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m BO GIAO DUC VA OAO TAO TRAN VAN HAG (Tong Chu bien) - VU TUAN (Chu bien) D A O NGOC NAIVI - LE VAN TI^N VU VIET YEN DAI SO VA GIAITiai (Tdi bdn ldn thit ba) NHA X U A T B A N G I A O DUC VIET NAM I HIEU DilNG TRONG SACH Phan hoat dong cua hoc sinh Tuy doi tuong cu the ma giao vien sii dung Ket thuc chihig minh hoac ldi giai Ban qayin thu6c Nha xuit ban Giao due Viet Nam - B6 Giao due va Dao tao 01 - 2010/CXB/566 - 1485/GD Ma s o : CHIO ITG wm sd LUDDG GiflC Vfl / f+^uunG mlnfi LUDTIG Giflc ^^^:^ ^ j k> * Tiep tgc phan gia tri lugng giac va cac cong Mc lugng giac duoc hoc chuong cuoi cua Oal sd 10, chuong cung cap kien thuc ve ham so lugng giac va each giai phuong trinh lugng giac day chi yeu cau giai thao cac phuong trinh co ban va nhung phuong trinh bae nhat va bae hai doi vol mot ham sd luong giac Khae voi nhirng ham soda duoc hoc trudc day, cac ham sdy=sin;c,)'=cos.x; y=tan.x; \/ay=catx\a nhung ham sd tuan hoan Cac ham sd gap nhieu cae mon khoa hoc ling dung (Vat If, Hoa hoc, ) H A M S LtfONG GIAC I - DINH NGHIA Tnldfc hd't, ta nhde lai bang cac gia tri luong giac eua eae eung dae biet ^v,^^ Cung Giatrj\ luong g i a c ^ \ sinjc COSA; n 'A 2 tanx n V3 S I n 2" 2 ^/2 2 J3 COtJC 71 0 a) Sijrdung may tfnh bo tui, hay tfnh sinx, cosx vdi x la cac so sau : 5;^;1,5;2;3,1;4,25;5 b) Tren di/dng tron lUOng giac, v6i diem goc A, hay xae ^nh cac diem M ma so cCia r\ cung AM bang x (rad) ti/ong Lfng da cho d tren va xae dmh sinx, cosx (lay n = 3,14) Ham so sin va hanfi so cosin a) Ham so sin d ldp 10 ta da bie't, eo the dat tuong ling mdi s6' thue x vdi mdt dilm M rv nha't tren duofng tron luong giac ma sd cua eung AM bang x (rad) (h.la) Dilm M co tung dd hoan toan xae dinh, dd ehfnh la gia tri sinx Bilu diin gia tri eua x tren true hoanh va gia tri eiia sinx tren true tung, ta duoc Hinh lb 7- O a) X b) Hinh Quy tac dat tuong ling mdi sd thuc x vdi sd thue sinx sin: R ^ R ( > J = sinx X— duoc goi la ham so sin, ki hieu la y = sinx Tap xae dinh eua ham sd' sin la R b) Ham sd cosin M" cosx O a) b) Hinh Quy tac dat tuong dng mdi sd' thuc x vdi sd thuc cosx cos : R -^ R X I—» J = eosx duge goi la hdm so cosin, ki hidu la j = eosx (h.2) Tap xae dinh eiia ham sd cdsin la R X H a m so tang va ham so cotang a) Ham sd tang Ham sd tang la ham sd dugc xae dinh bdi cdng thdc sinx (eosx^t 0), y eosx kl hidu la J = tan x Vi cos X ^ va chi x 9^n— + ^7t (^ e Z) nen tap xae dinh ciia ham sd'j = tanxla D=R\l- + kn,keZ\ b) Ham so cotang Ham sd cotang la ham sd dugc xae dinh bdi cdng thde cosx (sinx ^ 0), y= — sinx kl hieu Ik y - cot x Vi sinx ;^ va chi x ^ kn {k e Z) nen tap xae dinh ciia ham sd y = cotx la D= M.\{kn,ks Z } ^2 Hay so sanh cac gia trj sinx va sin(-x), cosx va cos(-x) NHAN XET Ham sd J = sin X la ham sd le, ham sd y = cos x la ham sd' chSn, tiit dd suy cac ham sd 3^ = tan x va j = cot x diu la nhiing ham sd le II - TINH TUAN HOAN CUA HAM SO LUONG GIAC Tim nhOhg sd r cho/(x+r) =fix) vdi moi x thuoc tap xae djnh cOa cac ham sd sau : a) f{x) = sin X; b) f{x) = tan x Ngudi ta chiing minh duge rang T = 2n Ik sd duong nhd nhSit thoa man dang thde sin(x + T) = sinx, Vx e R (xem Bai dgc them) Ham sd j = sin x thoa man dang thde trdn dugc ggi la hdm so tudn hodn vdi chu ki 2n Tuong tu, ham sdy = eosx la ham sd tuSn hoan vdi chu ki 2n Cae ham sd j = tan x va y = cot x cdng la nhihig ham sd tuSn hoan, vdi chu ki n Ill - SUBIEN THIEN VA DO THI CUA H A M S L U O N G GIAC Ham s o J = sinx , Tii dinh nghia ta tha'y hkm sd y = sinx : • Xae dinh vdi mgi x e R va -1 < sinx < ; • La ham sd le ; • La ham sd tudn hoan vdi chu ki 27t Sau day, ta se khao sat su bidn thidn cua ham s6y = sinx a) Su bie'n thien va thi ham sd^' = sinx tren doan [0 ; n] Xetcae sd thuc Xj,X2, ddO0 5'(sin3x) + 181 DAP S6 - HUdNG D A N CHUGNG1 §2 Si- a) X = arcsin + k2n, a) tanx = tai v e (-Jt, 0,7t} ; b) tan v = I tai v X = Jt - arcsin n 5n '4' 3JI E + ^2ji, ^ e Z ; c) tan.v> b ) x = - + A-—,A:e Z ; 3n V e ! - T t ; — u 0;— l u i n; K 2J [ 2).{ c)x=- d) tanx x = - - + ;t27C, k e 71 d ) x = ±—+ ^71, x = ±— + kn,k 7t x = — + kK,ke Z a)x = 45° + A180° A e Z ; •^ 57t , 71 , _ b ) x = - + — + k—,k e Z ; 18 l2 571 , 47t , „ - + k—,A:eZ ; 18 = e Z r» c ) x = —+ A:—, X = kn, K e Z d) X = k—, X = — + ^7t, k ^ 3m ; it, m e Z X € {k2n ; Ji + k2K), k e Z a ) < c o s x < l , y v < , y^^^ = X = A-2jt, A e Z : b) V ^ l Vn.ax=l 271 x= —^ + A2jt, A e a) X = - +arcsin —+ A2jt, ke v = Jt-1-arcsin — + A2JI, A e Z b) X = ±—+ AJI, X = ± — + ATI , A e c) x = + — + A27t,A e Z ; d) x = — + A—, A e Z 144 12 a) X = A27t, X = ± - + A27t A e Z ; JI b) X = — + Ajt, X = arctan h Ajt, A e Z 15 c ) x = A27t, A e Z ;x = 7t-2a+A27i, A e , (voi cos a = —^ ; sin a = —j=) s s d) DiSu kien sinx^O, x = ± hA27t b) x = —+ —+ A — , A e Z 3 4^ (voi c o s a = — ; s i n a = —) 5 c ) x = — +A27t,x= - — +A27c,Ae Z ; 12 12 CHUONG II §11 a ) ; b ) ^ = ; c ) = ; 2.42 a) 24 ; b) 576 d) x = - - - + A7i, A e Z ; 12, (voi s i n a = — ; cosa = — ) 13 13 §2 a) 6! ; b ) x ! ; c ) 4 2.10!; 3.210 ,360 a) 60 ; b) 10 20 60 183 §3 12 n = 28 5.-1 a), b) Gffi y Khai tri^n l l ' ° = (10 + 1)'", 10l''' = (100+l)'ô Đ4 a) n = [SSS, SSN, NSS, SNS, NNS, NSN, SNN, NNN) b) A= {SSS, SSN, SNS SNN] ; B= [SNN, NSN, NNS] ; C= [SSN,NSS,SNSNNSNSN,SNN,NNN] a ) Q = | ( ; , ) ; ! < ; , ; < 6) a ) = ( { l , ) , {1,3}, (1,41, {2,3}, |2,4), {3,4}} b) A={{1,3),{2,4}}; S= {{1,22^, {1,4}, {2,3}, {2,4}, {3,4}} a) A = A m A2 ; B = Aj n /ij ; C = (A, n A ) u ( A i n A ) ;Z) = A,uA2 b) HD D la bi^n c6 "Ca hai ngitod diu ban truot" a ) Q = {1,2, , 10} ; b) A = {1,2, 3,4, 5} ; B = {7,8,9, 10} ; C = (2,4, 6, 8, 10} a) n = {5, NS, NNS, NNNS, NNNN} ; b) A = [S, NS, NNS]; B = [NNNS, NNNN] a) Q g6m cac chinh hop chap cua chO $61,2,3,4,5; b) A ={(1,2), (1,3), (1,4), (1,5), (2, 3), (2, 4), (2, 5), (3, 4), (3, 5), (4, 5)) ; B={(2, 1),(4,2)};C = §5 a) D6c lap ; b) — ; c) — • •'^ 25 25 n t$p chirong II a) 1176 ; b) 420 a) = 0,1 ; b) 0,2 a) - ^ , b) — 105 210 « 0,4213 a) - ; b ) - ; c ) - 5 1 a) - ;b) - ' 4 CHirONG III §1 a) 5, = - , S , = - , S , = - §2 b) u„ = Vn + vdi w e N a) T&y s6giam; b) Day sd tang ; c) Day s6 khdng tang cung khdng giam ; d) Day sd giam a) Day sd hi chan dudi vl M„ > ; b) Day sd bi chan vi < ô /l9 x(x + Av) a ) ; b ) - - : c ) - 4 HD Chiing minh / gian doan tai v = Tix suy / khdng co dao ham tai diem a) v = 3x + ; b) v = 12x- 16: c) y = 3x + va y = 3.v - a) y = -4(x - 1) ; b) y = -ix + 2); 186 '.il 7(7^77 a)Av = 2A.v ^ = ; ^x Av Av d) Av = — x(x + Av) Av :x-5 ;b)2V2-5X-X' c)i:i^=^;d) Đôã a ) / ( ) - / ( l ) = ; b)/(0 ) - / ( l ) = -0.271 n vH""? a ) x - -sTx CHUONG V Av - 1+Vi9' a) 5cos X + 3sin x : b) c) cotx- (sinx-cosx) ' Sin" V d) (v cosX - sin x) ^v e) ;f) COS XVl + 2tanx sm X x c o s v x +1 ^y' + a) -2(2r'' - ftv" + 1) + (dv^ - 18.v) (9 - 2x); b ) | - ^ + l (7x-3) + 7f6V7- :) 777i+4i^ ; -, 2tanx 2x \ d) T~ + — ^ T T ; s) -22 COS X sm X d) y x ^sin(1 + x)^ 1+-1 9.v->/x-6x 2>/7 + 2x- 1 e) y' = i v'^(l-VI)-4v l0x+15 Oy' = - 71 (X 3.V)- 1- A27t, (A e Z) vdi cos = — a) X = ^ + X = 7t + A47t b) , Jl , 47t a- e Z) x = — + A— 3 a) \' = a): (\'A- + l)xsinx + (2x2\/x + l)cosA ^ -3( 2.V +1) sin X - cos v b) y' (2.V + 1)-) ( s i i u - r cosf-2 a ) ( - ^ ; ) u ( ; + » ) ; b ) ( - x ; ) u ( l ;+oo) c) y' = §4 siii'r l.a) dx: 2{a + b)4x -7 d) y': b) (2v+4Xx2-4x)+{r +4x+l)x 2v dv; +sin V e) y' = cos" v(sin.v + 2) 2tanx (x - l ) s i n x + 2xcosx dv a) — — d x - ; b) COS X (l-.v^)^ b) / " | - ^ | = - ; / " ( ) = 0; /" - 2%/x oy=- §5 a) 622 ; 3.2 + r -v cot X V-v f2VI-l)- x-3 6.-I ( ± ; ± ) _ a)y = -2x + ; 18 b)y = - x - ; a).3-" = - ^ ; b ) y " = —pl= (1-x)" 4^/(l-x)•' WO ' c) y" = sinx COS d) y" = - 2cos 2x c) y = -2.V + = 0, y = 2v - a ) - m / s : b) 12m/s2 ; c) 12m/s2; d)-12m/s X a) v= — p r x + ^ / , v = N/2X • ^/2 • b)90° n t^p chuong V a) y' = x^ - X + ; 15 ^ 24 On tap cuoi nam c) y' = 3x^-7 4x2 l.b) V - -^^x rV3 c) R 187 a) 65 15 HD Xet ham sd / ( x ) = x"* - 3x^ + x - va 113 haisd-1 ;0 -70cos2x ,^ b) y' = (6 + 7sin2x)^ 7t c) , 7t , i - A j t ; — + A71 ke a) - + - n ; ( - l ) * - + A27t,«, A( (jt b) {—a; + A27t, A e Z } vdi c o s a =— ; (2 j s i n a = — c) - + A27t;/27t,A,/eZ ; d ) • (7t 27t,— ; 17 a) „1 e Z >; 6sin3x cos 3x b) N -x{vx^ +1 sin Vx^ +1 + cos Vx^ +1) V(7T? s "' X^ a)A^Q = 1560;b)40C39 18 a) y" = a) - | - ; b ) 1J-.3 ""-10 c ) x smx ; d)(cos x + x s i n x ) e) Vd nghidm 210 , 71 16 a) {—+ A —; —arcsm— + n7t; (4 2 I I , n arcsm — i - mn, k.n,m 2 b) { A - ; ± - + /27t, A , / e ; ' pj ^10 _ , 2.9! ^, 2.8! a) ; b) 10! 10! tti=5, rf = 186 2 ; b) y" = — + — ^ ; x" ( - x ) " (1 + x)" c) y" = -a 19.b = ,c sinax ; d) y" = 2cos2x = 0,d= 20.a)y = x + l ; X = A7l 10.a)4;b)i;c)^;d) 2 11 a) -^ 13 a) ; b) — ; c) - ^ ; d) ^ » ; 16 b) X = arcsin+ ô27i ã x = 7t-arcsmi-w27i c)5 e) ; f) - ; g) -HX) 188 (m,/i,AeZ) ; BANG TRA CLOJ ilk, ,• :- THUAT N G Q Bit phuong trinh luong giac Bie'n cd Bidn cd chdc chdn Bidn cd ddi Bidn cd khdng Bidn cd xung khae Bidn ed ddc lap C^ sd cdng Cdp sd nhan Chinh hop Cdng bdi Cdng sai Cdng thurc cdng xae sudt Cdng thiic nhan xdc sudt Cdng thiic nhi thiic Niu-ton Cudng dd tiic thdi ciia ddng didn Day sd Day sd bi chdn Day sd c6 gidi han hChi han Day sd c6 gidi han Day sd cd gidi han vd cue Day sd giam Day sd huu han Day sd khdng ddi Day sd Phi-bd-na-xi Day sd tdng Dao ham Dao ham bdn phai Dao ham ben trai Dao ham cd'p hai Dao ham cdp n Dao ham ciia ham hop Dao ham mdt bdn Dao ham tai mdt didm Dao ham tren mdt doan Dao ham trdn mdt khoang Dudng hinh sin Gia tdc tiic thdi ciia chuydn ddng Giao ciia hai bidn cd Gia thidt quy nap Gidi han b d n ^ a i ciia hdm sd Gidi han bdn trai ciia ham sd Gidi han h&u han ciia day sd THUAT NGUT _, , JRAMOL 37 61 61 62 61 62 72 93 98 49 98 93 69 72 55 153 85 90 113 112 117 89 85 9l 91 89 145 154 154 • 171 171 161 154 146 155 153 10 172 62 80 126 126 112 189 THUAT NGi; f^f^M^l,- Gidi han hiru han ciia ham sd tai mot diem Gidi han hiiti han ciia ham sd tai vo cue „.,., , sinx Gioi han lim —— 'iM Pl^Pl^l 123 127 163 v->0 X Gidi han mdt ben Gidi han vd cue (ciia day sd) Gidi han vd cue eiia ham sd Ham sd gian doan Ham sd hop Ham sd lien tuc tai mdt diem Ham sd lidn tuc tren mdt doan Ham sd lien tuc tren mot khoang Ham sd luong giac Ham so tudn hoan He thiic truy hdi Hinh hoc Fractal Hodn vi Hop ciia hai bidn cd Kdt qua thudn loi cho bien cd Khdng gian mSu Phep thiJr Phep thit ngdu nhien Phucmg phap quy nap toan hoc Phuong phdp truy hdi Phuong trinh bdc hai ddi vdi mdt ham sd luong giac Phuong trinh bdc nhd't ddi vdi mot ham sd luong giac Phuong trinh bdc nhd't ddi vdi sinx va eosx Phuang trinh luong giac eo ban Phuong trinh tiep tuye'n Quy tdc cdng (trong td hop) Quy tdc nhdn (trong td hop) Sd hang tdng qudt ciia day sd Tam giac Pa-xcan Tdn suat Tie'p diem Tie'p tuydn Tdng ciia cd'p sd nhdn liii vd han Tdhop Vdn tdc tiie thdi ciia chuyen ddng Vi phdn Xdc sud't ciia bidn cd Y nghla hinh hoc ciia dao ham Y nghla vdt li eua dao ham 190 126 117 129 136 161 135 136 136 14 "" 87 104 46 62 61 60 59 59 80 87 31 29 35 18 152 43 44 85" 57 ' 75 151 151 116 51 147 170 65 150 153 MUC LUC Trang Chuang I nku SO LLlCpNG GIAC VA PHUONG TRINH L U O N G GIAC §1 Ham soli/ong giac §2 Phuong trinh li/gng giac co ban §3 M6t so phuong trinh iLfpng giac thirdng gap n tap chiftfng I ChuangII TO HOP 18 29 40 x A c SUAT §1 Quy tac dem §2 Hoan vj - Chinh hpp - Td hpp §3 Nhi thurc Niu-ton ' §4 Phep thijr va big'n c6 §5 Xae suat cua bi§'n co 6n tap chi/0ng II Chuang III D A Y SO 43 46 55 59 65 76 CAP SO C O N G VA CAP SO NHAN §1 Phirong phap quy nap toan hoc §2 Daysd §3 Cap so cpng §4 Cap s6 nh§n On tap chiTdng III 80 85 93 98 107 Chuang IV G l l HAN § GiPi han cua day s6' §2 GiPi han cua ham sd §3 Ham so lien tuc On tap chifdng IV 112 123 135 141 Chuang V DAO HAM §1 Ojnh nghTa va y nghTa cua dao ham §2 Quy tac tfnh dao ham §3 Dao ham cOa ham s6 li/png giac §4 Vi phan §5 Dao ham cap hai On tap ChiTdng V d n tap cuoi nam 146 157 163 170 172 176 178 191 Chiu track nhiem xudt bdn : Chi tich HDQT kiem Tdng Giam ddc N G TRAN AI Pho Tdng Giam ddc kifim Tdng bien tap N G U Y £ N QUt THAO Bien tap ldn ddu PHAM BAO KHU£ - NGUYfiN XUAN BINH Bien tap tdi bdn : Lfi THANH HANG Bien lap kl thuat vd trinh bdy : TRAN THUt HANH - NGUYEN THANH THU^ Trinh bdy bia : BUI QUANG TUAN S«a ban in Lfe THANH HANG Che bdn C N G TY CP THIET KE VA PHAT HANH SACH GIAO DUC DAI SO VAGIAI TICH 11 Ma so :CH 101 TO In 35.000 cuon; (QDIO/GK); kho 17x24cm In tai Cong ty co ph&i In Mc Giang So in: 05 So xuSft ban: 01-2010/CXB/566-1485/GD In xong va nop liru chieu thang 06 nam 2010 m •UM g 11 A L I r \