iu chnh, b sung nm 2011 Lu hnh ni b GIO KHOA & PP GII TON 11 GV: NGUYN THANH NHN 1 eW: 0987. 503.911 MC LC CNG THC LNG GIC .................................................................... 4 1. v radian .......................................................................................... 4 2. Cc h thc c bn ................................................................................. 4 3. Cc h qu cn nh ................................................................................ 4 4. Cc cung lin kt ................................................................................... 5 5. Cc cng thc bin i ........................................................................... 6 HM S LNG GIC ............................................................................ 8 1. Cc hm s lng gic ........................................................................... 8 2. Tp xc nh ca hm s ........................................................................ 9 3. Tm gi tr nh nht, gi tr ln nht ca hm s ..................................... 9 4. Xt tnh chn, l ca hm s ................................................................... 9 PHNG TRNH LNG GIC ........................................................... 10 1. Phng trnh lng gic c bn............................................................ 10 2. Phng trnh bc hai i vi mt hm s lng gic ............................ 12 3. Phng trnh bc nht i vi sinx v cosx ........................................... 12 4. Phng trnh ng cp bc hai i vi sinx v cosx .............................. 13 5. Phng trnh i xng, phn i xng ................................................. 13 6. Phng trnh lng gic khc............................................................... 13 I S T HP ....................................................................................... 14 1. Php m ............................................................................................. 14 2. Hon v ................................................................................................ 14 3. Chnh hp ............................................................................................ 14 4. T hp ................................................................................................. 15 5. Cch phn bit t hp v chnh hp ...................................................... 15 NH THC NEWTON .............................................................................. 15 1. Khai trin nh thc Newton .................................................................. 15 2. Tam gic Pascal ................................................................................... 15 3. Gii phng trnh ................................................................................. 16 XC SUT ................................................................................................. 16 DY S ...................................................................................................... 17 1. Tnh n iu ca dy s ..................................................................... 17 2. Tnh b chn ca dy s ........................................................................ 17 CP S CNG .......................................................................................... 18 1. nh ngha ........................................................................................... 18 2. Tnh cht.............................................................................................. 18 3. Tng n s hng u tin ca cp s cng .............................................. 18 CP S NHN .......................................................................................... 18 1. nh ngha ........................................................................................... 18 GIO KHOA & PP GII TON 11 GV: NGUYN THANH NHN 2 eW: 0987. 503.911 2. Tnh cht.............................................................................................. 18 3. Tng n s hng u tin ....................................................................... 18 GII HN CA DY S ......................................................................... 19 1. nh ngha ........................................................................................... 19 2. Tnh cht.............................................................................................. 19 3. Mt s gii hn c bn ......................................................................... 19 4. Cch tm gii hn ................................................................................. 19 GII HN CA HM S ........................................................................ 20 HM S LIN TC ................................................................................. 22 1. Xt tnh lin tc ca hm s( ) y f x =ti 0x........................................ 22 2. Tm m hm s( ) y f x =lin tc ti im ch ra .......................... 22 3. Chng minh phng trnh c nghim ................................................... 22 O HM CA HM S ........................................................................ 22 1. Bng cc o hm ................................................................................ 22 2. Cc qui tc tnh o hm ...................................................................... 23 3. o hm cp cao .................................................................................. 23 TIP TUYN CA NG CONG ........................................................ 23 CC PHP BIN HNH TRONG MT PHNG .................................... 26 I. Cc php bin hnh ............................................................................... 26 II. V nh ca mt hnh qua php bin hnh ............................................. 27 III. Tm phng trnh ca nh .................................................................. 27 NG THNG V MT PHNG........................................................ 28 1. Tm giao tuyn ca hai mt phng ........................................................ 28 2. Tm giao im ca ng thng d v mt phng (P) ............................. 28 3. Chng minh 3 im thng hng ............................................................ 28 4. Tm thit din ...................................................................................... 29 QUAN H SONG SONG ........................................................................... 29 I. Cc nh ngha...................................................................................... 29 II. Cc tnh cht ....................................................................................... 29 III. Chng minh hai ng thng song song ............................................. 30 IV. Chng minh ng thng song song mt phng ................................. 30 V. Chng minh hai mt phng song song ................................................. 31 VI. Chng minh hai ng thng cho nhau ............................................ 31 QUAN H VUNG GC.......................................................................... 31 I. Chng minh hai ng thng vung gc ............................................... 31 II. Chng minh ng thng vung gc mt phng .................................. 32 III. Chng minh hai mt phng vung gc ............................................... 32 GC ........................................................................................................... 33 1. Gcgia hai ng thng a, b ......................................................... 33 GIO KHOA & PP GII TON 11 GV: NGUYN THANH NHN 3 eW: 0987. 503.911 2. Gcgia ng thng a v mt phng (P)........................................ 33 3. Gcgia hai mt phng (P) v (Q)................................................... 33 KHONG CCH ...................................................................................... 33 1. Khong cch t im O n ng thng a .......................................... 33 2. Khong cch t im O n mt phng (P)........................................... 33 3. Khong cch gia ng thng a // (P) ................................................. 34 4. Khong cch gia hai mt phng (P) // (Q) ........................................... 34 5. Khong cch gia hai ng thng cho nhau ...................................... 34 H THC LNG TRONG TAM GIC ............................................... 34 1. nh l c sin ....................................................................................... 34 2. nh l sin ............................................................................................ 35 3. Cng thc tnh din tch tam gic ......................................................... 35 4. Cc h thc lng trong tam gic vung .............................................. 36

GIO KHOA & PP GII TON 11 GV: NGUYN THANH NHN 4 eW: 0987. 503.911 CNG THC LNG GIC 1. v radian: ( )0180 ( ) rad = ; 01180=(rad); 01801( ) rad| |= |\ . 2. Cc h thc c bn: *( )sintan cos 0cos = = ; *( )coscot sin 0sin = =* 2 2sin cos 1, + = ; * 2211 tan ,2 cosk k | |+ = = + e |\ .Z* 2211 cot ( , )sink k + = = eZ*tan .cot 1 ,2kk | |= = e |\ .Z . 3. Cc h qu cn nh: * 4 4 21sin cos 1 sin 22x x x + = * 6 6 23sin cos 1 sin 24x x x + = sin( 2 ) sin ; cos( 2 ) costan( ) tan ; cot( ) cotk kk k + = + =+ = + = tanxc nh khi,2k k Z = + ecot xc nh khi, k k Z = e1 sin 11 cos 1 s s s s GIO KHOA & PP GII TON 11 GV: NGUYN THANH NHN 5 eW: 0987. 503.911 Du cc gi tr lng gic: Gc phn t GTLG IIIIIIIV sino ++ coso++ tano++ coto++ 4. Cc cung lin kt: a. Cung i:v b. Cung b:v c. Cung ph:v 2 d. Cung hn km nhau :v + e. Cung hn km nhau 2:v 2 + sin( ) sin ; cos( ) costan( ) tan ; cot( ) cot = = = = cos( ) cos ; sin( ) sintan( ) tan ; cot( ) cot = = = = sin cos ; cos sin2 2tan cot ; cot tan2 2 | | | | = = ||\ . \ .| | | | = = ||\ . \ . tan( ) tan ; cot( ) cotsin( ) sin ; cos( ) cos + = + =+ = + = sin cos ; cos sin2 2tan cot ; cot tan2 2 | | | |+ = + = ||\ . \ .| | | |+ = + = ||\ . \ . GIO KHOA & PP GII TON 11 GV: NGUYN THANH NHN 6 eW: 0987. 503.911 5. Cc cng thc bin i: a. Cng thc cng: b. Cng thc nhn i: * Cng thc tnh theotan2xt =22 2 22 2 1tan ;sin ; cos1 1 1t t tx x xt t t= = = + + c. Cng thc h bc: Lu :* 21 cos 2cos2xx + =* 21 cos 2sin2xx =d. Cng thc bin i tch v tng: -sin(a b) = sina cosb cosa sinb -cos(a b) = cosa cosbsina sinb -tan(a b) = tan tan1 tan tana ba b

-cot(a b) = 1 tan tantan tana ba b -sin2a = 2 sina.cosa -cos2a = cos2a sin2a = 2cos2a 1 = 1 2sin2a -tan2a = 22tan1 tanaa ;cot2a = 2cot 12cotaa cos2a = 1 cos22a +;sin2a = 1 cos22a ;tan2a = 1 cos21 cos2aa+ GIO KHOA & PP GII TON 11 GV: NGUYN THANH NHN 7 eW: 0987. 503.911 e. Cng thc bin i tng v tch: Ch : *sin cos 2sin 2 cos4 4x x x x | | | |+ = + = ||\ . \ . *sin cos 2 sin 2 cos4 4x x x x | | | | = = + ||\ . \ . sina.cosb = 1[sin( ) sin( )]2a b a b + + cosa.cosb = 1[cos( ) cos( )]2a b a b + + sina.sinb = 1[cos( ) cos( )]2a b a b + -sinA + sinB = 2sin cos2 2A B A B + -sinA sinB= 2cos sin2 2A B A B + -cosA + cosB = 2cos cos2 2A B A B + -cosA cosB = 2sin sin2 2A B A B + -tano tan| = sin( )cos .cos ; ,2k k | |= + e |\ .Z GIO KHOA & PP GII TON 11 GV: NGUYN THANH NHN 8 eW: 0987. 503.911 f. Gi tr lng gic ca cc cung c bit: Gc 00 300 450 600 900 1200 1350 1500 1800 0 6 4 3 2 23 34 56 sin0 12 22 32 1 32 22 120 cos1 32 22 120 12 22 32 1 tan0 13 1 3|| 3 1 13 0 cot|| 31 13 0 131 3|| sc< HM S LNG GIC 1. Cc hm s lng gic: sin y x =cos y x =- TX: D= - L hm s l - Hm tun hon vi chu k2- Tp gi tr:1;1 T( = - Hm s ng bin trong 2 ; 22 2k k | | + + |\ . - Hm s nghch bin trong 32 ; 22 2k k | |+ + |\ . - TX: D= - L hm s chn - Hm tun hon vi chu k2- Tp gi tr:1;1 T( = - Hm s ng bin trong ( )2 ; 2 k k + - Hm s nghch bin trong ( )2 ; 2 k k + GIO KHOA & PP GII TON 11 GV: NGUYN THANH NHN 9 eW: 0987. 503.911 tan y x = cot y x =- TX: D= \2k + ` )- L hm s l - Hm tun hon vi chu k- Tp gi tr:T = - Hm s ng bin trong ;2 2k k | | + + |\ . - C cc ng tim cn 2x k = + - TX: D= \2k + ` )- L hm s l - Hm tun hon vi chu k- Tp gi tr:T = - Hm s nghch bin trong ( ); k k + - C cc ng tim cnx k =2. Tp xc nh ca hm s: a) ( )( )PxyQx=xc nh khi ( )0 Qx =b) ( )y Px =xc nh khi ( )0 Px >c) ( )( )PxyQx=xc nh khi ( )0 Qx >d) ( ) ( )sin ; cos y f x y f x = =xc nh khi ( )f x xc nh. e) ( )tan y f x =xc nh khi ( )2f x k = +f) ( )cot y f x =xc nh khi ( )f x k =3. Tm gi tr nh nht, gi tr ln nht ca hm s: a) p dng cc tnh cht ca bt ng thc, v vi mi x ta c: 2 21 sin 1; 1 cos 1; 0 sin 1; 0 cos 1 x x x x s s s s s s s sb) Gi tr nh nht, gi tr ln nht ca hm ssin cos y a x b x c = + +x e ta c 2 2 2 2cos a b ainx b x a b + s + s + 2 2 2 2sin cos c a b a x b x c c a b + s + + s + +4. Xt tnh chn, l ca hm s: Cho hm s y = f(x) xc nh trn D. GIO KHOA & PP GII TON 11 GV: NGUYN THANH NHN 10 eW: 0987. 503.911 * Hm s y = f(x) c gi l hm s chn nu ( ) ( )x D x Df x f xe e = * Hm s y = f(x) c gi l hm s l nu ( ) ( )x D x Df x f xe e = sc< PHNG TRNH LNG GIC 1. Phng trnh lng gic c bn: a) Phng trnhsin x m =* iu kin c nghim:1 m s*Tmgcasaochosina m = (sdngMTCT: 1sin a m= ).Ta c:sin sin x a =v p dng cng thc: ( )2sin sin2u v ku vu v k k = += = + e

Hay00 0360180 360u v ku v k

= +

= +

nu trong phng trnh c cho . * Trng hp c bit: sin 0 u u k = = sin 1 22u u k = = + sin 1 22u u k = = + * Nu khng phi l gi tr c bit th c th s dng cng thc: arcsin 2sinarcsin 2u m ku mu m k = += = +arcsin2 2m | | s s |\ . * ( )sin sin ; cos sin ; cos sin2 2u u u u u u | | | | = = = ||\ . \ . b) Phng trnhcos x m =* iu kin c nghim:1 m s*Tmgcasaochocosa m = (sdngMTCT: 1cos a m= ).Ta c:cos cos x a =v p dng cng thc: GIO KHOA & PP GII TON 11 GV: NGUYN THANH NHN 11 eW: 0987. 503.911 ( )2cos cos2u v ku vu v k k = += = + e

Hay00360360u v ku v k

= +

= +

nu trong phng trnh c cho . * Trng hp c bit: cos 02u u k = = + cos 1 2 u u k = = cos 1 2 u u k = = + * Nu khng phi l gi tr c bit th c th s dng cng thc: arccos 2cosarccos 2u m ku mu m k = += = +arcsin2 2m | | s s |\ . * ( )cos cos ; sin cos ; sin cos2 2u u u u u u | | | | = = = + ||\ . \ . c) Phng trnhtan2x m x k| |= = + |\ . * Tm gc a sao chotana m =(s dng MTCT: 1tan a m= ) Ta c:tan tan x a = v p dng cng thctan tan u v u v k = = +Hay0180 u v k = +nu trong phng trnh c . * c bit: tan 0tan 14u u ku u k= == = + * Nu m khng phi l gi tr c bit c th s dng cng thc: tan arctan arctan2 2u m u m k m | |= = + < < |\ . * ( )tan tan ; cot tan ; cot tan2 2u u u u u u | | | | = = = + ||\ . \ . d) Phng trnh ( )cot x m x k = =* Tm gc a sao chocot a m =(s dng MTCT: 11tan am| |= |\ .) GIO KHOA & PP GII TON 11 GV: NGUYN THANH NHN 12 eW: 0987. 503.911 Ta c:cot cot x a = v p dng cng thccot cot u v u v k = = +Hay0180 u v k = +nu trong phng trnh c . * c bit: cot 02tan 14u u ku u k= = += = + * Nu m khng phi l gi tr c bit c th s dng cng thc: ( )cot arccot 0 arccot u m u m k m = = + < c) Phng php gii: Chia hai v ca phng trnh cho 2 2a b +Ta c phng trnh: 2 2 2 2 2 2sin cosa b cx xa b a b a b+ =+ + + GIO KHOA & PP GII TON 11 GV: NGUYN THANH NHN 13 eW: 0987. 503.911 t 2 2 2 2cos sina ba b a b = =+ +. Ta c phng trnh: ( )2 2 2 2sin cos sin cos sinc cx x xa b a b + = + =+ + (*) (*) l phng trnh dng c bn. 4. Phng trnh ng cp bc hai i vi sinx v cosx a) Dng: ( ). . . . 2 2a sinx b sinx cosx c cosx d 1 + + =b) Phng php gii: * Kim tra cosx = 0 c tho mn hay khng? Lu : cosx = 02sin 1 sin 1.2x k x x = + = =* Khicos 0 x = , chia hai v phng trnh (1) cho 2cos 0 x =ta c: 2 2.tan .tan (1 tan ) a x b x c d x + + = +* t: t = tanx, a v phng trnh bc hai theo t: 2( ) . 0 a dt b t c d + + =5. Phng trnh i xng, phn i xng:a) Dng:.( ) . . a sinx cosx b sinx cosx c 0 + + =b) Phng php gii: * t:cos sin 2.cos ; 2.4t x x x t | |= = s |\ . 2 211 2sin .cos sin .cos ( 1).2t x x x x t = = *Thayvophngtrnhcho,tacphngtrnhbchaitheot. Gii phng trnh ny tm t tha 2. t sSuy ra x. Ch :*cos sin 2 cos 2 sin4 4x x x x | | | |+ = = + ||\ . \ . *cos sin 2 cos 2sin4 4x x x x | | | | = + = ||\ . \ . 6. Phng trnh lng gic khc: giimt phng trnhlnggic cha phil ccdngquen thuc ta cn s dng cc php bin i lng gic a phng trnh v dng quen GIO KHOA & PP GII TON 11 GV: NGUYN THANH NHN 14 eW: 0987. 503.911 thuc, c th phn tch phng trnh cho v dng phng trnh tch hoc p dng tnh cht bt ng thc a v h phng trnh gii. Cc phng php gii phng trnh lng gic thng s dng: * Bin i phng trnh cho v mt trong cc dng phng trnh c bn bit (a v cng mt cung hoc cng mt hm s lng gic,...). * Bin i phng trnh cho v dng tch: 0. 00AA BB == = *Biniphngtrnhvdngcthtnsph(ixng,t tan2xt = ,) sc< I S T HP 1. Php m: a) Qui tc cng: Gi s hon thnh hnh ng (H) ta c th thc hin qua cc trng hp A hoc B hoc C ... (mi trng hp u hon thnh cng vic) NuA cm cch, B c n cch, C c p cch th cm n p + + ... cch hon thnh (H). b) Qui tc nhn: Gi s hon thnh hnh ng (H) ta phi qua nhiu cng on (bc) A, B, C lin tip nhau. CngonAcmcch,cngonBcncch,cngonCcp cch... Khi hon thnh (H) th c. . m n p ... cch 2. Hon v:a) Hon v:Cho tp A c n phn t, mi cch sp th t n phn t ca A gi l mt hon v. b) S cc hon v n phn t:!nP n =Ch : Giai tha * ( )! . 1 ...3.2.1 n n n = * Qui c:0! 1 =3. Chnh hp:a) Chnh hp: GIO KHOA & PP GII TON 11 GV: NGUYN THANH NHN 15 eW: 0987. 503.911 ChotpAcnphnt,mibspthtgmkphntlytrongn phn t ca A ( , 0 k k n e < s ) gi l mt chnh hp chp k ca n. b) S cc chnh hp chp k ca n: ( )( ) ( )!. 1 ... 1!knnA n n n kn k= = + 4. T hp:a) T hp:ChotpAcnphnt,mitphpcongmkphntcaA ( , 0 k k n e s s ) gi l mt t hp chp k ca n. b) S cc t hp chp k ca n: ( )!! !knnCk n k= c) Tnh cht: 0 1 111n k n k k k kn n n n n n nC C C C C C C + ++= = = + =5. Cch phn bit t hp v chnh hp:* Chnh hp c tnh n th t ca k phn t. * T hp khng tnh n th t ca k phn t. sc< NH THC NEWTON 1. Khai trin nh thc Newton: ( )0 1 1 2 2 2 1 1... ...nb n n k n k k n n n nn n n n n na b Ca C a b Ca b Ca b C ab Cb + = + + + + + + + S hng tng qut th k+1 ca khai trin: 1k n k kk nT Ca b+=2. Tam gic Pascal: (cho bit gi tr ca knC ) n \ k0123456 01 111 2121 31331 414641 515101051 61615201561 Mun tm knCta tm s dng n, ct k. V d: 3620 C =(dng 6, ct 3) GIO KHOA & PP GII TON 11 GV: NGUYN THANH NHN 16 eW: 0987. 503.911 3. Gii phng trnh: gii phng trnh ta cn t iu kin cho n s v p dng cng thc hon v, t hp, chnh hp a v phng trnh i s gii. Ch ch ly nhng nghim tha mn iu kin. sc< XC SUT 1. Tp hptt c cc kt qu c th xy ra ca php th c gi l khng gian mu. a) Gieo n con sc sc th6n=b) Gieo n ng tin th2n=c) Ly k vin bi trong hp c n vin bi th knC =d) Hp 1 c m vin bi, hp 2 c n vin bi. Ly k vin hp 1 v h vin hp 2 thk hm nCC =2. Mt bin c A lin quan ti php th T l A c . Bin c A xy ra khi v ch khi kt qu ca T thuc A . Mi phn t ca Agi l kt qu thun li cho A. 3. Hai bin c A, B gi l xung khc nu A, B khng ng thi xy ra. 4. Hai bin c A, B gi l c lp nu vic xy ra hay khng xy ra ca bi c nay khng nh hng n xc sut xy ra ca bin c kia. 5. Xc sut ca A l ( )APA=6. 1 2, ,...,kA A Al cc bin c i mt xung khc th ( ) ( ) ( ) ( )1 2 1 2... ...k kPA A A PA PA PA= + + +7. 1 2, ,...,kA A Al cc bin c c lp th ( ) ( ) ( ) ( )1 2 1 2... ...k kPA A A PA PA PA =8.Al bin c i ca bin c A th: ( )( )1 PA PA = 9. X l bin ngu nhin ri rc vi tp gi tr l { }1 2, ,...,nx x xa) K vng ca X l ( )1ni iiEX x p== vi ( ), 1,2,3,...,i ip PX x i n = = = GIO KHOA & PP GII TON 11 GV: NGUYN THANH NHN 17 eW: 0987. 503.911 b) Phng sai ca X l ( ) ( )21ni iiV X x p == hay ( )2 21niiV X xp == trong ( ), 1,2,...,i ip PX x i n = = =v ( )EX =c) lch chun: ( ) ( )X EX =sc< DY S 1. Tnh n iu ca dy s: a) nh ngha: Cho dy s ( )nunu* n eta c: * 1 n nu u+< th dy s ( )nul dy s tng. * 1 n nu u+> th dy s ( )nul dy s gim. * Mt dy tng (hay gim) gi l dy s n iu. b) Cch xt tnh n iu ca dy s: xt tnh n iu ca mt dy s ta c th p dng tnh cht bt ng thc suy trc tip. Hoc xt hiu 1 n nT u u+= * Nu0, * T n > eth ( )nul dy s tng. * Nu0, * T n < eth ( )nul dy s gim. Nu0,nu n > e ta c th xt 1nnuu+ * 11nnuu+>th ( )nul dy s gim. * 11nnuu+th dy s ( )nub chn di. * Dy s va b chn trn va b chn di gi l dy s b chn. GIO KHOA & PP GII TON 11 GV: NGUYN THANH NHN 18 eW: 0987. 503.911 CP S CNG 1. nh ngha: ( )nul mt cp s cng nu* n etn ti s d sao cho 1 n nu u d+= +d: cng sai nu : s hng tng qut th n. 2. Tnh cht:a) S hng tng qut th n: ( )11nu u n d = + b) ( )nul cp s cng 1 12n n nu u u + + = ,1 n >3. Tng n s hng u tin ca cp s cng: ( )( )112 12 2nnn u n dnu uS (+ + = =sc< CP S NHN 1. nh ngha: ( )nul mt cp s nhn nu* n etn ti s q sao cho 1.n nu u q+=q: cng bi nu : s hng tng qut th n. 2. Tnh cht: a) S hng tng qut: 11.nnu uq=b) ( )nul cp s nhn 21 1.n n nu u u + ( = ,1 n >3. Tng n s hng u tin: *1 q =th 1.nS n u =*1 q =th 11.1nnqS uq= * CSN li v hn l CSN c cng bi1 q f)lim 0lim 0n nnnu vuv< =`=) 3. Mt s gii hn c bn: a) 1lim 0n= b) ( )*limn = + ec) 0, 1lim, 1nqqq

e) 31lim 0n =4. Cch tm gii hn: a) t tha s chungnly tha caonht trong c t svmu s, sau n gin tha s chung ri p dng cc tnh cht v cc gii hn c bn tnh. b)Khitronggiihnccnthctacthnhnchiachobiuthclin hp. sc< GIO KHOA & PP GII TON 11 GV: NGUYN THANH NHN 20 eW: 0987. 503.911 GII HN CA HM S 1. ( )lim lim limx a x a x au v u v = 2. ( )lim . lim.limx a x a x au v u v =3. ( )limlim lim 0limx ax a x ax auuvv v | |= = |\ . 4. ( )lim lim lim 0x a x a x au u u = >5. ( ) ( ) ( )lim ( )lim( ) lim( )x ax a x agx f x hxf x Lgx h x L s s == = 6. 1lim ( ) lim 0( )x a x af xf x = + =7. Qui tc tnh gii hn: lim ( )lim ( ). ( ) ( )lim( )x ax ax af xf x gxgx L = ( = = (ty theo du calim ( )x a f x vL . 8. Hm s lin tc: Hm s( ) y f x =lin tc ti alim ( ) ( )x a f x f a =9. Hm s( ) y f x =lin tc trong( ; ) a bv( ). ( ) 0 f a f b th 222..b cx a khi xxxax bx cb cx a khi xxx+ + ++ + = + + c) Dng v nh v0. Phng php: Thc hin php bin i a v dng 00 hoc sc< GIO KHOA & PP GII TON 11 GV: NGUYN THANH NHN 22 eW: 0987. 503.911 HM S LIN TC 1. Xt tnh lin tc ca hm s( ) y f x =ti 0x* Tnh 0( ) f x(nu0( ) f xkhng tn ti th hm s khng lin tc) * Tm 0lim ( )x xf x, khi cn c th tnh gii hn 1 bn. * So snh 0( ) f xv 0lim ( )x xf x kt lun. 2. Tm m hm s( ) y f x =lin tc ti im ch ra Phng php: * Tnh( ) f av tmlim ( )x a f x * Hm s lin tc tix a = lim ( ) ( )x a f x f a = . T iu kin ny tm m, khi cn c th tm gii hn 1 bn. 3. Chng minh phng trnh c nghim: Phng php:* t( ) f xl v tri ca phng trnh,( ) f xlin tc trong D. * Tm hai s a, b eD sao cho( ). ( ) 0 f a f b* /, MMkhc pha i vi I nu0 k