tài liệu bồi dưỡng học sinh giỏi toán 12

ThS. LE HOANH PHO Nhd gido Uu tu C c c Mil BOIDU8NG 7 HOC SINH GIOI TOAN DAI SO-GIAI TICH - Ddnh cho HS lop 12 on tap & nang cao kinang lam bai. - Chudn bi cho cdc ki thi quoc gia do Bo GD&DT to choc Mil NHA XUAT BAN DAI HQC QUOC GIA HA NQI Bin DUSNG , HQC SINH GO TOAN OAI SO -GIAI TICH Boi duQng hoc sinh gioi Toan Dai so 10-1. Boi duQng hoc sinh gioi Toan Dai so 10-2. - Boi duQng hoc sinh gioi Toan Hinh hoc 10. - Boi duOng hoc sinh gioi Toan Dai so 11. Boi duQng hoc sinh gioi Toan Hinh hoc 11. Bp de thi tif luan Toan hoc. Phan dang va pht/Ong phap giai Toan So phtfc. Phan dang va phucing phap giai Toan To hop va Xac suat. 1234 Bai tap tir luan dien hinh Dai so giai tich 1234 Bai tap ta iuan dien hinh Hinh hoc li/ong giac ThS. LE HOANH PHO Nha gido iCu tu BOIDUQNG , HOC SINH GIOI TOAN DAI SO-GIAI TICH 12 * - Danh cho HS lap 12 on rflp & ndng cao kfndng lam bai. - Chudn bj cho cdc ki thi qudc gia do Bo GD&DT td choc. Ha Npi NHA XUAT BAN DAI HQC QUOC GIA HA NQI NHA XUAT BAN DAI HOC QUOC GIA HA N0I 16 Hang Chudi - Hai Ba Trirng Ha Npi Dien thoai: Bien tap-Che ban: (04) 39714896; Hanh chinh: (04) 39714899; Tdng bien tap: (04) 39714897 Fax: (04) 39714899 Chiu trach nhiem xuat bdn: Giam ddc PHUNG QUOC BAO Tong bien tap PHAM THI TRAM Bien tap noi dung THUY NGAN Sda bdi NGOC HAN Che bdn CONG TI ANPHA Trinh bay bia SON KY Ddi tdc lien ket xudt bdn CONG TI ANPHA SACH LIEN KET BOI DUONG HQC SINH GIOI TOAN DAI SO GIAI TICH 12 -TAP 1 Ma so: 1L-177DH2010 In 2.000 cuon, khd 16 x 24 cm tai cong ti TNHH In Bao bi Hung Phu So' xua't ban: 89-2010/CXB/11-03/DHQGHN, ngay 15/01/2010 j Quyet dinh xua't ban sd: 177LK-TN/XB In xong va nop ltiu chieu quy II nam 2010. Ldi N6I DAU De giup cho hoc sinh lap 12 co them tai lieu tu boi duong, ndng cao va ren luyen ki ndng gidi todn theo chuong trinh phdn ban mdi. Trung tdm sdch gido due ANPHA xin trdn trong giai thieu quy ban dong nghiep vd cdc em hoc sinh cuon: "Boi dudng hqc sinh gioi todn Dai so' Gidi tich 12 " nay. Cuon sdch nay nam trong bo sdch 6 cuon gom: - Boi ducmg hoc sinh gidi todn Hinh hoc 10. - Bdi ducmg hoc sinh gidi todn Dai so' 10. - Boi dudng hoc sinh gidi todn Hinh hoc 11. - Boi dudng hoc sinh gidi todn Dai so - Gidi tich 11. - Bdi dudng hoc sinh gidi todn Hinh hoc 12. - Boi dudng hoc sinh gidi todn Gidi tich 12. do nhd gido uu tu, Thac sTLe Hoanh Phd to'chirc bien soan. Ndi dung sdch duoc bien soan theo chuong trinh phdn ban: co bdn vd nang cao mdi ciia bd GD & DT, trong dd mot so van de duoc md rdng vdi cdc dang bdi tap hay vd kho dephuc vu cho cdc em yeu thich mud'n ndng cao todn hoc, cd dieu kien phdt trien tot nhat kha nang ciia minh. Cuon sdch la su ke thira nhung hieu bii't chuyen mdn vd kinh nghiem gidng day ciia chinh tdc gid trong qua trinh true tiep dirng ldp bdi dudng cho hoc sinh gidi cdc ldp chuyen todn. Vdi ndi dung sue tich, tdc gid da co'gang sap xep, chon loc cdc bai todn tieu bieu cho tirng the loai khdc nhau ung vdi ndi dung cua SGK. Mdt sd'bdi tap cd the khd nhung cdch gidi duqc dua tren nen tdng kien thuc vd ki nang co bdn. Hqc sinh can tu minh hoan thien cdc ki nang ciing nhu phdt trien tu duy qua viec gidi bdi tap cd trong sdch trudc khi ddi chieu vdi led gidi cd trong sdch nay, cd the mdt soldi gidi cd trong sdch con cd dong, hqc sinh cd thetu minh lam rd hon, chi tie't hon, ciing nhu tie minh dua ra nhung cdch lap ludn mdi hon. Chung tdi hy vong bd sdch nay se la mdt tdi lieu thie't thuc, bo ich cho ngudi day vd hqc, dqc biet cdc em hqc sinh yeu thich mdn todn vd hqc sinh chuan bi cho cdc ky thi qudc gia (tot nghiep THPT, tuyen sinh DH - CD) do bq GD & DT to chirc sap tdi. Trong qua trinh bien soqn, cudn sdch nay khdng the tranh khoi nhirng thieu sdt, chung tdi ra't mong nhdn duqc gop y ciia ban dqc gdn xa debq sdch hoan Men hon trong lah tdi ban. Moi y kien dong gop xin lien he: - Trung tam sach giao due Anpha 225C Nguyen Tri Phuong, P.9, Q.5, Tp. HCM. - Cong ti sach - thiet bj giao due Anpha 50 Nguyen Van Sang, Q. Tan Phii, Tp. HCM. DT: 08. 62676463, 38547464 . Email: alphabookcenter@yahoo.com Xin chan thanh cam on! MUC LUC Chuong I: tTng dung dao ham de khao sat va ve do thi cua ham so § 1. Tinh don dieu cua ham so 5 Dang 1: Dong bien, nghich bien, ham hang 5 Dang 2: Ung dung tinh don dieu 17 §2. Cue tri ciia ham so 37 Dang 1: Cue dai, cue tieu , 37 Dang 2: Ung dung ciia cue tri 48 §3. Gia tri Ion nhat va gia tri nho nhat 58 Dang 1: Tim gia tri ldn nhat, nho nhat 58 Dang' 2: Bai toan lap ham so 69 Dang 3: Ung dung vao phuong trinh 77 §4. Duong tiem can cua do thi ham so 88 Dang 1: Tim cac tiem can 88 Dang 2: Bai toan ve tiem can 96 §5. Khao sat va ve ham da thuc 103 Dang 1: Ham bac ha 104 Dang 2: Ham trung phuong 113 §6. Khao sat va ve ham hOu ti 126 Dang 1: Ham so v = ax + k (c * 0 va ad be * 0) 126 cx + d 2 i Dang 2: Ham s6 v = &X + (a * 0. a' * 0) 135 a'x + b' §7. Bai toan thuong gap ve do thi 148 Dang 1: Tuong giao, khoang each, goc 149 Dang 2: Tiep tuyen. tiep xuc 159 Dang 3: Yeu to co dinh. doi xung - quy tich 170 Chirong II: Ham so luy thua ham so mu va ham so logarit § 1. Quy tac bien doi va cac ham so 186 Dang 1: Bien doi luy thua - mu - logarit 188 Dang 2: Cac ham so mu. luy thua, logarit 200 Dang 3: Bat dang thuc va GTLN, GTNN 212 CHUONG I: UNG DUNG DAO HAM OE KHAO SAT VA VE DO THj CUA HAM SO §1. TINH DON DIEU CUA HAM SO A. KIEN THLTC CO BAN • Dinh nghTa: Ham so f xac dinh tren K la mot khoang, doan hoac nira khoang. - f dong bien tren K neu vdi moi Xi, X2 6 K: X] < X2 => f(xi) < f(x2) - fnghich bien tren K neu vdi moi xi. xi e K: Xi<X2=>f(xi)>f(x2). • Dieu kien can de ham so don dieu Gia sir ham so co dao ham tren khoang (a; b) khi do: - Neu ham so f dong bien tren (a; b) thi f'(x) > 0 vdi moi x e (a; b) - Neu ham so f nghich bien tren (a; b) thi f'(x) < 0 vdi moi x e (a; b). • Dieu kien du de ham so don dieu - Gia sir ham so f co dao ham tren khoang (a; b) Neu f'(x) > 0 voi moi x e (a; b) thi ham so f dong bien tren (a; b) Neu f'(x) < 0 voi moi x e (a; b) thi ham so nghich bien tren (a; b) Neu f'(x) = 0 vdi moi x e (a; b) thi ham so f khong doi tren (a; b). - Gia sir ham so f co dao ham tren khoang (a; b) Neu f '(x) > 0 (hoac f '(x) < 0) vdi moi x e (a; b) va f '(x) = 0 chi tai mot so huu han diem cua (a; b) thi ham so dong bien (hoac nghich bien) tren khoang (a; b). B. PHAN DANG TOAN DANG 1: B6NG BliN, NGHICH BIEN, HAM HANG • Phuong phap xet tinh don dieu: - Tim tap xac dinh - Tinh dao ham, xet dau dao ham, lap bang bien thien - Ket luan Chii y: - Dau nhi thuc bac nhat: f(x) = ax + b, a ^ 0 x -00 -b/a +co f(x) trai dau a 0 ciing dau a - Dau tam thuc bac hai: f(x) = ax 2 + bx + c, a * 0 Neu A < 0 thi f(x) luon ciing dau vdi a Neu.A = 0 thi f(x) luon cung dau vdi a, trir nghiem kep Neu A > 0 thi dau "trong trai - ngoai ciing" X -CO X] X 2 +00 f(x) ciing dau a 0 trai dau a 0 ciing diu a -BDHSG DSGT12/1- Vi du 1: Xet chieu bien thien ciia ham sd: a) y = x 2 - 6x + 5 c) y = x 3 - 2x 2 + x + 1 b)y= -x 3 3 2x 2 + x - 3 d) y = -x 3 + 4x 2 - 7x + 5 Giai a) Tap xac dinh D = R. Ta co y' = 2x - 6. Cho y' = 0 » 2x - 6 = 0 » x = 3. Bang bien thien X —oo 3 +00 y' - 0 + y — Vay ham so nghich bien tren (-oo; 3), dong bien tren (3; +oo). b) D = R. Ta cd y' = 4x 2 - 4x + 1 = (2x - l) 2 > 0 vdi moi x y' = 0 o x = —. Vay ham so dong bien tren R. 2 c) D = R . Ta co y' = 3x 2 - 4x + 1 Cho y' = 0 o 3x 2 - 4x + 1 = 0 <=> x = - hoac x = 1. J 3 BBT X —00 1/3 1 +00 y' + 0 0 + y ^* —^ Vay ham so dong bien tren moi khoang (-co; —) va (1; +oo), nghich bien 3 tren khoang (—; 1). 3 d) D = R Ta cd y' = -3x 2 + 8x - 7 ViA' = 16-21<0 nen y' < 0 vdi moi x do do ham so nghich bien tren R. Vi du 2: Xet chieu bidn thien cua cac ham so sau: a) y = x 4 - 2x 2 - 5 b) y = x 4 + 8x 2 + 9 Giai a) D = R. Ta co y' = 4x 3 - 4x = 4x(x 2 - 1) Cho y' = 0 <=> 4x(x 2 - 1) = 0 <=> x = 0 hoac x = ±1 BBT X —00 -1 0 1 +00 y' - o •+ 0 - 0 + y ^ ^* ^ ^ Vay ham sd nghich bien tren moi khoang (-co; -1) va (0; 1), ddng bidn tren moi khoang (-1; 0) va (1; +=»)• 6 -BDHSG DSGT12/1- b) D = R. Ta co y' = 4x 3 + 16x = 4x(x 2 + 4),y' = 0ox = 0. y' > 0 tren khoang (0; +co) => y dong bien tren khoang (0; +co) y' < 0 tren khoang (-co; 0) => y nghich bien tren khoang (-co; 0). Vi du 3: Xet su bien thien cua ham so: a) y = x + — x b)y c)y a) Tap xac dinh D = R\ {0} _3_ ,.2 Ta co y' = 1 BBT: Giai , y' = 0 <» X x 2 -3 3x-8 1-x :V3" d)y 1 (x-4) 2 X -co QN/3 +00 y' + 0 - - 0 + y Vay ham so dong bien tren khoang (-co; -^3 ) va (J3 ; +oo), nghich bien tren m6i khoang (-S ; 0) va (0; V3 ). b) D = R\ {0}. Tacdy' = 1 > 0 vdi moi x ^ 0 nen ham so dong bien tren moi khoang (-oo; 0) va (0; +co). -5 c) D = R \ {1}. Ta cd y' = - < 0 vdi moi x -t- 1 nen ham so nghich 3 (1-x) 2 bien trong cac khoang (-co; 1) va (1; +oo). d) D = R\ {4}.Tacdy'= ———- (x-4) 3 y' < 0 tren khoang (4; +co) nen y nghich bien tren khoang (4; +co). y' > 0 tren khoang (-co; 4) nen y ddng bien tren khoang (-co; 4). Vi du 4: Tim cac khoang don dieu ciia ham so: , x-2 2x a) y = -5 b) y X + X + 1 a) D = R. Ta cd: y' x 2 -9 Giai -x 2 + 4x + 3 (x 2 + x + l) 2 y' = 0 e> x 2 - 4x - 3 = 0 <=> x = 2 ± ^7 BBT: X -co 2-V7 2+V7 +°o y' - 0 + 0 - y —^—* - * -BDHSG DSGT12/1- 7 Vay ham so dong bien tren khoang (2 - yfl ; 2 + V7 ) va nghich bien tren cac khoang (-co; 2 - 77 ) va (2 + ; +oo). b) D = R\{-3;3}.Tac6y'=^^ < 0 ,Vx*±3. (x 2 -9) 2 Do do y' < 0 tren cac khoang (-co; -3), (-3; 3), (3; +oo) nen ham so da cho nghich bien tren cac khoang do. Vi du 5: Xet su bien thien cua ham sd: a) y = V4-x 2 c)y Vl6-: b)y = Vx 2 -2x + 3 d)y x + 2 Giai a) Dieu kien 4-x 2 >0<=>-2<x<2 nen D = [-2; 2] Vdi -2 < x < 2 thi y' BBT: V4~ ,y' = 0<=>x = 0. X -2 n 2 y' + 0 - y ^ ^ — ^ Vay ham so dong bien tren khoang (-2; 0) va nghich bien tren khoang (0; 2). Do ham so f lien tuc tren doan [-2; 2] nen ham so dong bien tren doan [-2; 0] va nghich bien tren doan [0; 2]. b) Vi A' = 1 - 3 < 0 nen x 2 - 2x + 3 > 0, Vx => D = R. •p , 1 2x-2 x-l , . , Ta co y = — = = =. y = 0 o x = 1. 2vx 2 -2x + 3 Vx 2 -2x + 3 y'>0ox>l,y'<0ox<l nen ham so nghich bien tren khoang (-co; 1) va dong bien tren khoang (1; +00). c) DK: 16 - x 2 > 0 o x 2 < 16 o -4 < x < 4. D = (-4; 4). Ta co v' = 16 > 0, Vx e (-4; 4). (16-x 2 )Vl6-x 2 Vay ham so dong bien tren khoang (-4; 4). d) D = [0; +00). Vdi x > 0, y' = X _, y' 2^y^(x + 2) 2 BBT: X 0 2 +00 y + 0 y ^ Vay ham sd ddng bien tren (0; 2) va nghich bien tren (2; +00). 8 -BDHSG DSGTu/1- [...]... vd nghiem Xet 0 < x < 1 thi f(x) = x + ( l - x ) > 0: vd nghiem Xet x < 0 thi: f '(x) = 13x - 6x + 12x - 6x = 13x - 6x(x - l ) > 0 nen f dong bien Bang bien thien: x —O C 0 y' + 1 y —O O Nen f(x) = 0 cd nghiem duy nhat x < 0 Vay phuong trinh cho cd nghiem duy nhat 6 7 2 13 12 22 2 2 5 3 3 12 2 -BDHSG DSGT12/1- V i du 16: Chung minh rang phucmg trinh 2 x V x - 2 = 11 cd mot nghiem duy nhat Giai Xet ham... tinh dao ham Bai 8: Tim tham sd de ham sd: a ) y = 1 1 3 — x - — (sin a + cos a)x + — sm 2a.x ddng bien tren R 3 2 b) y = (m - 3)x - (2m + l)cosx nghich bien tren R DS: a) — + kn < a < — + krc 12 12 32 -BDHSG DSGT12/1- Bai 9: T i m tham so de ham sd: a) y = nghich bien tren (-°o;l) x+a b) y = 2x + 3x + 6(m + l ) x nghich bien tren (-2; 0) DS: b) m < - 3 Bai 10: T i m tham sd de ham sd: a) y = x + 3x +... 4 - x 3 2 b)Vx + l + 2Vx + 6 > 20 - 3Vx + 13 -BDHSG DSGT12/1- 33 3 DS: a) 1 < x < 4 b) x > 3 1 1 ^ [cot x - cot y = x - y x— =v — y | l 5 x + 7y = rc;.x,y £ (0, rc) 2y = x + 1 Bai 18: Giai he: a) x 3 1 + V5 DS: a)x = y = l ; x = y X= b) y= 7 1 22 T C 22 Bai 19: Chung minh phuong trinh: x + x + 12x - V3 = 0 co nghiem 3 2 H D : Ham y = x + x + 12x - V3 don dieu tren R 3 2 Bai 20: Chung minh phuong trinh... trinh cd mot nghiem duy nhat T C 5 Con vdi moi x e [0; — ], ta cd 1 < f(x) < — nen phuong trinh khong co 3 4 nghi?m suy ra dpcm V i du 18: T i m sd nghiem cua phuong trinh x - 3x - 9x - 4 = 0 -BDHSG DSGT12 /123 3 2 Giai Xet ham so y = x - 3x - 9x - 4, D = R 3x - 6x - 9, y' = 0 co x = - 1 hoac x = 3 BBT X —O G -1 3 + 00 3 2 + y' 0 y + 0 - +00 1^ r —O G * -31" Dua vao BBT thi phuong trinh y = 0 co dung 3... kien -3 < x < 1 PT co — —— =m 4Vx + 3 + 3V1 - x + 1 Ta cd (Vx + 3) + (Vl-x) = 4 nen dat: i 2t / „ 1-t Vx + 3 = 2sin(p = 2 - , V I - x = 2coscp = 5 1+t 1+t Vdi t = tan ^.0 c > d > 0 Xem ve trai la ham so f(a), a > 0 f '(a) = 4a + 2bcd - 2a(b + c + d ) f "(a) = 12a - 2(b + c + d ) > 0 nen f' dong bien tren (0; +co): a > b => f '(a) > f '(b).Vi f '(b)... 2v/Tl > 0 vdi moi Phuong trinh (1) cd dang f(x) = f(2) nen (1) co x = 2, thay vao (2) ta duoc y = 1 Vay nghiem cua phuong trinh la (x; y) = (2; 1) x - 12x + 35 < 0 2 V i du 13: Giai he bat phuong trinh: (1) x - 3x + 9x + - > 0 (2) 3 Giai: Ta cd (1) o x - 12x + 35 < 0 co 5 < x < 7 u 2 2 Xet (2): Dat f(x) = x - 3x + 9x + - , D = R 3 f (x) = 3x - 6x + 9 > 0, Vx e R nen f(x) ddng bien: x > 5 =o f(x) > 286/3... va (-2; +oo) x -2x-5 b) D = R \ { - l } T a c d y ' = ~ < 0 vdi moi x * - l (vi A' = 1 - 5 < 0) (x + 1) ' ' Vay ham so nghich bien tren mdi khoang (-oo; - 1 ) va ( - 1 ; +oo) V i du 12: Chung minh ham so: 2 2 -BDHSG DSGT12/1- v 1 1 a) y - i + dong bien trong khoang ( - 1 ; 1) va nghich bidn trong cac x 2 khoang (-co; -1) va (1; +oo) , sin(x + a) , , _ , , ) y~ ( ^ b + krt; k e Z) don dieu Uong mdi... kien x * 1; —, phuong trinh trd thanh: 2 ( 2 x - 5 ) - - J _ = (x-l) |2x-5| |x-l| 2 2 Xet f(t) = t - i vdi t > 0 2 Ta co: f '(t) = 2t > 0 nen f dong bien tren (0; +oo) Phuongtrinh:f(|2x-5|) = f(|x - l|)o 12x- 51 = |x-l| < > 4x - 20x + 25 = x - 2x + 1 < > 3x - 18x + 24 = 0 = = c ^ x - 6 x + 8 = 0 c o x = 2 hoac x = 4 (chon) V i du 5: Giai bat phuong trinh: 4 | 2x - 11 (x - x + 1) > x - 6x + 15x - 14 Giai... 2) o I 2x - 11 > x - 2 Xet x - 2 < 0 thi BPT nghiem dung Xet x - 2 > 0 thi 2x - 1 > 0 nen BPT o 2 x - 1 > X - 2 < = > Vay tap nghiem la S = R 2 2 2 2 2 3 2 3 2 3 3 3 2 18 X > - 1 : £) ng U -BDHSG DSGT12/1- V i du 6: Giai bat phuong trinh: Vx + 1 + 2Vx + 6 < 20 - 3VX + 13 Giai Dieu kien: x > - 1 BPT viet lai: Vx + 1 + 2%/x + 6 + 3Vx + 13 > 20 Xet f(x) la ham so ve trai, x > - 1 Ta co: 1 1 3 A f ' . NQI Bin DUSNG , HQC SINH GO TOAN OAI SO -GIAI TICH Boi duQng hoc sinh gioi Toan Dai so 10-1. Boi duQng hoc sinh gioi Toan Dai so 10-2. - Boi duQng hoc sinh gioi Toan Hinh . vd cdc em hoc sinh cuon: "Boi dudng hqc sinh gioi todn Dai so' Gidi tich 12 " nay. Cuon sdch nay nam trong bo sdch 6 cuon gom: - Boi ducmg hoc sinh gidi todn . ducmg hoc sinh gidi todn Dai so' 10. - Boi dudng hoc sinh gidi todn Hinh hoc 11. - Boi dudng hoc sinh gidi todn Dai so - Gidi tich 11. - Bdi dudng hoc sinh gidi