Hm s FB: http://www.facebook.com/VanLuc168 V TIP TUYN CA TH HM S Chuyờn : Hm s A Túm tt lớ thuyt & phng phỏp gii toỏn I KIN THC C BN Dng 1: Vit phng trỡnh tip tuyn vi th (C):y = f(x) ti im M (x ; y ) (C) y (C): y=f(x) y0 M x x0 Phng phỏp: Phng trỡnh tip tuyn vi (C) ti M(x0;y0) cú dng: y - y0 = k ( x - x0 ) hay y f '(x )(x x ) f(x ) Trong ú: x0: honh tip im y0: tung tip im v y0 = f(x0) k: h s gúc ca tip tuyn v c tớnh bi cụng thc : k = f'(x0) 2x x Vớ d: Cho hm s y cú th l C Vit phng trỡnh tip tuyn ca C ti cỏc giao im ca C v ng thng y x Bi gii Phng trỡnh honh giao im: 2x x x (1) iu kin: x Khi ú: (1) 2x ( x 3)( x 1) x2 2x x x Suy ta cỏc giao im l A 0; , B 2; Ta cú: y ' x Phng trỡnh tip tuyn ti A l y y '(0)( x 0) y y Phng trỡnh tip tuyn ti B l y y '(2)( x 2) x v y x Vy cú hai tip tuyn tha bi l y NGUYN VN LC 0933.168.309 x x SP Toỏn K35 - H Cn Th Hm s FB: http://www.facebook.com/VanLuc168 BI TP T LUYN Cõu Vit phng trỡnh tip tuyn ca th hm s y x3 3x ti im M(1;2) (C) : y x 3x y 3x x ti im M(1; 2) ta cú: y (1) PTTT: y x Cõu 2: Cho hm s y x x cú th (H) Vit phng trỡnh tip tuyn ca (H) ti A(2; 3) y x x y ( x 1)2 Ti A(2; 3) k y (2) PTTT : y x Cõu 3: Cho hm s f ( x) x3 3x Lp phng trỡnh tip tuyn ca th hm s ti im M(1; 2) f ( x ) x 3x f ( x ) 3x f (1) PTTT: y Cõu 4: Cho hm s y 3x 1 x cú th (C) Vit phng trỡnh tip tuyn ca (C) ti im A(2; 7) y 3x 1 x y , k y (2) ( x 1)2 PTTT: y x 15 Cõu 5: Cho hm s y x x2 cú th (C) Vit phng trỡnh tip tuyn ca (C) x ti im M(2; 4) y x x2 x2 2x y' k f (2) x ( x 1)2 x0 2, y0 4, k PTTT : y x NGUYN VN LC 0933.168.309 SP Toỏn K35 - H Cn Th Hm s FB: http://www.facebook.com/VanLuc168 Cõu 6: Cho hm s y x 3x cú th (C) Vit phng trỡnh tip tuyn ca (C) ti im I(1; 2) y x 3x y ' 3x x k f (1) x0 1, y0 2, k PTTT : y 3x Cõu Cho hm s y x x cú th (C) Vit phng trỡnh tip tuyn ca (C) ti im cú honh bng x0 y0 y x x k y (1) Phng trỡnh tip tuyn l y = 2x + Cõu Cho hm s: y x3 7x (C) Vit phng trỡnh tip tuyn ca th (C) ti im cú honh x = y x3 7x y ' x Vi x y 3, y (2) 17 PTTT : y 17x 31 0 Cõu Cho (C): y x3 3x Vit phng trỡnh tip tuyn ca (C) ti cỏc giao im ca (C) vi trc honh Cho (C): y x3 3x y 3x x Giao ca ( C) vi trc Ox l A(1; 0), B 3; , C 3; Tip tuyn ti A(1; 0) cú h s gúc l k = nờn PTTT: y x Tip tuyn ti B 3; cú h s gúc l k = nờn PTTT : y x Tip tuyn ti C 3; cú h s gúc l k = nờn PTTT : y x Cõu 10: Vit phng trỡnh tip tuyn ca th hm s yx x ti giao im ca nú vi trc honh y x x y x2 Cỏc giao im ca th hm s vi trc honh l A 1; , B 1; Ti A(1; 0) tip tuyn cú h s gúc k1 nờn PTTT: y = 2x +2 Ti B(1; 0) tip tuyn cng cú h s gúc k2 nờn PTTT: y = 2x NGUYN VN LC 0933.168.309 SP Toỏn K35 - H Cn Th Hm s FB: http://www.facebook.com/VanLuc168 Cõu 11 Cho hm s: y (C ) 2x x Vit phng trỡnh tip tuyn ca (C ) ti im trờn cú tung bng Ta cú: y0 f (x ) 2x x0 (2 1)2 2x 5x x0 3(x 2) y Phng trỡnh tip tuyn cn tỡm: y Cõu 12 Cho hm s y 3x 11 2x Vit phng trỡnh tip tuyn ca th (C) bit x tip im cú tung bng Go i tiờ p iờ m la M ( x0 ; y0 ) , ta co y0 2x x0 x M (2; 3) Suy ra, hờ sụ goc k cua tiờ p tuyờ n la: k y '(2) Do o phng trinh tiờ p tuyn cõ n lõ p la: y 1(x 2) hay y x Cõu 13 Cho hm s y x +3x Lp phng trỡnh tip tuyn ca (C) ti cỏc giao im ca th vi trc honh th ct trc honh ti cỏc im A(0;0) v B(3;0) Phng trỡnh tip tuyn ca th ti A(0;0) l: y Phng trỡnh tip tuyn ca th ti B(3;0) l: y y , 3x x 27 Vy tip tuyn cn tỡm l y v y x 27 Cõu 14 Cho hm s y x3 3x (C ) Gi giao im ca th (C ) v ng thng y x l M , vit phng trỡnh tip tuyn vi th (C ) ti im M y x3 3x Ta ca M l nghim ca h y x y x y x M (1; 2) x 3x x x Phng trỡnh tip tuyn vi (C) ti M l y f '(1)( x 1) y 9( x 1) y x NGUYN VN LC 0933.168.309 SP Toỏn K35 - H Cn Th Hm s FB: http://www.facebook.com/VanLuc168 Dng 2: Vit phng trỡnh tip tuyn vi th (C): y = f(x) bit tip tuyn cú h s gúc k cho trc y (C): y = f(x) y0 M x x0 Phng phỏp: Ta cú th tin hnh theo cỏc bc sau Bc 1: Gi M ( x0 ; y0 ) (C ) l tip im ca tip tuyn vi (C) Bc 2: Tỡm x0 bng cỏch gii phng trỡnh : f ' ( x0 ) k , t ú suy y0 f ( x0 ) =? Bc 3: Thay cỏc yu t tỡm c vo pt: y - y0 = k ( x - x0 ) ta s c pttt cn tỡm 2x x Vớ d: Cho hm s y cú th l C Vit phng trỡnh tip tuyn ca C , bit h s gúc ca tip tuyn bng Bi gii Gi M ( x0 ; y0 ) (C ) l tip im ca tip tuyn vi (C) Ta cú: y ' x 2 H s gúc ca tip tuyn bng Vi x0 y0 Vi x0 y0 3 y '( x0 ) x0 x0 x0 5 : M (1; 3) pttt: y 5x : M (3; 7) pttt: y x 22 Vy cú hai tip tuyn tha bi l y 5x v y x 22 Bi tng t Cho hm s y 2x x cú th l C Vit phng trỡnh tip tuyn ca C , bit h s gúc ca tip tuyn bng x 2; y x 10 ỏp s: y NGUYN VN LC 0933.168.309 SP Toỏn K35 - H Cn Th Hm s FB: http://www.facebook.com/VanLuc168 Chỳ ý : i vi dng ngi ta cú th cho h s gúc k di dng giỏn tip nh : tip tuyn song songtip tuyn vuụng gúc vi mt ng thng cho trc y y (C): y=f(x) ka y ax b (C): y=f(x) x x O k / a : y ax b Khi ú ta cn phi s dng cỏc kin thc sau: nh lý 1: Nu ng thng ( ) cú phng trỡnh dng : y = ax + b thỡ h s gúc ca ( ) l: k a nh lý 2: Trong mp(Oxy) cho hai ng thng (1 ) vaứ ( ) Khi ú: // k k k k Vớ d 1: Cho hm s y x3 3x 2 cú th l C Vit phng trỡnh tip tuyn ca C , bit tip tuyn song song vi ng thng ( ) : y x Bi gii Ta cú: y ' 3x x Do tip tuyn song song vi ng thng ( ) nờn h s gúc ca tip tuyn l k Gi M ( x0 ; y0 ) (C ) l tip im ca tip tuyn vi (C) H s gúc ca tip tuyn k y '( x0 ) 3x02 Vi x0 Vi x0 y0 : M ( 1; 2) y0 : M (3; 2) x0 x0 x0 pttt: y x pttt: y x 25 Vy cú hai tip tuyn tha bi l y x v y x 25 NGUYN VN LC 0933.168.309 SP Toỏn K35 - H Cn Th Hm s FB: http://www.facebook.com/VanLuc168 Bi tng t Cho hm s y x3 3x 3x cú th l C Vit phng trỡnh tip tuyn ca C , bit tip tuyn song song vi ng thng ( ) : y x ỏp s: y 3x x x Vớ d 2: Cho hm s y 2 cú th l C Vit phng trỡnh tip tuyn ca C , bit tip tuyn vuụng gúc vi ng thng ( ) : y Bi gii Ta cú: y ' x x 2 Do tip tuyn vuụng gúc vi ng thng ( ) nờn h s gúc ca tip tuyn l k Gi M ( x0 ; y0 ) (C ) l tip im ca tip tuyn vi (C) H s gúc ca tip tuyn k y '( x0 ) Vi x0 Vi x0 y0 y0 x0 x0 x0 x0 2 x0 x0 : M (0; 1) pttt: y x : M ( 4;3) pttt: y x Vy cú hai tip tuyn tha bi l y x v y x 7 Bi tng t Cho hm s y 2x x cú th l C Vit phng trỡnh tip tuyn ca C , bit tip tuyn vuụng gúc vi ng thng ( ) : x y x 1; y x ỏp s: y NGUYN VN LC 0933.168.309 SP Toỏn K35 - H Cn Th Hm s FB: http://www.facebook.com/VanLuc168 BI TP T LUYN Cõu Cho hm s y x3 3x2 (C) Vit phng trỡnh tip tuyn ca th (C) bit h s gúc ca tip tuyn k =3 Ta cú: y ' 3x2 x Gi M ( x0 ; y0 ) l tip iờ m Tip tuyn ti M cú h s gúc k f ' ( x0 ) 3x02 x0 Theo gi thit, h s gúc ca tip tuyn k = - nờn: 3x02 x0 x02 x0 x0 Vỡ x0 y0 M (1; 2) Phng trinh tip tuyn cn tỡm l y 3( x 1) y 3x Cõu Cho hm s: y x3 7x (C) Vit phng trỡnh tip tuyn ca th (C) cú h s gúc k = y x3 7x y ' x x Gi ( x0 ; y0 ) l to ca tip im Ta cú: y ( x0 ) x02 x0 Vi x0 y0 PTTT : y x Vi x0 y0 PTTT : y x Tip tuyn song song vi ng thng d Cõu Cho hm s y tuyn song song vi d: y x x d: y y x 2 ( x 1)2 x Vit x x y phng trỡnh tip tuyn ca th hm s bit tip ( x 1) cú h s gúc k 2 y ( x ) ( x0 1)2 TT cú h s gúc k Gi ( x0 ; y0 ) l to ca tip im Ta cú x x0 + Vi x0 y0 PTTT: y x NGUYN VN LC 0933.168.309 SP Toỏn K35 - H Cn Th Hm s FB: http://www.facebook.com/VanLuc168 + Vi x0 y0 PTTT: y x x 3x (1) Vit phng trỡnh tip tuyn ca th hm x s (1), bit tip tuyn ú song song vi ng thng d: y x Cõu Cho hm s f ( x ) f (x) x 3x x f (x) Tip tuyn song song x2 2x ( x 1)2 vi d: y x nờn tip tuyn cú h s gúc Gi ( x0 ; y0 ) l to ca tip im Ta cú: f ( x0 ) x02 k x0 ( x0 1)2 x x0 Vi x0 y0 PTTT: y x Vi x0 y0 12 PTTT: y x 22 Cõu Cho hm s f(x) = -x3 + 3x + (cú th (C)) Lp phng trỡnh tip tuyn ca th (C) bit tip tuyn song song vi ng thng d: y = -9x -15 Tip tuyn // d: y = -9x -15 nờn phng trỡnh tip tuyn cú dng y = -9x + m, m -15 x 3x x m (1) iu kin tip xỳc: h 3x (2) cú nghim x m 15 (2) m 17 x m 17 Vy phng trỡnh tip tuyn l: y = -9x +17 Cõu Cho hm s y x ( x 1) cú th (C) Vit phng trỡnh tip tuyn ca th (C), bit tip tuyn song song vi ng thng d: y x Vỡ tip tuyn song song vi d: y x nờn tip tuyn cú h s gúc l k = Gi ( x0 ; y0 ) l to ca tip im x0 y '( x0 ) 3x x0 x x0 x 2 Vi x0 y0 PTTT: y 5x Vi x0 y0 50 27 PTTT: y x NGUYN VN LC 0933.168.309 175 27 SP Toỏn K35 - H Cn Th Hm s FB: http://www.facebook.com/VanLuc168 x x Cõu Cho hm s y cú th (H) Vit phng trỡnh tip tuyn ca (H) bit tip tuyn song song vi ng thng y x y x x y ( x 1)2 Vỡ tip tuyn song song vi ng thng y x nờn h s gúc ca tip tuyn l k Gi ( x ; y0 ) y ( x0 ) k l to ca tip im x ( x0 1)2 16 ( x0 1) x0 2 1 x 3 x0 y0 PTTT : y x Vi x0 y0 PTTT : y Vi Cõu Vit phng trỡnh tip tuyn ca th hm s y x bit tip tuyn song song vi ng thng y x y x y ( x 0) x Vỡ tip tuyn song song vi ng thng y x nờn tip tuyn cú h s gúc k=4 x0 Gi ( x0 ; y0 ) l to ca tip y ( x0 ) x0 x Vi x0 y0 PTTT : y x Vi x0 y0 PTTT : y x Cõu Cho hm s: y x3 3x x Vit phng trỡnh tip tuyn ca th hm s, bit tip tuyn ú song song vi ng thng d: x y 50 y x x x y x x Vỡ tip tuyn song song vi ng thng d: x y 50 nờn tip tuyn cú h s gúc k = ( x ; y0 ) Gi l to ca tip im Ta cú: x02 x0 x02 x0 x0 NGUYN VN LC 0933.168.309 SP Toỏn K35 - H Cn Th Hm s FB: http://www.facebook.com/VanLuc168 Khi ú y0 phng trỡnh tip tuyn l y ( x 1) y x Cõu 10 Cho hm s y x3 3x 2 cú th l C Vit phng trỡnh tip tuyn ca C , bit tip tuyn song song vi ng thng ( ) : y x Ta cú: y ' 3x x Do tip tuyn song song vi ng thng ( ) nờn h s gúc ca tip tuyn l k Gi M ( x0 ; y0 ) (C ) l tip im ca tip tuyn vi (C) H s gúc ca tip tuyn k y '( x0 ) 9 3x02 x0 x0 x0 Vi x0 Vi x0 y0 y0 : M ( 1; 2) pttt: y x : M (3; 2) pttt: y x 25 Vy cú hai tip tuyn tha Cõu l y x v y x 25 x x Cõu 11 Cho hm s y 2 cú th l C Vit phng trỡnh tip tuyn ca C , bit tip tuyn vuụng gúc vi ng thng ( ) : y Ta cú: y ' x x 2 Do tip tuyn vuụng gúc vi ng thng ( ) nờn h s gúc ca tip tuyn l k Gi M ( x0 ; y0 ) (C ) l tip im ca tip tuyn vi (C) H s gúc ca tip tuyn k y '( x0 ) 1 Vi x0 Vi x0 y0 y0 x0 x0 x0 x0 2 x0 : M (0; 1) pttt: y x : M ( 4;3) pttt: y x v y x Vy cú hai tip tuyn tha Cõu l y NGUYN VN LC 0933.168.309 x0 x SP Toỏn K35 - H Cn Th Hm s FB: http://www.facebook.com/VanLuc168 Cõu 12 Vit phng trỡnh tip tuyn ca th hm s y x3 3x2 (C) Bit tip tuyn ú song song vi ng thng y = 9x + Ta cú: y ' 3x2 x Gi M ( x0 ; y0 ) l tip im Tip tuyn ti M cú h s gúc k f ' ( x0 ) 3x02 x0 Theo gi thit, tip tuyn ú song song vi ng thng y = 9x + +6 tip tuyn x0 M (1; 3) x0 M (3;1) cú h s gúc k = 3x02 x0 x02 x0 Phng trinh tip tuyn ca (C) ti M(-1;-3) l: y 9( x 1) y x (loi) Phng trinh tip tuyn ca (C) ti M(3;1) l: y 9( x 3) y x 26 Cõu 13 Cho hm s y x3 3x (C) Vit phng trỡnh tip tuyn ca (C) bit tip tuyn ú vuụng gúc vi ng thng y x Ta cú y ' 3x2 Do tip tuyn ca (C) bit tip tuyn ú vuụng gúc vi ng x nờn h s gúc ca tip tuyn k = 9 Do ú y ' k 3x x x thng y +) Vi x = y Pttt ti im cú honh x = l: y 9( x 2) y x 14 +) Vi x y Pttt ti im cú honh x = - l: y 9( x 2) y x 18 Vy cú hai tip tuyn c (C) vuụng gúc vi ng thng y x l: y =9x - 14 v y = 9x + 18 Tip tuyn vuụng gúc vi ng thng d Cõu 14 Cho hm s y x x (C) Vit phng trỡnh tip tuyn ca (C) vuụng gúc vi d: x y d: x y cú h s gúc kd Tip tuyn cú h s gúc k Gi ( x0 ; y0 ) l to ca tip im Ta cú: y ( x0 ) x03 x0 x0 ( y0 ) PTTT: y 2( x 1) y x NGUYN VN LC 0933.168.309 SP Toỏn K35 - H Cn Th Hm s FB: http://www.facebook.com/VanLuc168 Cõu 15 Vit phng trỡnh tip tuyn ca th hm s y x3 3x vuụng gúc vi ng thng d: y x (C) : y x 3x y 3x x Tip tuyn vuụng gúc vi d: y x Tip tuyn cú h s gúc k Gi ( x0 ; y0 ) l to ca tip im x0 x0 Ta cú: y ( x0 ) 3x02 x0 x02 x0 Vi x0 y0 PTTT: y x Vi x0 y0 PTTT: y x 25 Cõu 16 Cho ng cong (C): y x3 3x Vit phng trỡnh tip tuyn ca (C), bit tip tuyn vuụng gúc ng thng y x y x 3x y ' 3x x Vỡ tip tuyn vuụng gúc vi ng thng y x nờn tip tuyn cú h s gúc l k = x Gi ( x0 ; y0 ) l to ca tip im x02 x0 x02 x0 x0 Vi x0 y0 PTTT: y x y 3x Vi x0 y0 PTTT: y x y 3x Cõu 17 Cho hm s y x2 x x Vit phng trỡnh tip tuyn ca th hm s bit tip tuyn vuụng gúc vi ng thng y x TX: R \ Cú f '( x) x 2x ( x 1) Phng trỡnh tip tuyn ca th hm s ti M ( x0 ; f ( x0 )) l: ng thng y f ' ( x0 )( x x0 ) f ( x0 ) 4 y x cú h s gúc k= 3 Vỡ tip tuyn vuụng gúc vi ng thng y NGUYN VN LC 0933.168.309 x nờn f '( x0 ) 3 SP Toỏn K35 - H Cn Th Hm s FB: http://www.facebook.com/VanLuc168 x0 x 2x x02 x0 3x02 x0 x02 x0 ( x0 1) x0 3 3 Vi x0 ta cú f (1) tip tuyn y x 4 Vi x0 ta cú f (3) tip tuyn y x 4 3 Vy cú hai tip tuyn tha Cõu: y x v y x 4 4 Cõu 18 Cho hm s y x3 x 3x Lp phng trỡnh ng thng i qua im cc i ca th (C) v vuụng gúc vi tip tuyn ca th (C) ti gc ta + im Cc i ca ( C ) l M(1;4/3) +T.T ca ( C ) ti gc to cú h s gúc k= y(0)=3 +ng thng cn tỡm i qua im M v cú h s gúc k= -1/3 nờn cú pt: y= - 1/3(x-1)+4/3=-1/3x+5/3 Phng trỡnh tip tuyn dng c bit Cõu 19 Cho hm s : y x3 x x Tỡm trờn trc honh nhng im m t ú k c cỏc tip tuyn vi (C), cho ú cú hai tip tuyn vuụng gúc M a;0 l im cn tỡm.Tip tuyn ca (C) k t M l ng thng t : y k x a k tha: x x x x 12 x x a x x x k x a 2 x 12 x k x 12 x k x x x 3ax 3a x 3ax 3a * Lp lun i n (*) cú hai nghim phõn bit x1 , x2 : k x1 k x2 9a 24a 82 82 Vy M ;0 a 27 27 27a 81 Cõu 20 Cho hm s y mx , Cm xm Gi I l giao im hai ng tim cn ca th Cm Tip tuyn ti im bt kỡ ca Cm ct tim cn ng v tim cn ngang ln lt ti A v B Tỡm m din tớch tam giỏc IAB bng 12 Vi mi m , th hm s cú tim cn ng I m; m x m , tim cn ngang y m , m2 Gi s M x0 ; m Cm , phng trỡnh tip tuyn ti M ca xm NGUYN VN LC 0933.168.309 Cm : SP Toỏn K35 - H Cn Th Hm s y FB: http://www.facebook.com/VanLuc168 m x0 m Tỡm x x0 m c m , x0 m x0 m 2m A m; m , x m B x0 m; m , t ú suy m2 IA , x0 m IB x0 m S IAB IA IB m 12 m 2x Vit phng trỡnh tip tuyn ca (C), bit khong x cỏch t im I(1;2) n tip tuyn bng Cõu 21 Cho hm s y *Tip tuyn ca (C) ti im M (x ; f (x )) (C ) cú phng trỡnh y f '(x )(x x ) f (x ) Hay x (x 1)2 y 2x 02 2x (*) *Khong cỏch t im I(1;2) n tip tuyn (*) bng 2x (x 1) gii c nghim x v x *Cỏc tip tuyn cn tỡm : x y v x y x2 (C) Cho im A(0;a) Tỡm a t A k c hai tip x tuyn ti (C) cho hai tip im tng ng nm v hai phớa trc honh Cõu 22 Cho hm s y k: , PT ng thng d qua A v cú hsg k cú dng: d tip xỳc vi (C) h pt sau cú nghim Thay (2) vo (1) ta c: t qua A k c tip tuyn Theo viet ta cú: cú nghim phõn bit v tip im nm v phớa ca trc honh NGUYN VN LC 0933.168.309 SP Toỏn K35 - H Cn Th Hm s FB: http://www.facebook.com/VanLuc168 T (*) Kt hp vi iu kin (1) ta c: Vi tha bi toỏn Cõu 23 Cho hm s y x3 x x (1) cú th (C) Chng minh rng trờn (C) khụng th tn ti hai im cú honh ln hn cho hai tip tuyn vi (C) ti hai im ú vuụng gúc vi Gi s trờn (C) cú hai im A( x1 ; y1 ), B( x2 ; y2 ) vi x1, x2 > cho tip tuyn vi (C) ti hai im ny vuụng gúc vi Khi ú, ta cú: y '( x1 ) y '( x2 ) (3x12 12 x1 9)(3x22 12 x2 9) x1 x1 x2 x2 (*) Do x1 > v x2 > nờn VT(*) > Do ú (*) vụ lớ Vy: Trờn (C) khụng th cú hai im cho tip tuyn vi (C) ti hai im ny vuụng gúc vi Cõu 24 Cho hm s y x2 (C) Vit phng trỡnh tip tuyn vi (C) bit rng 2x tip tuyn ct trc honh ti A, trc tung ti B cho tam giỏc OAB vuụng cõn ti O, õy O l gúc ta Ta cú: y ' (2 x 3) Vỡ tip tuyn to vi hai trc ta mt tam giỏc vuụng cõn nờn h s gúc ca tip tuyn l: k Khi ú gi M x0 ; y0 l tip im ca tip tuyn vi th (C) ta cú y ' ( x0 ) x0 (2 x0 3) x0 Vi x0 thỡ y0 lỳc ú tip tuyn cú dng y x (trng hp ny loi vỡ tip tuyn i qua gúc ta , nờn khụng to thnh tam giỏc OAB) Vi x0 thỡ y0 lỳc ú tip tuyn cú dng y x Vy tip tuyn cn tỡm l y x Cõu 25 Cho hm s y = 2x cú th (C) Lp phng trỡnh tip tuyn ca x th (C) cho tip tuyn ny ct cỏc trc Ox, Oy ln lt ti cỏc im A v B tha OA = 4OB NGUYN VN LC 0933.168.309 SP Toỏn K35 - H Cn Th Hm s FB: http://www.facebook.com/VanLuc168 Gi s tip tuyn d ca (C) ti M ( x0 ; y0 ) (C ) ct Ox ti A, Oy ti B cho OA 4OB OB 1 H s gúc ca d bng hoc OA 4 x ( y ) 0 1 H s gúc ca d l y ( x0 ) ( x0 1) ( x0 1) x ( y 5) Do OAB vuụng ti O nờn tan A y ( x 1) y x 4 Khi ú cú tip tuyn tha l: y ( x 3) y x 13 4 NGUYN VN LC 0933.168.309 SP Toỏn K35 - H Cn Th Hm s FB: http://www.facebook.com/VanLuc168 Dng 3: Vit phng trỡnh tip tuyn vi (C): y = f(x) bit tip tuyn i qua im A(xA;yA) y (C ) : y f ( x) A( x A ; y A ) x O : y y A k(x xA ) y k(x xA ) y A Phng phỏp : Ta cú th tin hnh theo cỏc bc sau Bc 1: Vit phng trỡnh tip tuyn (d) vi (C) ti im M0(x0;y0) (C ) (d ) : y f '( x0 )( x x0 ) f ( x0 ) (*) Bc 2: nh x0 (d) i qua im A(xA;yA) Ta cú: (d) i qua im A(xA;yA) y A f '( x0 )( x A x0 ) f ( x0 ) (1) Bc 3: Gii pt (1) tỡm x0 Thay x0 tỡm c vo (*) ta s c pttt cn tỡm Vớ d 1: Cho hm s y x3 3x 2 cú th l C Vit phng trỡnh tip tuyn ca C , bit tip tuyn i qua im A 2; Bi gii Ta cú: y ' 3x x Gi M x0 ; y0 C vi y0 x03 l tip im v 3x02 l tip tuyn vi C ti M0 Phng trỡnh : y y0 y '( x0 )( x i qua im A 2; 2 ( x03 x03 x0 Vi x0 :y y ( x03 x0 ) 3x02 x02 2) 12 x0 2 2x x0 3x02 (3x02 2) (3x02 x0 )(2 x0 )( x x0 ) x0 ) x0 x0 2 NGUYN VN LC 0933.168.309 SP Toỏn K35 - H Cn Th Hm s FB: http://www.facebook.com/VanLuc168 Vi x0 x :y Vy cú hai tip tuyn tha bi l y x x Vớ d 2: Cho hm s y 2 v y x cú th l C Vit phng trỡnh tip tuyn ca C , bit tip tuyn i qua im A 6;5 Bi gii Ta cú: y ' x Gi M x0 ; y0 C Phng trỡnh vi y0 : y y0 y x0 x0 2 x0 x0 l tip im v y '( x0 )( x x0 ) x0 i qua im A 6;5 (x x0 x0 x0 ) 2 x02 x0 x0 x0 Vi x0 :y x Vi x0 :y x 4 x0 2 ( x0 ) Vy cú hai tip tuyn tha bi l y NGUYN VN LC 0933.168.309 l tip tuyn vi C ti M x v y x SP Toỏn K35 - H Cn Th Hm s FB: http://www.facebook.com/VanLuc168 BI TP T LUYN Cõu Cho hm s : y x 2x (C) Vit phng trỡnh tip tuyn vi (C), bit tip tuyn ú i qua giao im ca ng tim cn v trc Ox Giao im ca tim cn ng vi trc Ox l Phng trỡnh tip tuyn () qua A cú dng () tip xỳc vi (C) A ,0 y k x x 1 2x k x / x k coự nghieọm 2x x 1 2x k x (1) k (2 ) 2x 12 Th (2) vo (1) ta cú pt honh tip im l x x 2x 2x 1 (x 1)(2x 1) 3(x ) v x x 2 x Do ú k 12 1 Vy phng trỡnh tip tuyn cn tỡm l: y x 12 Cõu Cho hm s y 2x x (C ) Cho hai im A(1; 0) v B (7; 4) Vit phng trỡnh tip tuyn ca (C ) , bit tip tuyn i qua im trung dim I ca AB Gi qua I 3; cú h s gúc iu kin tip xỳc (C) k : y k ( x 3) 2x k ( x 3) x k ( x 1) Gii h x k Vy phng trỡnh tip tuyn : : y x NGUYN VN LC 0933.168.309 SP Toỏn K35 - H Cn Th Hm s FB: http://www.facebook.com/VanLuc168 Cõu Cho th (C): y x 3x 1, vit phng trỡnh tip tuyn vi (C) bit tip tuyn i qua im A(-2; -1) Ta cú: y ' 3x2 Gi M x0 ; x03 3x0 l tip im H s gúc ca tip tuyn l y '( x0 ) 3x02 Phng trỡnh tip tuyn vi (C) ti M l : y x03 3x0 (3x02 3)( x x0 ) qua A(-2;-1) nờn ta cú: x03 3x0 (3x02 3)(2 x0 ) x03 3x02 x0 y0 ( x0 1)( x02 x0 4) x0 y0 Vy cú hai tip tuyn cn tỡm cú phng trỡnh l: : y 1; : y x 17 NGUYN VN LC 0933.168.309 SP Toỏn K35 - H Cn Th