TÓM TẮT LÝ THUYẾT và GIẢI NHANH TOÁN 12

Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn TểM TT Lí THUYT V GII NHANH TON 12 http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Nguyn Chin - Nguyn Hng Quõn PHặN HM S S NG BIN NGHCH BIN CA HM S nh nghùa x1, x K , x x ( K l khoõng hoc on hoc na khoõng) f x1 f x y f x ng bin trờn K th i lờn t trỏi sang phõi f x1 f x y f x nghch bin trờn K th i xung t trỏi sang phõi Chỳ ý: + N u f x 0, x a;b hm s f x ng bi n tr n khoõng a;b + N u f x 0, x a; b hm s f x nghch bi n trờn khoõng a;b h ng i trờn khoõng a;b + N u f x ng bi n trờn khoõng a;b f x 0, x a;b + Nu f x nghch bi n trờn khoõng a;b f x 0, x a;b + N u f x 0, x a;b hm s f x http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Quy tc v cụng thc tớnh ọo hm Quy tc tớnh o hm: Cho u u x ; v v x ; C : l hỡng s Tng, hiu: u v Tớch: u.v u v u .v v .u C u C u u u .v v .u C C u , v v2 u2 v u o hm hm hp: Nu y f u , u u x yx yu ux Thng: http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Bõng cụng thc tớnh ọo hm: ọo hm ca hm s cỗp C x (C l hỡng s) x (x 0) x x x x x sin x cos x ọo hm ca hm hp x u x u u u u u u u u u0 u cos x sin u sin x cos u Nguyn Chin - Hng Quõn: 0973.514.674 u .cos u u .sin u Page | www.facebook.com/tailieupro ng ký thnh viờn nhn ti liu thng xuyờn Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn tan u cosu tan x cos1 x http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ cot x ex ax sin2 x cot u ex e u u .e u a x ln a au ln x log a x x ln u x ln a u u sin2 u u .a u ln a loga u u u u u.ln a Cụng thc tớnh nhanh o hm hm phõn thc: a b ax b ad bc ; cx d cx d x2 a c x d f ax bx c d e dx ex f dx ex f b c e f http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ o hm cp : + nh nghùa: f x f x + í nghùa c hc: Gia tc tc thi cỷa chuyn ng s f t tọi thi im t l: a t0 f t0 * Mt s chỳ ý: Nu hm s f x v g x cựng ng bin (nghch bin) tr n K thỡ hm s f x g x cỹng ng bin (nghch bin) tr n K Tớnh chỗt ny cũ th kh ng ỳng i vi hiu f x g x http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Nu hm s f x v g x l cỏc hm s dỵng v cựng ng bin (nghch bin) tr n K thỡ hm s f x g x cỹng ng bin (nghch bin) tr n K Tớnh chỗt ny cũ th kh ng ỳng cỏc hm s f x , g x kh ng l cỏc hm s dỵng trờn K Cho hm s u u x , xỏc nh vi x a;b v u x c;d Hm s f u x cỹng xỏc nh vi x a;b Quy tc xột tớnh n iu ca hm s Giõ s hm s f cũ ọo hm trờn K Nu f ' x vi mi x K v f ' x chợ tọi mt s hu họn im x K thỡ hm s f ng bin trờn K Nu f ' x vi mi x K v f ' x chợ tọi mt s hu họn im x K thỡ hm s f nghch bin trờn K Nguyn Chin - Hng Quõn: 0973.514.674 Page | www.facebook.com/tailieupro ng ký thnh viờn nhn ti liu thng xuyờn Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn Chỳ ý: http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ * i vi hm phõn thc hu tợ y ax b d x thỡ dỗu " " xột dỗu ọo cx d c hm y khụng xõy Giõ s y f x ax bx cx d f x 3ax 2bx c Hm s ng bin trờn f x 0; x Hm s nghch bin trờn a a b c f x 0; x a a b c Trỵng hp thỡ h s c khỏc vỡ a b c thỡ f x d (ỵng thợng song song hoc trựng vi trýc Ox thỡ kh ng n iu) * Vi dng toỏn tỡm tham s m hm s c a n iu mt chiu trờn khoõng cũ di bng l ta giõi nh sau: http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Bỵc 1: Tớnh y f x ; m ax bx c Bỵc 2: Hm s n iu trờn a x1; x y cú nghim phõn bit * Bỵc 3: Hm s n iu trờn khoõng cũ di bỡng l x1 x l x1 x 2 4x1x l S2 4P l ** Bỵc 4: Giõi * v giao vi * * suy giỏ tr m cổn tỡm http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ CC TR HM S nh nghùa Giõ s hm s f xỏc nh tr n tờp K v x K + x0 l im cc tiu cỷa hm s f nu tn tọi mt khoõng a; b cha x cho a; b K v f x f x , x a;b \ x Khi ũ f x0 ỵc gi l giỏ tr cc tiu cỷa hm s f + x l im cc ọi cỷa hm s f nu tn tọi mt khoõng a;b cha x cho a; b K v f x f x , x a;b \ x Khi ũ f x ỵc gi l giỏ tr cc ọi cỷa hm s f + im cc ọi v im cc tiu gi chung l im cc tr + Giỏ tr cc ọi v giỏ tr cc tiu gi chung l cc tr + im cc ọi v im cc tiu ỵc gi chung l im cc tr ca hm s v im cc tr phõi l mt im tờp hp K Nguyn Chin - Hng Quõn: 0973.514.674 Page | www.facebook.com/tailieupro ng ký thnh viờn nhn ti liu thng xuyờn Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn + Giỏ tr cc ọi v giỏ tr cc tiu ỵc gi chung l giỏ tr cc tr (hay cc tr) ca hm s http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ + Nu x0 l im cc tr cỷa hm s thỡ im x ; f (x ) ỵc gi l im cc tr ca th hm s f iu kin cổn hm s ọt cc tr nh lớ 1: Giõ s hm s y f x ọt cc tr tọi im x Khi ũ, nu y f x cũ ọo hm tọi im x thỡ f x Chỳ ý: ọo hm f x cú th bỡng tọi im x0 nhỵng hm s f kh ng ọt cc tr tọi im x0 Hm s cú th ọt cc tr tọi mt im m tọi ũ hm s kh ng cũ ọo hm Hm s chợ cú th ọt cc tr tọi mt im m tọi ũ ọo hm cỷa hm s bỡng hoc tọi ũ hm s kh ng cũ ọo hm iu in hm s ọt cc tr nh lớ 2: Giõ s hm s f ọt cc tr tọi im x Khi ũ, nu hm s f cũ ọo hm tọi http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ im x thỡ f ' x0 N u f x tr n khoõng x h; x v f x trờn khoõng x ; x h thỡ x l m t i m cỵc cỷa hm s f x N u f x trờn khoõng x h; x v f x trờn khoõng x0 ; x0 h thỡ x l m t i m cỵc ti u cỷa hm s f x Quy tc tỡm cc tr Quy tc 1: Bc 1: Tỡm tờp xỏc nh Tỡm f x Bc 2: Tỡm cỏc im x i i 1;2; m tọi ũ o hm ca hm s bng hoc hm s liờn tc nhng khụng cũ o hm http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Bc 3: Lờp bõng bin thiờn hoc bõng xột dỗu f x Nu f x i du i qua x i thỡ hm s ọt cc tr tọi x i nh lớ 3: Giõ s y f x cú ọo hm cồ p khoõng x h; x h vi h Nu f x 0, f x thỡ hm s f ọt cc ọi tọi x Nu f x0 0, f x thỡ hm s f ọt cc tiu tọi x T nh lớ trờn, ta cũ mt quy tc khỏc tỡm cc tr ca hm s Quy tc 2: Bc 1: Tỡm tờp xỏc nh Tỡm f x Bc 2: Tỡm cỏc nghim x i i 1;2; cỷa phỵng trỡnh f x Bc 3: Tớnh f x v tớnh f x i Nu f x i thỡ hm s f ọt cc ọi tọi im x i Nu f x i thỡ hm s f ọt cc tiu tọi im xi Nguyn Chin - Hng Quõn: 0973.514.674 Page | www.facebook.com/tailieupro ng ký thnh viờn nhn ti liu thng xuyờn Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn MT S DNG TON LIấN QUAN N CC TR HM S http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ I CC TR CA HM A THC BC BA: Tỡm iu kin hm s cú cc i, cc tiu tha honh cho trc i to n t ng quat: Cho hm s y f x ; m ax bx cx d Tỡm tham s m hm s cú cc ọi, cc tiu tọi x 1, x thúa iu kin K cho trỵc Phng ph p: c 1: Tờp xỏc nh: D 2 ọo hm: y 3ax 2bx c Ax Bx C c 2: Hm s cú cc tr (hay cú hai cc tr, hai cc tr phõn bit hay cú cc ọi v cc tiu) y cú hai nghim phõn bit v y i dỗu qua nghim ũ phỵng trỡnh y cú hai nghim phõn bit A 3a a m D1 y B 4AC 4b 12ac b 3ac c 3: Gi x 1, x l hai nghim cỷa phỵng trỡnh y http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ B 2b x x A 3a Khi ũ: C c x x A 3a c 4: Bi n i i u ki n K v da ng t ng S v ti ch P T ú giõi tỡm ỵc m D2 c 5: K t luồn cỏc giỏ tr m thúa món: m D1 D2 * Chỳ ý: Hm s bờc ba: y ax bx cx d a Ta cú: y ' 3ax 2bx c iu kin Kt lun Hm s kh ng cũ cc tr Hm s cũ hai im cc tr http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ b 3ac b 3ac iu kin hm s cú cc tr cựng du, trỏi du Hm s cú cc tr trỏi du phỵng trỡnh y cú hai nghim phõn bit trỏi dỗu ac Hm s cú hai cc tr cựng du y phỵng trỡnh y cú hai nghim phõn bit cựng dỗu C P x 1.x A Hm s cú hai cc tr cựng du dng y B phỵng trỡnh y cú hai nghim dỵng phồn bit S x x A C P x x A Nguyn Chin - Hng Quõn: 0973.514.674 Page | www.facebook.com/tailieupro ng ký thnh viờn nhn ti liu thng xuyờn Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn Hm s cú hai cc tr cựng du õm http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ y ' B phỵng trỡnh y cú hai nghim õm phõn bit S x x A C P x x A Tỡm iu kin hm s cú hai cc tr x 1, x tha món: x1 x x1 x x1 x Hai cc tr x 1, x thúa x1 x x1 x x1.x x1 x Hai cc tr x 1, x thúa x1 x x x2 x x x1 x x x x1 x Hai cc tr x 1, x thúa x1 x http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ x x2 x x x1 x x x x1 x Phỵng trỡnh bờc cú nghim lờp thnh cỗp s cng cú nghim l x b d , cú nghim lờp thnh cỗp s nhõn cú nghim l x 3a a Tỡm iu kin th hm s cú c c im cc i, cc tiu nm cựng phớa, khỏc phớa so vi mt ng thng http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ i tri tng i gia iờm vi ng th ng: Cho i m A x A; yA , B x B ; yB v ỵng thởng : ax by c N u ax A byA c ax B byB c thi hai im A, B nởm v hai phớa so vi ỵng thởng N u ax A byA c ax B byB c thi hai im A, B nởm cu ng phớa so vi ỵng thợng Mt s trng hp c biờt: + Cỏc im cc tr cỷa th nỡm cựng v phớa i vi trc Oy hm s cú cc tr cựng dỗu phỵng trỡnh y cú hai nghim phõn bit cựng dỗu + Cỏc im cc tr cỷa th nỡm cựng v phớa i vi trc Oy hm s cú cc tr trỏi dỗu phỵng trỡnh y cú hai nghim trỏi dỗu + Cỏc im cc tr cỷa th nỡm cựng v phớa i vi trc Ox phỵng trỡnh y cú hai nghim phõn bit v yC yCT Nguyn Chin - Hng Quõn: 0973.514.674 Page | www.facebook.com/tailieupro ng ký thnh viờn nhn ti liu thng xuyờn Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn c bit: + Cỏc im cc tr cỷa th nỡm cựng v phớa trờn i vi trc Ox http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ y y phỵng trỡnh y cú hai nghim phõn bit v C CT yC yCT Cỏc im cc tr cỷa th nỡm cựng v phớa di i vi trc Ox y y phỵng trỡnh y cú hai nghim phõn bit v C CT yC yCT + Cỏc im cc tr cỷa th nỡm v phớa i vi trc Ox phỵng trỡnh y cú hai nghim phõn bit v yC yCT (ỏp dung khụng nh m c nghiờm v vit c phng trỡnh ng thng i qua hai im cc tr ca th hm s) Hoc: Cỏc im cc tr cỷa th nỡm v phớa i vi trc Ox th cớt trýc Ox tọi im phõn bit phỵng tri nh honh giao i m f x co nghi m phõn bi t (ỏp dung nh m c nghiờm) Phng trỡnh ng thng qua c c im cc tr http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ 2c 2b y.y y .y bc hoc g x 9ay hoc g x y g x x d 3y 9a 9a Khoõng cỏch gia hai im cc tr ca th hm s c l AB b 3ac 4e 16e vi e a 9a II CC TR CA HM BC TRNG PHNG y ax bx c a0 MT S KT QU CặN NH Hm s cú mt cc tr ab Hm s cú ba cc tr ab a b a Hm s cũ ỳng mt cc tr v cc tr l cc ọi b a Hm s cú hai cc tiu v mt cc ọi b a Hm s cú mt cc tiu v hai cc ọi b http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Hm s cũ ỳng mt cc tr v cc tr l cc tiu Giõ s hm s y ax bx c cú cc tr: A(0;c), B b b ; ,C ; 2a 4a 2a 4a tọo thnh tam giỏc ABC thúa d kin: ab Nguyn Chin - Hng Quõn: 0973.514.674 Page | www.facebook.com/tailieupro ng ký thnh viờn nhn ti liu thng xuyờn Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn MT S CễNG THC GII NHANH http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ y Tng quỏt: b cot 8a A O x B C Cụng thc tha ab D kin Tam gi{c ABC vuụng c}n ti A b 8a b 24a 32a (S )2 b Tam gi{c ABC u Tam gi{c ABC cú din tớch S ABC S Tam gi{c ABC cú din tớch max (S ) S0 Tam gi{c ABC cú b{n kớnh ng trũn ni tip b5 32a b2 http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ rABC r0 Tam gi{c ABC cú b{n kớnh ng trũn ngoi tip r b3 a 8a b 8a RABC R R Tam gi{c ABC cú d|i cnh BC m0 am02 2b Tam gi{c ABC cú d|i AB AC n0 16a 2n02 b 8ab Tam gi{c ABC cú cc tr B,C Ox Tam gi{c ABC cú gúc nhn 8ab b 4ac b(8a b ) Tam gi{c ABC cú trng t}m O Tam gi{c ABC cú trc t}m O b 6ac b 8a 4ac b 2ac b 8a 4abc b 8a 8abc b k 8a(k 4) http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Tam gi{c ABC cựng im O to th|nh hỡnh thoi Tam gi{c ABC cú O l| t}m ng trũn ni tip Tam gi{c ABC cú O l| t}m ng trũn ngoi tip Tam gi{c ABC cú cnh BC kAB kAC Trc ho|nh chia tam gi{c ABC th|nh hai phn cú din tớch bng b ac Tam giỏc ABC cũ im cc tr cỏch u trýc honh b 8ac th hm s C : y ax bx c cớt trýc Ox tọi im phồn bit lờp thnh cỗp s cng nh tham s hỡnh phợng gii họn bi th C : y ax bx c v trýc honh cũ din tớch phổn tr n v phổn dỵi bỡng b2 100 ac b2 36 ac c y c b 4a b 4a 2 Phỵng trỡnh ỵng trủn ngoọi tip ABC : x y Nguyn Chin - Hng Quõn: 0973.514.674 Page | www.facebook.com/tailieupro ng ký thnh viờn nhn ti liu thng xuyờn Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn GI TR LN NHầT - GI TR NH NHầT http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ I nh nghùa Cho hm s y f x xỏc nh trờn tờp D f (x ) M , x D x D, f (x ) M S M gi l giỏ tr ln nht cỷa hm s y f x trờn D nu: Kớ hiu: M max f ( x) xD f (x ) m, x D x D, f (x ) m S m gi l giỏ tr nh nht cỷa hm s y f x trờn D nu: Kớ hiu: m f (x ) x D Phng phỏp tỡm GTLN,GTNN * Tỡm GTLN, GTNN ca hm s bng cỏch khõo sỏt trc tip Bc 1: Tớnh f x v tỡm cỏc im x1, x 2, , x n D m tọi ũ f x hoc hm s kh ng cũ ọo hm + Bc 2: Lờp bõng bin thi n v ri suy giỏ tr ln nhỗt, giỏ tr nhú nhỗt cỷa hm s * Tỡm GTLN, GTNN ca hm s tr n mt oọn Bc 1: http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Hm s ó cho y f x xỏc nh v liờn týc tr n oọn a;b Tỡm cỏc im x1, x 2, , x n trờn khoõng a;b , tọi ũ f x hoc f x kh ng xỏc nh Bc 2: Tớnh f a , f x1 , f x , , f x n , f b Bc 3: Khi ũ: max f x max f x , f x , , f x n , f a , f b a ,b f x f x , f x , , f x n , f a , f b a ,b http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ * Tỡm GTLN, GTNN ca hm s tr n mt hoõng Bc 1: Tớnh ọo hm f (x ) Bc 2: Tỡm tỗt cõ cỏc nghim x i (a;b) cỷa phỵng trỡnh f (x ) v tỗt cõ cỏc im i (a;b) lm cho f (x ) kh ng xỏc nh Bc Tớnh A lim f (x ) , B lim f (x ) , f (x i ) , f (i ) x a Bc x b So sỏnh cỏc giỏ tr tớnh ỵc v kt luờn M max f (x ) , m f (x ) (a ;b ) (a ;b ) Nu giỏ tr ln nht (nhú nht) l A hoc B thỡ kt lun khụng cũ giỏ tr ln nht (nhú nht) f x f a a ;b + N u y f x ng bi n trờn a;b thỡ f x f b max a ;b f (x ) f b a ;b + N u y f x nghich bi n trờn a;b thỡ f (x ) f a max a ;b Nguyn Chin - Hng Quõn: 0973.514.674 Page | www.facebook.com/tailieupro ng ký thnh viờn nhn ti liu thng xuyờn Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn NG TIM CN CA TH HM S http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ ng tim cn ngang Cho hm s y f (x ) xỏc nh trờn mt khoõng vụ họn (l khoõng dọng a; , ;b hoc ; ) ỵng thợng y y0 l ỵng tim cn ngang (hay tim cờn ngang) cỷa th hm s y f (x ) nu ớt nhỗt mt cỏc iu kin sau thúa món: lim f (x ) y0, lim f (x ) y0 x x ng tim cn ng ỵng thợng x x ỵc gi l ỵng tim cn ng (hay tim cờn ng) cỷa th hm s y f ( x) nu ớt nhỗt mt cỏc iu kin sau ỵc thúa món: lim f (x ) , lim f (x ) , lim f ( x) , lim f ( x) x x x x Lu ý: Vi th hm phồn thc dọng y ngang l y x x0 ax b cx d x x0 c 0; ad bc lu n cũ tim cờn a d v tim cờn ng x c c http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ KHO ST S BIN THIấN V V TH HM S S hõo sỏt hm s Cho hm s y f x Tỡm xỏc nh ca hm s S bin thi n Chiu bin thi n i Tớnh y ' ii Tỡm cỏc nghim cỷa phỵng trỡnh y ' v cỏc im tọi ũ y ' khụng xỏc nh iii Xột dỗu y ' v suy cỏc khoõng bin thi n cỷa hm s Tỡm cc tr (nu cũ) Tỡm cỏc gii v cc; cỏc gii họn tọi , v tọi cỏc im m hm s http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ kh ng xỏc nh Tỡm cỏc ỵng tim cờn cỷa hm s (nu cũ) Lờp bõng bin thi n th Lit k cỏc im c bit ( im cc ọi, im cc tiu, tồm i xng,) Xỏc nh giao im cỷa (C) vi Ox, Oy (nu cũ) V th Nguyn Chin - Hng Quõn: 0973.514.674 Page | 10 www.facebook.com/tailieupro ng ký thnh viờn nhn ti liu thng xuyờn Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn MT PHNG http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ n kh{c v| cú gi{ vuụng gúc mp(P) c gi l| vộc t ph{p tuyn ca (P) (k 0) cng l| vộc t ph{p tuyn ca (P) Nu n l| vộc t ph{p tuyn ca (P) thỡ kn Phng trỡnh tng quỏt ca mp(P): qua M (x ; y0 ; z ) cú vộc t ph{p tuyn n (A; B;C ) l|: A(x x ) B(y y0 ) C (z z ) Khai trin ca phng trỡnh tng quỏt: Ax By Cz D (A,B,C khụng ng thi bng 0) Nhng trng hp ri ng ca phng trỡnh tng quỏt: (P) qua gc ta D=0 (P) song song hoc trựng (Oxy) A=B=0 (P) song song hoc trựng (Oyz) B=C=0 (P) song song hoc trựng (Ozx) A=C=0 (P) song song hoc cha Ox A=0 (P) song song hoc cha Oy B=0 (P) song song hoc cha Oz C=0 (P) ct Ox ti A(a;0;0), ct Oy ti B(0;b;0) v| ct Oz ti C(0;0;c) http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ (P) cú phng trỡnh x y z a b c Khong cỏch t im n mt phng: Cho M x ; y0 ; z v| (P ) : Ax By Cz D ; d(M ,(P )) Ax By0 Cz0 D A2 B C Chựm mt phng Tp hp tt c cc mt phng qua giao tuyn ca hai mt phng v| ( ) c gi l| mt chựm mt phng Gi d l| giao tuyn ca hai mt phng d : A x B y C z D v| : A x B y C z D Khi ú nu P l| mt phng cha d thỡ mt phng P cú 1 1 2 2 http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ P dng P : m.(A1x B1y C1z D1 ) n.(A2x B2y C 2z D2 ) 0, m n C[C DNG TO[N THNG GP VN 1: Vit phng trỡnh mt phng lp phng trỡnh mt ph ng ta cn xỏc nh mt im thuc v mt VTPT ca nú Dng 1: i qua im M x ; y0 ; z cú TPT n A; B;C : : A x x B y y0 C z z Dng 2: i qua im M x ; y0 ; z cú cp TCP a , b : Khi ú mt VTPT ca l n a ,b Dng 3: i qua im M x ; y0 ; z Nguyn Chin - Hng Quõn: 0973.514.674 v| song song vi mt Page | 68 www.facebook.com/tailieupro ng ký thnh viờn nhn ti liu thng xuyờn phng Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn : Ax By Cz D 0: http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ : A x x B y y0 C z z Dng 4: i qua im khụng thng h|ng A, B,C Khi ú ta cú th xỏc nh mt VTPT ca l: n AB, AC Dng 5: i qua mt im M v| mt ng thng d khụng cha M : Trờn d ly im A v VTCP u Mt VTPT ca l: n AM , u Dng 6: i qua mt im M , vuụng gúc vi ng thng d : VTCP u ca ng th ng d l mt VTPT ca Dng 7: i qua ng thng ct d1, d2 : Xỏc nh cỏc VTCP a , b ca cỏc ng th ng d1, d2 Mt VTPT ca l: n a ,b http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Ly mt im M thuc d1 hoc d2 M Dng 8: cha ng thng d1 v| song song vi ng thng d2 ( d1, d2 chộo ) : Xỏc nh cỏc VTCP a , b ca cỏc ng th ng d1, d2 Mt VTPT ca l: n a ,b Ly mt im M thuc d1 M Dng 9: i qua im M v| song song vi hai ng thng chộo d1, d2 : Xỏc nh cỏc VTCP a , b ca cỏc ng th ng d1, d2 Mt VTPT ca l: n a ,b http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Dng 10: i qua mt ng thng d v| vuụng gúc vi mt mt phng : Xỏc nh VTCP u ca d v VTPT n ca Mt VTPT ca l: n u, n Ly mt im M thuc d M Dng 11: i qua im M v| vuụng gúc vi hai mt phng ct , : Xỏc nh cỏc VTPT n , n ca v Mt VTPT ca Dng 12: l: n u , n i qua ng thng d cho trc v| c{ch im M cho trc mt khong k cho trc: Gi s () cú phng trỡnh: Ax By Cz+D A2 B C Ly im A, B d A, B Nguyn Chin - Hng Quõn: 0973.514.674 ( ta c hai phng trỡnh , ) Page | 69 www.facebook.com/tailieupro ng ký thnh viờn nhn ti liu thng xuyờn Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn T iu kin khong cỏch d(M,( )) k , ta c phng trỡnh http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Gii h phng trỡnh , , (bng cỏch cho giỏ tr mt n, tỡm cỏc n cũn li) Dng 13: l| tip xỳc vi mt cu S ti im H : Gi s mt cu S cú tõm I v bỏn kớnh R Mt VTPT ca l: n IH VN 2: V tr tng i ca hai mt phng Cho hai mt phng P : Ax By Cz D v| P : Ax By Cz D Khi ú: P ct P P // P P P A : B : C A : B : C A B C D A B C D A B C D A B C D http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ P P n P n P n P n P AA BB CC VN 3: Khong cỏch t m t im n m t mt phng Khong c{ch gia hai mt phng song song Hỡnh chiu ca mt im tr n mt phng im i xng ca mt im qua mt phng Khong cỏch t im M x ; y0 ; z n mt ph ng ( ) : Ax By Cz D d M 0,( ) Ax By0 Cz D A2 B C Khong cỏch gia hai mt ph ng song song bng khong cỏch t mt im bt kỡ trờn mt ph ng ny n mt ph ng Chỳ ý: Nu hai mt ph ng khụng song song thỡ khong cỏch gia chỳng bng http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ MH , n cung phuong im H l hỡnh chiu ca im M trờn P H (P ) im M ' i xng vi im M qua P MM 2MH VN 4: Gúc gia hai mt phng Cho hai mt ph ng , cú phng trỡnh: : A1x B1y C 1z D1 : A2x B2y C 2z D2 Gúc gia , bng hoc bự vi gúc gia hai VTPT n1, n2 cos ( ),( ) Chỳ ý: 00 ( ),( ) 900 ; n1.n2 n1 n2 A1A2 B1B2 C 1C A12 B12 C 12 A22 B22 C 22 ( ) ( ) A1A2 B1B2 C1C Nguyn Chin - Hng Quõn: 0973.514.674 Page | 70 www.facebook.com/tailieupro ng ký thnh viờn nhn ti liu thng xuyờn Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ VN 5: V tr tng i gia mt phng v mt cu Phng trỡnh mt phng tip xỳc vi mt cu Cho mt ph ng : Ax By Cz D v mt cu S : (x a )2 (y b)2 (z c)2 R2 v S khụng cú im chung tip xỳc vi S d(I ,( )) R d(I ,( )) R l tip din tỡm to tip im ta cú th thc hin nh sau: Vit phng trỡnh ng th ng d i qua tõm I ca S v vuụng gúc vi Tỡm to giao im H ca d v H l tip im ca S vi ct S theo mt ng trũn d(I ,( )) R xỏc nh tõm H v bỏn kớnh r ca ng trũn giao tuyn ta cú th thc hin nh sau: Vit phng trỡnh ng th ng d i qua tõm I ca S v vuụng gúc vi Tỡm to giao im H ca d v H l tõm ca ng trũn giao tuyn ca S vi http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Bỏn kớnh r ca ng trũn giao tuyn: r R2 IH NG THNG I Phng trỡnh ca ng thng: 1) Vect ch phng ca ng thng: énh ngha: Cho ng thng d Nu vect a v| cú gi{ song song hoc trựng vi ng phng d thỡ vect a c gi l| vect ch phng ca ng phng d Kớ hiu: a (a1;a2 ;a3 ) Chỳ : TCP ca d thỡ k a (k 0) cng l| 1) a l| TCP ca d http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ ) Nu d i qua hai im A, B thỡ AB l| mt TCP ca d 3) Trc Ox cú vect ch phng a i (1; 0; 0) 4) Trc Oy cú vect ch phng a j (0;1; 0) 5) Trc Oz cú vect ch phng a k (0; 0;1) 2.Phng trỡnh tham s ca ng thng: Phng trỡnh tham s ca ng thng () i qua im M (x ; y0 ; z ) v| nhn a (a1;a2 ;a3 ) l|m TCP l| : z a ( ) M0 O x x ta M ( x, y, z ) y () : y y ta z z ta t x Phng trỡnh ch nh tc ca ng thng: Nguyn Chin - Hng Quõn: 0973.514.674 Page | 71 www.facebook.com/tailieupro ng ký thnh viờn nhn ti liu thng xuyờn Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn Phng trỡnh chớnh tc ca ng thng () i qua im M (x ; y0 ; z ) v| nhn a (a1;a2 ;a3 ) http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ l|m TCP l| : () : x x0 a1 y y0 a2 z z0 II V tr tng i ca ng thng v mt phng : 1.V tr tng i ca ng thng v mt phng : PP HèNH HC ( ) M a n a3 a ( ) n n M a M a a a ( ) x x a t (1) nh l : Trong Kg Oxyz cho: ng thng () : y y a2t (2) cú TCP a (a1;a2 ;a3 ) z z a t (3) v| qua M (x ; y0 ; z ) v| mt phng ( ) : Ax By Cz D cú TPT n (A; B;C ) http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Khi ú : () cat ( ) a.n Aa1 Ba2 Ca a.n Aa Ba2 Ca Ax By Cz D M (P ) a.n Aa Ba2 Ca () ( ) Ax By Cz D M (P ) a ( ) ( ) a v| n cựng phng () // ( ) c bit: n a1 : a2 : a3 A : B : C a http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ pt() tỡm x, y, z pt( ) PP I S: Mun tỡm giao im M ca v| ta gii h phng trỡnh: Suy ra: M x, y, z Th , , v|o phng trỡnh mp P v| rỳt gn da v dng: at b (*) d ct mp P ti mt im Pt d song song vi P Pt d nm P Pt * cú mt nghim t * vụ nghim * cú vụ s nghim t d vuụng gúc P a v| n cựng phng V tr tng i ca hai ng thng: PP HèNH HC V tr tng i ca hai ng thng khụng gian Cho hai ng thng: i qua M v| cú mt vect ch phng u1 Nguyn Chin - Hng Quõn: 0973.514.674 Page | 72 www.facebook.com/tailieupro ng ký thnh viờn nhn ti liu thng xuyờn Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn i qua N v| cú mt vect ch phng u2 http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ u1 , u2 u1 , MN // u , u ct u , u MN v| chộo u1 , u2 MN u , u u , MN pt(1 ) tỡm pt(2 ) PP I S: Mun tỡm giao im M ca (1 ) va ( ) ta gii h phng trỡnh : x, y, z Suy ra: M x, y, z 3) V tr tng i gia ng thng v mt cu: x x a t (1) y y a t (2) v| mt cu S : (x a )2 (y b)2 (z c)2 R2 cú Cho ng thng d: z z a t (3) t}m I (a;b;c) , b{n kớnh R http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ PP HèNH HC B Tớnh khong c{ch t t}m I ca mt cu S n ng thng d l| h d (I ,d ) IM a a B So s{nh d(I , d ) vi b{n kớnh R ca mt cu: Nu d(I , d ) R thỡ d khụng ct S Nu d(I , d ) R thỡ d tip xỳc S http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Nu d(I , d ) R thỡ d ct S ti hai im ph}n bit M , N v| MN vuụng gúc vi ng kớnh (b{n kớnh) mt cu PP I S: Th , hai theo t , v|o phng trỡnh S v| rỳt gn a v phng trỡnh bc * Nu phng trỡnh * vụ nghim thỡ d khng ct S Nu phng trỡnh * cú mt nghim thỡ d tip xc S Nu phng trỡnh * cú hai nghim thỡ d ct S ti hai im ph}n bit M , N Chỳ ý: é tỡm ta M , N ta thay gi{ tr t v|o phng trỡnh ng thng d Nguyn Chin - Hng Quõn: 0973.514.674 Page | 73 www.facebook.com/tailieupro ng ký thnh viờn nhn ti liu thng xuyờn Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn n1 ( A1 ; B1 ; C1 ) III Gúc khụng gian: Gúc gia hai mt phng: http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ n2 ( A2 ; B2 ; C ) nh l : Trong Kg Oxyz cho hai mt phng , x{c nh bi phng trỡnh : ( ) : A1x B1y C 1z D1 ( ) : A2x B2y C 2z D2 Gi l| gúc gia hai mt phng ( ) & ( ) ta cú cụng thc: a 0 90 b cos A1A2 B1B2 C 1C A B C A B C 2 2 2 ( ) 2 a (a; b; c) n ( A; B; C ) Gúc gia ng thng v mt phng: Cho ng thng () : x x0 y y0 z z0 a b c a v| mt phng ( ) : Ax By Cz D Gi l| gúc gia hai mt phng () & ( ) ta cú cụng thc: 0 90 http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ sin Aa Bb Cc A2 B C a b c 3.Gúc gia hai ng thng : Cho hai ng thng : (1 ) : (2 ) : x x0 a x x0 a' y y0 b y y0 b' a1 (a; b; c) z z0 c z z0 c' a ( a ' ; b' ; c ' ) http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ 0 90 Gi l| gúc gia hai mt phng (1 ) & (2 ) ta cú cụng thc: cos aa ' bb ' cc ' a b c a '2 b '2 c '2 IV Khong cỏch: Khong cỏch t m t im n m t mt phng: Cho mt phng ( ) : Ax By Cz D v| im M (x ; y0 ; z ) Khong c{ch t im M n mt phng ( ) c tớnh bi : M ( x0 ; y ; z ) d(M ; ) a H Nguyn Chin - Hng Quõn: 0973.514.674 Ax By0 Cz D A2 B C Page | 74 www.facebook.com/tailieupro ng ký thnh viờn nhn ti liu thng xuyờn Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn Khong cỏch t m t im n m t ng thng: http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Cho ng thng () i qua im M (x ; y0 ; z ) v| cú TCP u (a;b;c) Khi ú khong c{ch t im M1 n () c tớnh bi cụng thc: M1 u ( ) d(M 1, ) M M ; u M ( x0 ; y ; z ) H u Khong cỏch gia hai ng thng chộo nhau: nh l : Trong Kg Oxyz cho hai ng thng chộo : (1 ) co VTCP u (a;b; c) va qua M0 (x ; y ; z ) (2 ) co VTCP u ' (a ' ;b ' ; c ' ) va qua M0' (x 0' ; y 0' ; z 0' ) u, u ' M M ' 0 Khi ú khong c{ch gia (1 ) va ( ) c tớnh bi cụng thc d (1, ) u; u ' u M0 http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ M ' u' C[C DNG THNG GP VN 1: L p phng trỡnh ng thng lp phng trỡnh ng th ng d ta cn xỏc nh mt im thuc d v mt VTCP ca nú Dng 1: d i qua im M (x ; y0 ; z ) v| cú TCP a (a1;a2 ;a3 ) : x x a t o (d ) : y yo a2t z z a t o ( t R) http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Dng 2: d i qua hai im A, B : Mt VTCP ca d l AB Dng 3: d i qua im M (x ; y0 ; z ) v| song song vi ng thng cho trc: Vỡ d / / nờn VTCP ca cng l VTCP ca d Dng 4: d i qua im M (x ; y0 ; z ) v| vuụng gúc vi mt phng P cho trc: Vỡ d P nờn VTPT ca P cng l VTCP ca d Dng 5: d l| giao tuyn ca hai mt phng P , Q : Cỏch 1: Tỡm mt im v mt VTCP (P ) (vi vic chn giỏ (Q ) Tỡm to mt im A d : bng cỏch gii h phng trỡnh tr cho mt n) Tỡm mt VTCP ca d : a nP , nQ Cỏch 2: Tỡm hai im A, B thuc d , ri vit phng trỡnh ng th ng i qua hai im ú Dng 6: d i qua im M (x ; y0 ; z ) v| vuụng gúc vi hai ng thng d1, d2 : Nguyn Chin - Hng Quõn: 0973.514.674 Page | 75 www.facebook.com/tailieupro ng ký thnh viờn nhn ti liu thng xuyờn Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn Vỡ d d1, d d2 nờn mt VTCP ca d l: a ad , ad http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Dng 7: d i qua im M (x ; y0 ; z ) , vuụng gúc v| ct ng thng Cỏch 1: Gi H l hỡnh chiu vuụng gúc ca M trờn ng th ng H M 0H u Khi ú ng th ng d l ng th ng i qua M 0, H Cỏch 2: Gi P l mt ph ng i qua A v vuụng gúc vi d ; Q l mt ph ng i qua A v cha d Khi ú d P Q Dng 8: d i qua im M (x ; y0 ; z ) v| ct hai ng thng d1, d2 : Cỏch 1: Gi M1 d1, M2 d2 T iu kin M , M1, M th ng hng ta tỡm c M1, M T ú suy phng trỡnh ng th ng d Cỏch 2: Gi P (M 0, d1 ) , Q (M 0, d2 ) Khi ú d P Q Do ú, mt VTCP ca d cú th chn l a nP , nQ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Dng 9: d nm mt phng P v| ct c hai ng thng d1, d2 : Tỡm cỏc giao im A d1 P , B d2 P Khi úd chớnh l ng th ng AB Dng 10: d song song vi v| ct c hai ng thng d1, d2 : Vit phng trỡnh mt ph ng P cha v d1, mt ph ng Q cha v d2 Khi ú d P Q Dng 11: d l| ng vuụng gúc chung ca hai ng thng d1, d2 chộo nhau: MN d1 Cỏch 1: Gi M1 d1, M2 d2 T iu kin , ta tỡm c M , N Khi ú, d l MN d2 ng th ng MN Cỏch 2: Vỡ d d1 v d d2 nờn mt VTCP ca d cú th l: a ad , ad http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Lp phng trỡnh mt ph ng P cha d v d1, bng cỏch: + Ly mt im A trờn d1 + Mt VTPT ca P cú th l: nP a , ad Tng t lp phng trỡnh mt ph ng Q cha d v d2 Khi ú d P Q Dng 12: d l| hỡnh chiu ca ng thng l n mt phng P : Lp phng trỡnh mt ph ng Q cha v vuụng gúc vi mt ph ng P bng cỏch: Ly M Vỡ Q cha v vuụng gúc vi P nờn nQ a , nP Nguyn Chin - Hng Quõn: 0973.514.674 Page | 76 www.facebook.com/tailieupro ng ký thnh viờn nhn ti liu thng xuyờn Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn Khi ú d P Q http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Dng 13: d i qua im M, vuụng gúc vi d1 v| ct d2 : Cỏch 1: Gi N l giao im ca d v d2 T iu kin MN d1, ta tỡm c N Khi ú, d l ng th ng MN Cỏch 2: Vit phng trỡnh mt ph ng P qua M v vuụng gúc vi d1 Vit phng trỡnh mt ph ng Q cha M v d2 Khi ú d P Q VN 2: V tr tng i gia hai ng thng xột VTT gia hai ng th ng, ta cú th s dng mt cỏc phng phỏp sau: Phng phỏp hỡnh hc: Da vo mi quan h gia cỏc VTCP v cỏc im thuc cỏc ng th ng Phng phỏp i s: Da vo s nghim ca h phng trỡnh cỏc ng th ng http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ VN 3: V tr tng i gia ng thng v mt phng xột VTT gia ng th ng v mt ph ng, ta cú th s dng mt cỏc phng phỏp sau: Phng phỏp hỡnh hc: Da vo mi quan h gia VTCP ca ng th ng v VTPT ca mt ph ng Phng phỏp i s: Da vo s nghim ca h phng trỡnh ng th ng v mt ph ng VN 4: V tr tng i gia ng thng v mt cu xột VTT gia ng th ng v mt cu ta cú th s dng cỏc phng phỏp sau: Phng phỏp hỡnh hc: Da vo khong cỏch t tõm mt cu n ng th ng v bỏn kớnh Phng phỏp i s: Da vo s nghim ca h phng trỡnh ng th ng v mt cu VN 5: Khong cỏch http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Khong cỏch t i m M n ng thng d Cỏch 1: Cho ng th ng d i qua M v cú VTCP a M M , a d(M , d ) a Cỏch 2: Tỡm hỡnh chiu vuụng gúc H ca M trờn ng th ng d d M , d MH Cỏch 3: Gi N x ; y; z d Tớnh MN theo t (t tham s phng trỡnh ng th ng d) Tỡm t MN nh nht Khi ú N H Do ú d M , d MH Khong cỏch gia hai ng thng chộo Cho hai ng th ng chộo d1 v d2 Bit d1 i qua im M1 v cú VTCP a1 , d2 i qua im M Nguyn Chin - Hng Quõn: 0973.514.674 Page | 77 www.facebook.com/tailieupro ng ký thnh viờn nhn ti liu thng xuyờn Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn d(d1, d2 ) a1, a2 M1M a1, a2 http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ v cú VTCP a Chỳ ý: Khong cỏch gia hai ng th ng chộo d1, d2 bng khong cỏch gia d1 vi mt ph ng cha d2 v song song vi d1 Khong cỏch gia hai ng thng song song Khong cỏch gia hai ng th ng song song bng khong cỏch t mt im thuc ng th ng ny n ng th ng Khong cỏch gia mt ng thng v mt mt phng song song Khong cỏch gia ng th ng d vi mt ph ng song song vi nú bng khong cỏch t mt im M bt kỡ trờn d n mt ph ng VN 6: Gúc G c gia hai ng thng Cho hai ng th ng d1, d2 ln lt cú cỏc VTCP a1, a2 Gúc gia d1, d2 bng hoc bự vi gúc gia a1, a2 http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ cos a1, a2 a1.a2 a1 a2 G c gia mt ng thng v mt mt phng Cho ng th ng d cú VTCP a (a1;a2 ;a3 ) v mt ph ng cú VTPT n (A; B;C ) Gúc gia ng th ng d v mt ph ng bng gúc gia ng th ng d vi hỡnh chiu d ' ca nú trờn sin d,( ) Aa1 Ba2 Ca A2 B C a12 a22 a 32 MT CU http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ I Phng trỡnh mt cu: Phng trỡnh ch nh tc: Phng trỡnh ca mt cu S t}m I a;b;c , b{n kớnh R l|: (S ) : (x a )2 (y b)2 (z c)2 R2 Phng trỡnh c gi l| phng trỡnh chớnh tc ca mt cu 2 2 c bit: Khi I O thỡ (C ) : x y z R Phng trỡnh tng quỏt: 2 Phng trỡnh : x y z 2ax 2by 2cz d vi a b2 c d l| phng trỡnh ca mt cu S cú t}m I a;b; c , b{n kớnh R a b c d 2 II Giao ca mt cu v mt phng: Cho mt phng ( ) v| mt cu S cú phng trỡnh : ( ) : Ax By Cz D (S ) : (x a )2 (y b )2 (z c )2 R2 Nguyn Chin - Hng Quõn: 0973.514.674 Page | 78 www.facebook.com/tailieupro ng ký thnh viờn nhn ti liu thng xuyờn Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn Gi d(I ; ) l| khong c{ch t t}m mt cu S n mt phng http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Cho mt cu S I ; R v| mt phng P Gi H l| hỡnh chiu vuụng gúc ca I l n P d IH d I , P dR dR dR Mt phng tip xỳc mt cu Mt phng ct mt cu theo P l| mt phng tip din Mt cu v| mt phng khụng cú im chung thit din l| ng trũn cú ca mt cu v| t}m I v| b{n kớnh H: tip im r R2 IH http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Dng 1: S cú t}m I a;b;c v| b{n kớnh R : S : (x a )2 (y b)2 (z c)2 R2 Dng 2: S cú t}m I a;b;c v| i qua im A : Phng phỏp: Khi ú b{n kớnh R IA Dng 3: S nhn on thng AB cho trc l|m ng kớnh: Phng phỏp: T}m I l| trung im ca on thng AB : x I xA xB ; yI y A yB ; zI zA zB AB Dng 4: S i qua bn im A, B,C , D ( mt cu ngoi tip t din) B{n kớnh R IA http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Phng phỏp: Gi s S cú dng: x y z 2ax 2by 2cz d * Thay ln lt to ca c{c im A, B,C , D v|o * , ta c phng trỡnh Gii h phng trỡnh ú, ta tỡm c a,b, c, d Phng trỡnh mt cu S Dng 5: S i qua ba im A, B,C v| cú t}m I nm tr n mt phng P cho trc: Phng phỏp: Gii tng t nh dng Dng 6: S cú t}m I v| tip xỳc vi mt cu T cho trc: Phng phỏp: {c nh t}m I v| b{n kớnh R ' ca mt cu T S dng iu kin tip xỳc ca hai mt cu tớnh b{n kớnh R ca mt cu S ( ột hai trng hp tip xỳc v| ngo|i) Nguyn Chin - Hng Quõn: 0973.514.674 Page | 79 www.facebook.com/tailieupro ng ký thnh viờn nhn ti liu thng xuyờn Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn Chỳ : i phng trỡnh mt cu S : x y z 2ax 2by 2cz d http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ vi a b2 c d thỡ S cú t}m I a; b; c v| b{n kớnh R a b c d 2 Cho hai mt cu S1 I 1, R1 v| S2 I 2, R2 S1 , S2 I 1I R1 R2 I 1I R1 R2 S1 , S2 ngo|i I 1I R1 R2 S1 , S2 tip xỳc I 1I R1 R2 S1 , S2 tip xỳc ngo|i R1 R2 I 1I R1 R2 S1 , S2 ct theo mt ng trũn Dng 7: it phng trỡnh mt cu S cú t}m I a;b;c , tip xỳc vi mt phng P cho trc Phng phỏp: B{n kớnh mt cu R d I ; P cú t}m I a;b;c , ct mt phng P it phng trỡnh mt cu S Dng 8: cho trc theo giao tuyn l| mt ng trũn tho iu kin a ng trũn cú din tớch cho trc b ng trũn cú chu vi cho trc c ng trũn cú b{n kớnh cho trc Phng phỏp: http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ T cụng thc din tớch ng trũn S r hoc chu vi ng trũn P r ta tỡm c b{n kớnh ng trũn giao tuyn r Tớnh d d I , P Tớnh b{n kớnh mt cu R d r Kt lun phng trỡnh mt cu 2 cú t}m I a;b;c , ct mt phng P it phng trỡnh mt cu S Dng 8: cho trc theo http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ giao tuyn l| mt ng trũn tho iu kin Phng phỏp: Ta cú b{n kớnh mt cu R d I ; P Kt lun phng trỡnh mt cu it phng trỡnh mt cu S Dng 10: tip xỳc vi mt ng thng cho trc v| cú t}m I a;b;c cho trc Phng phỏp ng thng tip xỳc vi mt cu S ta cú R d I, Dng 11: it phng trỡnh mt cu S tip xỳc vi mt ng thng ti tip im M xo , yo , zo thuc v| cú t}m I thuc ng thng d cho trc Phng phỏp it phng trỡnh mt phng P i qua im M v| vuụng gúc vi ng thng To t}m I P l| nghim ca phng trỡnh Nguyn Chin - Hng Quõn: 0973.514.674 Page | 80 www.facebook.com/tailieupro ng ký thnh viờn nhn ti liu thng xuyờn Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn Kt lun v phng trỡnh mt cu S Dng 12: it phng trỡnh mt cu S cú t}m I a;b;c v| ct ng thng B{n kớnh mt cu R IM d I, http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ ti hai im A, B tho iu kin: a d|i AB l| mt hng s b Tam gi{c IAB l| tam gi{c vuụng c Tam gi{c IAB l| tam gi{c u Phng phỏp {c nh d I , IH , vỡ IAB c}n ti I n n HB AB a B{n kớnh mt cu R IH HB IH b B{n kớnh mt cu R sin 45o IH c B{n kớnh mt cu R sin 60o MT S DNG GII NHANH CC TR KHễNG GIAN http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Cho P v hai im A, B Tỡm M P MA MB + Nu A v B trỏi phớa so vi P ? M , A, B thợng hng M AB P + Nu A v B cựng phớa so vi P Tỡm B ' l i xng cỷa B qua P M , A, B ' thợng hng M AB ' P Cho P v hai im A, B Tỡm M P MA MB + Nu A v B cựng phớa so vi P max ? M , A, B thợng hng M AB P Tỡm B ' l i xng cỷa B qua P + Nu A v B trỏi phớa so vi P http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ MA MB ' AB ' Cho im M x M ; yM ; z M kh ng thuc cỏc trýc v mt phợng ta Vit phỵng trỡnh P qua M v cớt tia P : x y z 3x M 3yM 3z M Ox,Oy,Oz lổn lỵt tọi A, B,C cho VO ABC nhú nhỗt? Vit phỵng trỡnh mt phợng P cha ỵng thợng d , cho khoõng cỏch t im M d n P l ln nhỗt? Vit phỵng trỡnh mt phợng P qua A v cỏch M mt khõng ln nhỗt Nguyn Chin - Hng Quõn: 0973.514.674 Qua A d P : n P u d , AM , u d Qua A P : n P AM Page | 81 www.facebook.com/tailieupro ng ký thnh viờn nhn ti liu thng xuyờn Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn Vit phỵng trỡnh mt phợng P cha Qua A d P : n P u d , u u d http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ ỵng thợng d , cho P tọo vi ( kh ng song song vi d ) mt gũc ln nhỗt l ln nhỗt ? Cho / / P Vit phỵng trỡnh Lỗy A gi A l hỡnh chiu vu ng gũc cỷa ỵng thợng d song song vi v cỏch mt khoõng nhú nhỗt ? A trờn P Vit phỵng trỡnh ỵng thợng d i qua im A cho trỵc v nỡm mt phợng P cho trỵc cho Qua A d: u d u Qua A d d: u d n P , AM khoõng cỏch t im M cho trỵc n d l ln nhỗt ( AM khụng vuụng gúc vi P ) ? Vit phỵng trỡnh ỵng thợng d i qua im A cho trỵc v nỡm Qua A d d: u d n P , AM , n P http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ mt phợng P cho trỵc cho khoõng cỏch t im M cho trỵc n d l nhú nhỗt ( AM khụng vuụng gúc vi P ) ? Vit phỵng trỡnh ỵng thợng d i qua im A P d nỡm cho trỵc , cho P v tọo vi ỵng Qua A d d: u d n P , AM , n P thợng mt gũc nhú nhỗt vi cớt nhỵng kh ng vu ng gũc vi P ? http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ Nguyn Chin - Hng Quõn: 0973.514.674 Page | 82 www.facebook.com/tailieupro ng ký thnh viờn nhn ti liu thng xuyờn ... au ln x log a x x ln u x ln a u u sin2 u u .a u ln a loga u u u u u.ln a Cụng thc tớnh nhanh o hm hm phõn thc: a b ax b ad bc ; cx d cx d x2 a c x d f ax bx c d e ... i dỗu qua nghim ũ phỵng trỡnh y cú hai nghim phõn bit A 3a a m D1 y B 4AC 4b 12ac b 3ac c 3: Gi x 1, x l hai nghim cỷa phỵng trỡnh y http://www.tailieupro.com/ http://www.tailieupro.com/... ti liu thng xuyờn Truy cp website www.tailieupro.com nhn thờm nhiu ti liu hn MT S CễNG THC GII NHANH http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/ http://www.tailieupro.com/