CHUYÊN đề đạo hàm

TRUNG TM O TO T HC WTS a ch: Tng s nh 403 ng Nguyn Khang, Cu giy, H Ni Hotline: 0986 035 246 Email: trungtamdaotaotuhocwts@gmail.com Website: wts.edu.vn /nguyenvanson.vn CHUYấN : O HM KIN THC CN NH nh ngha o hm ti mt im y = f ( x) 1.1 1.2 nh ngha : Cho hm s hm s ti im x0 l : Chỳ ý : a ; b) x a ; b) xỏc nh trờn khong ( v ( , o hm ca f ' ( x0 ) = lim f ( x ) f ( x0 ) x x0 x x0 Nu kớ hiu x = x x0 ; y = f ( x0 + x ) f ( x0 ) thỡ : f ' ( x0 ) = lim f ( x0 + x ) f ( x0 ) x x0 x x0 Nu hm s y = f ( x) í ngha hỡnh hc: Cho hm s f ' ( x0 ) y x x = lim cú o hm ti x0 thỡ nú liờn tc ti im ú í ngha ca o hm 2.1 y = f ( x) C cú th ( ) C y = f ( x) M x ,y C l h s gúc ca tip tuyn th ( ) ca hm s ti ( 0 ) ( ) Phng trỡnh tip tuyn ca th hm s y = f ( x) ti im y = f ' ( x0 ) ì( x x0 ) + y0 2.2 M ( x0 , y0 ) ( C ) l : í ngha vt lớ : Vn tc tc thi ca chuyn ng thng xỏc nh bi phng trỡnh : s = s ( t ) ti thi im t0 l v ( t0 ) = s ' ( t0 ) Cng tc thi ca in lng Q = Q ( t ) ti thi im t0 l : I ( t0 ) = Q ' ( t0 ) Qui tc tớnh o hm v cụng thc tớnh o hm u = u ( x) ; v = v ( x) ; C : 3.1 Cỏc quy tc : ( u v ) ' = u ' v ' Cho ( u.v ) ' = u '.v + v '.u 3.2 l hng s ( C.u ) = C.u C.u u u '.v v '.u C , v = ( ) ữ= ữ v2 u2 v u ( ) Nu Cỏc cụng thc : y = f u , u = u ( x) yx = yu ux ( C ) = ; ( x) = 1 TRUNG TM O TO T HC WTS a ch: Tng s nh 403 ng Nguyn Khang, Cu giy, H Ni Hotline: 0986 035 246 Email: trungtamdaotaotuhocwts@gmail.com Website: wts.edu.vn /nguyenvanson.vn ( x ) = n.x ( x ) = 1x n ( ) = n.u n un , ( x > 0) n u , ( n Ơ , n ) ( u ) = 2uu , ( u > 0) ( sin x ) = cos x ( sin u ) = u. cos u ( cos x ) = sin x ( cos u ) = u .sin u ( tan x ) = ( tan u ) = cos x ( cot x ) = sin x u cos u u ( cot u ) = sin u Vi phõn 4.1 nh ngha : y = f ( x) Cho hm s Cho hm s df ( x0 ) = f ( x0 ) x y = f ( x) y = f ( x) 4.2 y = f ( x) cú o hm ti x0 vi phõn ca hm s ti im x0 l : cú o hm f ( x) thỡ tớch f ( x ) x c gi l vi phõn ca hm s df x = f ( x ) x = f ( x ) dx Kớ hiu : ( ) hay dy = y.dx Cụng thc tớnh gn ỳng : f ( x0 + x ) f ( x0 ) + f ( x0 ) x o hm cp cao 5.1 o hm cp : nh ngha : f ( x ) = f ( x ) í ngha c hc: Gia tc tc thi ca chuyn ng s = f ( t) a t = f ( t0 ) ti thi im t0 l ( ) n n1 f ( ) ( x ) = f ( ) ( x ) , ( n Ơ , n ) 5.2 o hm cp cao : CC DNG TON THNG GP : Tỡm o hm theo nh ngha 5.3 Phng phỏp : tỡm o hm theo nh ngha ta cú cỏch sau : Cỏch : Theo quy tc o Bc : Cho x mt s gia x v tỡm s gia y tỡm y = f ( x + x ) f ( x ) y Lp t s x TRUNG TM O TO T HC WTS a ch: Tng s nh 403 ng Nguyn Khang, Cu giy, H Ni Hotline: 0986 035 246 Email: trungtamdaotaotuhocwts@gmail.com Website: wts.edu.vn /nguyenvanson.vn y Bc : Tỡm gii hn x x lim o x x0 x x0 Cỏch : p dng cụng thc: 5.4 f ( x ) f ( x0 ) f ' ( x0 ) = lim Cỏc vớ d minh : Vớ d Tỡm o hm ca cỏc hm s sau theo nh ngha ti cỏc im ó ch ra: f ( x ) = x3 2x + 2x x + ti x0 = f ( x) = x =2 ti a) ; b) Vớ d Tỡm o hm ca cỏc hm s sau theo nh ngha ti cỏc im ó ch ra: x x x f ( x ) = 3x + f ( x) = 10 x 16 x =3 ti a) ; b) Vớ d Tỡm o hm ca cỏc hm s sau theo nh ngha : a) y = x x + 5.5 ; b) x < x = ti y = f ( x ) = x2 3x + Bi ỏp dng : Bi Tỡm o hm ca cỏc hm s sau theo nh ngha ti cỏc im ó ch : a) f ( x ) = x 3x + f ( x) = x =3 ti ; b) x 3x + x =4 x+2 ti f ( x) = 2x x2 x =1 ; ti f ( x ) = cos x x0 = c) ; d) ti Bi Xột tớnh liờn tc v s tn ti o hm v tớnh o hm ca cỏc hm s sau õy trờn Ă ; x2 4x + x > x + a x f ( x) = x f ( x) = 3 x x x + bx x > ; a) ; b) f ( x ) = x 3x + c) ; d) Bi Tỡm o hm ca cỏc hm s sau theo nh ngha : a) f ( x ) = x3 3x + x + x f ( x) = x +1 c) ; b) f ( x) = x ; d) Bi Tỡm o hm ca cỏc hm s sau theo nh ngha : a) f ( x ) = x 4x c) f ( x ) = x + 3x ( C) : y = x ; f ( x) = sin x ; ; ; d) Bi Cú bao nhiờu tip tuyn ca x sin x + cos x x > f ( x) = x x + b) ; f ( x) = f ( x ) = tan ( x + 1) 3x + x cú h s gúc õm ? 5.6 Cỏc vớ d minh : TRUNG TM O TO T HC WTS a ch: Tng s nh 403 ng Nguyn Khang, Cu giy, H Ni Hotline: 0986 035 246 Email: trungtamdaotaotuhocwts@gmail.com Website: wts.edu.vn /nguyenvanson.vn 1Tỡm o hm ca cỏc hm s sau : y = 2x4 x3 + x a) b) y = (x 2)(1 x ) ; Vớ d Tỡm o hm ca cỏc hm s sau : y= 2x + 1 3x y= x2 3x + x1 ; a) ; b) Vớ d Chng minh cỏc cụng thc tng quỏt sau a b a) x2 + 1+ x x2 x + x2 a c b c x+ a1 c1 b1 c1 ax + bx + c a1 b1 = ữ ữ a x + b x + c 1 a1x + b1x + c1 ( c) y= ) ; b c a a x + a b x + 1 ax + bx + c a1 b1 = ữ ữ a1x + b1 ( a1x + b1 ) b) ; ( a , b , c , a1 , b1 , c1 l hng s) ( a , b , c , a1 , b1 l hng s) Vớ d Tỡm o hm ca cỏc hm s sau : a) y = (x + x + 1) ; b) Vớ d Tỡm o hm ca cỏc hm s sau : y= (x + 1)2 y= (x 1)3 ; c) (x2 2x + 5)2 ( ) b) y = (x 2) x + ; c) y = 1+ 2x a) y = 2x 5x + ; Vớ d Tỡm o hm ca cỏc hm s sau : a) y = sin x cos x ; b) y= + tan x y= tan x ; c) sin x + cos x sin x cos x Chỳ ý : Khi gp cỏc hm s phc nu cú th ta hóy rỳt gn hm s ri hóy i tớnh o hm , c bit l i vi cỏc hm s cú cha cỏc hm s lng giỏc Vớ d Tỡm o hm ca cỏc hm s sau : a) y = (sin x + cos x) ; b) y = tan x + cot x ; ( ) y = tan2x + tan3 2x + tan5 2x c) ; y = tan sin cos3 x d) y = f ( x) = x x + mx + Tỡm m : f ( x ) > , x ( 0; + ) Vớ d 10 a) c) Cho hm s : f ( x ) x Ă f ( x ) < , x ( 0; ) ; ; d) b) f ( x ) , x ( ; ) ; m m f ( x ) = x x + ( m ) x + 5m + Vớ d 11 Cho hm s : Tỡm m : f ( x ) < , x Ă f ( x) = a) ; b) cú hai nghim cựng du TRUNG TM O TO T HC WTS a ch: Tng s nh 403 ng Nguyn Khang, Cu giy, H Ni Hotline: 0986 035 246 Email: trungtamdaotaotuhocwts@gmail.com Website: wts.edu.vn /nguyenvanson.vn Bi Tỡm o hm ca cỏc hm s sau : y= a) y= c) y= x x + 4 x x x 3 x + y= x + 4x ; 1 x + x 0,5 x b) ; d) y = x x + x x ; ; 2 x x b a2 + +c x + b a x ( a , b , c l hng s) e) Bi Tỡm o hm ca cỏc hm s sau : a) y = (2 x 3)( x x) y= d) ; b) y = x (2 x 1)(3 x + 2) ; 2x x ; 2x 4x + 2x + g) ; h) Bi Tỡm o hm ca cỏc hm s sau : a) y = (2 x x x + 1) y = x +1 x + ; i) y= ; c) ( b) ) x +1 1ữ x ; x + x y= 2x e) y= y= y= ; f) 5x y= x + x +1 y= ; k) x ; x + x +1 x2 x + ( x x + 1)5 2 c) y = ( x x + 1)3 ( x + x + 1) ; e) y = g) y= + x x2 x+ x+ x d) y= x ữ x ; f) y = ; h) y = x2 + x2 ; x3 x + ; 2x y=3 ữ x+3 i) ; ; k) ( y = x + x2 + ) Bi Tỡm o hm ca cỏc hm s sau : y= a) c) sin x x + x sin x sin x + cos x y= sin x cos x y= sin x + cos x ; d) y = 4sin x cos x.sin x ; ; b) ; sin x + cos x sin x cos x x +1 y = tan g) e) y= sin x + cos3 x y= ; f) ; sin x x cos x cos x x sin x ; h) y = tan x cot x ; TRUNG TM O TO T HC WTS a ch: Tng s nh 403 ng Nguyn Khang, Cu giy, H Ni Hotline: 0986 035 246 Email: trungtamdaotaotuhocwts@gmail.com Website: wts.edu.vn /nguyenvanson.vn y= + tan x ; k) y = cot l) y = cos x + sin x ; m) n) ; o) i) tan x x2 + ; y = sin x cos x 3 y = (sin x + cos x) y = sin ( cos3 x ) ; ; x y = cot cos ữ y = sin cos ( cos3x ) x + p) ; q) cos x f ' ( 0) ; f ' ( ) ; f ' ; f ' f ( x) = + sin x Tớnh Bi 10 a) Cho hm s cos x f ữ f ' ữ = y = f ( x) = + sin x Chng minh: b) Cho hm s Bi 11 Tỡm o hm ca cỏc hm s sau : a) ( ) ( y = sin x + cos x sin x + cos6 x y = cos x ( 2cos x 3) + sin x ( 2sin x ) b) d) y= ; sin x + 3cos x sin x + cos6 x + 3cos x ; ; y = cos x + cos + x ữ+ cos xữ e) sin x + sin x + sin 3x + sin x cos x + cos x + cos3 x + cos x g) ; Bi 12 Cho hm s chng minh : y = x sin x y= a) ; y = ( sin x cos x ) + ( cos x 2sin x ) + 6sin x c) ) xy ( y ' sin x ) + x ( 2cos x y ) = x tan ữ ( + sin x ) y= sin x ; f) ; y = + + + 2cos x , x ; ữữ h) ; y' x = tan x b) cos x 4 6 Bi 13 Cho cỏc hm s : f ( x ) = sin x + cos x , g ( x ) = sin x + cos x Chng minh : f ' ( x ) g ' ( x ) = Bi 14 a) Cho hm s y = x + + x Chng minh : + x y ' = y b) Cho hm s y = cot x Chng minh : y '+ y + = TRUNG TM O TO T HC WTS a ch: Tng s nh 403 ng Nguyn Khang, Cu giy, H Ni Hotline: 0986 035 246 Email: trungtamdaotaotuhocwts@gmail.com Website: wts.edu.vn /nguyenvanson.vn Bi 15 Gii phng trỡnh y ' = bit : b) y = cos x + sin x ; a) y = sin x cos x ; c) y = 3sin x + cos x + 10 x ; d) y = ( m 1) sin x + 2cos x 2mx y = x3 ( 2m + 1) x + mx Bi 16 Cho hm s Tỡm m : a) y ' = cú hai nghim phõn bit ; b) y ' cú th vit c thnh bỡnh phng ca nh thc ; c) y ' , x Ă ; d) y ' < , x ( ; ) e) y ' > , x > ; y = mx + ( m 1) x mx + 3 Bi 17 Cho hm s Xỏc nh m : a) y ' , x Ă b) y ' = cú hai nghim phõn bit cựng õm ; c) y ' = cú hai nghim phõn bit tha iu kin : y= x12 + x22 = mx + x ( ; + ) x+2 Xỏc nh m hm s cú y ' 0, x Bi 18 Cho hm s Bi 19 Tỡm cỏc giỏ tr ca tham s m hm s: y = x + 3x + mx + m cú y ' trờn mt on cú di bng Bi 20 Cho hm s bit \ y = mx4 + ( m2 9) x2 + 10 ( 1) ( m laứtham soỏ) Xỏc nh m hm s cú y ' = cú nghim phõn Vit phng trỡnh tip tuyn ca ng cong 5.7 Phng phỏp : Khi bit tip im : Tip tuyn ca th ( C ) : y = f ( x ) ti M ( x0 ; y0 ) , cú phng trỡnh l : y = f ' ( x0 ) ( x x0 ) + y0 (1) Khi bit h s gúc ca tip tuyn: Nu tip tuyn ca th ( C ) : y = f ( x ) cú h s gúc l k thỡ ta gi M ( x0 ; y0 ) l tip im f ' ( x0 ) = k (1) Gii phng trỡnh (1) tỡm x0 suy y0 = f ( x0 ) TRUNG TM O TO T HC WTS a ch: Tng s nh 403 ng Nguyn Khang, Cu giy, H Ni Hotline: 0986 035 246 Email: trungtamdaotaotuhocwts@gmail.com Website: wts.edu.vn /nguyenvanson.vn Phng trỡnh tip tuyn phi tỡm cú dng : Chỳ ý : H s gúc ca tip tuyn ti y = k ( x x0 ) + y0 M ( x0 , y0 ) ( C ) l k = f ( x0 ) = tan Trong ú l gúc gia chiu dng ca trc honh v tip tuyn Hai ng thng song song vi thỡ h s gúc ca chỳng bng Hai ng thng vuụng gúc nu tớch h s gúc ca chỳng bng Bit tip tuyn i qua im A ( x1 ; y1 ) Vit phng trỡnh tip tuyn ca Vỡ tip tuyn i qua : y = f ( x) ti M ( x0 ; y0 ) : y = f ' ( x0 ) ( x x0 ) + y0 ( 1) A ( x1 ; y1 ) y1 = f ' ( x0 ) ( x1 x0 ) + f ( x0 ) ( *) Gii phng trỡnh(*) tỡm x0 th vo (1) suy phng trỡnh tip tuyn 5.8 Cỏc vớ d minh : ( C ) : y = f ( x ) = x3 3x Vit phng trỡnh tip tuyn ca ( C ) M ( ; 2) a) Ti im ; ( C ) v cú honh x0 = ; b) Ti im thuc ( C ) vi trc honh c) Ti giao im ca A ( ; ) d) Bit tip tuyn i qua im 1Cho ng cong 2Cho ng cong ( C) : y = cỏc trng hp sau : 3x + 1 x ( C ) bit tip tuyn song song vi ng thng ( d ) : x y 21 = ; ( C ) bit tip tuyn vuụng gúc vi ng thng ( ) : x + y = ; b) Vit phng trỡnh tip tuyn ca ( C ) bit tip tuyn to vi ng thng : c) Vit phng trỡnh tip tuyn ca a) Vit phng trỡnh tip tuyn ca x y + = mt gúc 300 y = x + 3x x + Vớ d 12 Cho hm s s gúc nh nht y= x+2 2x + ( C ) Trong tt c cỏc tip tuyn ca th ( C ) , hóy tỡm tip tuyn cú h ( 1) Vớ d 13 Cho hm s Vit phng trỡnh tip tuyn ca th hm s (1), bit tip tuyn ú ct trc honh, trc tung ln lt ti hai im phõn bit A, B v tam giỏc OAB cõn ti gc ta O (Khi A 2009) Vớ d 14 Cho hm s Tỡm cỏc im thuc th m qua ú k c mt v ch mt tip y = x + 3x ( C ) tuyn vi th ( C) ( C) (Hc vin Cụng ngh Bu chớnh Vin thụng, 1999) TRUNG TM O TO T HC WTS a ch: Tng s nh 403 ng Nguyn Khang, Cu giy, H Ni Hotline: 0986 035 246 Email: trungtamdaotaotuhocwts@gmail.com Website: wts.edu.vn /nguyenvanson.vn ( C) Vớ d 15 Cho l th ca hm s y = x x Chng minh tip tuyn ti mt im bt kỡ ca tung ti mt im cỏch u gc ta v tip im 5.9 ( C) ct trc Bi ỏp dng: Bi 21 Cho hm s ( C ) : y = x x + Vit phng trỡnh tip vi ( C ) : x =2 ; a) Ti im cú honh b) Bit tip tuyn song song vi ng thng : x y = ; c) Vuụng gúc vi ng thng : x + y 2011 = ; d) Bit tip tuyn i qua im A( ; 0) 3x + y= x Bi 22 Cho hm s : ( C) ( C ) ti im M ( ; 1) ; ( C ) ti giao im ca ( C ) vi trc honh; b) Vt phng trỡnh tip tuyn ca ( C ) ti giao im ca ( C ) vi trc tung ; c) Vit phng trỡnh tip tuyn ca ( C ) bt tip tuyn song song vi ng thng ( d ) : x y + = ; d) Vit phng trỡnh tip tuyn ca ( C ) bit tip tuyn vuụng gúc vi ng thng ( ) : x + y = e) Vit phng trỡnh tip tuyn ca a) Vit phng trỡnh tip tuyn ca Bi 23 Cho hm s : ( C) y = x3 x a) Vit phng trỡnh tip tuyn ca th ( C) ti im b) Chng minh rng cỏc tip tuyn khỏc ca th Bi 24 Cho hm s y = x x a) Ti im cú honh x0 = y = x3 + 3mx + ( m + 1) x + Tỡm cỏc giỏ tr ca m Bi 26 Cho hm s y= (1) ti im ( C ) khụng i qua I ( C ) Tỡm phng trỡnh tip tuyn vi ( C ) : ; ( d ) : x + 2y = b) Song song vi ng thng : Bi 25 Cho hm s I ( ; 2) ( 1) , m l tham s thc tip tuyn ca th ca hm s (1) ti im cú honh 3x + x +1 M ( ; ) x = i qua im A( ; 2) (D b A1 - 2008) Tớnh din tớch ca tam giỏc to bi cỏc trc ta v tip tuyn ca th ca hm s ( 1) (D b D1 - 2008) TRUNG TM O TO T HC WTS a ch: Tng s nh 403 ng Nguyn Khang, Cu giy, H Ni Hotline: 0986 035 246 Email: trungtamdaotaotuhocwts@gmail.com Website: wts.edu.vn /nguyenvanson.vn Bi 27 Cho hm s ( d) : y = 3x3 + ( C ) 3y x + = Bi 28 Cho hm s Vit phng trỡnh tip tuyn ca th y= Bi 29 Cho hm s vi ng thng IM bit tip tuyn to vi ng thng gúc 30 y = x3 x + x ( C ) ln nht ( C) 2x x ( C) Gi Trong tt c cỏc tip tuyn ca th I ( ; 2) Tỡm im M ( C) cho tip tuyn ca (D b B2 - 2003) Bi 30 (*) Cho hm s y= 2x ( C) x +1 giỏc OAB cú din tớch bng Tỡm im M ( C) ( C) , hóy tỡm tip tuyn cú h s gúc , bit tip tuyn ca ( C ) ti ( C) ti M vuụng gúc M ct hai trc ta ti A , B v tam (Khi D - 2007) x y= x Bi 31 (*) Cho hm s : ( C) Vit phng trỡnh tip tuyn ( d1 ) : x = ; ( d2 ) : y = ct to thnh mt tam giỏc cõn Bi 32 Cho hm s y = x+ ú vuụng gúc vi Bi 33 (*) Cho hm s ( C) x +1 Chng minh rng qua im y = x3 x + 3x ( C ) Qua im A ( 1; 1) 4 A ; ữ ( ) ca ( C ) cho k c hai tip tuyn vi ( ) ( C) v hai ng (D b D2 - 2007) v hai tip tuyn cú th k c my tip tuyn n th ( C) Vit phng trỡnh cỏc tip tuyn y Bi 34 (*) Cho hm s y= x2 + x + (C ) I ( ; ) x +1 Gi Chng minh rng khụng cú tip tuyn no ca ( C) i qua im I Bi 35 (*) Cho hm s tuyn vi th ( C) y = x4 + x2 ( C ) (D b B2 - 2005) Tỡm tt c cỏc im thuc trc tung cho t ú cú th k c ba tip Tỡm vi phõn ca hm s v tớnh gn ỳng nh vi phõn 10 TRUNG TM O TO T HC WTS a ch: Tng s nh 403 ng Nguyn Khang, Cu giy, H Ni Hotline: 0986 035 246 Email: trungtamdaotaotuhocwts@gmail.com Website: wts.edu.vn /nguyenvanson.vn 5.10 Phng phỏp : Da theo nh ngha v cụng thc sau : Cho hm s y = f ( x ) cú o hm f ( x ) thỡ tớch f ( x ) x c gi l vi phõn ca hm s y = f ( x) Kớ hiu : df ( x ) = f ( x ) x = f ( x ) dx hay dy = y.dx f ( x0 + x ) f ( x0 ) + f ( x0 ) x 5.11 Cỏc vớ d minh : 1Tỡm vi phõn ca cỏc hm s sau : x 3x + x y= a) ; 2Tỡm vi phõn ca cỏc hm s sau : b) y= (x + 1) ( x3 x ) sin x x + x sin x y = tan x cot 3x a) ; b) 3Tớnh gn ỳng cỏc giỏ tr sau (ly ch s thp phõn kt qu) : a) 8,99 ; b) cos 46 ; y= 5.12 Bi c) tan 59 45' ỏp dng: Bi 36 Tỡm vi phõn ca cỏc hm s sau : y= a) 2x + y= b) y = ( x x ) ; + cos x y= ữ cos x ; d) y = sin(cos x) + cos(sin x) f) x2 + ; x c) e) ; 32 x 5x + ; y = cot (2 x + ) sin x cos3 x + sin x.cos x Bi 37 Cho hm s Chng minh ng thc : y.dy cos x.dx = y= Bi 38 Tớnh gn ỳng cỏc giỏ tr sau (ly ch s thp phõn kt qu) : a) 4,02 ; b) tan 44 30' ; c) 7,97 o hm cp cao 5.13 Phng phỏp : Da theo cỏc nh ngha sau : 11 TRUNG TM O TO T HC WTS a ch: Tng s nh 403 ng Nguyn Khang, Cu giy, H Ni Hotline: 0986 035 246 Email: trungtamdaotaotuhocwts@gmail.com Website: wts.edu.vn /nguyenvanson.vn o hm cp : f ( x ) = f ( x ) n n1 f ( ) ( x ) = f ( ) ( x ) , ( n Ơ , n ) o hm cp cao : Chỳ ý : tỡm cụng thc tớnh o hm cp n ca mt hm s ta tỡm o hm cp , , sau ú d oỏn cụng thc tớnh o hm cp n v chng minh cụng thc ú bng phng phỏp quy np 5.14 Cỏc vớ d minh : Tỡm o hm cỏc cp ó ch ca cỏc hm s sau : x x + 5x2 4x + a) Tỡm y , y ; x y= ( 4) x + Tỡm y , y , y b) ; c) y = 3x x Tỡm y y= Vớ d 16 Chng minh cỏc h thc sau vi cỏc hm s c ch ra: a) y y + = y = ( b) x y x + y ) ( + y ) = ; y = x.tan x * Chng minh bng quy np cỏc cụng thc sau ỳng n Ơ : Vớ d 17 a) 2x x2 ( sinax) ( n) n = an sin ax + ữ ( n) ữ ax + b c) Vớ d 18 ; b) ( cosax) ( n) n = cos ax + ữ ; ( 1) a n! n+1 ( ax + b) n n = Tỡm cỏc o hm cp n ca cỏc hm s sau : y= a) Vớ d 19 4x +1 2x ; b) y= x 3x + x +1 Tỡm cỏc o hm cp n ca cỏc hm s sau : 4 a) y = sin x + cos x ; b) y = 8sin x.cos3 x.cos x Chỳ ý : Khi tỡm o hm cp n ca mt hm s , nu c ta hóy bin i hm s ó cho thnh tng ca cỏc hm ; sinax ; cosax s cú mt cỏc dng : ax + b ri ỏp dng cỏc cụng thc vớ d trờn , d oỏn cụng thc o hm cp n ca hm s ó cho v chng minh li bng quy np (nu cn) 5.15 Bi ỏp dng: Bi 39 Tỡm o hm cỏc cp ó ch ca cỏc hm s sau : a) y = x.cos x tỡm y c) y= ( x + 1) ( 5) tỡm y b) y = sin ; y= ; d) 2x x + 3x + x2 tỡm y ; ( 4) tỡm y 12 TRUNG TM O TO T HC WTS a ch: Tng s nh 403 ng Nguyn Khang, Cu giy, H Ni Hotline: 0986 035 246 Email: trungtamdaotaotuhocwts@gmail.com Website: wts.edu.vn /nguyenvanson.vn Bi 40 Chng minh cỏc ng thc sau : a) xy ( y ' sin x ) + xy " = nu y = x sin x ; b) 18( y 1) + y" = nu y = cos x ; c) y"+ y = nu y= sin3 x + cos x sin x cos x ; ( ) [ 4] + xy y = 40 y = x2 y d) nu ; x3 y= x+4 ; e) y ' = ( y 1) y" nu ( ) f) x + y"+4 x y ' y = nu y = ( + x ) y "+ xy ' k g) 2 y=0 x + 1+ x2 ; ) ( ( k Ơ) nu y = x + x + , Bi 41 Tỡm o hm cp n ca cỏc hm s sau : a) y= y= 2x x+2 x+2 x2 x + ; x2 5x + y= 2 x 3x + d) y = sin x + cos x ; y( f) Cho y = cos3 x Chng minh 2n ) k y= x2 x ; b) ; d) y = 8sin x.sin x.sin x = ( 1) 32 n y ; ; c) e) n Dựng nh ngha o hm tỡm gii hn 6.1 Phng phỏp : f ' ( x0 ) = lim x x0 f ( x ) f ( 0) x x0 Ta cú th s dng nh ngha ca o hm : tớnh cỏc gii hn cú dng vụ nh Bng cỏch vit gii hn cn tỡm thnh dng : lim x x0 f ( x ) f ( 0) x x0 , sau ú tớnh o hm ca hm f ( x ) ti im x0 ri ỏp dng nh ngha o hm suy kt qu ca gii hn 6.2 Cỏc vớ d minh : 1Tỡm cỏc gii hn sau : lim a) x + 4x x ; x3 x2 + lim x2 b) x 13 TRUNG TM O TO T HC WTS a ch: Tng s nh 403 ng Nguyn Khang, Cu giy, H Ni Hotline: 0986 035 246 Email: trungtamdaotaotuhocwts@gmail.com Website: wts.edu.vn /nguyenvanson.vn Vớ d 20 Tỡm cỏc gii hn sau : x + x2 + L + xn n lim x1 a) x1 Vớ d 21 a) 6.3 ; x n nx + n lim x ( x 1) b) ; sin x lim sin x x Tỡm cỏc gii hn sau : lim tan x tan x x b) Bi ỏp dng: Bi 42 Tỡm cỏc gii hn sau : lim a) x x+8 lim x + 2x 2x + + sin x lim x 3x + x c) lim b) ; 3 x3 24 + x + x lim x2 d) x2 ; n x e) Bi 43 Tỡm cỏc gii hn sau : x a) x a ; x , ( a 0) 2a ; b) f) x cos5 x cos3x x.sin x c) x0 ; cos x e) x x sin x x lim x 1+ 2x 1 + 3x ; lim lim g) lim m ; 2x + x + sin x ; lim lim lim x x ; x lim(a x)tan 3x 3 x d) ; x2 + 4x2 + 1 cos x ; h) f) lim x x lim x x + 2x tan( x 1) ; cos x + + sin x + sin 3x + tan x + sin x x3 ; x2 + + 2x2 + 4x + 19 3x2 + 46 i) x1 x2 Tớnh cỏc tng cú cha t hp (NNG CAO DNH CHO NHNG BN Cể TR THI I HC) 7.1 Phng phỏp : Trong phn i s t hp ỏp dng nh thc Newton tớnh cỏc tng cú cha cỏc cụng thc t hp ụi ta phi bit ỏp dng khộo lộo vic ly o hm cỏc cp ca cỏc v ta s tớnh c tng cn tớnh 7.2 Cỏc vớ d minh : 1Tớnh cỏc tng sau : a) S1 = Cn1 + 2Cn2 + 3Cn3 52 + L + nCnn 5n1 ; 14 TRUNG TM O TO T HC WTS a ch: Tng s nh 403 ng Nguyn Khang, Cu giy, H Ni Hotline: 0986 035 246 Email: trungtamdaotaotuhocwts@gmail.com Website: wts.edu.vn /nguyenvanson.vn b) S2 = 2.1.Cn2 2n2 3.2.Cn3 2n3 + L + ( 1) n ( n 1) Cnn c) S3 = C + C + C + L + n C d) n 7.3 n 2 n n n n ; S = 2Cn0 + 5Cn1 + 8Cn2 + + ( 3n + ) Cnn Bi ỏp dng: Bi 44 Rỳt gn cỏc tng sau : n n S = C + C + L + ( n 1) C + nC n n n n a) ; n S = Cn + 2Cn + 3Cn + + nCn + (n + 1)Cnn ; b) S = 2C + 5C + 8C + + ( 3n + ) Cnn n n n c) Bi 45 (*) Rỳt gn cỏc tng sau : 99 100 198 199 1 99 100 a) S1 = 100C100 ữ 101C100 ữ + L L 199C100 ữ + 200C100 ữ 2 2 18 17 20 S2 = 2.1.C20 3.2.C20 + L + 380.C20 b) c) S3 = C d) S = 3C 5C + 7C + 4023C 2009 n C n 2009 + C 2009 2009 L + 20092.C2009 n 2010 2010 Bi 46 Cho s nguyờn n tha ng thc An3 + Cn3 = 35, ( n 3) ( n 1) ( n ) S = C C + L + ( 1) n C 2 n n n Tớnh tng : n n b B1 2008) Bi 47 Chng minh rng vi n l s nguyờn dng , ta luụn cú : (D n.2n.Cnn + ( n 1) 2n1.Cn1 + ( n ) n2.Cn2 + L + 2.Cnn1 = 2n.3n1 (D b D1 2008) Bi 48 Tỡm s nguyờn dng n cho : C21n+1 2.2C22n+1 + 3.22C23n+1 4.23C24n+1 + + ( 2n + 1) 22nC22nn++11 = 2011 ( Cnk l s t hp chp k ca n phn t ) Phn ln chỳng ta cõn nhc quỏ nhiu v cỏi giỏ phi tr cho s thay i m ớt chu cõn nhc v cỏi giỏ phi tr nu khụng thay i. 15 TRUNG TM O TO T HC WTS a ch: Tng s nh 403 ng Nguyn Khang, Cu giy, H Ni Hotline: 0986 035 246 Email: trungtamdaotaotuhocwts@gmail.com Website: wts.edu.vn /nguyenvanson.vn THễNG BO HC PH Kớnh gi ph huynh em: Lờ c Tun Minh Lp 11 Mụn hc:kl o Toỏn S bui hc: 12 bui Hc phớ: 1.200.000 VN ( Ghi bng ch ): Mt triu hai tram ngn ng Kinh mong quý ph huynh to iu kin cho em úng tin ỳng hn ( t ngy 25/03/2016 5/04/2016 ) theo quy nh ca Trung tõm em c hng nhng quyn li xng ỏng Xin cm n! Ch ký PHHS Trung tõm o to T hc WTS 16