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 nh ngha : Cho hm s f ' ( x0 ) = lim ( a ; b) xỏc nh trờn khong x0 ( a ; b ) v x0 , o hm ca hm s ti im l f ( x ) f ( x0 ) x x0 : x x0 1.2 Chỳ ý : x = x x0 ; y = f ( x0 + x ) f ( x0 ) Nu kớ hiu thỡ : f ' ( x0 ) = lim f ( x0 + x ) f ( x0 ) x x0 x x0 y x x = lim y = f ( x) Nu hm s x0 cú o hm ti thỡ nú liờn tc ti im ú í ngha ca o hm y = f ( x) 2.1 í ngha hỡnh hc: Cho hm s ( C) cú th f ' ( x0 ) ( C) l h s gúc ca tip tuyn th ca hm s Phng trỡnh tip tuyn ca th hm s ti M ( x0 , y0 ) ( C ) y = f ( x) M ( x0 , y0 ) ( C ) y = f ( x) ti im l : y = f ' ( x0 ) ì( x x0 ) + y0 2.2 í ngha vt lớ : 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 s = s( t) Vn tc tc thi ca chuyn ng thng xỏc nh bi phng trỡnh : Q = Q( t) ti thi im ti thi im l I ( t0 ) = Q ' ( t0 ) t0 Cng tc thi ca in lng v ( t0 ) = s ' ( t ) t0 l : Qui tc tớnh o hm v cụng thc tớnh o hm u = u ( x) ; v = v ( x) ; C : 3.1 Cỏc quy tc : Cho l hng s ( u v ) ' = u ' v ' ( C.u ) = C.u ( u.v ) ' = u '.v + v '.u C.u u u '.v v '.u C , v ( ) ữ= ữ = 2 v u v u y = f ( u) , u = u ( x) yx = yu u x Nu 3.2 Cỏc cụng thc : ( C) = ; ( x) = ( x ) = n.x n n ( x ) = 1x ( ) = n.u un , ( x > 0) n ( u ) = 2uu u , ( n Ơ , n ) , ( u > 0) ( sin x ) = cos x ( sin u ) = u. cos u 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 ( cos x ) = sin x ( cos u ) = u .sin u ( tan x ) = cos x ( cot x ) = sin x ( tan u ) = u cos u ( cot u ) = u sin u Vi phõn 4.1 nh ngha : y = f ( x) Cho hm s y = f ( x) x0 cú o hm ti vi phõn ca hm s x0 ti im l : df ( x0 ) = f ( x0 ) x y = f ( x) Cho hm s f ( x) cú o hm df ( x ) = f ( x ) x = f ( x ) dx f ( x ) x thỡ tớch y = f ( x) c gi l vi phõn ca hm s Kớ hiu : dy = y.dx hay 4.2 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 ) 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 s = f ( t) í ngha c hc: Gia tc tc thi ca chuyn ng 5.2 o hm cp cao : ti thi im n n1 f ( ) ( x ) = f ( ) ( x ) , ( n Ơ , n ) a ( t0 ) = f ( t0 ) t0 l 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 x mt s gia y = f ( x + x ) f ( x ) y v tỡm s gia tỡm Lp t s y x y x x lim o Bc : Tỡm gii hn f ' ( x0 ) = lim f ( x ) f ( x0 ) x x0 x x0 5.4 Cỏch : p dng cụng thc: 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 + f ( x) = x0 = 2x x+2 x0 = a) ti ; b) ti 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 + a) f ( x) = 10 x 16 x0 = ti ; b) x < x0 = ti 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 Tỡm o hm ca cỏc hm s sau theo nh ngha : y = f ( x ) = x 3x + y = x3 x2 + a) ; b) 5.5 Bi ỏp dng : Bi Tỡm o hm ca cỏc hm s sau theo nh ngha ti cỏc im ó ch : f ( x ) = x 3x + a) c) f ( x) = 2x x2 x0 = ; ti x 3x + f ( x) = x+2 b) ; ti x0 = d) ti Bi Xột tớnh liờn tc v s tn ti o hm v tớnh o hm ca cỏc hm s sau õy trờn x 4x + x > f ( x) = x x x ; ti f ( x ) = cos x x0 = x0 = Ă ; a) x + a x f ( x) = x + bx x > ; b) f ( x ) = x 3x + 2 f ( x) = x c) ; d) Bi Tỡm o hm ca cỏc hm s sau theo nh ngha : f ( x ) = x3 3x + x + a) f ( x) = f ( x) = x ; x x +1 b) ; f ( x) = sin x c) ; d) Bi Tỡm o hm ca cỏc hm s sau theo nh ngha : f ( x ) = x 4x a) ; b) ; ( C) : y = x 5.6 sin x + cos x x > f ( x) = x x + ; c) Bi Cú bao nhiờu tip tuyn ca ; f ( x ) = tan ( x + 1) f ( x ) = x + 3x ; d) 3x + x cú h s gúc õm ? 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 y = (x3 2)(1 x2) ; a) Vớ d Tỡm o hm ca cỏc hm s sau : b) x2 3x + y= x1 2x + y= 3x y= ; a) ; b) Vớ d Tỡm o hm ca cỏc hm s sau : y= y = (x + x + 1) c) (x + 1)2 y= (x 1) ; a) ; b) Vớ d Tỡm o hm ca cỏc hm s sau : y = 2x2 5x + a) ; b) x + x2 (x 2x + 5)2 ; c) sin x + cos x sin x cos x ; c) y = ( 1+ 2x ) a) ; b) Vớ d Tỡm o hm ca cỏc hm s sau : y= 1+ x x2 c) y = (x 2) x2 + y = sin x cos x + tan x y= tan 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 : y = (sin x + cos x) y= ; a) ; ( y = tan2x + tan3 2x + tan5 2x c) ; d) Vớ d Cho hm s : m Tỡm f ( x ) x Ă c) ) y = tan sin cos3 x y = f ( x ) = x3 x + mx + a) tan x + cot x b) : f ( x ) > , x ( 0; + ) ; f ( x ) < , x ( 0; ) ; d) ; b) f ( x ) , 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 f ( x) = Vớ d 10 m m x x + ( m ) x + 5m + Cho hm s : f ( x ) < , x Ă f ( x) = a) ; m Tỡm : cú hai nghim cựng du b) Bi Tỡm o hm ca cỏc hm s sau : y= x + 2 3 x x y= x + 4x a) ; y= x 4 x 3 + x 2 ; x y = x 4x + 2x x d) ; x b a + +c x + 3b a x a ,b ,c e) ( Bi Tỡm o hm ca cỏc hm s sau : l hng s) a) ; 2x x ; e) 2x 4x + b) ; y = x +1 2x + g) ; h) Bi Tỡm o hm ca cỏc hm s sau : y = (2 x 3x x + 1) a) y= ; ( ) x +1 1ữ x c) x + x y= 2x y= ; y= y = x (2 x 1)(3 x + 2) y = (2 x 3)( x x) y= x + x 0,5 x b) ; y= d) c) y= y= x +1 x f) ; i) ; 5x x + x +1 ( x x + 1) b) y= x ữ x y = ( x x + 1)3 ( x + x + 1) c) ; d) ; y = + x x2 e) y= ; f) x2 + x2 ; 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= sin x x + x y= sin x a) ; sin x + cos x y= sin x cos x c) y= e) sin x + cos x ; y = 4sin x cos x.sin x ; sin x + cos x d) y= sin x cos x y = tan b) sin x + cos3 x ; f) ; sin x x cos x cos x x sin x ; x +1 y = tan x cot x g) ; f ( x) = Bi a) Cho hm s cos x + sin x y = f ( x) = h) ; f ' ( ); f ' ( ) ; f ' ; f ' Tớnh f ữ f cos x + sin x ' ữ = 3 b) Cho hm s Chng minh: Bi 10 Tỡm o hm ca cỏc hm s sau : ( ) ( y = sin x + cos x sin x + cos6 x a) ; y = cos x ( 2cos x 3) + sin x ( 2sin x ) b) ) ; y = ( sin x cos x ) + ( cos x 2sin x ) + 6sin x 8 6 c) ; Bi 11 Cho hm s y = x sin x chng minh : xy ( y ' sin x ) + x ( 2cos x y ) = a) b) ; y' x = tan x cos x Bi 12 Cho cỏc hm s : f ( x ) = sin4 x + cos x f ' ( x) 2g ' ( x) = g ( x ) = sin6 x + cos x , Chng 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 y = x + 1+ x2 Bi 13 a) Cho hm s + x y' = y Chng minh : y '+ y + = y = cot x b) Cho hm s y' = Bi 14 Gii phng trỡnh Chng minh : bit : y = cos x + sin x y = sin x cos x a) ; c) b) ; y = ( m 1) sin x + 2cos x 2mx y = 3sin x + cos x + 10 x ; d) y = x3 ( 2m + 1) x + mx Bi 15 Cho hm s m Tỡm y' = a) : cú hai nghim phõn bit ; y' b) c) d) cú th vit c thnh bỡnh phng ca nh thc ; y ' , x Ă ; y ' < , x ( ; ) ; y ' > , x > e) y = mx + ( m 1) x mx + 3 Bi 16 Cho hm s Xỏc nh y ' , x Ă a) : y' = b) m cú hai nghim phõn bit cựng õm ; x12 + x22 = y' = c) cú hai nghim phõn bit tha iu kin : y= Bi 17 Cho hm s mx + x x+2 m Xỏc nh hm s cú y ' 0, x ( ; + ) Bi 18 Tỡm cỏc giỏ tr ca tham s m hm s: y = x + x + mx + m y' cú trờn mt on cú di bng 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 = mx4 + m2 x2 + 10 ( 1) ( m laứtham soỏ) Bi 19 Cho hm s bit y' = m Xỏc nh hm s cú cú nghim phõn Vit phng trỡnh tip tuyn ca ng cong 5.7 Phng phỏp : ( C ) : y = f ( x) Khi bit tip im : Tip tuyn ca th M ( x0 ; y0 ) ti , cú phng trỡnh l : y = f ' ( x0 ) ( x x0 ) + y0 (1) ( C ) : y = f ( x) Khi bit h s gúc ca tip tuyn: Nu tip tuyn ca th cú h s gúc l k M ( x0 ; y0 ) thỡ ta gi l f ' ( x0 ) = k tip im (1) y0 = f ( x0 ) x0 Gii phng trỡnh (1) tỡm suy y = k ( x x0 ) + y0 Phng trỡnh tip tuyn phi tỡm cú dng : Chỳ ý : M ( x0 , y0 ) ( C ) H s gúc ca tip tuyn ti trc honh v tip tuyn k = f ( x0 ) = tan l Trong ú l gúc gia chiu dng ca 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 A ( x1 ; y1 ) Bit tip tuyn i qua im : 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 y = f ( x) Vit phng trỡnh tip tuyn ca M ( x0 ; y0 ) ti y = f ' ( x0 ) ( x x0 ) + y0 ( 1) : A ( x1 ; y1 ) y1 = f ' ( x0 ) ( x1 x0 ) + f ( x0 ) ( *) Vỡ tip tuyn i qua x0 Gii phng trỡnh(*) tỡm 5.8 th vo (1) suy phng trỡnh tip tuyn Cỏc vớ d minh : ( C) ( C ) : y = f ( x ) = x 3x 1Cho ng cong Vit phng trỡnh tip tuyn ca M ( ; 2) a) Ti im ; ( C) b) Ti im thuc x0 = v cú honh vi trc honh A ( ; ) d) Bit tip tuyn i qua im 2Cho ng cong ; ( C) c) Ti giao im ca cỏc trng hp sau : 3x + ( C) : y = x ( C) a) Vit phng trỡnh tip tuyn ca b) Vit phng trỡnh tip tuyn ca c) Vit phng trỡnh tip tuyn ca x 2y + = 30 ( C) ( C) ( d ) : x y 21 = bit tip tuyn song song vi ng thng bit tip tuyn vuụng gúc vi ng thng ; bit tip tuyn to vi ng thng : mt gúc y = x + 3x x + Vớ d 11 Cho hm s s gúc nh nht ; ( ) : 2x + y = (C) ( C) Trong tt c cỏc tip tuyn ca th , hóy tỡm tip tuyn cú h 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 y= x+2 2x + ( 1) Vớ d 12 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 13 Cho hm s tuyn vi th y = x3 + 3x ( C ) ( C) Tỡm cỏc im thuc th ( C) y = 6x x2 Vớ d 14 Cho l th ca hm s tung ti mt im cỏch u gc ta v tip im Bi m qua ú k c mt v ch mt tip (Hc vin Cụng ngh Bu chớnh Vin thụng, 1999) ( C) 5.9 ( C) Chng minh tip tuyn ti mt im bt kỡ ca ct trc Bi ỏp dng: ( C) ( C ) : y = x2 2x + Bi 20 Cho hm s Vit phng trỡnh tip vi x0 = a) Ti im cú honh ; 4x y = b) Bit tip tuyn song song vi ng thng : ; x + y 2011 = c) Vuụng gúc vi ng thng : ; A( ; 0) d) Bit tip tuyn i qua im Bi 21 Cho hm s : 3x + y= x : ( C) a) Vit phng trỡnh tip tuyn ca b) Vt phng trỡnh tip tuyn ca c) Vit phng trỡnh tip tuyn ca d) Vit phng trỡnh tip tuyn ca e) Vit phng trỡnh tip tuyn ca ( C) ( C) ( C) ( C) ( C) M ( ; 1) ti im ti giao im ca ti giao im ca ( C) ( C) ; vi trc honh; vi trc tung ; bt tip tuyn song song vi ng thng bit tip tuyn vuụng gúc vi ng thng ( d ) : 4x y + = ( ) : 4x + 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 ( C) y = x3 x Bi 22 Cho hm s : ( C) I ( ; 2) a) Vit phng trỡnh tip tuyn ca th ti im ( C) b) Chng minh rng cỏc tip tuyn khỏc ca th y = x x khụng i qua I ( C) ( C) Bi 23 Cho hm s Tỡm phng trỡnh tip tuyn vi : x0 = a) Ti im cú honh ; ( d ) : x + 2y = b) Song song vi ng thng : y = x + 3mx + ( m + 1) x + Bi 24 Cho hm s ( 1) , m l tham s thc A ( ; 2) Tỡm cỏc giỏ tr ca m tip tuyn ca th ca hm s (1) ti im cú honh x = i qua im (D b A1 - 2008) Bi 25 Cho hm s (1) ti im y= 3x + x +1 M ( ; ) ( 1) Tớnh din tớch ca tam giỏc to bi cỏc trc ta v tip tuyn ca th ca hm s (D b D1 - 2008) ( C) y = 3x3 + ( C ) Bi 26 Cho hm s ( d) : Vit phng trỡnh tip tuyn ca th 3y x + = bit tip tuyn to vi ng thng 300 gúc y = x 3x + x ( C ) Bi 27 Cho hm s ln nht Trong tt c cỏc tip tuyn ca th 2x x y= Bi 28 Cho hm s vi ng thng IM ( C) I ( ; 2) Gi ( C ) , hóy tỡm tip tuyn cú h s gúc M ( C) Tỡm im ( C) cho tip tuyn ca ti M vuụng gúc (D b B2 - 2003) Bi 29 (*) Cho hm s y= 2x M ( C) ( C) M A, B ( C) x +1 Tỡm im , bit tip tuyn ca ti ct hai trc ta ti v tam 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 giỏc OAB cú din tớch bng (Khi D - 2007) y= Bi 30 (*) Cho hm s : x x ( d1 ) : x = ; ( d2 ) : y = ( C) Vit phng trỡnh tip tuyn ( ) ( C) ca ( ) cho v hai ng ct to thnh mt tam giỏc cõn (D b D2 - 2007) y = x+ ( C) A ( 1; 1) ( C ) v hai tip tuyn x +1 Chng minh rng qua im k c hai tip tuyn vi y= 4 A ; ữ x x + 3x ( C ) ( C ) Vit Qua im cú th k c my tip tuyn n th Bi 31 Cho hm s ú vuụng gúc vi Bi 32 (*) Cho hm s phng trỡnh cỏc tip tuyn y 14