–xg}Vgs’¥gi¥vwqh‘CYegzž g”2k xgiyu   eu sh h† eed e hu s e  e d q”h Wxg”e ¦ Yegz˜vjvhvu  ˜ s˜ sž h zP™o ¦ ef es Vgzž ‚i xPgfž  qj}„sg‚¦qk—hxgi¥qwEW˜s h e ‚eh ¦ s h ‚ ™qw¤q”vh„h” ¡ g}¥os ¦ x˜ “d ¡ |xgiqeEvi™vwh šWgsCf o § vjqh—“d e h d h h h  e hf a∈[a,b] ¦ h (vjvhvu ¤  ¦ v”” d h ” h ¥qh —t‘“V…  qe—v”h  ™¥v™gw’d „kgWs o w” }j• h fo h e “ s 4 eb " 4 8 qh d f ” s f ” ˜ s ”hh‚ }e o s k w o s j } ”e   ” s f ˜ w • eh o o k o k ˜e 4 X 9% 3 2 Y—™W¢qv„” ¡ g¥Ÿ™x˜ SvYs gŽ—6PxV“—q™x˜ 3xi ” 3xz ž  C 4 Q es h e d † Vgzž ¦ vfh v¥ –‘¢0”Žg}Yes ¦  ‘™3#‘xqi—vfvu  ‚ jok f j h h M ≥ f (xnk ) > M − ¦ sx¥¥” y•“™¥¥” i–o h ˜ dd i–o fd ‚id ‚ e– “g—x“’„s‘gj—” h ‚ e– x˜“vgjg—” ‚ j h ” Y  e ˜ f ¡ ” s s e s o ‚ i  d xb " 4 8 ” gw(ev¥všxgiC¤fVq…hCf™V… ¦ x—gfž 4 {xnk } 1 n f™¥v™gw’d } ¡ ¦ ohe s • ”gWs g} v…h ¦ f n ¦ Žs k} es ž e  i u ¦ ef Vgztvk„hg‚šx‘¤fs v…hs gsP™o ¦ 0 < δ = δ(ε, x) |f (x) − f (xi )| < ε = 1 ∆y = f (x0 + ∆x) − f (x0 ) f (xnk ) → f (c) c 1 M−n M − lim c ∈ [a, b] x ∈ (xi − δi , xi + δi ) f (x0 ) x0 ∆y ∆x 1 →M n dy dx ε = 1 x 1 + maxi f (xi ) > f (x) > 1 − maxi f (xi ) [a, b] f (x) |f (x) − f (x )| < ε M f (c) = M f (x) i 1 + f (xi ) > f (x) > 1 − f (xi ) dy dx |x0 = [a, b] xn ∈ [a, b] f (x0 ) f (x0 ) ¦ δi = δ(εi xi ) x ∈ [a, b] {xn } |x − x| < δ α   Y ˜ ¢  W˜gs s (sin x) = cos x • – lim sin(x + h) − sin(x) h = sin x cos h + cos x sin h − sin x = h 1 − cos h sin h + cos x · = lim − sin x · h→0 h h 1 − cos h = −0 · + cos x · 1 = cos x h h→0 h→0 lim = (cos x) = − sin x • – lim cos(x + h) − cos x h = = limh→0 limh→0 = cos x cos h − sin x sin h − cos x = h sin h cos h − 1 − sin x · cos x · h h sin x h→0 (tan x) = 1 cos2 x • – tan(x + h) − tan x h tan x + tan h − tan x(1 − tan x tan h) h(1 − tan x tan h) (1 + tan2 x) tan h · = = h 1 − tan x tan h 1 = 1 + tan2 x = cos2 x = (0 < a = 1) (ax ) = ax ln a • – ah − 1 ax+h − ax = ax · h h ah −1 h h→0⇔t→0 t = ln(a ) = h ln a h e =a → ln a t – – – h ah − 1 et − 1 = ln a h t t → 0 ⇔|r| → ∞ 1 r t = ln(1 + 1 r) = et − 1 1+ – 1 r =e – t " ¥ # ¦ 2 & £ ¡( ©¦   ¨ ‚xPgfž i g‚|h  ¢ e ‚i xPgfž ¦   ‚ hu s} ‘yv˜‘’ŽE˜ ¦ WsgPf™o ¦ e  h˜ s hu s} 2xSxv˜‘6€|h  slq  – e −1 t t = = 1 1 = ln(1 + r ln(1 + r ) 1 1 −→ =1 1 r =− − r→∞ ln e ln(1 + r ) 1 r) 1 r = et −1 t =1 – 1 x ln a (a < a = 1) (loga x) = loga (x + h) − loga x h t = loga (x + h) − loga x loga x t+loga x t loga (x + h) − loga x 1 = −− −→ t − 1) t→0 x ln a h x(a x g(x), f (x) (c · f (x)) = c · f (x) g(x) = 0 (f (x)g(x)) = f (x)g(x) + f (x)g (x) f (x) g(x) = f (x)g(x)−f (x)g (x) (g(x))2 (f (x) + g(x)) f (x + h) + g(x + h) − (f (x) + g(x)) = h f (x + h) − f (x) g(x + h) − g(x) = lim + = h→0 h h = f (x) + g (x) = h→0 lim (f (x)g(x)) = = = h→0 f (x + h)g(x + h) − f (x)g(x) = h g(x + h) − g(x) lim f (x + h) + g(x) h→0 h lim h→0 f (x + h) − f (x) h = lim (f (x + h)) g (x) + g(x)f (x) ¡ n ‰ ”g‘g’d ¢ e e r ‰ q (f (x) ± g(x)) = f (x) ± g (x) h q”ghe 2 ¥ 3 ¦ 2 n ¥        & £ ¡( ¦   „†v“™go’š„}vgjg—”  h ed ‚ e– ¦ v˜—gf”} ¢ hes x·a =a ·a =a h→0⇒t→0 t t =a loga (x+h) ‚i ž kd ” ‚ xPgf~} ¡ “#—gf(e” ¦ t =x+h ∗ h = x(at − 1) ∗ ¦  k ‘u      • – –  –     f j s e iw ‘Ž}˜gdH™“d 4 d xb " 48 v‚’gd’s ¡ gdH™“d “q•—” ¦ h ž e iw j ‚ x0 ¦ „kv“h v}˜gdH™“d f j s e iw y q‚“gd”—sgz¥‘C’šCgk¥gw#3k ¢ h ž “ eo“kd } e ž ”} —kq“#” gw(ev¥v” h ‚jh ¨    05 " © 2 z = g(y) = g (f (x)) = (g ◦ f ) (x) d —v”h „†v“™š} ¡ ¦ ’q”h ho” ed x0 e h o” ‚ e– fd h —”v3k‘—†q“h™š„}‘gjgP” ¦ ’‘qjvhvu ¢ „†v“™š„}v¥ –—” ho” ‚ j e  y0 = f (x0 ) x0 = (g(x))2 g (x) Q ¥ 3 ¦ 2 & £ ¡( ¦ £   §A $ ¢  '¡ ¦ ¤   #¨&  ¨ ( ¥  e– u‚fk o r §gjgy“P¥C’0‚„†“ ’d ef˜df hu Px2—xv˜‘šz} ̉¤   h→0  ¨ ¥   ¦ ¦ ¦ § ¡ ¢  ¢ & q”gh’Ce ¡ ed … † }e s … ”    I ¡ &¢  ¢  vuvŽgY‡s gs   F¤ —†q“™~—}vgjg—” ¦ „k“ ho” ‚ e– x f (x) o w” ‚ e– ¥qh —€„}vgjg—”  x lim (f (x + h)) g (x) + g(x)f (x) f (x + h) − f (x) = h→0 lim f (x0 + ∆x) z (x0 ) = g (f (x0 )f (x0 ) = g (y0 )f (x0 ) ∆z ∆z ∆x 1 g(x+h) = g(y0 − ∆y) − g(y0 ) g(y0 + ∆y) − g(y0 ) ∆y = · = ∆y ∆x = g (y0 ) · f (x0 ) ∆y h h→0 lim (f (x + h)) = f (x) − dz dy dz dz = · = dy dy dx dx f (x + h) − f (x) · h = f (x) · h → 0 h 1 g(x) = f (x0 + ∆x) + y0 = y0 + ∆y = f (x0 + ∆x) − f (x0 ) = = h→0 lim − z = g(y) = f (x)g (x) + g(x)f (x) g(x)g(x + h) f (x+h)−g(x) h f (x) ∆x y = f (x) 1 g(x) z f (x0 + ∆x) = f (x0 ) + ∆xf (x0 ) + ∆x · α(∆x) ∆x → 0 f (x0 + ∆x) − f (x) − f (x) ∆x ∆x → 0 ⇒ α → 0 α (∆x) = f (x0 + ∆x) = ∆z ∆z ∆x = xa = a ln x d a (ln y) = dx x d d dy (ln y) = (ln y) · = dx dy dx 1 dy = · y dx 1 dy a · = y dx x dy a a = · y = · xa = a · xa−1 dx x x y ln y ln x −(−x)n d y dx e d y (e ) = dx 0‚gq–™o †h k Px˜ o v”3¨gw(ev¥v™W˜s yz˜2™“”3™x€Cx˜ o hk ‚j h” ˜ ekwd skw i k  vdvgj ež ¢ v‡gw(e‘¥q” „†v“™~„}vgjg—” ¦ „kq“¦vu ‘Ž}˜gdH™“d v”” h ku ‚jh ho” ‚ e– h k f j s e iw h h h˜ h” } e k x~} v…šg”‘” ¢ ‘u e ¡ 77 ∆x = 0 ∆y = 0 ∆x x0 x0 f (x) ∆x α(∆x) f (x0 ) + ∆x · f (x0 ) + ∆x · α(∆x) ∆z ∆x = g(y0 + ∆y) − g(y0 ) = = ∆y · g (y0 ) + ∆y · α(∆y) ∆y ∆y = g (y0 ) + α(∆y) = ∆x ∆x = f (x0 )g (y0 ) + f (x0 ) · 0 = = f (x0 )g (y0 ) (xa ) = axa−1 • – x<0 y = x ln x d y dy (e ) = dy dx dy dy = xx = = ey · dx dx = xx · (ln x + 1) ‚id s e} x’gWEz˜‘‡…s – (xx ) • – ¨ ¥  ¦ 2 & £ ¡( ©¦ ' §©¦   ¨ &¨ #¨ ‚ vj  } h–” —qf¥—Pe ¡ } eo”‚ ‚ ¥g¥’gz(e”   @ ~vAs ¦ ‚ ‘j  v…I™˜ ¡ v”¦gfˆe¥¥o  @ f h h” †– s‰q y = xx (xx ) = xx · (ln x + 1) – ln y = x ln x d (ln y) = ln x + 1 dx 1 dy · = ln x + 1 y dx dy = y(ln x + 1) = xx (ln x + 1) dx (uv ) x v = v(x) u = u(x) y = uv – ln y y y = v ln u = v ln u + v (ln u) = = v ln u + v · u u v·u v ln u + u y = uv (uv ) = uv v ln u + v·u u – x(x x ) dy 1 = dx dx dy y = sin x x = arcsin y 1 d arcsin y = = dy cos x cos x = 1 y2 − 1 sin2 x − 1 y x x = ax = loga y 1 = y ln a y = f (x) x = f −1 (y) †—g“¥•“d eo k d ™0ov†hs ¢  ¥  ¦ 2 h q”ghe & £ ¡( 1)§A $ ¢§©'§¨ ¥     ¦   #¨&        # ‚i xPgfž g”‘vg‚Hi ’d ¢ —v”” e” e w h h  ¦  „†v“™go’š} ¡ ’s xg} hed  ˜  – • ¢   ¢ ¡   4 qƒ k e h ff ˜eo w” } i h} ff w• e– d h C 4 X 9BQ ¨ 2 vu gf(eg‚’z•(ev‚’d —x¥¥qh —§xŽv}’d „kq“h —x˜ “˜gjgyu ¦ q”” ϕ (y0 ) = 1 f (x0 ) x0 ∆x → 0 f ∆y → 0 o hw”e e h ff ˜ ¥v™PPgf(eg‚’—•v‚’d —x€} ¡  y0 ‚˜ h k yCqkV§ “d ‚id s x’gW‡…s „kv“h „kv“€¥3—qfghpz˜2k § s ¡ h –h ž e Yegz˜vjqh‘#0”Žg}Yes ¦ WgsC™o ¡ „”0‚„†™“™˜s „}‘gjgP” ¦ —xz˜ e—egh ež s ž h u † ef “od ‚ e– e” y0 ϕ ϕ(y0 + ∆y) − ϕ(y0 ) ∆y = = CxVo k˜ k Cx˜ o hq”v‘u h”† h h”† q”v‘u ¦ vh |h o¥v™~} ¡ šWgsP™o ¢ hw”  ef o hw” ¥q™~} ¡  ¢ x0 y0 f ϕ ‚ w v…Yh¦“¤™gw„ev¥v—” ™gw’s s kd ” ‚j h ”” e”  ef ~WgsC™o ¦ 4 d xb " 48 d v”h gdH™“˜“—•” ¦ xPgfž e iwd j ‚ ‚ i WgsP™o ¢ ef x = ϕ(y) ¨    05 2 © 2 ek d o w” 2vu —v”h¥qh —š} ¡ ¦ „†„kq“v“h h —¦„†„}q“h‘—go j €—}–”e ” ¡ !¡¦ e qeq”h o” ‚ h ϕ y = f (x) y0 ϕ  „‚v‚h  |h f (x0 ) = 0 ϕ(y) x0 + ∆x = ϕ (y0 + ∆y) ∆x = ϕ(y0 + ∆y) − ϕ(y0 ) ∆y d arcsin y dy ∆x → 0 ⇔ ∆y → 0 ∆x = 0 ⇔ ∆y = 0 f (x0 ) + ∆x) = y + ∆y d loga y dy ∆x = 0 ⇔ ∆y = 0 = f (x0 + ∆x) − f (x0 ) = f (x0 + ∆x) − y0 dx |y = dy 0 y0 = f (x0 ) x0 ϕ (y0 ) = f f1 0 ) y0 (x = = 1 f (x0 ) dy dx |x0 ∆x = f (x0 + ∆x) − f (x0 ) 1 1 1 − y2 1 y ln a 1 f (x) ∆y → 0 ∆x → 0 ∆x = 0 f (x0 +∆x)−f (x0 ) ∆x ∆x → 0 ∆y → 0  ku hfd ” ‚j h” ” e %vƒqjvhP“¥™gw(e‘¥qE—” gw” q– ∆y = f (x0 )∆x + ∆x · α(∆x) x0 f x0 f f ‚q“gd”—sxi ‘C’š¥g™’d ¢ h ž ‚  “kd } eo —–h ™„o¥v3Žv}hqs“~} ¡ v¥vt‘u „†q“h —€} ¡  w ‚ hk hd kueoj k o” 4 ¨    T %  2 s „wq‚§zž ¦ ™C„kg‚E0˜‘#q”h V gz ež Ÿ‘u „†v“™~} ¡ —kq“h q–™™„o¥v3q}hvs“d  h ”o e † u ” w ‚  k ho” hw ‚ hk h WgsP™o ¦ ef  f (x0 + ∆x) − f (x0 ) =A ∆x f ∆x→0 f k gCgkž ¦ vu k Cy˜ o g”2§xŽv}ŽEW˜s ¡ l ¦ g}™… ¦ e ek }i h}” je 4 G E C 4 ¨ IHFD9¨  2  ¦ ‚kfo h ‚ j h 3—™§v¨gw(ev¥v” ˜ †–od } i h}” —gfˆe¥¥“€xq}™W˜s       q”C¨q–™™„o¥v3q}hvs“š} ¡ ¦ ‘u vjqh‘#x™™o Y ˜ ¡ hk hw ‚ hk hd k h u f iw       ¦¥ ¦¥ 2 ¡  # ( ¢$      ¨ ¦ &              }3vk„h¥ –y¥v” ¢ ‚ejh k Px˜ o f e ze  f (n) (x0 ) = f ((n−1) (x) x0 k ho f iwo w… h u ‚ e– h † ho } ‚ e j h vu „†v“™” x™™‡VPvfv~—}vgjg—” „kv“2ˆ‚¥q–™~3vk„h¥ –y¥v” ( ¥ ) ¦ 2 & £ ¡( ©1')( ¥   £   ¨   ¨ ¦   ¡ A (f (x) · g(x))(n) x0 x2 sin x (cf (x)) (f ± g)(n) (sin x) f (x0 + ∆x) = f (x0 ) + A∆x + ∆x · α(∆x) (n) (n) (k) ∆x→0 lim = = f (n) ± g (n) = = = = c (f (x)) = k=0   k ∈ {2n} sin x  2 k / 2 ∈ {2n} , k ∈ {2n} − sin x (n ∈ N)    k=0 n 0  k   k    k  k      n 2 =1 =2 =3 =4 x2 (sin x)(n) + n k f (x) · g (n−k) (x) k n k x0 (n) cos x − sin x − cos x sin x x2 α(∆x) − − → 0 −− k (sin x) n 1 x0 n−k x2 (sin x)(n−1) = f (x) A f (2) = f , f (3) = f A = f (x0 ) ∆x → 0 ∆x dx ∆x dx = ∆x dy ∆y ∆y = f (x0 ) · dx = = dy · dx dx dy · dx dx dy · dx + α(dx) · dx dx dy = f (x0 )dx dx ∆y dy x0 = 1, y0 = ln 1 = 0, dx = 0.1 y = ln x ln 1.1 dy ln 1.1 − ln 1 = ∆y ln 1.1 0.1 = f (x0 )dx = 1 · dx = 0.1 dy = 0.1 sin 46o = sin π 4 + π 180 y = sin x sin π π π − sin + 4 180 4 π √ 180 2 π π sin 46 = sin + 4 190 = ∆y 1 π √ √ + 2 180 2 (2.20) 3 1 (1.3)3 = 2.197 3 3 3 3 (2.20) = 1.3 + = 1.3 1 + 1000 2197 1 f (1) = 1 f (1) = , 3 3 1 3 1 dx = ∆y dy = f (1) · dx = · = 2197 3 2197 2197 1 1 1/3 ∆y = 3 1+ 2197 1 3 − 13 1 1+ 1 2197 0 ≤ 0 ≤ 2π, y = sin t, x = cos t (exPgfž ‚i hg –e €|h … qxz˜Ž} he ˜} } h e} k y§vvj™go‘¦“d ¦ v”h d ¦  –ˆe—3”x|˜PgfH˜ ¢ †fk s h e g“‘g‚Hi pq˜g‚‘—d’•qj™goŽ1x¥¥¨3xz˜ eFž¤ “gdH™“d e e ” e ‚d h e} f i–o ok s e iw ¢ £ ‚i ž …d” k xPgf¨“Ÿ“d  –ˆePCvg‚H™“d ¢ †fk e iw  ¦ 2 ¥ s †f s‚ ž sh ¥ –ˆePCk ¦ f 0…†‘“gd e¨…3¥—–h ¢ #¨& $   § '¢§  & ¦ ¢§!$ (   # (  ¦ # ¦  ¨&   ¨ (  …3¥—–h  sh ‚˜} } h e ‘‘x§vvj™goŽ} h˜ s˜} k x6‘x¦“d  k j ž f e Cy˜ o “z•¦3gkg–  q›  £œq f (−1) f (x0 + ∆x) ≤ f (x0 ) f (x− ) = 0 d2 y dx2 dy dx lim ∆x → 0 ∆x < 0 f (x0 ) = 0 f (−4) = = = = f (x) f (1) f (4) x = = = = = = = ? = ˙ ψ(t) ϕ(t) ˙ dy dx +2 0 = −2 ±1 ¡ ¨ £   ¢ h ˜  ˜} kd ”fj h e xSs xgŸ“¤—vxvi—gfyu  f (x0 ) ⇓ = 0 ‚ ‘gj’#‘xqi—gfy¤vgjPqfh ¦ „kv“‡Žg}™o “z•ž x™vwv3v”” ‚ e–d fj h eu f e h … e j fi hjh h 4 xb " 4 8 ‚„vh  d dq”h —†q“h™”š} ¡ ¦ vu v”3’vgjg’Ÿ‘x“• Pgfv#vgjPvf’d ¦ —kq“h o k hk‚ e–d kj i o eu f e h  f w• wof x˜ “‡™——”    5 ¥ ¨ 2 ¤ ¥  ¦ 2 ¥  £    ¨ ¡ § (   ¡ ¨ ¡ F'# ¦ 0 '00 ¨# ¢¢!¦(   ¥  £ ( &   ¤¢ F#   £¡ ¨ £ „kq“h˜vjhvPf“¤” gwe(v¥vh” —” gw——eCk¨¥v™” ¦ h d ‚ j ” e ” h o hw e 2„k“ „‚q‚‘ gfe(g‚’z•(e‚v’d —fx€‘“•V… e—q”h3¨¥v™”  h e h f ˜ } j ¦ k o hw d v”h }‘“V… j•  „”“gd(eš¥g¥¨w P“• vo“Ìd¤ ‚ ‚” } eo” of h ¨    5 (  2  o d ˜s ‚ j h )„”0‚„†“ ’EW‡gw(ev¥v” f (x0 + ∆x) − f (x0 ) f (x+ ) = lim 0 ∆x ∆x → 0 ∆x > 0 y = ψ(t) = ψ · ϕ d dt d dx ¨ ˙ ˙ ¨ ¨˙ ˙¨ ψ(t)ϕ(t) − ψ(t)ϕ(t) ψφ − ψϕ = (ϕ(t)) ϕ(t) ˙ ˙ ϕ3 ˙ dy dt dx dt f (x0 + ∆x) − f (x0 ) ∆x f (x0 + ∆x) − f (x0 ) ∆x ˙ ψ(t) ϕ(t) ˙ −64 + 12 = −52 64 − 12 = 52 dt · = dx |∆x| d dx −1 3x2 − 3 t=ϕ ˙ ψ(t) ϕ(t) ˙ (x) d dt dx dt −1 ˙ ψ(t) ϕ(t) ˙ ≥ ≤ ≤ ≥ = (x) 0 0 0 (∆x > 0) 0 (∆x < 0) α≤t≤β = f (x) = x3 − 3x, −4 ≤ x ≤ 4 x0 x0 x0 Yegzž s x = ϕ(t), y = ψ(t) f (x) ϕ(t) ¢ £    –ˆe—3k ¢ †f ” e” s †–od ” h hk ed xz˜ e—”0”g™’¢¥—–™w „oq‚¥CŽv}hqsgh’ ¤   ¡ ¦ 7e —4” fxv9—%¤”4‚—jCi q” 32h Ex˜‚ ¢Pgft—x˜ “¨0s™„o„‚“ ¨ ¦ k i ž }f w• † X o hw” ¥q™~} ¡ ˆe„†v“™~} ¡ Vgz’™qw¨goYe3x˜ ho” ¦es ž s h sh (x − a) (a, b) [a, b] „kv“h q‚“vd„h“šg“} ¥g¥” h ‚d s”d } eo ˜} g”  F (x) F (b) = F (a) = 0 F (c) = 0 c 4 d xb " 48 0‚¥—–™‡gw(e‘¥q” † ho ‚ j h [a, b] d ” … ˆe„†v“™™go e ™“gd ho” } e ”w e f (x) = 0  h hf ¦„kq“•qjvhP“d ˜} g”    5 ¤ ¨  2 o h ” } j •… e ¥v™w~‘“V§ˆig†™o f w x˜ “• ˆs† „o„‚“ ¨ c M ¦ „kv“2x˜ hh ˜} g” sPxW˜s ¦ „}‘gjgP‡„wv‚—qf“d C2„k6™qw¨™—” ¦ f ‚ e–” h h e “ s h wof  k “ gke “v}qs‘~„}vgjg—‡„wv‚Pvf“d  fdj hu ‚ e–” h h 4 eb " 4 8 „qh d ¦ „kv“h s  s ž “o gWs CYegz#—”ˆ‚—†™“d s gs  —˜v‚h#vj™vohvu ¢ f o hw” s f j sd f j h e e h e } j• ¥v™PYegzž ‘Ž}“#vxvi—gfyu Pqf„hv‚—gfyu ‘“V…  4 eb " 4 8 d d v”h ¥qh w—€v“VE0ig†¥o se—†q“™š™go ee  ™“gd ¦ „kt‘u o ” } j •… e ho” } ”w e “k f w• e x˜ “¨goVs    5 ¢ ¨  2 0<θ <1 f (c) = 0 c = x0 + θ∆x a0 jž h “z•˜vjvhvu hh —v”” = e v ¦ i ho” s  ’vd™‡gYB˜ ¡ q   Ìk„‚q‚‘”PxW˜s  ¤ h sf n vu  k —kq“’s qw¨™——” h h wof c=a a 0) F (a + ∆x) − F (a) ∆x F (a+∆x)−() ∆x = −−→ −− ∆x→0 <0 f (b − ∆x) − f (b) −µ −∆x f− (b) − µ 0 < ∆x < δ ∆x > 0 F (c) = 0 = f (c) − µ f (x) c=a f (a + ∆x) − f (a) −µ ∆x f+ (a) − µ < 0 [a, b] (g(x) − g(a)) f+ (b) < f− (b) (a, b) [a, b] [a, b] [a, b] g(x) f (x) f (c) = µ F (x) f (x) F (x) Un‰ s gzž qk„h—‚—’vgjg’tqk¨¥v™gw’™W˜s ‚”‚ e–d h o h ed x0 lim f (x) x → x0 x < x0 f (x0 )  f (x0 ) = f− (x0 ) = h k u e h h u † e d @s s ”f w 4 d xb 4 8 vtv¤q”gh•qjvhPvfv§—˜v‚h 0”Žg}Yegs’#„”|‚A—#W˜#—x˜ “´•¤ „kv“h fi ‚ sk hfd ‚j h xv„j"” ¢ C˜qjvhP“‡gw(e‘¥q” —˜0‚„†™“#v”3k “od” h v‘vgjg –’¦vk#™¥v™gw’pW˜s g”txgiˆe¨3xz˜ ež  h ‚ e ed h ”o h ed ž f † ok WgsCf o ¦ ‘gjg –‘tvk€¥v™gw’™W˜s g”ž Pvf¥txgise§—˜v‚h  e ‚ eed hohed d h– f † sk h 3•qjvhvu 7 „kq“t‘u vgjg’¦vk€¥v™gw’EW˜s h k ‚ e–d h o h ed −−→ −− ∆x→0 „k‡™w g’¦k ¥t0”Žg}Yegs’#” ¥§ggq–vu “ s ed – † ed – – h d ” „†v“™~v“Y•… ho” } j     05 ¨ © 2 —vh Yegz˜vjqh‘#0”Žg}Ye(ev}s sž hu † s ‚j  ‚˜df˜ }f w † ¦ ˜ ef y“—x€—x˜ “• 0s™„o„‚“ ¨ BCgkP™o ¢ f (x0 ) = f (x) 4 xb " 4 8 d ” … —” ¢ PxV““gz evv”0sg}Vgs’d ¥€¥g—–‘’s ¡  o ”f ˜w • ‚ ž † u † e e – – h u „kv“h vjqh‘3Hz˜ e—’s ¡  hue e” x0 f (x0 )     05 Q © 2 d q”h b o hw” } i h} ho” } i h} ¦ o• h ‚ e– …f  k h u † ¥v™€xq}“d ˆe„†v“™€xq}” w “¦gkYevs’„svgjg—” ¥—” •‘u vjqh‘¤0”Žg}Yes x0 x0 s sk‚ e–d h e e i ho gzž 3’vgjg’¤vf„hq‚—gfyu ¦ (eg“p“vd™”  F (b − ∆x) − F (b) > 0 −∆x F (b − ∆x) − F (b) < 0 F (b − ∆x) < F (b) Yegz˜vjqh‘u sž h ˜… e e s Vg}¥go”gWs hh q”” [x] f (x0 + ∆x) − f (x0 ) ∆x f (x) f (x0 ) = f+ (x0 ) = f (x0 ) = lim lim f (x) x → x0 x > x0 x→x0 f (x0 + ∆x) − f (x0 ) ∆x→0 ∆x = lim f (x) = f (x) = [x] lim f (x) x → x0 x < x0 lim f (x) x → x0 x > x0 0 < ∆x < δ x0 f (x0 + θ∆x) f+ (x0 ) ∆x > 0 f− (x0 ) ∆x < 0 [a, b] f (x) f (x) δ >0 f (x)   © § ¦ ¨ ¥ cÎ g@´Î ¾ ´Ã” e Î€É€È V¾ £¼VY¼‘ ¾ ˆ»Ì¿6{ÍÆ £¼È ª ¿Í  ¿ Í È Î Â ¡ ¥ £ ¤ ¡ ¢ F (h) ≥ F+ (0) qj}„sg‚y‘€|h h e†u es h u † Vgzž qjvhv#0”g}Yes kvu qj}„sg‚~|h h e ‚i xPgfž k„v“h …e g}¥o  e “ †–od ”j ed fj sd k ” eo 3„k#0”¥¥“#v—qfh go‘¤vY}“¦“d ¥g¥” ¢ ∆x > 0 4 G E C 4  IHFDD © 2 „kq“6gs hs „kv“tv“v‘ h ku‚j ¦ „ex’gW~} ¢ ‚id s 4 eb " 4 8 q” d h ”h k j eo” } j … “ } s ‚ e–” h h } j •…h ˜ o d h žef ˜f s —gf¥~v“Y•¦—kšg’vgjg—”„wv‚Pvf“d v“V2y“0‚„†“ ’¢qf—vfgh3Cy—3k’s ¡   5 X 6VT 4B( © 2 ‘u j g…¥tvtvkh xz˜ eqegh ež „”v‚At“™V  eo” ku ” @ s f d ” w j eo” } j Pgf™—§v“Y•… ¦ ˆ”¥¥” ¢ †–o w goHz˜e W˜gW™2vd‘‘gjg –‘d e s s ˜ h‚ ee hh q”” ¢ £ i h £†„ £rH¥ {uP†yx‡ „‡ „ „  ¦  ¦   x0 f (x0 + t∆x) ≤ (1 − t)f (x0 ) + t · f (x0 + ∆x) f (x0 + ∆x) − (1 − t)f (x0 ) f (x0 + ∆x) − f (x0 ) ≥ − f (x0 ) = t f (x0 − t∆x) − f (x0 ) ≥ = t f (x0 − t∆x) − f (x0 ) ∆x · t f ((1 − θ)x1 + θx2 ) ≤ (1 − θ)f (x1 ) + θf (x2 ) (1 − t)x1 + tx2 x2 = x0 + ∆x x1 = x0 F+ (0) = a ≤ x 1 ≤ x2 ≤ b f (x0 + h) − f (x0 ) ≥ f+ (x0 ) h F (h) = = (1 − t)x0 + t(x0 + ∆x) = = x0 + t∆x = f+ (x0 ) f (x0 + h) − f (x0 ) h x0 lim F (h) = h→0 h>0 0 F (h1 ) [a, b] h>0 f (x) f ↓ F (h) 0 < h 1 < h2 f (x) †‰‰ ” … —” ¢ PxV““gz evv”0sg}Vgs’d ¥~¥g—–‘u  o ”f ˜w • ‚ ž † u † e e – – h x→a ¢ ¤ ¥   „kv“h ‰  qjvhvu h  h es s e d e h e ‚ e #v˜gzY”v”gh‘§gi¥vw•gk¦vk„hxgi¥vwÌh¤ q n …e Žg}¥o xq}”  }i h} d ” h h ‚ „†q“h go‘§xq}“d ™“tgkYeqs’s ¡ C‘qjvh—gf‘¨xq}#„”g‚vŽ“E˜s ¢ e d } i h} wo• h ef h e d } i h}” e j}d     05 © 2 —vv„” s0Žg}Yegs’t—¥g™—” † ed f” eo qkgh#ˆ”¥¥¨xz˜’” ¦ q‚“gd‡—¥g2¦gk‘gd6s v…“v•‘š} „…“ ’€WYs6gs¥qw“W•s  ž †–o” w e• h ž sf ek e ž h hu od s e h ih —Š €Š3ˆ9 Ši Š „ r% s gs  f (y) − f (x) ≥0 y−x —kq“h ‚˜ h kd yCqkV§ “” k Cx˜ o y>x y→x „evY}¨“” ‚j s…d f    T 4  %¥  2 2™„k}e ™“• ¦ ‘u w“ o k qjvh—’d v“Y•2se‘u Pgf™—˜—kq“h hf } j … † j eo” s }j• hh˜ e } s †‚ ee gWs v“V… ˜vyy§„sg‚¤gxg“ˆe9vgjg –’d x Yegz”gW¨gw(ev¥v¥Pgf¥” s ž s s ‚j h” j eo } ež ef 3gk¥gw€ –C” f y>x WgsCf o ¦ vu yg”Pe ¡ —gf¥t„kv“h e k s ˜ e j eo” s gWs x™vwv˜“z•ž  fihj j h>0 x 0 a0 ε>0 x0 = a + η a < x < x0 g(x) → ∞ a < x < x0 Yegzž ¦ gs s s e Y…§zž s ‚d ž …d” “Žg}(eg‚¨“š} ¢ ‚i x—gfž s “ e d @s ‚ j s s ™vwh „”g”—egh’#„”|‚A’‘Ž}™˜gWs  –ˆe—3k ¢ †f f “ e d @s e}… h s „”g”‘¤„”v‚A“g‚‘V€|”gWs ku‚ h• } ‚ h ‘’v}“€“„d3vk™˜ ¡ 4 d xb " 48 ‚i x—gfž s f w e hs ’™vwh x˜ “• gjxv˜”gWs fd h ’‘qjvhvu ¢ ‚j s ˜ h v}¥‘vjqh‘u fd h “vvjvhvu ¢ fd h “vvjvhvu vjvhvu h f“‘jqvhvu d h g‚W6„”v‚h ~}qs¤ˆ”}“d gsvsgh’™˜ e ss  h † s ” ed Y W˜gWs s ‰– Y ˜ ¢ fd h “vvjqh‘u s gWs  [a, b] f, g g(a) = f (a) = 0 g (a) = 0 x→a lim f (x) f (a) = g(x) g (a) x→a+ lim g(x) − g(a) x−a = x→a+ lim g(x) = g (a) = 0 x−a a 10 2 ec 10−4 < (n + 1)! 2 e<3 10−4 3 ≤ (n + 1)! 2 60, 000 k=0 −4 n+1 = 9 9! = 9 · 8! = 40, 320 · 9 > 60, 000 1 1 1 1 1 1 1 1 e = 1 + + + + + + + + + R8 (1) 1! 2! 3! 4! 5! 6! 7! 8! 3 10−4 0 < R8 (1) < < 9! 2 f (k) (0) ≤ x0 x0 |Rn (x)| an n→∞ n! lim ≤ = 0 f (n) (x) ≤ 1 ‚ xgŸ“#—x™™o ¥’¥—–ghv™W˜s r i ™go’¦‘™qoh —”vgjg—” ‘‘j   ˜} kd ”f iw …– s e j w ed k ‚ e– ‚ e x0 f (x) x k = 0, 1, 2, . . . (x − x0 )n+1 −−→ 0 −− n→∞ (n + 1)! x ∈ (−∞, ∞) x0 = 0 f (x) = sin x ¨ "    2  –se—” ¢ ‚ ’‘•“d †f wd k s gWs (    05 (  2 ¡ ¨ £   ©¦   #¢F£ ¨ £ ¨ eY‘gusk Ws x„†q“i hq}—“”}oh d ™¥v¦ ™gw„k’v“e h •gWs ¥¥šxq}“¦—”vgjg—” } }o h d f i–o } i h}d ‚ e– &¨  '©  ¦ ih v}Ž} ¦ ¡  ¨    A( ¡ ¢   ¦ ¦ zVs} ¡ ¦ 2‘u Wt„”0‚„†™go’d e s ek s “e e¡    s gs −∞ < x < ∞ f (2n) (0) = 0 Rn (x) → 0 f (2n+1) (x) = (−1)n cos x f (2n+1) (0) = (−1)n 1 · x x3 x5 (−1)n x2n+1 sin x = − + −+...+ + R2n+2 (x) 1! 3! 5! (2n + 1)! π R2n+2 ( ) 18 = π (−1)n−1 sin c · 18 (2n + 3)! 1 2n+3 5 2n+3 |R2n+2 | ≤ 10−4 2 n=1 R2n+2 (2π) = n = 12 > (2n + 3)! 1 2n+3 5 (2n + 3)! 1 10−4 < 5! 120 · 625 · 5 2 2n+3 (−1)n−1 sin c · (2π) 72n+3 10−4 < < (2n + 3)! (2n + 3)! 2 = 1 5 5 ln(x + 1) Rn (x) → 0 x −1 ≤ x ≤ 1 x0 = 2 1 x3 f (x) = x−3 f (x) f (x) f (n) (x) = −3x−4 , f (3) = −3! −3 = 5 24 2 4! 6 = −3 · −4x−5 = −3 · −4 · 2−5 = = f (n) (2) = 1 x3 Rn (x) = = = k=0 (n+3)! (−1)n+1 c−(n+4) (x 2 (n + 2)!(−1)n x−(n+3) 2 (n + 2)! · (−1)n 2n+4 n (n + 2)!(−1)k (x − 2)k + Rn (x) 2k+4 k! − 2)n+1 = (n + 1)! (n + 2)(n + 3)(−1)n+1 (x − 2)n+1 2 · cn+4 00 y <0 y  –ˆe—3k †f f (x) = lim h→0 (h > 0) f (x + h) − f (x) ≥0 h 4 d xb " 48 ¥    5 2 ¦ 2 4 C 4 X D¥" ¥  2 k vu f (x) g…Ye” s x1 gfŽ Cgk‘gdŸgk¥gwž e } e e e sh † ™vw60s™„o„‚“ ¦¨ q}ghž e ¡ ¦ gzž s ‚ ‘j  " 48 f (x) > 0 Wxg”ž ¦ s˜ s gs ¦ —kq“h x f (x) 4 d xb • k vu g…YetPy˜  s” ff f (x2 ) − f (x1 ) = f (c) ≥ 0 x2 − x 1 f (x2 ) ≥ f (x1 ) −f (x) ⇐⇒ „†v“™~} ¡ ho” eY32gh¥¥’d ¦ „kv“h sk eo– s gWs  gh™g’d Yegzž Px‘„jg‚’d vj gw‘S0sŽv}§gw„ev¥vgh‘2gh™g –’Ÿgk”3¨g…Vgs’d eo– ¦ s ”f i e d e d † h ‚ j e d eo ed e sk e e x f (x) ≥ 0 f (x0 ) y(1) = 1 e f (x) f (x0 ) = 0 x0 x0 x→−∞ lim xe−x = lim y y = x3 = 3x2 ≥ 0 = xe−x = e−x (1 − x) x<1 x→−∞ f (x) x>1 x=1 x = −∞ ex x0 x 0 x0 f (x) f (x) ≥ f (x0 ) f (x) ≤ f (x0 ) x0 x0 (a, b) x0 x2 f (x) x0 f (x) f (x) f (x) x0 ¡ x1 < x 2 x0 c x0 x0 f (x0 )  • •   ¢ 2≥x≥0 • }¡ ¡   q l   ¦  –ˆe—f3k ¢ † k‘’s™gwe(‘j¥q” u ‚ h ¦ —”ˆ—†™“d f qsh —§viPgfVg…‘™€™V)vgjg—” ‚ “o  w” h ž } e}o” –o ‚ e– ” ok ž s} hjd kji of gWYss ” 3xz˜ e’¥—–‘“¦‘x“• ——” ¢    5  £ 2 —kq“h ‘gj’¦vx“• —tvu ‚ e–d kji of” k x0 o hw” ¥v™~} ¡  x0 2 " 3 "  2 ¡    #AI¦ ¢ ¦ ( ) ¨ k ‘u • „k‘‘u “† „kv“h ¤ ¢ ¤ k v”3Ÿxvi—™“z•’€} ¡  ¥‘u hk k hfw e•” 4 G E C 4 ¥ IHF9% ¦ 2 k hfw e ed k uh xviP™“z•’z•’§‘2x´˜¤ qwIG gdYeE s ¦ 9B ¦¥ 2  v„f‚ gjv‚g’‘dh–q €vwu “|h  šYeh gd „}s ‘g“‚ v’gj–‚ee gj g—qw– tgd¤”” xgwh (e§V¥Ž}h…jf v €¥vvfo —™Ìhwvf ” ¤go ’x‘—v”™go’™§gkxq”™Yw ¤  v¨„”0‚„†™“d ¥v™” £„k6„}‘gjg‘d h “ C4 d h } ‚ o h eh e¨ ”“ ˜Yes ˜ v ” € j™ e d s j h e d  e s h k u “ o o h w ¦ “ ‚ e – 4 ‚ eed h vgjg –’€vw“gdYes  kj i of fj h e e h eu f e h e ‚ e e ˜} ” ‚j h” †–o h ef ˜f ¦ e s” ‚ e e ‘x“• P”  ‘xqi—gfyu gk¨vf„hv‚PgfvpvgjPvfqh‘u kgs”vgjg –’d gg”ž ™gw(e‘¥qƒgfse¥¥“d —qfghž Px3ks 3Yk’‘gjg –‘d 4 IG E 9%¥ ¦ 2 C 4 ¥ ‚ eed h vgjg –’˜vj¥g ež  d gCvf˜—qf™go’Cvj v…™go’d  e h j hede he d gCvf‡…qshPqe™vog—”  e h s” h h– ‚ e ed ”dkw e vgjg –’t“3™xgi’d  }} h jd ‚ j h }– @ w h u kd ”–} h u ”} h ¥—–‘“§gw(e‘¥q” ¦ g™vo¤#f ™yv˜‘Ÿ“¤¥Ž™vov¤3vkvu ¢ " " # "  2 ¦ ¥¨& )§¨ ¦ ¡ A     ¡ y −∞ −∞ y = mx + b y =0 • • • • • f (x) m+ b f (x) − x x (f (x) − (mx + b)) − − → 0 −− y y x0 m b = x2 e−x = e−x (2x − x2 ) = = e−x x(2 − x) → 0 = = (mx+b)−f (x) −−→ −− x x→∞ x→∞ x→∞ f (x) x lim (f (x) − mx) lim x→∞ 0 (mx + b) − f (x) − − → 0 −− y x y y y   l   y = f (x) x→∞ x≤0   • ” ‚j h” s ™gw(e‘¥q¨g…Ye—” ” ‚j h” eo– ™gw(e‘¥q2gh¥¥’d ¡ ll  ¢ ¢    05 2 ¥ 2 d q”h „†v“™” VPvfvš} ¡ ¦ Pvvjqh—“d ho w… h u ef hf x0 n ¢ ¦ £¡¦ £ ©¡   ¢ ¦¤¢ £¨¡§¥£¡    –se—3k †f ¦ xt‘x“• Pš} ¡ ˜} kji of” e˜ o} – ” f ˜ f s ¡   k s h˜ i h} gzž ¦ xEv}Ž” YeW˜gss zž gh¥g¥’d x3k’s Evu f (x0 +h) h x0 „kv“h Yegzž s g…Ye” qf—qfgh‡s ¡ v3vu s h ž ek x0 ¢ £ ” gw(ev¥v’gh™g’d  ‚ j h” eo– y =0 y = e−x ((x − 2)2 − 2) √ x−2=± 2 √ x = 2± 2 √ √ 2, x ≥ 2 + 2  4 xb " 4 8 „vh ”Pys d ‚‚  s f ˜ k vu ¦ —kq“h ‘gjg‘¤vxviPgfvu  ‚ e–d fj h e k vu „kv“h ‘gj’#vf„hv‚Pgfvu  ‚ e–d h e dq”h —†q“™‡w —vfqh‘š„}vgjg—” ¦ e ¡ ho” … h u ‚ e–  ”  s s ” ˜‚h s } –h j d k j i • o f  "  5 A  2 gWY¤x—v6gqv“tvx“VP” ”u‚ eed h e v’vgjg –’€vw“gdVs  s ” ‚ j h” j eo” }d e h gzž ™gw(e‘¥q¥Pgf™—€’‘gCvfvu f (x0 ) = 0 x0 f (x0 ) > 0 x0 x0 • √ √ x = 2 − 2, 2 + 2 • f (x0 ) < 0 f (x0 ) > 0 f (x) • • e od hk j eo 2k  ’¤q”3¥Vg…¥”  h<0 f (x0 ) = lim f (x) f (n) f (x0 ) = f (x0 ) = . . . = f (n−1) (x0 ) = 0 y y h→0 (x0 ) = 0 = 1 + 2 cos x = 0 1 cos x = − 2 2π 4π 2π 4π x= , , + 2π, − 2π 3 3 3 3 = −2 sin x √ 2π =− 3<0 y 3 √ 4π y = 3>0 3 √ 2π y + 2π = − 3 < 0 3 √ 4π + 2π = 3 > 0 y 3 f (x0 + h) > 0 f (x + h) f (x0 + h) − f (x0 ) = lim h→0 h h x≤2− h>0 [−π, 3π) >0 h=0 f (x) f (x) y = x + 2 sin x f (x0 + h) < 0 x≥2 • ¦ „kq“h ¨ ¢ n vu g“ˆeq†t„kv“h —gf¥~} ¡ vu  k h j eo” k ¢    05 ¨ ¥ 2 e eo” k vj g…¥š} ¡ vu x0 eYv”™V ¡ vu s hw k „kv“€xq}“d h }i h} e¡ x→x0 —‚‚qh €vh „†v“™” w —qf‘~} ¡ ¦ (e„‚v‚h   ho … h u ¦ „k“g‚‘V™˜ ¡ “ e}… ¦ |“@ k xŽv}Ž“d “vvjqh‘u ¦ }i h} fd h q‚“gd”„”„“‘——xqi—gfž W˜s h ž s ‚”e” h h h” h e q”tvk™W˜vsYhvsh ¦ 2„k“ ¦ ¦ gsP™o ¡ Vg…¥”   ef j eo }i h} xq}“d  x0 Wxg”e ’™3€|#” ‘j ™3xz˜ e™W˜s ™qw“gd™z˜ e—’s ¡ s ˜ ‚ok” h s ”ok ž ” h e e” (x − x0 )2 f (x0 ) + (x − x0 )2 α(x) 2 f (x) ¦ „kv“h ¢ x0 …’” ¦ „‚v‚h §v”s ¡ vjqh—“#„”0‚„†™“d d  h hfd “o k j eo” ¦ ‚o” vu Pgf¥~} ¡ 1™¥€|h x0 f (x) x0 f ‘‚ x€“gdŸ|•Yhgs™€3xz˜ e§q}} … ˜” w e eo ok ž i h }i h}” xŽv}—EW˜s  (x, f (x0 )) x0 f ekd ”f h j ” h e… e xz˜ e2“¤—x|˜v~} ¡ ¢ x|˜™gw™¤g”2k —0”™V§ g…Ž¥š¥V e †o e}– –o ¦ yz˜ e2“¤—xv˜‘#v”3k f ekd ”f hj h „}IHgFDDQ P‚ ¥2f 3tv”“d “d  —†P™gfh}q“ g}S„}vgjgx—´˜–se ¤” 0s™VP§eo gfsex¥“dhoi– q} xW˜q}sh “d v3“—C™PEo˜ ¡ |h Vg…¥—qoje f ™~}o”h S} ¡ v¤‡x|˜xv|˜hf ~e0s™V—§‚o ¥„}‘gjgfe P’0s™V§  ‚ – ‘gje P”  k — fh  oŽ” t‚” † }¥ } } ™ }  ‚  k ‚ f k o ™ j ” ¡ k u vu —f — jvj } † f ‚ … – …s” † o ež d † ”i f k”” h 4 G E C 4 ¢ £ Q " 3 "  2 n  ( ¨ ¦ A ¡ ( ©§ A )¢   ¨  ¨ ¦ & ¨ ¡ „kv“h ‘gj’#vf„hv‚Pgfvu  ‚ e–d h e x0 f (n) k h vu g“ˆev†—e ¡ e‘—v™™˜—sg‚e ¦ }f jo” k Px˜ o  ” e   ‚ e ˜ k † “ h o w —vfvš} ¡ „kv“€xq}“d v”h … hu h }i h} d 4 xb " 4 8 g‘„g™C‘u „v™” d k‘u vk„hg“ˆeq†t„kv“h vk„h„‚’vgjg’Ÿ‘x“• Pgfvu  h ‚”‚ e–d kj i o e kvu e ¡ g“sev†t„kv“h vgjg’d vxviPgfvu  h ‚ e– fj h e kvu e ¡ g“sev†t„kv“h vgjg’d qf—hq‚—gfyu  h ‚ e– h e k ‘u g“ˆeq†h ¦ —kq“h ‘gjg‘¦vx“VPgfvu  ‚ e– d k j i •o e x0 x0 x0 x0 n n n f x0 x0 f (n) (x0 ) < 0 f (n) (x0 ) > 0 n n • • • • y = f (x) x0 f (x) = f (x0 ) + (x − x0 )f (x0 ) + ¢ f (x) ≤ g(x) f (x) ≥ g(x) f (x) = f (x0 ) + x0 x0 f (n) (x0 ) < 0 g(x) = y f f (x0 )(x − x0 ) f (x0 )(x − x0 )2 + + 1! 2! f (n) (x0 )(x − x0 )n n + ...+ + (x − x0 ) α(x) n! f (x0 ) = f (x0 ) = . . . = f (n−1) (x0 ) = 0 f (x) − g(x) = (x − x0 ) y − f (x0 ) = f (x0 )(x − x0 ) g(x) f x0 = f (x0 ) + f (x0 )(x − x0 ) x0 x0 f (x) − f (x0 ) = (x − x0 )n f (x) − g(x) f (x0 ) + α(x) 2 f (x) − g(x) ≤ 0 f (n) (x0 ) > 0 x0 x0 f (x0 ) g(x) ≤ f (x) x0 n (x0 ) n! f (n) (x0 ) > 0 0 > f (x0 ) f (x) f + α(x) f α(x) x0 α(x) − − → 0 −− α(x) − − → 0 −− x→x0 n  x0 ¦ ¢ j Žg}e …  d” ¦‚kf 73—™o ˜ ¡ 4 G E C 4 IHFDD¥  2 qh eff h h u “od … ” h P—xv˜g|v€} „…™“‡™Yo q}—hq‚vhvu  ¢ ¥  ¥¨¨ §¢$ ¦ kus v‡gWs fd h ’‘qjvhvu e¡ ¢ £ 4 G E C 4 ¢ IHFD9 £ 2 2 " 3 2  2  '# £ ¦ we j h gfse¥¥“~‘“V… CV“qd‘—Ÿgk’gd§…¨¥Pgfe ’0‚¥—–h o †–o d } j • ef… h j” e e s …  † 4 deb " 4 8 ”™g(‚‘¥q” ˜ f e s } ˜ k ¥v™~v“Y•E0ig†¥o CV’vdv¦“d …¨¥Pgf¦„kv“3x“vgjg’E“d  o hw” } j … e ef… h j” k s … e hh ˜ ‚ e– d ˜} 4 X D4B(  2 |h—g”ŽEC‘u C d —v”h gw(e‘¥q” gfˆe¥¥“¨…˜“Y•… ‘vgjg—” vj 3Ÿ‘u ‚j h †–od s j ‚ e– d ok k  & ¨ ¦ A )¡ §  £ [a, b] ¦ ' ¡F#¢¨ £ ) £   &   ¨ £   o¥g—e’} g¥¦“d h s eo k k ”wd e vu ™“™go ež g”3k ¦  ˜ ¡ e” ‚j h e hd }j• —™gw(e‘¥q” gf(eg‚’z•„eq‚“~‘“V…  gogvv”” … qoYh “d ehus k ¦ §z€¥gqeE˜ ¡ q”™¥¥“˜gjH|˜~‘“V…  …s ž o h d h” i–o d e h } j • [a, b] xn = xn−1 − xn f (xn−1 ) , n = 1, 2, . . . f (xn−1 ) |c − xn | f (a) < 0 < f (b) e} ‚ ej h Ž3vk„h¥ –y¥v” ¢ [a, b] x x x1 Ž…e g}s ™„f “vdh‚vo ¦“gd 2k yv|˜} h“g…j V¥…e• o ” ¡ W˜—’vvvj‚fs” vhgjv’d ” q“gdV¦gg}(e“e ™¤voo vqsh u h” #‘0f} “q–V™…h• t„ku v“¢gfgCqk‘#“3vgjg’¢‘q“gde  ˜š—x|˜vj s ¡ v“V… e u —k ¥ e e s  j¥  qo‘ h e  h u k d ‚ e – d ” h s ” f h } j • † h sf h g–u h e  e " " # 2  2  A( ¢ ¨ ¦ ¨ $  ¨ & †h¤£h —†yv©‹2p i r „ r ¦  e zž ¦ „kv“h f (x) = (f (x) + (x − x0 )f (x0 )) + + f (n) (x0 )(x − x0 )n + (x − x0 )n α(x) n! „kt™vw’gde •|•qhhgs¥€„}vgjg—” ¢ “ f h  eo ‚ e– 4 dxb " 4 8 q ‘u qk2g“ˆev†h ¦ „kv“h vgjg’€“gdYe— WgsP™o xŽv}Ž3—vf„hq‚“#g”3§Vg…¥PŽx3kqs“¥—gf¥Ÿgkt™Yw ¤   k h ‚ e–d w s ef ¦ } i h}”h hd ek j eoe} ˜f hd j eo e ”” x0 x0 f (n) (x0 ) = 0 n K>1 x0 |f (x) − f (y)| ≤ K |x − y| x0 f (x0 ) = x0 f (x) − g(x) = (x − x0 )n x, y ∈ [a, b] I g(x) = f (x) − x g(a) = f (a) − a ≥ 0 g(b) = f (b) − b ≤ 0 x1 = x 0 − f (x0 )(x − x0 )2 f (n−1) (x0 )(x − x0 )n−1 +... + + 2 (n − 1)! x0 x0 f (x0 ) f (x0 ) f (n) (x0 ) + α(x) n! I I f (x) = 0 [a, b] f f : [a, b] → R f f (x) f (x) f (c) α(x) − − → 0 −− x→x0 ¢ ¢ ku} ” v~3vkh xz˜ eqegh ež „”|‚“xPqkV§ @ s ‚ ˜ h ‚j vY}s sg¡s zž  ¦ gWs qjvhvu Y Cvu s h ˜k n→∞ N (ε) ε>0 c ¦ hh v”” s gzž h " 4 8 v”h v”3™¥¥“¥gjH|˜h ¦ Yegzt“g—x“’0sg}Vs  hk i–od e s ž fd ‚ id † e ≤ K |f (xn ) − f (xn−1 )| ≤ K 2 |f (xn−1 ) − f (xn−2 )| . . . = K n |x1 − x0 | c m0 K <1 {xn }n=0 f x0 ∈ [a, b] ξ F1 (x0 ) = f (x0 ), F2 (x0 ) = f (f1 (x0 )) , . . . Fn (x0 ) = f (fn−1 (x0 )) h •} v”h Vq…h  n |f (xn+1 ) − f (xn )| x1 = f (x0 ), x2 = f (x1 ) . . . , xn = f (xn+1 ) |xn − c| ≤ f (x) − f (y) = |f (ξ)| ≤ K x−y x=y K := sup |f (x)| < ∞ Kn |x1 − x0 | − − → 0 −− n→∞ 1−K = |f (f (xn )) − f (f (xn−1 ))| x |xn − c| f (x) [a, b] K f (x) f : [a, b] → [a, b] c = f (c) xn − − → c −− n→∞ f {xn } 4 xb d |xn − xm | = |(xn − xn−1 ) + (xn−1 − xn−2 ) + . . . + (xm+1 − xm )| ≤ |xn − xn−1 | + |xn−1 − xn−2 | + . . . + |xm+1 − xm | ≤ K n−1 |x1 − x0 | + K n−2 |x1 − x0 | + . . . K m |x1 − x0 | = K m |x1 − x0 | 1 + K + K 2 + . . . + K n−m−1 Km |x1 − x0 | = 1−k xn |xn − xm | ≤ ε |c − xm | ≤ Km 1−k n > m ≥ N (ε) |x1 − x0 | i–od e h }j• eo s ž fd ‚ i ¥¥“§gjHv˜š‘“V… 0ig†¥CYegz•’gqx’d ƒl sf ek wd ž sf ek —¥g2§“™go e¨—xz˜ e2”  –ˆe—3k †f n = 100 ˜ Cvk„hg‚e h ž hk ’gdE(˜v†3”   e f h j …–e e s ” wo  h h WgsCf o ¦ ¥ –qeqh‘‡gCzVs#xi ™” ” Pvfx|˜“d  |xn − c| ≤ 1− √ 3 2 √ 3 2 n ‚ Ž}  o } π 4 sk e} sf w ¦ 3•z•Ž”—x˜ “• vh ¦ k Ž}s e ek } i– —g”2€xg™o  gogvv¨—–ghv‡g6s ¡ ™™go’Ÿ‘™qo¢„”vgjg—” ™qoYh e h u e j …– iw ed k h ‚ e– ¦  a = 1, x0 = 1 4 k ’} ™o ‚   f } ‚ h˜ ej  h s sx ye ˜ ‚ – d x” w ˜ • “ ¦ V h V’ ”g– “ ˜ } ™—–¨“gd2Pede˜ z P3„k“„}¦ ‘™gsj gvwe h‘t˜x“g§d „}£™¦  voxB‘gj˜‚ v‘E…Y•d g’d ” f   }j ef v“V¦• … WgsC™o ¦ [0, a] sž ‚jh gzŸ” gw(ev¥v” „kv“¤™gw„ev¥v—” h ” ‚jh q”3¤” “vtgz eg“~v“Y•…  hk …dj” f ed }j [0, a] |c − d| – ˆe—3k ¢ q”h †f h fd huoj k “vvjqh‘‡v¦vu c π 2 ¦ e i ho (eg“p“vd™” ¦ Yegzž s d  ‚ e–d ” ˜} hk h–” h ku† ‚ e–d ˜} ‘gjg‘¤x“d v”33h v¥P§|Ÿ‘vvu vgjg’E“d e2gkeP„f‘u ¦ ¦ P™¥v™gw’d ¦ ‚ efo h e  X 4 VT ¥"  2 5 6 ¤ cos x 0 ≤ x ≤ a , |f (x)| = |− sin x| = sin x ≤ sin a = a>1 = sin 1 (sin 1)n |1 − cos 1| |xn − c| ≤ 1 − sin 1 10−4 (sin 1) (1 − cos 1) < 1 − sin 1 2 n = 64 d = |f (c) − f (d)| ≤ K |c − d| = f (d), c = f (c) [cos a, cos 0] ⊆ [0, a] c = cos c c ¦ f (x) = K n √ π 3 K = sin = 3 2 1 sin x x · (1 − cos 1) :x=0 :x=0 c ← f (xn ) = xn+1 → c f 1a x= sin x x 10−5 2  K >1 > c = f (c) 1≤0≤ xn → c x0 = 0 a= [0, a] k ‚u … hj” k P—fv¦f “qd‘˜’d ¡   fd ‚id “g—x“’s ¡  s gzž c g(x) f hfd kd ‚ f w•‚ e–d ” ˜} hdo s ž k u‚ vvjvhP“¦’•„d3vkh x˜ ““vgjg’#xg’d ““™‡…§zŸ‘“vj  d q”h es Vgzž Wxq”3k  s˜h f ed …u j gz eg“”v¥Žg}e … d g (x) = 1 − f (x) m ≤1− M M g(x) ¤ s ”gzž gf(eg‚’z•(ev‚’¨g…Ye”  e hd s 4 xb " 4 8 d e „k¡ v“¨V™“(e…”f gw qdv‘¥—vjjh‚ ” “d “gde  ¥P¦ gfe ¦ Ye… g}¥gose z’s ¡ y’vvjgjqh—’“§g’Ÿk “Pqeh vg—” v“Y•¦ … Yegzž šWgsP” ™“¦…f o všgz eg“@‘u ” ‚ h h¦k s … ž h ˜ f ‘ e h – f d d ˜ } d  d h – } j s  e d j ” f e  d … q  vjvhPvfvu e ¡ Y ˜ ¡ hh s gWs  n d ” ¥v™—ˆe„†v“™~v“Y•… ’„‚v‚h €|h o hw” ho” } j      05 2  2 —vh ‚™C—E|˜PgfHSPxV“‡…3¥q–~ —˜™“gd¥txz˜ e3—t¥g¥” ok” h e ˜ sf ˜w • sh h ‚w eo f ek” f” eo  gogvvu ¥—–ghv™W˜s ™™go’¦k ™vo#„”vgjg—”   e h } e j iw ed  h ‚ e– √ 2 2 π ‚ vj  ¦ „k“ Y•„…” ‚ f ed gz eg“~} ¡  |f (x)| = cos |tan x − x| x cos x − sin x = 2 x x2 √ √ 3/2 2/2 , π/4 π/3 ‚  kd ”j• vj ¦“¤‘“V… f (x)  m K = 1− M f (c) = 0 ⇐g(c) = c m g(x) 0 ≤ g (x) ≤ 1− M (< 1) [a, b] c max π |f (x)| = f π 4 a = f (x) = 0 π 3 1− 1 m n M m M m n −M = m √ 3 3 3 − π < 0.313 2π 2 x |f (x0 )| f (x0 ) = M π π 4, 3 Y§v”3k ¦ „kv“#x¥¥”  ˜ h h ” i–o [a, b] 0 < m ≤ f (x) ≤ m n=9 (1 >)1 − ⊇ λ= g 1 M g (x) = 1 − λf (x) = 1 − c = 0.87673 [0, π ] 2 m M x1 = f (x0 ) = =K g(x) = x−λf (x) [a, b] ←−− −− f f (x) f (a) < 0 < f (b) m π π 4, 3 5 M f (x) M a≤x≤b x0 = g(x) c π 4 g  ¦ f (x) = x3 + x − 1 f (1) = 1 m = 1 ≤ f (x) = 3x2 + 1 ≤ 4 = M 1 1 λ = = M 4 3 m = K = 1− M 4 [0, 1] K= 3 4 f (0) = −1 g(x) = x − n 1 4 x3 + x − 1 x0 = 1 2 xn = g(xn−1 ), n = 1, 2, 3 . . . 1 1 3 3 + −1 = · 1 8 2 4 8 1 1 1 1 19 = g(x0 ) = − + −1 = 2 4 8 2 32 = 0.642 = 0.666 = 0.675 = 0.679 = 0.681       |xn − c| ≤ x1 x2 x3 x4 x5 x6 3 n 4 c = 0.68232780382801932738 . . . 0.42 · 10−19 f (x) = 0 x1 y = f (x) y − f (x1 ) x − x1 y = f (x1 ) = f (x1 ) + f (x1 )(x − x1 ) x2 0 = f (x1 ) + f (x1 )(x2 − x1 ) f (x1 ) x2 = x 1 − f (x1 )       xn+1 = xn − f (xn ) , n = 1, 2, 3 . . . f (xn ) [a, b] f (x) f (a) < 0 < f (b) ‚ vj  ‚ ž …–  h ej gz e§¥€™—–¨—–ghvpW˜s Q " ! 2 # 2  –ˆe——” †f f ek” e fd w id ‚ e– xz˜ e2’g“Ÿ“3k x’€„}vgjg—”  4    05 ¨ ¤  2 f ekd ”f hj sf ek C 4 X DBQ  2 xz˜ e3’¢Pyv˜‘¨—xz˜ e2” n £ ¤ ¦ ¦ #™gw„ev¥v—”  ” ‚jh ¦ ¨ ¦ ¨ $   # '( ¨ $ F£ ¨    & ¦  ¨ d —v”h „†v“™¨VPvfvhv€‘“V… ho” w… h u } j •  sk … ˜” } “o h e} ‚ ”3’‚ xšV„…™“d vj™go‘š—}V   „}‘gjgP” v”3k ‚ e– h ef hf Pvvjqh—“d f ed …u j e gz eg“‡v˜g}™…     ef… h j” k CV’vdv¦“d s… V¥Pgfe  ©   = (xn+1 − xn ) f (ξ) 2 k ” e ” † h o e d e” skw i s V “gdzž W˜s xgiˆe™qovhvu ” Žg}e g…‘pz˜ eP63™xPYegzž Yegzž s ¦ e h ed ‚ e– ¦ e Cvfh v¥ –‘¤„”vgjg—” 1g‚Ž} … f( ) f( ) c f  gf(eg‚’z•(ev‚“§g…YetPx´˜¤ yg‚‘d ’” ™voV ’d e hd s” ff  e o s … h k = lim xn+1 = lim f (xn ) → f ( ) n→∞ n→∞ Yegzž s ‚ok”  o h e ‚ j ™3E˜ ¡ Ÿf ¥q™gw’d v}s efo h ed † CC™¥v™gw’E˜s vu Yegzž s  x¥¥” xgiC” f™V¨…3¥q–h ¦ Yegz¦’gqx’S0sŽg}Yes  ” i–o e ef so… sh s ž f d ‚ i d †  |h ¤ es )Ygzž v”gh‘gde ¡ e ¦‚j} )‘Žs ¦ ef e  C„”g‚Hi Ck WgsC™o ef Yegzž s  ef s … xgiC” f™o ‡|…@ h  c „k¦——xz˜ e2” “ f”f ek ξ  e DgdV ”}” ¥—EW˜s Pxz˜ e3—6s v…YhÌs¤ “‘Ž}s ¢ }f ek” fdj 0 = f (c) = f (xn ) + f (xn ) · (c − xn ) + f (c) = 0 xn k ‚ e–” Px˜ o vgjg—€v}ghž e ¡ ˜v”h “™go e¦Pxz˜ e3—”  d wd ž ”f ek 4  xn d xb " 48 d ” i–o @ f h ed eo …– i–o ok ž i h v”h x¥¥” „”v‚As t™vw“gd’•|•qhhgs¥‡¥p¥¥§3xz˜ e§q}} = a 0 vjhq‘u~}¥—–ghvpz˜¥y¦k ¥– v“Y•… h ej eo ˜  }j s gWs  a≤x≤b |xn+1 − c| {xn } f (xn+1 ) = [f (xn ) + (xn+1 − xn )f (xn )] + f (x) = f (xn ) + f (xn )(x − xn ) + (0 = f (c) <) f (xn ) f (xn ) − f (xn )xn (xn+1 − c) f (xn ) = x1 = b |xn+1 − c| < M m = {xn } f M |xn+1 − xn |2 2m xn − = f (xn )(xn − xn+1 ) = −f (xn )xn+1 sup f (x) inf f (x) [a, b] f (ξ)(x − xn )2 2 2 f (xn ) f (xn ) xn+1 > c c f (ξ)(x − xn )2 2! † f (xn+1 − xn )2 f (ξ) 2 f (ξ)(c − xn )2 2 f (xn ) → f ( ) = − =c 0 = f (c) < f (xn ) xn+1 − c > 0 xn+1 = xn − c xn ↓ f (x) > 0 x xn f (xn ) f (xn ) xn f ,f > 0 xn > xn+1 f ≥0 f( ) = 0 x = xn+1 =c f↑ ‚ h} } h e e ‘¥q~vvj™goŽ} |•qhhgs¥o f (xn+1 ) − f (c) = f (η) xn+1 − c c < η < xn+1 f (xn+1 ) = f (η)(xn+1 − c) f (η)(xn+1 − c) = |xn+1 − c| = ≤ (xn+! − xn )2 f (ξ) 2 f (ξ) (xn+1 − xn )2 f (η) 2 M |xn+1 − xn |2 2m x=1 f (x) = x3 + x − 1 x1 x2 x3 x5 x6 = 1 = 0.75 = 0.6860       = 0.6823278039 = 0.68232780382801932738 ‚ h} ‚j ¥q€} ¡ (ev}s † …e Žg}™o  –ˆePš¥ –ˆePCk †f” } †f k s h Ž}6s qwSˆs† „o„‚“ ¨ f hf xgi—evhP“d  ‚ s… “d vYh‡‘u „evY}s ¢ ‚j £rƒ