Phần 1 tài liệu Giải bài tập giải tích 12 nâng cao do Nguyễn Đức Trí biên soạn cung cấp cho người đọc kiến thức cần nhớ và phương pháp giải các bài tập ứng dụng đạo hàm để khảo sát và vẽ đồ thị của hàm số, Mời các bạn cùng tham khảo.
515.076 GI-103B ^BlfCCHI NANG r DVL.013442 m Nha xuat ban Dai hoc Quoc gia Ha Noi NGUYEN oCfC CHI ^UJOI bai tdfL I A I T I C H 12 XAXG CAO THU VlifJ l\m BINH TH'JAN NHA XUAT BAN DAI HOC Q U c GIA HA LCJl NOI DAU duac bien soan v6i muc dich giiip hoc sinh doi chieu va kiem tra lai cac ket qua thtfc hien giai cac bai tap sach giao khoa Muon the, cac em hay danh thcri gian nhat dinh de lam cac bai tap sach, sau doi chieu va kiem tra lai ket qua thiTc hien G I A I B A I T A P D A I S O 12, phu huynh c6 the suf dung de kiem tra minh viec hoc tap va luyen tap cac kien thufc va ky nang CO ban G I A I B A I T A P D A I S O 12, GlAl B A I T A P D A I S O 12, cac dong nghiep c6 the suf dung de tham khao Mong diioc sii gop y chan cua ban doc gan xa TAC GIA TJTNG D U N G D A O H A M D E K H A O S A T V A V E D6 T H I C U A H A M S O §1 TiNH DdN DIEU CUA HAM SO ivdl DUI\ cAi^ I^IOf 1, D i n h l i : Gia s i l hkm so f c6 dao M m t r e n k h o a n g I • Neu f '(x) > 0, Vx e I t h i h ^ m so f dong b i e n t r e n k h o a n g I ã NĐ'u f '(x) < 0, Vx e I t h i h ^ m so' f nghich bien t r e n k h o a n g I • Neu f '(x) = 0, Vx e I t h i h ^ m so f k h o n g d6i t r e n k h o a n g I Chii y: K h o a n g I neu duoc t h a y b k n g mot doan hoac m o t niira k h o a n g t h i p h a i bd sung gia t h i e t " H a m so' l i e n tuc t r e n doan hoac niifa k h o a n g do" V i # c x e t c h i e u b i e n t h i e n ciia h a m so' c6 dao h a m c6 t h e chuyen ve viec xet dau dao h a m ciia h a m so' ^ B A I T A P (Bai trang SGK) a) y = 2x^ + x ^ + H a m so' xdc d i n h t r e n Ta c6; Gidi R y' = Gx^ + 6x y' = o 6x^ + 6x = o 6x(x + 1) = o X = x =- l Bang bien thien: X —00 + y' y -1 0 — ' — * — +0C + , — Vay h a m so' dong b i e n t r e n m o i k h o a n g (-oo; - ) va (0; +oo), n g h i c h bien t r e n k h o a n g ( - ; 0) b) y,= x^ - 2x^ + X + H a m so' xdc d i n h t r e n K T a c6: y' = 3x^ - 4x + y' = 3x^ - 4x + = o X = hoac x = - Bang bien thien +00 V§y hkm so d6ng b i e n t r g n m o i k h o d n g ( - 0 ; ~) vk ( ; Bang bien thign n g h i c h bien X tren khoang c) y = X + — X (Bdi trang GSK) • x-2 a) y = —7; • X + H a m so xacx d+i n2h- (x t r e-n 2) •Ta c6: y = (x + 2)^ H ^ m so xdc d i n h t r e n M \1 Ta y' = y'^Oc=>l CO 5- = c ^ x ^ = o x = ±73 Bang bien thien: X -73 —00 + y' II - • J3 + X -2 —00 + + X y = +1 + (-2x - 2)(x + 1) - ( - x ' - 2x + 3) +0C II y Vay hkm so dong b i e n t r e n m i k h o a n g ( - 0 ; 0) e) y = x^ - 2x^ - + (x + Bang bien thien X (0; + 0 ) H a m so xac d i n h t r e n M y' = 4x^ - 4x y' = 4x'' - 4x = o 4x(x'' - 1) = o • ^x = x = ±1 - x ' - 2x - 2x - + x ' + 2x - (x + l f _ - x ' - 2x - —00 (X —00 - CO f '(x) = 3x^ - 12x + 17 f '(x) > 0, Vx e R Vay h a m so' dong b i e n t r e n R Tac y - ^ ^ y' = » X = +x — - — • y V a y h a m so n g h i c h b i e n t r e n m§i k h o d n g ( - 0 ; - ) va ( - ! ; • + « ) , (Bdi trang SGK) Gidi a) f(x) = x'^ - Gx"^ + 17x + H a m so xac d i n h t r e n R Ta V a y h a m so dong b i e n t r e n m i khoang ( - ; 0) va ( ; + 0 ) , n g h i c h bien t r e n m o i k h o a n g ( - 0 ; - ) va (0; 1) + < 0, Vx e R\{-11 y' Bdng bien thien Oy = - x ' H a m so' xac d i n h t r e n [-2; 2] +00 H a m so xac d i n h t r e n M\{-11 CO CO J > 0, Vx e !R\1-21 (x + 2) -x'-2x + b) y = y' = + ^ > 0, Vx e M \1 x'' Bang bien thien Ta 1-2) y Vay h a m so d6ng bien t r e n m i khoang ( - 0 ; - ) va ( - ; + 0 ) X y Gidi y' H a m so \±c d i n h t r e n R \) X Bdng bien thien: +00 y - — ' ^ ~ — — — — — " — ' — — > V a y h a m so' dong b i e n t r e n m i khoang ( - 0 ; - 73 ) va (0; 73 ), nghich b i e n t r e n m o i k h o a n g ( - 73 ; 0) va (73 ; + 0 ) d) y = X Ta + y' y +2 -2 b) f{x) = x^ + X - cosx - H a m so xac d i n h t r e n R • Ta c6 f '(x) = 3x^ + + sinx > 0, Vx e R V a y h ^ m so dong b i e n t r e n R (Bdi trang SGK) ftx) = ax - x^ H a m so xAc d i n h t r e n R T a c6: f '(x) = a - Sx^ Gidi * N6u a < t h i f '(x) < 0, Vx e R H ^ m so nghich bi§'n t r g n x'' - 8x + x-5 H a m so xac d i n h t r e n R \} c) y = * Ng'u a = t h i f '(x) < 0, Vx e R DAng thiJc chi xay k h i x = Vay h a m so' n g h i c h b i e n t r e n R y = * Neu a > t h i f '(x) = cs- x = ± j V3 Bang bien thien x f(x) " if ~ V3 f '(x) — — — + +00 f '(x) = x^ + 2ax + 4, c6 A' = a^ - * Neu a^ - < hay - < a < t h i f '(x) > 0, Vx e R H ^ m so dong bien t r e n R - H a m so dong bien t r e n R H ^ m so dong b i e n t r e n M -_co d)y= Xi X2 f'(x) f(x) (Bdi trang GSK) Gidi y' y H a m so dong b i e n t r e n khoang (0; 1), nghich b i e n t r e n k h o a n g ( ; 2) H a m so xac d i n h t r e n R (do x^ - 2x + > 0) +00 , _ 2(x - ) ~ 2N/X' - X y' = O X = Bang bien thien X _ + x-1 V x ' - 2x + —00 +00 y' + V-— -> - 2x x +1 H a m so' xac d i n h t r e n R \) f)y = l i e n tuc tri§n R 4x + = (x - 2f > 0, Vx e K H ^ m so' dong b i e n t r e n R + 6x2-9x- y H a m so dong b i e n t r e n khoang ( ; +00), nghich b i e n t r e n k h o a n g (-00; 1) a) y = - x ^ - 2x^ + 4x - b) y = - l x ^ 1-X • TAP y' = x^ - , y = + H a m so n g h i c h b i e n t r e n ( x j ; X ) k h o n g thoa de t o a n H ^ m so' xdc d i n h , V2x - x ' y' = < » X = Bang bien thien X Vay - < X < t h i h a m so da cho dong bien t r e n M M LlJYEl^ N/2X-X' ' ) y = Vx' - x + • N e u a > hoac a < - t h i f '(x) = c6 h a i n g h i e m x i , X2 (xi < X ) Bang bien thien X + + Ta co: fix) = - x^ + ax^ + 4x + 3 H ^ m s6' x^c d i n h t r e n R • N e u a = - t h i f '(x) = (x - 2f > 0, Vx +00 y H a m so dong b i e n t r e n m o i khoang (-00; 5) va (5; +00) rr Gidi • N e u a = t h i f '(x) = (x + 2f > 0,\fx* >0,\fx^5 H a m so' xac d i n h t r e n [0; 2] V£iy v d i a < 0, hkm so nghich bien t r e n R (Bdi trang SGK) 5)^ (X - y' ' — " > (x-5)^ Bang bien thien —00 x - /—; I — Vay a > k h o n g t h o a dieu ki§n de toan H ^ m so' dong b i e n t r e n (x-5)^ x ' - l O x + 31 fa —00 (2x - 8)(x - 5) - ( x ' - 8x + 9) _ x ' - ISx + 40 - x ' + 8x - - - x ' - 4x - - x ' - 4x - (x + 1)' (x + D' < 0, Vx H ^ m so n g h i c h b i e n t r e n K M \I BAng b i e n t h i e n -1 +00 H ^ m so xdc d i n h va l i e n tuc t r e n R y' = -4x^ + 12x - = - ( x - 3)^ < 0, Vx € R G y >• H a m so n g h i c h b i e n t r e n m i khoang (-00; - ) ( - ; +00) iBdi trang SGK) Gidi c) X e t h ^ m so h(x) = x - — - sinx f(x) = cos2x - 2x + H a m so' xac d i n h t r e n R CO f '(x) = - s i n x - = - ( s i n x + 1) < 0, Vx e K h'(x) = - — - cosx Theo cau b) h'(x) < 0, Vx * f '(x) = sin2x = - o 2x = - ^ + 2k7t x = - ^ H a m so n g h i c h b i e n t r e n m o i doan Vay h a m so' n g h i c h b i e n t r e n R (Bai trang SGK) + k t ; - — + ( k + 1)71 +k7r, ke , k e Z x^ ^ X va Gidi sinx < => X f i x ) > f(0) Vx e ; ^ : f '(x) = - cosx > 0, Vx e , 0; V 2] N g h i a l a X - sinx > 0, Vx e Co f '(x) = cosx + cos^x cos^x - 2, Vx 0; Do h a m so f i x ) dong b i e n tr§n ( 0; V 2j hay X > sinx, Vx e 0;-, 2) Chufng m i n h tuong tyf sinx > x, Vx < Ta c6: f(x) > f(0;, Vx e H ^ m so l i e n tuc t r e n nufa k h o a n g [0; +ao), g'(x) = - s i n x + x Theo cdu a, g'(x) > 0, Vx > 0, d6 h ^ m so g dong big'n t r e n [0; + « ) , t +5 a) N a m 1980, t a c6 t = 10 ^^^^^^ g(x) > g(0), Vx > 10 + x' 270 15 = 18 (nghin) S6 d a n cua t h i t r a n n a m 1980 l a 18 n g h i n ngif^i 1>0, Vx>0 N a m 1995, t a f(25) = TCr d6 suy r a v d i m p i x < 0, t a c6: CO t = 25 26.25 + 660 25 + 30 = 22 So d a n cua t h i t r a n n a m 1995 l a 22 n g h i n ngKbi (-x)^ > hay cosx + x" cosx > - y , Vx Gidi 10 (Bai 10 trang SGK) x^ b) X e t hkm so g(x) = cosx + — - Vay > sinx, Vx < > 2, Vx e hay sinx + t a n x > 2x, Vx e cos{-x) + cos'x Vay x > sinx, Vx > N g h i a l a cosx + x^ > cos X + Vay sinx + t a n x - 2x > CO < sinx, Vx > Gidi M a t khdc x > sinx, Vx > — v i sinx < Va t a h ( x ) < h(0), Vx > Xet h a m so' f(x) = sinx + t a n x - 2x l i e n tuc t r e n niira k h o a n g Do cos^x + CO h(x) > h(0), Vx < hay x Do h a m so d o n g bi§'n t r e n Ta x^ J, (Bai trang SGK) a) X t h a m so' f(x) = x - sinx Hkm so l i e n tuc t r e n Do h a m so h nghich b i e n t r e n K 1>0, Vx 0, V t e (0; +00) c) Toe dp t a n g dan so vac n a m 1990 1^ 120 f ' ( ) = -7—=^ ~ 0,192 (20 + ) ' Toe t S n g dan so' v^o n&m 2008 1^ 120 f'(38)= * 0,065 (38 + 5)' 190 00 • Ta — = 0,125 o (t + 5)^ = « (t + 5)' 0,125 C O X, Xi f'(x) •f'(X) f(X) I f(x) f(Xi) cUc tieu t + » 31 * Qui tac • T i m f '(x) • T i m cac n g h i ^ m X i ( i = 1, 2, ) cua phupng t r i n h f '(x) = • T i m f "(x) v a t i n h f " ( X j ) + Neu f "(xi) < t h i h a m s6' dat cUe d a i t a i d i e m Xj + N e u f " ( X i ) > t h i h a m s6' dat cUc tieu t a i d i e m Xj t = 26 Vao n a m 1996 toc dp t a n g dan so ciia t h i tra'n l a 0,125 §2 CaC TR! CUA HAM SO J B A I T A P n.(Bai 11 trang 16 - 17 SGK) Gidi a ) f ( x ) = i x ^ + 2x2 + x - 1 D i n h n g h i a : G i a suT h a m so' f xac d i n h t r e n t a p H a m s6' xac d i n h t r e n K f ( X ) = x^ + 4x + 3; f ( X ) = o x^ + 4x + = o X = - hoac x = - c M va XQ e y * XQ dixac gpi l a m o t d i e m ciJ^c d a i ciia h a m so' f neu: T o n t a i m o t k h o a n g (a; b) chiJa XQ cho (a, b) c va f(x) < f ( x o ) v d i m p i x e (a, b) \ K h i f(xo) dupe gpi l a g i a t r i cufc d a i cua h a m so f * Xo difpc gpi l a m o t d i e m cxic t i e u cua h a m so' f neu: T o n t a i m o t k h o a n g (a; b) chufa XQ cho (a; b) e D va f(x) > f(xo) vdi mpi x e (a; b) Bdng bien thien Cdch CLCC CLTC -1 Vay h a m so da cho dat cUc d a i t a i d i e m x = - , gia t r i eUe d a i l a f ( - ) : - H k m so dat cUc tieu t a i d i l m x = - , gia t r i cUc tieu cua h a m so l a f l - ) = \ K h i f(xo) dupe gpi l a g i a t r i cij^c t i e u ciia h a m so' f * D i e m cifc d a i va d i e m cUc tieu gpi chung l a d i e m cu'c t r i Gia t r i cUe d a i va gia t r i cUc tieu gpi chung l a cu'c t r i * Neu X o l a d i e m t r i cua h a m so f t h i ( X Q ; f ( x o ) ) l a d i e m cu'c t r i ciia t h i h a m so f D i e u k i ^ n c a n de h a m so d a t ci/c t r i Gia sijf h a m so f dat t r i t a i diem X Q K h i d6, neu f c6 dao h a m t a i thi f '(xo) = D i e u k i ^ n d i i de h a m so d a t ci^c t r i Gik sijf h a m so f l i e n tuc t r e n k h o a n g (a; b) chiJa d i e m x o v a c6 dao h a m t r e n cae k h o a n g (a; X Q ) ( X Q ; b) K h i a) Neu f '(x) < v d i m p i x e (a; X Q ) v a f '(x) > v d i m p i x e ( x o ; b) t h i h a m so' f d a t cvtc t i e u t a i d i e m X Q b) Neu f '(x) > v d i m p i x e (a; X Q ) v a f '(x) < v d i m p i x e ( X Q ; b) t h i h a m so d a t cue d a i t a i d i e m X Q Q u i t S c t i m c\ic t r i * Q u i t ^ c 1: • T i m f '(x) • T i m cac d i e m X i ( i = 1, 2, ) t a i dao h a m cua h a m so' hhng hoac h a m so' l i e n tuc n h u n g k h o n g c6 dao h a m • X e t da'u f '(x) neu f '(x) ddi da'u k h i x qua d i e m Xj t h i h a m so' dat cUc t r i tai Xj +0C -3 1-00 X 2: Ham so xac d i n h t r e n M Ta c6: • f '(x) = x^ + 4x + 3; f '(x) = c : x ^ + 4x + = o x = - l hofic x = - ; f "(x) = 2x + =-1; V i f " ( - ) = ( - l ) + = > n e n h a m so d a t cUc t i e u t a i d i e m x XQ f(-l) = - = -3, f " ( - ) = ( - ) + = - < n e n h a m so d a t cUe d a i t a i d i e m x f(-3) = - b) f(x) = i x^ - x^ + x - 10 H a m so' xac d i n h t r e n R Ta C O f '(x) = x^ - 2x + > 0, Vx e K H a m so d o n g b i e n t r e n R , k h o n g c6 cUc t r i c) f(x) = X + — X H a m so xac d i n h t r e n f '(x) = - = R \1 x^-1 • f '(X) = « x ' - l = Oc:>X = ±l Bang bien thign -1 —oc X f'fx) f(x) + ^ Bang bien thien () — — + 00 + f (x) - \ ^ X >0 I (Bcii 12 trang 17 SGK) + x = -1 y' = ^ ~ V4-x' 0 f (x) + fix) 0« V2 x = ± 0 0 • +C0 , ~^ 78^ vdi moi x € 15 15 x • • 72 ] (-2 72; 72 ); y' = 0o x = 0 -2sl2 + 272 " 272 va h a m so • • c) A p dung qui tAc y = x - sin2x + H a m so xac d i n h t r e n R y' = - 2cos2x y' = 2cos2x = o x ' - 2x (x - ) ^ (x-1)^ f '(x) = c : > x - x = 0x = hoac x = -2 H a m so dat cifc d a i t a i x = 0, gia t r i cxic d a i : 72 Ta c6: x-1 H a m so' xac d i n h t r e n M \| ^0 Bang bien thien 28 x^ - 3x + (2x - 3)(x - 1) - ( x ' - 3x + 3) fi-1) ~ , gia t r i cUc d a i 1^ y 32 ^2— - y' Vay h a m so dat ciTc dai t a i x = - , gia t r i cvtc dai dat cUc t i e u t a i x = 1, gia t r i ciTc tieu: fil) = 72 + H a m so xac d i n h va l i e n tuc t r e n [-2 72; + 15 72 —* -2-—- b ) y = 78-x' y' = -1 -72 H ^ m so dat c\ic d a i t a i x = X = hoac x = ±1 32 f '(x) = = H a m so d a t cifc t i e u t a i x = - 72 , gia t r i c\ic t i e u l a 15 f) fix) = (-2; 2); y' e y' y —00 X -2 x _ Vay h a m so' dat cifc dai t a i x = - , gia t r i cUc d a i fi-1) = va dat ciic tieu t a i x = 0, gia t r i cifc t i e u fiO) = x^ x" e) fix) = — + H a m so' xac d i n h t r e n M X v6i m o i 2] Bang bien thien fix) f '(x) = x^ - x^ = x^ (x^ - 1) f '(x) = x = = ±— +k7t, ±|^ keZ o + k27x, k e Z * Vi y" — + kn = 4sin = 4sin = _ ^ I 3J f l a m so x a c d i n h t r e n K Vay h k m so dat ciTc t i e u t a i diem x = va f(0) = sinx = -1 + 2cosx = = 2cos b = f(x) = x"* + ax^ + bx + c sinx = + k27: h a m so f "(x) = - x + +2 CO t • V i y"(k7i) = 2cosk7r + 4cos2k7i = 2cosk7i + > 0, V k s Z - a - 2b = - K i e m t r a l a i k e t qua + - s i n - + k27I N g o a i r a y " = 2cosx + 4cos2x ± ^ a +b = l a = -2 Ta c6: y' = 2sinx + 2sin2x = 2sinx + 2.2sinxcosx = s i n x ( l + 2cosx) • Vi y" 3a + 2b = a? + 2b = 3a + 2b = y = - 2cosx - cos2x H a m so xac d i n h t r e n R y' = f(0) = = 4sin^ = ^ =273 >0 - + k t d) A p dung qui tic CO * Ta + + kn + — d = H a m so d a t cifc t i e u t a i d i e m x = 0, n e n : f (0) = => c = • Ta Do h ^ m so d a t cUc t i e u t a i cAc d i e m x = - + ku, k e Z Gid t r i ciTc t i e u y — + kn Giai 13 (BM 13 trang 17 SGK) f(x) - ax^ + bx^ + cx + d = - 4sin = + 2cos^ + c o s - = + 3cos^ = ^ 3 o - a + b + 12 = ] - a + b + 12 = 3a - 3b - - - a + 27 = a+b+c+l=0 +c+1= c = -4 -12 + b + 12 = b = b = a = a = a = K i e m t r a l a i k e t qua f ( x ) = X-' + 3x^ - f ' ( x ) = 3x? + 6x, f '(x) = o X = r\\ / n tt ^ ^ / { 2J = -2 PhiTcmg trtiih tiep tuyen v6i (•'i') tai diem B -2 hay y = I + - ^ y = -2x+2 d ) Ta CO fix) - g(x) = 2x'* + 3x^ + - (2x^ + 1) = 2x=' + B a n g x6t dau: 1^ X + — 2) X + — X fix) - 2J - g(x) + Tren khoang = x'^(2x + 1) + + b) T i m gia t r i ciia m Phuoug t r i n h ciia dirdng t h a n g (dm) y - = m(x + 2) hay y = mx + 2m + Hoanh dp giao diem ciia (dm) va di/cfng cong da cho la nghiem ciia phifoug trhili „ 2x mx + 2m + Z x +1 c> (mx + m + 2)(x + 1) = 2x - o m x " + 3mx + m + = 11) ^ = m - - m ( m + 3) = m^ - 12m * (dm) cdt diTcJng cong da cho t a i hai diem phan biet k h i va chi k l i i phJcfug fm ?t I'm ^ t r i n h (1) c6 h a i nghiem phan bi$t, nghia la •![A > •; m'^ - m > m < hoac m > 12 * H a i n h a n h ciia dudng cong n&m d h a i beia cua dudng t i | m can dufng X = - ciia t h i DiicJng t h a n g (dm) c^t diTdng cong da cho tai hai diem thuoc hai n h a n h ciia no k h i va chi k h i phUomg t r i n h (1) c6 hai nghiem X i , x^, va x, < - < Xv : {'f) n k m p h i a diTdi -oo; 2) + Tr§n ode k h o d n g V - -; y (Bai 58 trang 56 SGK) a) K h a o sdt su b i e n t h i e n 2x-l (0; +00): {T) n & m p h i a t r e n {.'/^ Giai ve t h i cua h ^ m s6' B a t X = t - 1, phifong t r i n h (I) \,va t h a n h : m(t - 1)- + 3m(t - 1) + 2m + = o m(t- - 21 + 1) + m t - m + 2m + = -1*) * v»n y = 2, n e n Axibng t h ^ n g y = la t i e m can ngang cua t h i hkm so (2) - m < +00) b) Svf b i e n t h i e n : Ta c6: 5- > 0, vdi m o i x * - i (x +1)^ - H ^ m so dong bien t r e n cac k h o a n g (-oc; - ) va ( - ; +•») c) B a n g b i e n t h i e n : _1 —00 X +00 h ( - l ) = (-1)- + ( - l ) + = - + = Vay A ( - l ; 2) thuoc ca ba diidng cong bieu dien cac h a m so' fix); g(x), h(x), ngoai r a ta c6: y' = + y' y f(-l) = -(-1)" + 3(-l) + = - - + = g ( - l ) = ( - ) ' - (-1)- + = - - + = f '(x) = - x + 3, g'(x) = 3x^ - 2x, h'(x) = 2x + va + f (-1) = - ( - l ) + = g'(-l) = ( - l ) ' - 2(-l) = + = h'(-l) = 2(-l) + = ^2 f''(-l) = g'(-l) = h'(-l) = nen Vay ba difcfng cong c6 chung tiep tuyen t a i A ( - l ; 2) Do t h i * Giao d i e m ciia t h i v^ true t u n g (0; - ) ^ * Giao diem cua t h i wk true ho&nh 60 (Bd, 60 trang 56 SGK) x' ••X N h a n xet: Do t h i n h a n giao d i e m I ( — ; 2) ciia h a i t i e m can cua t h i \h t a m do'i x i l n g Gidi Hoanh tiep diem cua hai dudng cong da cho la nghiem ciia he phucmg t r i n h — (I) 3x + —X = 2 + —X X + 3x X + 3x + —X = 2; X + — = x+ — j - (x + 2)' (1) (2) G i a i a) ~ + — 3x cs> (x^ + 3x)(x + 2) = 6x x + x(x + 3)(x + 2) - 6x = o » Vay vdi m o i a e x(x^ + 5x + - 6) = x = x(x^ + 5x) = x^(x + 5) = X = hai Parabol t i e p xiic Hoanh dp t i e p d i e m 1^: x = -5 Tung dp t i e p d i e m : y = - • X = - k h o n g t h a phifcfng t r i n h (2) H | phucfng t r i n h (I) c6 n g h i e m n h a t x = 0 ==> y = V$y h a i d u d n g cong t i e p xiic v d i t a i gS'c toa dp O Phifcfng t r i n h t i e p t u y e n chung cua h a i dudng cong t a i goc O l a y = f '(0).x hay y = - x 61 (Bdi 61 trang 56 + 57 SGK) Gidi H o a n h dp t i e p d i e m cua h a i Parabol la n g h i e m cua h | phJdng t r i n h (I) — ^ ( l 2v^^ Diem -(1 + t a n ' a ) x ' + x t a n a = — ^ x ' + 2v 2v 2g r ( l + tan^ a)x + t a n a = — ^ x ^ fl [ g t a n a ' 2g L o - (1) » (1) * vdi _ , X = gtana = vay x = l i m y = +00 va x-*-v (khi X l i m y - -oo, nen difdng t h ^ n g x = - la t i e m can dufng x->-r -V vk k h i x -> -1^) vk k h i x -> +oo) -oo b) Sir bien t h i e n tan a.x = t a n a x = -—-— gtana = (x + ir > 0, v d i moi x ^ - - H a m so' dong bien t r e n m o i k h o a n g ( - « ; - ) va ( - ; + Q O ) c) B d n g b i e n t h i e n X -1 -co (D +CC D6 t h i ''^.tana = - > + gtana 2g """^^^'o = A = v e ' p h d i cua (!') 2g 2g gtana Gidi ve t h i h a m so' x->±i ve t r ^ i cua (1') gHan^a 2v cung l a n g h i e m cua (1) 0;^ * l i m y = n e n dudng t h i n g y = 1^ t i $ m can ngang ciia 66 t h i h ^ m so' "0 tan^a, la tiep diem cua hai Parabol vdi moi a e ciia t h i h ^ m so ( k h i x ( + tan^a)x + tana = - 4" x - - ^ t a n V x ^ + xtana = ^ 2v^ 2g 2g t Su bien t h i e n ciia h a m so' a) G i d i h a n t a i v6 cifc, g i d i h a n v6 cUc, dudng t i e m can (2) x 2g _ ' 11 tan^aj^ 62 (Bai 62 trang 57 SGK) a) Kh^o sat sif b i e n t h i e n x-1 y tan a.x + tana = - v„0 2v^g^tan^a LUYEIV T A P G i a i phifcrng t r i n h (2) - _ 2g H a m so xdc d i n h t r e n K \1 ^ —^x' + 2v 2g + tan^a)x^ + x t a n a ' '0 ^ gtana 2gtan^a ' 2g Ta c6: f '(0) = + • 2v^ ^0 X = - - ^ ( + tan='a)x^ + x t a n a = -\x' 2wl 2v^ ——— gtana ^ * Giao diem cua t h j vk true tung (0; - ) * Giao diem ciia t h i va true hoanh (1; 0) N h a n xet: Do t h i n h a n giao diem I ( - l ; 1) cua hai tiem can cua t h i la tarn doi xuCng b) Chilng m i n h I ( - l ; 1) Ik tam doi xuiig ciia t h i Cong thtfc doi he toa theo phep t i n h tien vectcf |x = X - l ' ly = Y + l +00 PhUOng trinh thi do'i vdi h$ tea I X Y Do thi * Giao di^m cua thi vdi true tung (0; 2) X - l +l # Giao diem cua thi vdi true hoanh (-2; 0) hay Y = - l ^ Y = X Dat Y = f(X) = -2, X ~^ X Nhan x6t: Do thi nhan giao diem 11 - - ; ^ t i p xac dinh V = R \: V X e V, - X € (/ - cua 2j hai ti§m can cua thi lam tam d6'i xilng b) Durdng t h i n g (dm) di qua diem (xo; yo) IMm so Y = Ik hkm so IS n6n nhan goc toa dO I ( - l ; 1) lam tam dS'i xufng A Vay thi h^m so da cho nhan giao diem I ( - l ; 1) ciia hai difdng ti^m can cua thi la tam do'i xuTng 63 (Bdi 63 trang 57 SGK) y = Do thi di qua diem (XQ; yo) v(Ji moi m va chi k h i (1) nghi^m diing vdi moi m, nghia l a Gidi a) Khao sat s; i-ien thien va ve thi (.//) X va chi yo = mxo + m + m(xo + 1) - (1 + yo) = (1) x„ + = Xo = -1 yo+i = o y„ = -1 Vay vdi moi gia tri cua m, dubng t h ^ g (dm) luon di qua diim co dinh A ( - l ; - ) f —1 + Mat khac -1 = _ , < = > - ! = —- o - = - ( d i n g thiic diing) ( - l ) +1 -1 2x + l Vay A thuoc (.J^ hay du:dng t h i n g (dm) luon di qua diem co' dinh A ( - l ; - ) H a m so' xac dinh tren R \ — 2, cua (./(O k h i m bie'n t h i § n c) Hoanh giao diem ciia (dm) y = mx + m - va (.>lO l a n g h i | m phuTcfng trinh Sif bien thien cua h a m so' a) Gidi han tai v6 cifc, gidi han v6 cifc, difdng tiem can * lim y = - M lim y = +x, nen di/dng t h i n g x = - - mx + m - = la t i § m c^ln dufng x +2 2x + l o m(x + l)(2x + 1) - (2x + 1) - (X + 2) = m(x + l)(2x + 1) - 3(x + 1) = o (X + l)[2mx + m - 3] ciia thi h^m so (khi x -)• - X = va x -» - - ) Diem A thupc nhdnh trai cua (./(O vi XA = - < - - ^ -3 T < 0, v(Ji moi X (2x + 1)' • Dudng t h i n g (dm) da cho qua diem A ( - l ; - ) cua — nen (dm) c i t (.//) tai hai diem cimg mot nhanh cua (,//) klii va chi phu'ong trinh (2) co - H a m so nghich bie'n tren cac khoang va f — ; + oo nghiem x < - — v a x ; ^ - ! nghIa la 2 c) Bang bie'n thien x Hai nhanh cua (./rO n i m d hai ben ciia dudng ti^m can duTng x = - ^' - « va k h i x -> +oo) b) S u bie'n thien y' = (1) 2mx + m - = (2) • lim y = ^ ' J^^n dirdng thang y = ^ la tiem can ngang ciia thi ham so' (khi x -1 in 3-m +00 - x va k h i - H ^ m so' dat cifc d a i t a i x = 0, gia t r i ciic dai y(0) = - +x) H a m so' dat c\ic t i e u t a i x = 2, gia t r i eiie t i e u y(2) = b) Su b i e n t h i e n c) Bang b i e n t h i e n -2x^ + 4x - (x - If < 0, v d i m o i x # X +x - •y - + x —, -X *-x Do t h i * Giao d i e m ciia t h i va true t u n g (0; 0) - x ' + 3x = X = X = - ' ' -1' - - + +x~^ ^ ^ - \ +x —X 3- D6 t h i * Giao diem cua t h i va true tung (0; - ) \ ^ +x -X —00 y' + y c) B a n g b i e n t h i e n -X V' - Hkm so n g h i c h b i e n tr§n m o i k h o a n g (-00; 1) vk ( ; +00) CO b) Sii bien t h i e n , eiic t r i a) G i i h a n t a i v6 cifc, gidi h a n v6 cifc, du6ng t i e m can X ^ x-.r t h i h k m so ( k h i x ^> " va k h i x - * 1*) * Ta H a m so xdc d i n h t r e n R \} y = a) Gi 0) Bang bien X thien 10000 rr hay yw = 5XM - Vay diem M nkm tren diTdng t h i n g y = 5x - • Do m thoa dieu k i f n m < - x/6 hoac m > + 2^6 J6 • m < - 2j6 o 6x - < - S 6x < - 2^/6 x < ^/6 • m > + S o 6x - > + 2x^6 6x > + 2^6 x > + l2 ' Ta co: A,rv ^ n r^nrM M (x) = 0,0001 - M'(x) +00 + M(x) V4y tap hgp cac diem M trung diem ciia AB la phan ciia dudng thang y = 5x - vdi x < ^ 66 (Bdi 66, trang 58 SGK) hoac x > + — Gidi XSN' Diem M thuOc parabol (P) y = g(x) = 2x'^ + ax +- b va chi = ^ + ^a + b 2 hay a + 2b = (1) Ta CO fix) = - \ x fl ^ g'(x) = 4x + a ~-».2,2'^ Tii suy m i n = M(IOOOO) = 2,2 ^ = y = nen M - ; thu9C hypebol (//^ y = f(x) = - s6' g6c cua tiep tuyĐ'n vdi (,Jô0 tai M - ; la f {2) v2 J ~ minM(x) = M(IOOOO) = 2,2 = -4 Vay chi phi trung binh cho x cuo'n la tha'p nha't k h i x = 10000 (cuo'n) va chi phi cho moi cuo'n d6 la: 2,2.10000 = 22000 dong 2" a) Tong so' tien thti dUcfc k h i ban x cuo'n tap chi (x nguyen du'Ong) la 20000.x + 90000000 (dong) hay 2x + 9000 (van dong) So' tien lai ban x cuon la: L(x) = 2x + 9000 - T(x) = 2x + 9000 - (0,0001x^ + 0,2x + 10000) = -O.OOOlx^ + l,8x - 1000 b) De CO l a i ta phai c6 L(x) > « -0,0001x^ + l,8x - 1000 > c: 573,85 < x < 17426,14 0,9 - ToTn 0,0001 0,9 + yo,7j *C X 0,0001 Do X nguyen n e n < x < 17427 c) X e t h ^ m so' L(x) = -0,0001x^ + l , x - 1000 tr§n (0; +co) va t i m x > de h a m so L dat gia t r i Idn n h a t t r e n (0; +co), t a c6: 9000 L'(x) + 00 + L(x) Dao h a m y ' = u^ Dao h a m Do m a x L ( x ) = L(9000) = 7100 rv ON TAP CHaONG I ' y = k h o n g CO ciTc t r i -2x + y' = « dugc l a 7100 ( v a n dong) hay 71000000 (dong) X + H a m so f(x) l i e n tuc t r e n niia 0; o-i^ixi cos > 0, v6i m o i x x G juiig i^icii V4x - x^ () + y v a CO d a o h a m f ( x ) = -X + x= —00 y' Giai a) X e t h a m so f(x) = t a n x - x, x e +) >0 2V3x + 2V4x - x' Vay muon difoc l a i nhieu n h a t p h a i i n 9000 cuon k h i t i e n l a i thu khoang VSx + xac d i n h t r e n [- b) H ^ m so' y = v ' x - x ^ xac d i n h t r e n [0; 4] x>0 61 SGK) 0; Gidi H a m so d o n g b i e n t r e n Ta CO m a x L ( x ) = L(9000) = 7100 68 (Bdi 68, trang 61 SGK) a) H a m so y L'(x) = C5 X = 9000 — > , vdri m o i x e x" Vay tanx > x + -~-, vcJi m o i x e (Bai 69, trang L'(x) = - , 0 x + 1,8 x X hay tanx - x Vay so cuo'n i n de c6 l a i l a tiX 574 cuon den 17426 cuon ^) u l c i i +00 - v a u g i i i c i i u i e n i r e n Knoar - H a m so d a t cUc d a i t a i x = 2, gia t r i cifc d a i l a y(2) = c) H a m so y = X + 4^ xac d i n h t r e n [0; +oo) 2j Dao h a m y ' = + ^ = > , v d i m o i x e (0; +oo) 2Vx Do h a m so f d6ng b i e n t r e n nufa k h o a n g H a m so' dong b i e n t r e n (0; +oo), k h o n g c6 cUc t r i d) H a m so y = x - N/X xac d i n h t r e n [0; +oo) Tif f(x) > f(0), vdti m o i x e hay tanx - x > 0, v i m o i x e b) X e t h a m so' f(x) = t a n x - x hay tanx > x vdfi m o i x e X cos^x 2V^-1 24^ 2Vx => X = Bang bien thien va CO dao h a m - - x^ = tan x - x = (tanx - x)(tanx + x) > 0, vcri moi x e Vay h a m so f dong b i e n t r e n nufa k h o a n g y' = y' = vdi moi x G H a m so f(x) l i e n tuc t r e n nufa k h o a n g f (x) = Dao h a m 0;^ X 0; +00 y' 2y tir f(x) > f(0), v i moi r - H a m so n g h i c h b i e n t r e n 0; — ; dong bien t r e n 4; u • - H a m so d a t cUc t i e u t a i x = —, gia t r i cUc t i e u l a y 4" (Bdi 70,'trang Gidi 61 SGK) T h e tich h i n h t r u l a V = Tir^h d6 r l a ban k i n h d^y v a h \h chieu ca h i n h tru Di#n tich to^n phdn cua h i n h tru la: (Bdi 72, trang a) Khdo S = 27ir^h + 27irh = 27ir^ + i r — " vd ve thi ham s6 f(x) = - x ^ - 2x^ + — sdt sU biS'n thiin J H^ro so' f xAc d i n h Ttr- Gidi 62 SGK) 3 l i e n tuc tr§n R SU bien t h i e n ciia h ^ m so Ta t i m r > cho S c6 gia t r i nho S' = 47tr - 2V n r ' - 2V a) Gi ^ V J-P n cung dau • N e u m < v a m < t h i p h i t d n g t r i n h f ( x ) = c6 m o t n g h i e m d u y n h a t 1;' + — " f "(x) = t a i X = v a d o i d a u tiT a m s a n g ducfng n e n t h i n h a n d i e m U ( ; ( H i n h 2) • N e u m > va n > t h i p h u o n g t r i n h f ( x ) = c6 mOt n g h i e m d u y n h a t 1' A 1) l a d i e m uo'n /1 - G i a o d i e m ciia t h i v a t r u e t u n g (0; 1) Nhan ( H i n h 3) f "(x) = 6x; f "(x) = x = K h i d o phufdng t r i n h f ( x ) = c6 b a n g h i e m l a a , p, y ( H i n h 1) t r o n g Xo < +x D6 t h i b) N e u m n < t h i m > v a n < xo > ^-^ _-^3-~~^ _ - X c) N e u m n > t h i m -1 + fix) + p a < - J - f ; - f f < P < V • f'(x) xet: Do t h i n h a n d i e m uon U(0; 1) l a t a m d o i xuTng b) T a CO f (0) = 3.0^ - = - P h i r o n g t r i n h t i e p t u y e n c i i a t h i t a i U ( ; 1) l a y y = -3x + 1 = f'(0)(x - 0) {'f) hay It - il O -1" •3 \ / W ^ c) Phuang trinh dUdng th^ng (dm) qua U c6 he so' goc m 1^ y - = m(x - 0) h a y y = mx + PhUcfng trinh ho^nh dp giao diem ciia (dm) va Cf) la nghiem phifong trinh - 3x + = mx + (1) • fx = X' - 3x - mx = x[x'' - (m + 3)] = , [x' = m + (2) Difdng th^ng (dm) c^t {'f) tai ba diem phan biet k h i va chi phifang trinh (1) c6 ba nghiem phan biet nghia l a m + > < = > m > - 75 (Bai 75, trang 62, 63 SGK) Gidi a) Kh^o sAt su' bien thien va ve thi ham so' y = x^ - 3x^ + H k m so xjic dinh tren R Sir bien thien cua h^m so' a) Gidi han tai v6 cUc: l i m y = + Phi/Png trinh (1) c6 nghiem m > vk m * \k nghiem ciia phifpng trinh la x = - ; x = 1; x = -Jm , x = bien tren mSi khodng 72 d6'i xurng X + H a m so' nghich bien tren m6i kho^ng y 0) Nhdn xet: Hkm s6 da cho Ik hkm so chSn nen thi nhan true tung Ik true Do d6 (1) y' = x = hoac x = ± - / tai diem (- ; 0), ( - ; 0), (1; 0) v^ ( t = hoac t = m V6 -00 y = 0c=>x'*-3x^ + 2=:0x = ± l hoSc X = ± 72 Do t h i cdt true ho^nh Ta c6 phiTPng trinh t^ - (m + l ) t + m = (dang a + b + c = 0) b) Sif bien thien, c\ic tri y' = 4x' - 6x = 2x(2x2 - 3) x Do t h i c^t true tung tai (0; 2), 3l ;— ^ sl — ; — la cdc diem uo'n cua t h i Suy \ / \& \ ^ -1-2 / ^-^^4' * o 7rn - = (do m > 1) 7m = Vay vdi hay 7m = ~ thi difdng cong da cho c^t true hoknh tai [m = diem tao thanli ba doan bKng 76 (Bdi 76, trang 63 SGK) b) Gidi a) K h a o s ^ t s u b i e n t h i e n v a v e t h i h a m so' f ( y ) = x " x^ h o a n h v a l a y do'i xurng p h a n c i i a ( / ) Sif b i e n t h i e n c u a h a m so', nkm x^ n ^ m p h i a t r e n t r u e p h i a dufdi t r u e h o a n h qua true hoanh a) Gi(Si h a n t a i v ciTc l i m f ( x ) = +00 77 (Bdi 77, trang Gidi 62 S G K ) a) K h a o s a t sir b i e n t h i e n v a v e t h i i-'/fi) cua h a m so y = b) S u b i e n t h i e n , eye t r i f (x) = 4x'' - x = x ( x ^ - 1) x - 2x - H a m so' x a c d i n h t r e n K \ S y b i e n t h i e n cua h a m hoac x = ± X I f ( x ) I G i a n g u y e n p h a n c i i a t h i ("O h a m so' f ( x ) = x* - H a m so x a c d i n h t r e n K f (x) = Suy r a t h i c u a h ^ m so y = so' a) G i d i h a n t a i v c\ic, g i d i h a n v6 cUe, dUcmg t i e m c a n + H a m so' d o n g b i e n t r e n cac s/2 khoang ; va (42 - ; +00 • l i m y = +00 v a nghich x->r t h i ( k h i x ^ b i e n t r e n cac v^ khoang • 72 + H a m so' d a t ct/c t i e u t a i c^c d i e m x = ± — , giA t r i ciJc t i e u (khi v/2 y - f'(x) f(x) +x V2 +• l"") 2 - X va k h i x X c) B a n g b i e n t h i e n —00 va k h i x +oc) b) S i i b i e n t h i e n + H a m so d a t ciTc d a i t a i d i e m x = 0, g i a t r i eUc d a i f(Oi X l i m y = —, n e n d a d n g t h & n g y = - - l a t i e m c a n n g a n g cua t h i h a m so' t-*^^ ^/2 l i m y = -oo , n e n difcfng t h i n g x = l a t i e m c a n dufng ciia x->l* +« 0 = ^ , — > , voi moi X ?^ (2x-2)' H a m so' dong b i e n t r e n cac k h o a n g ( - « ; 1) v a ( ; +oo) c) B a n g b i e n t h i e n -00 X + + y' - ^ ^ + 1 D o t h i , —0 ^ ^ 2 D o t h i • f " ( x ) = 12x^ - 2, f " ( x ) = o X = ± 76 - G i a o d i e m c i i a t h i v d i t r u e t u n g ( ; 2) tai diem U , { I X ^/6 2' - ^^6 /6 qua 5] 36 j va U, V6 ' X 2/,.2 (x _5_ 36 l a d i e m uo'n Giao X =r hoac X = ±1 Do t h i cat true h o a n h t a i ( ; ) ; ( - ; 0) v a xet: cua t h i vdri true h o ^ n h (4; 0) Nhan I xet: 2j Do thi nhan giao diem cua h a i t i e m c a n c u a t h i b) G o i M ( x o ; yo) l a d i e m b a t k i c i i a m a t p h i n g t o a dp (.//„) d i q u a M k h i v a chi k h i m la n g h i e m ciia p h u o n g t r i n h ( ; 0) Nhan diem l a m t a m do'i xufng = 1) = o + • G i a o d i e m cua t h i v a t r u e t u n g (0; 0) ã y = ô x ^ - x ^ X = - H a m so da cho l a h a m so c h l l n n e n t h i n h a n true t u n g l a t r u e d o i x i l n g mxo - Xo - m = 2y„(mX(, mxg -1) x„ - m 2(mXo -1) = yo- *1 X(, - m = m x „ y „ - 2yo m X g ?t (1) m(2x„y,,+4)-(x„+2yo) = (2) + H ^ m so' n g h i c h b i e n t r e n Mpj dif&ng cong Cii^n,) v6i m ?t ± ^ deu d i qua M(xo; yo) k h i va chJ k h i he phUorng t r i n h t r e n nghiem dung vcfi mpi m -oo; 1^ ; dong b i e n t r e n —; — 2) / -I + H ^ m so dat cUc t i e u t a i x = - ; gid t r i cue tieu y +00 \ I2j ±— PhUicmg t r i i i l i (2) nghi|ni diing v6i moi m k h i va chi k h i 2x,y^ + - _ [xo + 2yo = r-4y^ + = H§ phuang t r i n h c6 h a i n g h i g m (xo; yo) = ( - ; l ) v^ (xo; yo) = ( ; K i e m t r a dieu k i e n (1) • V d i Xo = - , t a CO m 5t - X y -1) - Do t h i \h parabol d i n h S|^-; v(Ji m ^ ± ^ deu d i qua h a i d i e m c6 d i n h ^ , , 2mx - - 2m(x - 4m) - + Sm" Ta c6 y = — — = (2mx - 2? (2mx - 2)' He so g6c cua t i e p t u y e n v i 8m' - 8m' - T a i A y ' ( - ) = ( - m - 2f (4m + 2f 8ra' la h k n g so (4m - 2f T i c h cac h$ so g6c c^c t i e p tuyen t a i A v^ B Ik: ( m ' - 2) ( m ' - 2) y'(-2).y'(2) = (vdi m (4m + 2f •(4m - 2f Sif b i e n t h i e n ciia h a m so' a) G i d i h a n t a i v6 ciic, gidi h a n v6 ciic, diidng t i e m can • l i m t = - X va l i m y = +oo , nen diiong t h i n g x = - la t i | m can dilng cua 5t ± [(4m + 2)(4m - 2)]' (16m' - ) ' = — , vdi moi m thi -Vi^ (khi x t h i iJf) — Gidi - " va k h i x (khi X - X v^ k h i x +x) b) Sir b i e n t h i e n -1 < 0, vdi moi x (x + 1)' -1 + H a m so' nghich bien t r e i i cac khoang ( - x ; - ) va ( - ; + x ) X — — X y X->+oc ^ +x - —— +x ~~ b) Sii b i e n t h i e n , cufc t r i : - y' X = -1*) • l i m y = , nen diidng t h i n g y = (true hoanh) la t i | m can ngang ciia y = Sir b i e n t h i e n cua h ^ m so' a) G i d i h a n t a i v6 cijfc: l i m y = + » y' = 2x - 1; y ' = « 5^ ± — ) ± — ) , t i c h cac h | so g6c cdc tiep tuyen t a i A v^ B cua (,#„) 1^ m p t h^ng so 78 (Bai 78, trang 63 SGK) a) Ve t h i (P) y = x^ - x + 1 H ^ m so xAc d i n h t r e n K la ( ; 1) * Ve t h i (./O h a m so y = x + 1 H a m so xiic d i n h t r e n \ ( m ' - 2)' 16(4m' -1)= thang - Giao diem cua t h i va true tung (0; 1), d i e m doi xufng qua dUcfng thing X = - (8m'-2)' 4(4m' - D ' true doi xiJng 1^ diidng "=2-^ - (m.±-) -2 V$y k h i m b i e n t h i e n ( m »•+(» A ( - ; 1) v^ B(2; - ) c) Chiirng m i n h t i c h he so goc cac t i e p tuyen v d i T a i B y'(2) = + +00 D t h i — - (.X'm) +00 - y' V$y m o i dtrbng cong • • V d i Xo = 2, t a c6 m ^ —00 > - X " • »>0 Do t h i - Giao d i e m cua t h i v^ true tung (0; 1) N h a n xet: Do t h i n h a n giao diem I ( - l ; 0) cua h a i t i ^ m cSn cua t h i Ik tarn doi xufng b) H o a n h dp giao d i e m cua (P) - X + = c) B a n g b i e n t h i e n n g h i e m cua phuang t r i n h {.'ff) \h x +1 1)(X^ - X + l ) = l C : > X ^ + l = lX = + x = 0=>y=l la d i e m A(0; 1) (.'If) —00 ta CO g'(x) = „ , , , C) f(x) - g(x) {.'tf) X.X^ x+1 X+1 — X + _ f(x) - g(x) + + T r e n cac k h o a n g (-oo; X+1 y - f(xo) = —1 X — +oo), ^ x^ + hay + ^ (X - X(,) + 00 y = (x - X,,) + X,, + + parabol (P) n a m t r e n hypebol + T r e n k h o a n g ( - ; 0) parabol (P) n k m dufdi hypebol 79 (Bdi 79, trang 63 SGK) + - - ) va (0; Phtfong t r i n h t i e p tuyen ciia CO t a i M(xo; f(xo) l a : = —00 X f'(x) = - ^ ; f'(xo) - x' c6 t i e p tuyen chung t a i A (0; 1) = _^+'X) —00 x i e n ciia CO 1 X - x + —• + b) Ta CO ducfng t h i n g x = l a t i e m can dufng, dadng t h i n g y = x la t i e m can (x + 1)^ ro r ^ n g f (0) = g'(0) = - Vay h a i dirdng cong (P) va +C0 +x 06 t h i £)6 t h i n h a n giao d i e m ( ; 0) cua h a i t i e m can \h t a m do'i xufng D a t f(x) = x' - X + t a CO f'(x) = 2x - x +1 -^-2 y Giao d i e m cua (P) v& g(x) = + y' -1 — 00 x CS> ( X + f(Xo) = Xo + {.'If) (./lO — * (d) c i t ti§m can dufng x = t a i A , nen: Gidi a) Khao sat sif b i e n t h i e n v^ ve t h i i'f) h a m so' y = f(x) = x + — 1- yA (-Xo) + x„ + — = — X H a m so xac d i n h t r e n K \ Vay A 0; Sif b i e n t h i e n ciia h k m so a) G i d i h a n t a i v6 ciTc, g i d i h a n v6 cifc, diTdng t i ^ m can • l i m y = - 0 va l i m y = +oo * (d) cat t i e m can xien y = x t a i B, nen hoanh diem B la nghiem phiicfng trinh f( x - X o )^ + Xo+ — = x • l i m y = - va l i m y = +co , n e n difdng t h ^ n g x = (true tung) la t i e m can X -.0" 1 x - x„ + — + x„ + — = X o x->0' diifng cua t h i ( k h i x ^ 0^ vk k h i x 0*) • l i m ( y - x) = , n e n diidng t h i n g y = x la t i ^ m can x i e n cua t h i h ^ m so ( k h i X -> - X va k h i x +«) b) Sir bien t h i e n , ciic t r i , , X y' = o ^0 / X = '^ll ^x = ^0 '^o ^ o X = 2xo Suy r a y = 2xo Vay B(2xo; 2yo) x^-1 , T a CO X x' - = » X = ±1 + H a m so' dong bien t r e n cac khoang ( - x ; - ) va ( ; +oo) n g h i c h bien t r e n cac k h o a n g ( - ; 0) va (0; 1) + H a m so d a t eye dai t a i x = - , gia t r i cifc dai y ( - l ) = - H a m so' dat ciTc t i e u t a i x = 1, gia t r i cUc ti§'u y ( l ) = X + Xo ^ = + 2x„ = X,, = XM 2 " ma A , B, M t h i n g h ^ n g n e n M la t r u n g d i e m ciia A B * D i e n t i c h AOAB la 2x„ = , v d i m o i Xn * Vay d i e n t i c h AOAB k h o n g phu thupc vao v i t r i ciia d i e m M t r e n CO B A I TAP TRAC NGHlgM (Bdi 80, trang 64 SGK) g (Bdi 84, trang Chon (A) Hudng ddn Gidi ddn -00 X f(x)= i ^ _ i ^ - x + f ( x ) = x^ - X - y' y Bang bien thien —00 f'(x) -2 0 + 0 x = hoSc x = +00 + — — ~ ~ ~ ~ — f ' ( x ) = H a m so d o n g b i e n t r e n (Bdi 82, trang x^ - x - = H a m so d a t ciTc t r i t a i cAc d i e m x = ],x Bang bien thien X x ' - 3x + , x ' - 2x - — ; y = x - " (x-1)^ y' = > = X = h o a c X = Gidi dan y = ddn f ( x ) = 30\^ - 60x^ + 30x^ = 30x2(x^ - x + 1) = 30x^(x - if 65 SGK) Chon(B) —XJ + ' H a m so d a t ciTc t i e u t a i x = -1 0 ' + » ——•— | » x = ± - - + k27t k e Z o I; t X = ± - + kTt, k e Z y ' = t a i X = - — vk d o i d a u tii a m s a n g difcfng k h i x q u a x = n ^ m so d a t cuc t i e u t a i x = - — nen 93 (Bai 89, trang 66 SGK) Gidi Chon (D) Huang ddn y = -3 N / I - X H a m so' xac d i n h wh l i e n tuc t r e n ( - c o ; 1) y' = — , > 0, v6i moi x e Bang Bang bien bien tt h h ii ee n n x 1) Gidi SGK)] i • y = x - + 2x + t i | m can x i e n cua + >0 94 (Bai 94, trang 66 Chon (B) y —•— |- l i m [ y - (x - 2)] = nen dudng t h i n g y = x - 1^ SGK) Gidi H ^ m so x a c d i n h t r e n K \- - ; max y = y d ) = 1) Gidi (Bai 90, trang 65 SGK) • lim^y = 00 , difcfng t h i n g ^ = - ^ 14 t i $ m can dufng ciia (?D Chon (B) Gidi 91 (Bai 91, trang 65 SGK) • l i m y = 00 , duTcfng t h i n g x = 14 t i e m can dufng cua {f) X—»3 Chon (C) ddn f(x) = 2x'' + 3x2 _ + 2; fix) • l i m y = - —, dudng t h i n g y = - — l a t i ? m can ngang ciia {'f) 95 (Bai 95, trang 66 SGK) Gidi Chon (C) Hudng ddn = Gx' + 6x - 12 = ( x ' H- x - 2) = o X = hoac X = - fx) h i e nn Bang bien bie!n tthie X fix) VA/ 0 + -1 -2 —15_ fix) » - + + Hkm so x i c d i n h t r e n M \; • l i m y = 0 , diTdng t h i n g x = - l a t i e m c a n dufng c u a - max f ( x ) = f ( - l ) = 15 X I 1, • l i m y = 00 , diforng t h i n g x = Gidi 96 (Bdi 96, trang 66, 67 SGK) ddn H a m so' f(x) = V-x^ - 2x + xac d i n h va l i e n tuc t r e n [-3; 1] f'(x) = —^—1—, V-x^-2x + = o fix) Bang bb ii ee n n tt h Bang h ii ee nn X X :t; (f) l a t i ? m c§n n g a n g c u a CO- Gidi (B) ddn x + -1 -r 1) CO PhuoTng t r i n h h o a n h giao d i e m c u a h a i t h i -3 -—' f(-l) = max f(x) = X.I Huang = -1 + f(x) Chon v6i m o i x e (-3; 1) \K) Ik t i e m c a n dufng ciia • l i m y = - —, diTdng t h i n g y = - ~ Chon (A) I x->5 21 (Bai 92, trang 66 SGK) Huang 66 ''"*"2 V Huang trang l i m y = CO , dirdng t h i n g x = - — 1^ t i f m can dufng cua (T) (-oc; J x6(-»; 93, • H ^ m so xdc d i n h t r e n I N — X —00 (Bai Chon (D) Hudng ddn —»• x -^1 = o x - x + l = x - (Bdi 97, trang 67 SGK) Gidi Chon (D) Huang ddn Ve (90 y = + 3x2 H&m s6' x&c dinh trgn K Su bien thien h^m so • l i m y = +00 l i m y = -oo [bo (Bdi 100, trang 67 SGK) Chon (D) Hudng ddn • y' = 3x' + 6x = 3x(x + 2) y' = = o x = hoac x = - Hkm so dong bieii tr§n ckc khodng ( - » ; -2) vk (0; + o c ) , nghich bien tren khodng (-2; 0) Ham so' dat cifc dai t a i x = - , gid t r i c\ic dai H i m so' dat cUc tieu t a i x = 0, gid t r i cUc tieu • Bang bien thien X -00 -2 +00 0 Gidi =4x=' X Hai t h i tiep xiic k h i vk chi = (4x^)' 4x^ - 3x + l = 4x^ 3x + l = V = 8x (1) (2) , +00 - X Do t h i • y" = 6x + 6, y" = o X = - • y" = t a i X = - vk y" ddi da'u tU km sang dixang nen U ( - l ; 2) la diem uon cua t h i * Giao diem ciia t h i vk true tung (0;0) * Giao diem cQa t h i va true hoanh (0; 0), (-3; 0) • Ve (d) y = m • m = (d) c^t t a i diem phan bi^t 98 (Bdi 98, trang 67 SGK) Gidi Chon (A) Huang ddn The x = - vko (1), ta c6: •4 /fx / / • rf—1 / i l2j +1=0 c - g - g + l - (ddng thiJc dung) / V|iy hai t h i tiep xuc t a i d i l m M c6 h o i n h dp \k V - - •2/ i y =m \ -2 -1 0' X Rkm so xkc dinh tren R \ • l i m y = 00 , nen dudng th^ng x = - — la tiem can dufng ciia t h i hkm so • limy = — , nen dudng th^ng y = — la tiem can ngang cua t h i ham so x-.tx 2 ' 1 cua hai ti$m can cua t h i la tam do'i xiifng cua thj Giao diem I ^ 2' Gidi 99 (Bdi 99, trang 67 SGK) Chon (C) Huang ddn PhifOng t r i n h hoanh giao diem cua hai t h i x^ - x^ - 2x + = x^ - X + (1) o x^ - 2x^ - X + = c= x(x^ - 1) - 2(x''' - 1) = x(x^ - 1) - 2(x^ - 1) = o (x^ - l)(x - 2) = o x = hoac X = ±1 ... Giai 13 (BM 13 trang 17 SGK) f(x) - ax^ + bx^ + cx + d = - 4sin = + 2cos^ + c o s - = + 3cos^ = ^ 3 o - a + b + 12 = ] - a + b + 12 = 3a - 3b - - - a + 27 = a+b+c+l=0 +c +1= c = -4 -12 + b + 12 . .. +x) 16 ' B i e n so n e N* dUOc t h a y b k n g b i e n x e (0; + « ) ) f (x) = - x + 480 f'(x) = » - x + 480 = « Bang bien t h i e n X _ b f'(X) ^^^^ + X = 12 12 ?ssn f (12 ) = 480 .12 - 20 .12 ^ ... 0 ,19 2 (20 + ) ' Toe t S n g dan so' v^o n&m 2008 1^ 12 0 f'(38)= * 0,065 (38 + 5)' 19 0 00 • Ta — = 0 ,12 5 o (t + 5)^ = « (t + 5)' 0 ,12 5 C O X, Xi f'(x) •f'(X) f(X) I f(x) f(Xi) cUc tieu t + » 31