luyện giải đê trước kì thi 3 miền bắc tring nam môn toán

Nh^ gi^o uu tij - TTi^c siToAn hpc Gl^ng vi«n chfnh - Thac sITo^n hpc Ph6 gi^o su - mn si: NGUYEN VAN THONG NGUYEN VAN MINH NGUYEN VAN HIEU (Chu bifen) ••a LIIYeNGIIUIliniUIICKVTHIIilUHOCSMliN TOAN HOC ii@® Ddnh cho hpc sinh 12 luy^n thi BH-CB ^ Biin so$n theo n0i dung djnh hudng ^ i mdi cua BO Gi^o Due & B^o Tao THLT MIEN TiWH BINH THUAN MZ DVL J My^6' /I5 I NHAXUJTBiiNTONGHQPTHiiNHPHOHOCHlMINH Left noi ddu Chiing toi la cac anh em ruot thjl, muo'n vie't quyen sach cho the he sau on tap de chuan bi thi v^o dai hoc, vi bU'dc v^o triTdng dai hoc, ngtfcfi hoc sinh b^l dau mot ddi song mdi vk c6 tiTdng lai tiTdi sang chpn dtfdc mot triTcJng dai hoc tot, ch^c ch^n r^ng triT^ng se chon diem cao NhiT vay qua trinh on luyen cac em can mot tai lieu tUcfng thich Dieu se diTdc thoa man neu cAc em chiu kho on tap theo cac chuyen de mh chung toi bien soan, ch^c ch^n thi se dai diTpc diem cao I 'K^mdt m MJOl C«0 GI(3l HAN Mong muon ciia chiing toi l ^ m the nko de c^c em tif hpc to't mon Toan, va kha nang thi dau v^o cdc tri/dng Dai hpc Idn phai nh5 vko sir kien tri cua cic em "c6 cong mai s^t c6 ng^y nen kim" cd nhan day that diing vay Quyen sich cung 1^ ky nipm nim cic em hpc Toan vdi thay: Bich Lien, PhiTdng Thao, Anh ThiT, Mai HiTdng, Thiiy Hien, Th^o Uyen Cic em da giiip thay chinh sufa ban thao, mpt cich nhipt tinh trdch nhipm Chiic c^c em se th^nh cong my man ky thi s^p idi Chu bien: Nguyen VSn Thong (To trirdng To Todn trirdng chuyen Le Quy Don - Da NJng) Nhd sdch Khang Vi$t xin tran trgng giai thi^ tai Quy dgc gia va xin Idng nghe tnoi y kien dong gop, de cuan sdch ngdy cdng hay han, bo ich hart Thu xin giH ve: „,.^- _ Cty T N H H M p t Thanh Vien - Djch Van Hoa Khang Vi?t 71, D i n h Tien Hoang, P Dakao Qu^n 1, TP H C M Tel: (08) 39115694 - 39111969 - 39111968 - 39105797 - Fax: (08) 39110880 Hoac Email: khangvietbookstore@yahoo.com.vn z^r TWM I \ F M TRONG DI^M vA TfNH UfeN TUC C6A C a c d j n h If ve g i d i h a n M O T H A M S6 ••1 ( ' Gia sur lim f ( x ) = L, va lim g(x) = L j , X-^X() lim [f(x) X-+X,) + g ( x ) ] = lim f ( x ) + lim g(x) = L | + L x->X() lim "->"() lim "-+"() x-^xo [ f ( x ) g ( x ) ] = l i m f ( x ) lim g(x) = L|.L2 x^X() x-^xo f(x) lim X-^X() Neu g(x) lim f(x) lim g(x) X^X() , _ ^1 X-*X() =' 'v,V' ''->X() L2 vdi 3s > cho f ( x ) < g(x) V x € (xo - e; x,) + s) va ton tai lim f ( x ) , lim g(x) thi lim f(x) < lim g ( x ) x->X() a "-^^o [ f ( x ) - g ( x ) ] = lim f ( x ) - lim g(x) = L , - L x->X() bang xep Cic chuyen de chiing toi deu soan tilf can ban den nang cao, dp kho dii de cho cac hpc sinh khd va gioi tU' luyen, neu thong hieu taft ca, ch^c ch^n ring cac em se giai du'pc de thi Dai hpc mpt cdch de dang, nh\i chiing toi da tilfng luyen rat nhieu em thu khoa cic tru'dng Dai hpc danh tieng Dieu quan hdn nifa vdi each vie't ciia mOt gido vien day chuyen Todn lau nim, nen each gpi md, d i n d^t mang dam n6t t\i nang cao, hpc sinh se diTcJc phat trien kha nSng Toan hpc hpc quyen sdch n^y inx-r I T M TAT L I THUYET Quyen sdch gom 20 chu de diem theo cau true de thi cua Bo giao due hang nam, cac vi du duTa ttfcfng doi kh6 va c6 hiTdtng dan giai Tie'p theo la 28 bp de thi cho cac khoi A, B, D cua tri/cfng chuyen Le Quy Don - Da N^ng dilng de thi thijr tiTng nSm qua 66 danh gid diTdc hpc sinh, nit kinh nghiem giang day TriTdng Le Quy Don - Da NSng hang nam deu c6 ti 16 dau vao cac trirdng dai hpc la 100% Rieng nam 2010 c6 so hpc sinh dat diem cao diTcfc xep thu" hang cua Bp giao due va Dao tao mill ''-»X() x^XQ , li • x->X{) NguyinIfgidinank?p: Neu E > 0: f ( x ) < h(x) < g(x) V x e (x„ - E ; X„ + e) \} va lim f ( x ) = lim g(x) = L thi lim h(x) = L x^X() x-»X() x-»X() b Cdc dang gidi han dQc bi$t lim X->X() lim X->X() = ; lim (1 + x ) x = e X X^XO = e ; lim x-»X() I,,,,,.,.,.;,.,: e^-l X = ; lim ln(l + X) = x->X() G i d i h ^ n d a n g v d d j n h : — P(x):dathu'c, P(x„) = > P(x) a Dang I = lim — ^ vdi < x ^ x „ Q(x) [Q(x): da thiTc, Q(Xo) = PhiTdng ph^p: — - i' 1= Hm lim i ^ i ^ ^ i o l W ^ P ^ ^ P ^ x^xoQ(x) x->Xo(x-Xo)Qi(x) X^X()Q,(X) Qi(Xo) vdiQ,(x«)^0 , J I IWtg, „ Ne'u Pi(x„) = Qi(x,)) = thi phan tich tie'p • Qu^ |,W C6ngiyTNi:ii P,(x) = ( x - X o ) P ( x ) Q,(x) = ( x - X o ) Q ( x ) trinh khuT dang v6 dinh ^ la qua trinh k h u r cac nhan tuT chung ( x - X o ) ' ' s e difng l a i k h i nhan du'dc g i d i han xac dinh ttfc la Qk 5^ Khid6I= x->xo ' f ( X ( , ) = g(x„) vdi \ g(x) Ng'u p < q thi ton t a i g i d i han • N e u p > q t h i khong ton tai g i d i han ; W / / A / U ; ^ 0>t Gi6i han dang v6 dinh « - oo Phifdng phap: B i e n d d i diTa ve dang g i d i han — l i n - ^ = l i m A W ^ M ^ >'^''()Q(x) x ^ x „ Q , , ( x ) Qk(xo) b Dang l i m • : ' T i m g i d i han sau = O O f ( x ) , g ( x ) chtfa can thtfc dong bac lim (x + V x j - x = lim x^+«>^x + ^/^+^/x " Vx + V x - V x = l i m J x-++ x - - lim Vx^ + l - x = l i m X->+oo mil tioub fi? i m Vx^ +1 +: l + -^+l X A-B Gidi h^n dang vd djnh ham lUging giac 2"M/A+2n+^^ c 5.1 = ' PhtTdng phap: Suf dung cac ket qua g i d i han cd ban sau dSy: A + B hm vdi x^xog(x) [ f ( x ) chtfa can khong dong bac • , sinx , , X , hm = 1; h m =1 x->0 X x->() sinx • lim x->0 sin ax X = lim x-»0 sin ax I ax V , sinax = a h m x->() ax , sinax a => h m =a x^O X „ sin ax ^^"0 g(x) g(x) x^XQ h m mil g(Xo) = x->()sinbx Bi6n doi 1= lim x->xo sinax '?yu(v)-ci-rjyv(x)-c - lim g(x) •^uW-c 5!/v(x)-c g(x) g(x) X-+XO lim x-»o D e n day cac g i d i han diTdc tinh theo dang • Gi6i hain v6 djnh — 00 PhiTdng ph^p: X t I = tan ax x = — iUm iiii ax - ^ a , = — -—( a , b e R * ) x-+()sinbx b = ^^lim smax —:—-— — — —• 11111 x-»()bx sinbx bx b a sinax , tanax hm = a => h m =a x->0 X x^ocosax ax tanax , ax ax a Hm = hm — • =x->otanbx x->{)bx tanbx b tanax tanax , hm x->()tanbx a b bx lim ^ v d i P(x), Q(x) la cac da thi?c hoSc cic ham daj s6 G p i bac P(x) = p; bac Q(x) = q v^ m = min(p, q ) , k h i d6 chia ca mau cho x " ta c6 k e t luan sau: va x-»xo Sinax g(x) hm sinax x-*() tanbx , sinax , ax a x hmcosbx = hm—.cosbx x->() sinbx x->obx sin Dx bx ^ b r ,UI Luyfn gidi di truOc kp ihi DH miin Bdc, Trung, Nam Todn hoc - Nguyin Van ThOng Hifdng din giai Gidi han dang v6 djnh 1° Ta Phifdng phap: f a Sijfdung: lim(l + x ) " = e ; lim x->() = e ,,,, t'l • • • I-.,, lim ''1 " X-»X() ^JZ^ x-»3 =1 > X I- v l ^ - x->0 US;' fili'd i&ili ffH'T (u-ir X \ X^ Hifdng d§n giai (l + x ) - l Ta c6: lim •+ X — x-.(.^2 l + (u-l)"-' Bie'n doi lim '^^^^ = (X-3)X Bai Tim hm b Xct lim uCx)"*"' CO dang T X^X() lim x^3 1+ - x^+«> CO Gidi han d?ng v6 djnh cua ham mu va Idgarit: ^ 3/^1777+^/i77^^1 - - == lim iim , , = — ''-*"^(i+x2)+^/iT^+i Bai Tim hm ^^^ ' * x-»() X Hi/dng d i n giai Phifdng phap: S^rdung l i m ^ x-»() ^ = 1; lim X ^ x->() =1 Ta C O X II B A I T A P M I N H H Q A , (2N/rT7-2) + ( - ^ / ^ ) hm x-»() X , lim X x^O Bai Tim gidi han lim f ( x ) , vdi r(x) = x^-1 X->1 ,„;., Htfdng din sial Ta CO Vs-x^ - lim x^l = Hm x-^o \/x^ + - ^ x^-l x->() hm = lim X Va lim x->l X[4 + W ^ = hm Matkhac: X ^ l >/l + x + l ) Vx" + - X -1 = lim X->1 " - l / v ^ -1 - f x ^ + x + l) \ ' ^ / ^ ^ + 1^4 + 100 v2 x^+7 x^ +7 + 12 ^ ^ ^^100 ^ j ^ ^ N 100 + + + lim - = 100 100 (3) 19^ loo^ll^ + ^ + ^HX •a Bai Tim gidi han lim Thay (2), (3) vao (1) la difdc L = - - - — = - — 12 24 ™ , x'^ - x ^ +4x - B a i , Tim l i m — "-^^ x 3x ,^100 1+ X-»+QO 12 Hi/dng din giai Ta c6 x^l ""12 (x + 1)'"" + (x + 2)'"*' + ••• + (x + 9)'"" + ( X +100) Bai Tim lim X-»+oo x'"" + 10x'" + 100'" ^"'(x^-lWx^+vf+2^/?77 x^+7+4 = lim +^(8-x)2J ^ l(X) (2) x^-l = lim "->! + N/(8-X)^ (1) x^-l 1-x-^ fijsi 2x x-»l m 1-x"" , (m, n e N * , m 1-x" Hi^dng dan giai lim X->1 f_n m Al-x'" l-xj i_^] :; n) , , Luy(n gUU d6 trade lim thi DH miin Bdc Trung Nam Todn hoc - Nguyen Van ThOng m - ( l + x + x^+ + x'""') n - ( l + x + x^+ + x""') l-x"" 1-x" X-+1 lim sin ( l - x ) + ( l - x ^ ) + + ( l - x " ' " ' ) ( l - x ) + ( l - x ^ ) + + ( l - x " - ' ) (1 - X)(l + X + x^ + + x " ' " ' ) (1 - x ) ( l + x + x^ + + x"~') x-»l lim 2sm l + ( l + x) + + ( l + x + + x " ' ' ^ ) l + ( l + x) + + ( l + x + + x""^) l + x + x2+ + x " ' - ' + x + x^ + + x " ' t m(m-l) l + + + ( m - l ) n(n-l) = m-n 2 2 " ' sma 1- sm— , • cos = Hm smcx l-iVcosbx l-'^cosax (cx)2 x-*() A p dung (1): 2" C= (cx)2 u e B a i T i m g i d i han sau: I = Hm BaiS.Tim m 2c^ n m , a a a cos—.cos— cos2" lim n->-H» e-^" I 1- ^ a a a A = cos —.cos — c o s 2 2^^ sin ^ 2" 2^^ a a /sm — cos cos 2" 2" j ;a^0 •= Hm 3cos^ X x->() x->() e" B ^ i T i n h g i d i han I = l i m cos ,n-l 3x2 i• ' Sin X ^ I ; X - V c o s x + l n ( l + x^) ' •! x-»() a a ijji'j = 3-1=2 a = COS x" - Icos^x + c o s ^ x - l = Hm lAvtdng dSn giai A cos x - H i M n g d S n g i a i '"'^ + "Q^cosax + + \/cos"' ' a x aM n , l i m ^ i M L • = = x->o sm(tanx) tanx 1-cos ax l + !ycosbx+ + N/cos"-'bx jj,, —cosx x^O 1-cosbx bM , Hifdng d i n giai r x-X) ,, cos — cosx B a i T i m g i d i han sau: C = Hm — ; x-»o sin(tanx) (1) (cx)" C I ' , - j j ••• n a r = -^lim sma 2" III B A I T A P T i ; L U Y E N C O D A P S O (l-jycosbx)-(l-".ycosax) x^O i'.'Jir] sm- Hifdng d i n giai lim sma 2" I 1-cos ax 2".sin-^ 2" a ' „ %/cosax-Vcosbx ^, ,^ B a i T i n h L = l i m -^^ (a, b, c la cac so thifc khac khong) m , i "-»(' sin cx -H» lim y = lim (-x +2x + -5x ) = X->-QC X->- lv*i-. K;vM.!: • = l + m Won y'=0« [x = - m SB" 0* D i e u k i e n de h a m so c6 ctfc dai, cifc tieu la y ' = c6 nghipm phan biet PhiTdng trinh hoanh giao d i e m ciia (A) v^ (C) IS: -x^+2x^+-x x 3 • x = 0=>y = • x = 2=>y = o - x ^ ( x - ) = 0c^ om?[...]... x2+y2- = 3 (2) Bai 19 Cho he phiTdng trinh: Tim a de he phu'dng trinh c6 nghi0m duy nha't a log3 a o + log3 a = 1 + log3 5 log3 m b= ^ a (1) log, a(l + log3 m) = (1 + logj 5).log3 m a m = 5, he phiTcJng trinh trd th^nh b=5 a =5 a o a lb=i log3 a(l + log3 5) = (1 + log3 5) log3 5 log3 a = log3 5 Hifdng dSn giai PhiTcJng trinh (1) tiTdng diTcJng vdi f •* X - a + l o g 3 ( x - a) = l o g 3 ( y - 2a)... = 32 x' - 18x y' = 0 3 x = 0, x = ± - ii 0 m c:^ i Li^n COng ty TNHH giii di tniOc Aj> thi DH 3 miin Bdc, Trung, Nam Todn hQc - NguySn Van Thdng — 4 Va nghich bi6'n tren ( I Ncu 0 < 1 - m < 1 o 3 H a m so dong b i c n tren 3^ -co; ;+tx3 A',, Neu 0; — 4j 4; H a m so dat ciTc dai tai (0; 1) va dat ciTc tieu tai X -00 y' y +00 - 0 + r3 49 ^ ( 3 4 ' 32 J 49^ 4 ' " 32 Ta CO y " = 9 6 x ' - ^ 49 49 32 32 ... + 3) m^+2m-12 x +- uon G'd sur ho^nh do cua mot trong cac d i e m uon cua do t h i h ^ m so da cho 1^ x„ va day cung la nghiem cua phiTcfng trinh (*) hay X Q + 3axo + 3( a - 1)X(, - 1 = 0 ; khi d6, tung do trfdng uTng cua d i e m n ^ y chinh 1^ yo = —5-^ Xo + Xo Ta se t i m X Q + 3axf, + 3 a x „ + 3a - 1 = 3 x „ + 3a o + x„ +1) (x,) + 3a - l)(xo + x,, +1) = 3 ( X Q + a ) ^ XQ + 3( Xo + x„ +1) ' :d 3a... u v * Do thi ham so c^t true tung va true hoinh tai (0; 0) (ve h\nh) (X + 1 ) - = 1 o b) Goi X = 0 , X = 2 M(X(,; y„) la mot diem thuoc do thi Khi do tiep tuyen cua do thi tai M \k -3 Do do, hai diem cifc trj cua do thi chinh la (0; m + 1), (-2; m - 3) nen khoang y= K -3x -(X-X(,) + 4(Xo -3) 4(xo -3) 4xf)-9X() _, = y 4(Xo -3) ' 4(xo -3) ' each giffa hai diem nay la d = V(0 +2)^ + (m + 1 - m + 3) ^ - 2 y... ^ ( l + a) (l + log3 5).log3m r-* he phiTcfng trinh trc( thanh \P Do3x + 2 y < 5 n e n a < 5 1 + log3 m (l + loga5)logim , — - 2 ^ - ^ — £ ^ = logj a < logj 5 1 + log3 m < lx^-y^=a y = ^(i-a) Ket luan: Vdi moi a e (0; +oo)\ 1; 2}, h? phu'dng trinh c6 nghiem duy nha't (x; y) = b H? phtfdng trinh (1) tiTdng diTcing vdi: a = 2 (loai) ^Iog3nLll2g3£0 0 + 0 + VT(1) Vay nghiem cua bat phiTdng trinh la 0 < x < 2 hoSc x > 4 li-.^:-,:,,:.; ^^i '^11 ft) ? | ;0 4'''''"''U-.4l'-''''''U-4'''''''''U-.4 = x'' - 3x^... ciia do thi Ik (0; 0), diem cifc tieu cua do thi ham so' la ( - 1 ; -1), I'Vi o - 1 3 = -2xo+6xo-5 + 6xo+6xo-12x„-12xo o xf, - 3X() +2=0o X(, (1;-1) Bang bie'n thien = 1 V x„ = -2 +00 0 0 V d i M ( l ; - l ) thi phiTdng trinh tiep tuyen can t i m i a : +00 +00 y + l = 6 ( x - l ) c ^ y = 6x-7 - '* -1 -00 ' Ta CO cdc tung do tiTcJng iJng la y ( l ) = - 1 , y(-2) = 35 - iftj' Vdi M ( - 2 ; 35 ) thi phu'dng