Các phương pháp giải bài tập giải tích 12 nâng cao: Phần 2

Nối tiếp nội dung phần 1 tài liệu Giải bài tập giải tích 12 nâng cao, phần 2 giới thiệu tới người đọc kiến thức cần nhớ và phương pháp giải các bài tập hàm số lũy thừa, hàm số mũ và hàm số logarit; nguyên hàm - Tích phân và ứng dụng; số phức. Mời các bạn cùng tham khảo.

^jirmtg b) Tinh II« chat: Vdri a > 0, b > 0; m, n nguyen dUcfng va h a i so' p, q t u y y * 'Vab = HAM S O LtJY THIJTA HAM S O MtJ - HAM S O L O G A R I T '^.'l:a">a"m>n a) Dinh c) Sai Gidi = 4.3^ = 36 Cho m € Z, n e Z • a™ > b"" o Giai (Bai 3, trang 76 SGK) 14 = a" b" c) So sank cdc luy BAI L (Bdi 1, trang 75 SGK) a) Sai b) Dung (Bai 2, trang 75 SGK) Chon (C) a™ * — = a""" a" ^ Ghi chu: T i n h chat ciia luy thiTa vdi so mu hUu t i (cua so thi^c duong) c6 day dii t i n h chat nhuf luy thifa vdi so mu nguyen va m e Z, n € Z, t a c6: * a^.a" = a"""" ^ a" = d i n h bori: a° = 1; a" = — (a: cof so, n : so mu) a " b) Tinh chdt: V i a 7^ 0, b (trong m l a so W6\ nguyen duotng, can bSc n cua so thUc a, k i h i ^ u / 125 a) 81 -0,75 = 3-'' + s ^ l32j 33 + l2 Ghi chu: • Neu n le t h i m o i so' thuc a chi c6 mot can bac n • Neu n chSn t h i m o i so' thUc duone a c6 h a i can bac n la >/a va - N/^ 1 ,x3 ' 5-8 = i - = - « « 27 27 27 la so thirc b cho b " = a • N/EI = b b" = a (n nguyen difong) .5 ) 0,001 -(-2) ^643 -8 ? + ( " ) ^ = (10"'')"3 - ( - ) - ( ' ' ) -(2=) = 10 - (-2rl2^ + = 10 - - — + = - — = 16 16 111 16 +1^ - c) f 1^ + — -ojr) Cdc/i 2.- T a c6 + 5V2 = 1^ + 3.1^ 72 + 3.1.( N/2 )^ + ( >^ f = (1 + ^/3 12, - 5V2 = 1^ - 3.1.72 + 3.1(72)^ - (72)^ = (1 - 73)^ -5 7v + 572-+77-5v^ Dod6 -=9 + - = 12 = (2-')-^ - - + 19(-3)-' = I 27 12; = 2* - - a^ ^ a^ i ^ 27 b) a^ + a i a3(l-a') a ( - a " ) ^ (1 - a X l + a) aMl-a) a ( a + l) a) So sdiih N/2 va ( l - a ) ( l + a) ^'^ Giai b) >/3 + ^ va < ^ Vay 73 + ^ > V2 a-1 a) —3 =4 >^ o x - = ( d o x ^ + 2x + > ) o x = Vay ^ + 5>/2+^7-5>/2 = =—p= i ( i + l) = 7a - + = 7a == v'i + l i Va +1 Giai (Ta.Tb)" = (7a)".(7b)" = ab (vdi a > 0, b > 0, n nguyen duong) Theo dinh nghia cSn h&c n cua mot 86' ta suy ra: Tab = Ta.Tb 10 (Bdi 10, trang 78 SGK) =14 Giai a) Cdch 1: 74 + 273 - 74 - 273 = 7(73+1)' -7(73-1)' TO d6: x^ = - x o x^ + 3x - 14 = l + ^ • ^ suy ^ 7^ + 7b — — —_ + Vb - Va = ^b 7a - 7b 7a + 7b a-b a + b ^ (^f -(7b)' _ (^f + Cjhf 7^-7b 7^ + 7b 7a + 7b 7^ + 7b a + 7b) ^ (7i - 7b)(7^ + ^ + 7b^) (7^ + b ) ( ^ - i b + 7b^) ^ (^f > {Sf 7a-7b _ — T a c6: (>/2)' = ' = 8; ( ^ ) ' = ' = Do > nen (7^)^+7^' 7^-7b" c) = 7^^ + 7Sb + b ^ - ^ + ^/Sb-7b^= 2^'^ ^^^^-7dbl:(7^-7b)^ ,7a + 7b ) (7^)'+(7b)' - T i b : ( ^ - 27ib + 7b^) = + a - ( - a) = 2a (Bai 6, trang 76 SGK) (7^)^-(7b> 7^ +7b (7a" + 7b)(7i: - 7b) (vdi a > 0) Giai ^a+il^ ^if^-ilh = 16 - - = 10 a^b _5,3 27 a^b^ - = ab (vdi a > 0, b > 0) / ^ 7^-7b Giai 3u2 b) -27 8, ('Sdi 8, trang 78 SGK) a) (Bdi 5, trang 76 SGK) a) -(5^) 7(1 + 73)' +7(1-73)' = l + >,/3 + l - 73 = d) (0,5r* - ( p ^ - = = 73+ i | - - i | = 73+ 1-73+ = Cdch 2: Dat X = 74 + 273 - 74-273 > Ta c6 V l - 12 (do X > 0) = 8- Suy X = b) T a CO 3^°° = (3^)^°° = 27^°° 5400 ^ (52)200 ^ 25^°° 2^4 + 2V3 V4 - + 4- 2>/3 = + 2>/3 - Vay V4T2V3 - 4A-2S 8- ^/4 = Vay 3«"° > c) b) Cdc/i i ; Dat ,2 =2' = {2'')' 72.21" = 22.21" = 22'i" =21" =2 =a +b X = ^9 +VSO + ^9-^/80 = a^ + b % 3ab(a + b) = 72.2'" Vay T r o n g a ' + b^ = + N/80 + - 780 = 18 ab = ^9 + ^ - = 781 - 80 = Tii 343 10 256 10 = 18 + 3.1.x hay x^ - 3x - 18 = (x - 3)(x^ + 3x + ) = o Vay >4^" = (do x^ + 3x + > 0) c=> x X - =1 780 + ^9 - 780 = §2 Cdc/i 2.- L O Y THCTA V I S O M U T H U C Do ^9 + ^ / - - N e n neu ^ x/SO + 'iJ9~j80 = t h i cua phi/ong t r i n h Xi = va vttOla hai nghieni - X + = + 75 Cho a l a so' thifc diTOng va a l a so' v6 t i Xet day so' hufu t i r i , r2, T„, ma limrn = a K h i day so thuc a"' , a'', a'° c6 gidi h a n xac d i n h ( k h o n g phu thuoc vao d a y (rn) da chon) T a goi gidi h a n l a luy t h i i a ciia so' mu ot K i hieU a" 75 ^ 3-75 ^9-780 = (1) (2) a"= Ta chufng m i n h d i n g thiJc (1), t h a t vay '3 + ' ' Tifong tir 3^+3.3^75 + 3.3(75)'+(75)'^ - + 3275 8 11 (Bdi 11, trang a) (73)"6 C = A ( l + r)^' 78 SGK) =(32)" t r o n g = ^/g^Wlo _5_ ^3 Gidi 12 = V3 * = ( i Vay(73)"« = | - > | i - L u y thCra v d i so' mu va so' mu nguyen a m k h i co so' a - Luy thCfa vdi so' mu k h o n g nguyen t h i co so' a > C o n g t h i t c l a i k e p = ^9+780 lima^" Nhd: = + 4N/5 = + 780 T c r d o ^ I ^01 D U ] \ cAi\ K h a i n i ^ m 3-75 + hay ^/gTTsO =3 '2 A l a so' t i e n gijfi r la l a i suat m o i k i N l a so k i gufi C so' t i e n t h u difoc ca vo'n l l n l a i BAlTAP 12 (Bdi 12 trang Chon B (Bdi 13 trang Chon (C) (Bdi 14 trang Bieu k i e n < 81 SGK) Giai 81 SGK) Gidi 81 SGK) a 0) Dat t = 11 a 16 • Vdfi t = 1, ilx = c:> x = • V d i t = 2, = X = 16 42 (Bdi 22 trang 82 SGK) ,-2 t^2-2 X c) x'° > o |x| > d) x^ < g2 = ^ b) x'^ > « = a^ (do b ^ « > X < ^ 0) §3 LOGARIT (g72_t,73-,2 2a=^^ - 2a^b^^ I \ l D U I \ cAi^ I«i6' 1- D i n h n g h i a : Cho a > va a (a^ - b ^ ) ^ t = hoac t = Gidi a) x ^ < o < x ^ < V3 - V g3 +73-l-73 a ^ ^ - b^^ + a'^^" - a ^ b ^ + b^^ 23-^^(3^-b-^) C5> t = (do t > 0) > (vdi x > 0) ^ ^ a ^ ^ -b^^^ + ( a ^ -b^)=' — a la so thuc t u y y Vay phufOng t r i n h da cho c6 h a i n g h i e m x = va x = 16 ^-1-73 ,-2 ,V3-1 a = T a c6 phiWng t r i n h t^ - 3t + = o Gidi a ^ Ml) b) V i ' - V ^ + = '2^ — = a • Vdti t = 1, Vx = X = 1 ^1^1 ^ = a2 > Vay phiTOng t r i n h da cho c6 mot n g h i e m x = Va-^-' V3* 3'"' < 3^ » = o t ^ + t - = c : > t = l hoac t = - = a re ~16 =: a^e = a 19 (5di 19 trang 82 o zz T a CO phirang t r i n h t^ + t = (x>0) xi2 r lb 11 d) \|a^ja'•Ja x'^''2 ^* "> lb (2^ — = x2.x'2 '1 ra^ • N e u a ^ t h i (1) o Dat t = ^ SGK) 'a] a^ - a 21 (Bdi 21 trang 82 SGK) a) + i/x =2 M HJYfeM T A P 18 (Bdi 18 trang 81 a" + a"" - = a ' + a " - a a = b) 3'"' < 27 C = 15(1 + , f * 21,59 ( t r i # u dong) ! x" - y" I Gidi • Neu a = t h i (1) o Sau n a m ngifcfi fi'y t h u duac ca von l l n l a i l a : 2x"y" + y ' " = v'^x" - y " ) " " - SGK) a) - (a" + a-") = o ,V5-3 • 4-V5 V2 = , / x ^ + 2x''y" + y^" - 4x''y'' (x" + y " ) ' - 4" xy 16 2a 72 a^-b^'5 1, b > 0, So thUc a de a" = b duac goi la l o g a r i t ccf so' a cua b K i hieu logab, n g h i a l i : logab = a c=> a" = b Ta c6: logal = 0; logaa = logaa'' = b, Vb e K; a'°^-''=: b, Vb e R, b > ^ T i n h c h a t : C h o so' dUOng a va cac so dUcrng b, c * N e u a > t h i logab > log^c b > c * N e u < a < t h i logab > logaC b < c He qua: Cho so' difong a # • N e u a > t h i logab > o (Bdi 27 trang 90 SGK) • log.-,3 = • l o g s S l = loggS" = 41og33 = • log.a = cdc s o ' dUcfng b, c b > I • N e u < a < t h i logab > b < • logab = logaC C=> b = C log3 i = log.33-' = -21og33 = - • = log3 33 = ^ l o g 3 = • logs C a c q u i tAc t i n h l o g a r i t Cho so dircrng a = • loga = logab - lOgaC \c) • logab" = alogab logbC = , — • l o g o s i = logl o = ^ t a c6: log hay l o g a b l o g h C = logaC • logj36 = log, • log c = — logaC (a 0; c > 0) ^" a L o g a r i t t h a p p h a n L o g a r i t co s o 10 cua inQt s o dirong x difcfc goi la i o g a r i t t h a p p h a n cua x k i hieu Igx (hay l o g x ) * Logarit thap phan c6 day du cac t i n h chat cua logarit v d i ccf so lorn hem 23 (Bai 23 trang Giai 89 + 90 SGK) Gidi Chon d) 24 (Bai 24 trang a) Sai 25 (Bai 25 trang b) D i i n g 90 SGK) c) Sai d) Sai = - log, [6 b) loga— - logaX - logay (Dieu k i ^ n a > 0, a 1; x > 0, y > 0) y c) logax" = alogaX ( D i l u k i e n a > 0, a 1, x > 0) d) a'"^'" = b (Dieu k i e n a > 0, a * 1, b > 0) Gidi =-2 Gidi 90 SGK) • 3^'"''' = 3'°*^^''' = 2^ = 32 f-\ 125 {8^ / logn,5 ^^ log, Blog,2 log, ' (V 2) [2, J 30 (Bdi 30 trang 90 SGK) Gidi a) logsx = CO x = 5'' = 625 b) log2(5 - x ) = c = - x = 2^c=.x = - c) log3(x + 2) = X + = 3'^ X = 25 V Gidi a) loga (xy) = logaX + logay (Dieu k i e n a > 0, a 1; x > 0, y > 0) 26 (Bai 26 trang 90 SGK) a) logaX < logay < X < y Dieu k i e n a > b) logaX < logay x > y > Dieu k i e n < a < • 3'°*="* = 18 [32J • 89 SGK) 1=3 = 31og, hay logab.logba = B A IT A P =-3 VO 29 (Bdi 29 trang ^ = -31og i r , l o g i 125 = l o g , logab He qua: V d i a, b la h a i so difong khac 1, t a c6: • logab = ^ • log3-V log33"'.3-"2 = logs ^ = - f log.sS = - f 3V3 2 Gidi 28 (Bai 28 trang 90 SGK) va cAc so di/ong b, c, t a c6: • lOga(bc) = logab + logaC D o t cd so c u a l o g a r i t V d i a, b l a h a i s o difcJng khac va c la s o dMng, loff c 'Giai (1' d) l o g , (0,5 + X) = - o 31 (Bdi 31 trang log,25 = log,8 = 90 SGK) log? log5 0,5 + X = 1,65 1,29 l o g , , ^ - , log9 • logo, = ^ « -0,42 log0,75 Gidi ' = 2'* = 32 0,5 + X = X = 5,5 b) l o g l - log5 = l o g — = log2,4 32 (Bdi 32 trang 93 SGK) Gidi a) log8l2 - logglS + logg20 = logs~~ 15 b) ^ log736 - log7l4 - 31og7 log2.4 < log? = logglG = log^,, 2' = ^ o Vay l o g l - log5 < log? c) 31og2 + log3 = log2'^ + logs = log2^3 = log24 21og5 = logS^ = log25 ^ log24 < l o g = log? (6^)2 - log7l4 - l o g ( ^ ) ' = log72.3 - log72.7 - log73.7 = log72 + log73 - (log72 + log,?) - (Iog73 + log77) ^ l0g72 + l0g73 - log72 - - l0g73 - = - Cdch khdc ^ log736 - log7l4 - 31og7 v a y 31og2 + logs < 21og5 d) + 21og3 = loglO + logS^ = l o g i c s ^ = log90 Do log90 > log27 Vay + 21og3 > log27 Gidi 35 (Bdi 35 trang 92 SGK) ^ a) logax = log«a%' x/c = logaa-'.b' = l0g76 - I0g7l4 - l0g721 = l g _ ^ = l0g77^' = - 14.21 36 c) l " g r , - l o g , ^ ' " g ^ i ^ logs ^ log, log,3' 21og5 Cdchkhic l " g r - l o g , ^ l o g ^ =iog,3(d6icas6)= log.,3 = ilog33 logs logs " = logaa' + log„b^ + logaC^ = + 21og„b + - l o g c = + 2.3 + - ( - ) = b) logaX = l0g« ^ logaa^ + l o g , - log^ = + ^ log.b - SlogaC = + i - ( - ) = 11 3 36 (Bdi 36 trang 93 SGK) Gidi a) logsx = 4Iog3a + 71og3b = 6'"^'^^ + — Cdc/i khdc logsx = logaa" + log3b^ => logsx = logsa^.b'' => x = a*.b^ - 2'"^^'^ = 25 + - 27 = b) logox = 21og5a - 31og5b => logsx = logoa'^ - loggb^ = 6'"^-'' + 10'"«''^ 33 (Bdi 33 trang 92 SGK) a) log34 > log33 = Gidi 37 (Bdi 37 trang 93 SGK) ^ Gidi a) log^gSO = log 5.10 = 2[log3 + log3lO] B i e t log3l5 = log33.5 = loggS + log35 = + logaS l o g i = log43-' = -log43 < Vay log34 > l o g b) 3'°«»''' v& 7'°^«"'-'' Do l o g s l , ! > n e n ' * ' ' 2'°^-^^' = 25 + - 27 = logsX = l o g ^ => X = Suy r a log35 = a - Vay log ,| 50 = 2[a - + p] = 2a + 2p - b) Iog4l250 = log,, 5^2 = laogzS" + log22| = ^ [41og25 + 1| = 2a + > 3" (do > 1) hay 3'°^'-'' > 38 (Bdi 38 trang 93 SGK) va log60,99 < n e n 7'"'''''''^ < 7" (do > 1) Gidi ' r • ' a) l o g - + - l o g + 41og72 = log2^^ + I o g ( ' ) + log 2^ T' = log2^l2.2^ = l o g l hay 7'*"-^''< Vay 3'°^>i''' > y'nKsOSf 34 (Bdi 34 trang 92 SGK) a) log2 + log3 = log2.3 = log6 log6 > log5 V a y log2 + log3 > log5 Gidi l b ) l o g ^ + ^ l o g S + | l o g ^ = log(2^3-') + log(6')5 + log v2 3 V2 o-2^ = Iog(2'.S"') + log(2.3) + logCSl " ) = log(2l3-^.2.3.3^ 2"^) = log(3^.2.22 ) = log(18.72 ) | J c) log72 - 2\og~ + §4 SO e VA LOGARIT lU NHIEN logN/108 = l o g ( l ' ) - log + log(2^3^)2 = l o g ( l ' ) - log(3^2-'«) + log(2.32) v2 IVOI D I J I ^ G cAl^ NH€i = log ^ ' - f = I, So e: e = l i m log(2^° 3'^ ) = log2^" + log3'^ = 201og2 - | log3 Cong thiJc l a i kep l i e n tuc d) l o g - - log 0,375 + 21ogV0,5625 Dieu k i e n x > 0, x r: l a i sua't m i n a m = log — N : so n a m S: so t i e n t h u dugc ca vo'n I a n l a i Gidi L o g a r i t tu" n h i e n L o g a r i t cof so e cua mot so diTcfng a difoc goi la logarit t i i nhien (hay logarit n e - p e ) cua so a K i h i e u Ina a) logx27 = c o x ^ = c o x = b) l o g : , ! = - o X"' = ! o X " ' = 7"' o ^ c) log, V5 = - o X-' 40 r S d i 40 trang 93 SGK) M31 X = (VS) * L o g a r i t t i f n h i e n c6 day du t i n h chat cua l o g a r i t co so lorn hon X= ^ « J - i = >/5 X = » Gidi So cdc chuf so cua M k h i v i e t t r o n g he t h a p p h a n b^ng so cac chuf so cua " nen so cac chuf so cua M la [31.1og2] + = [9,31 + = 10 Tuang tiS, so cac chC so cua M = 2^^'' - k h i v i e t t r o n g he t h a p p h a n la: [127.1og2] + = 38 + = 39 Mi398269 k h i v i e t t r o n g he t h a p p h a n la [1398269.1og2] + = 420921 41 (Bai 41 trang 93 SGK) Gidi So t i e n ca von iSn l a i sau n qui \k S = 15(1 + 0,0165)" = 15.1,0165" ( t r i e u dong) Suy r a logS = l o g l + nlogl,0165 _ ^ 42 (Bdi 42 trang 97 SGK) Gidi Sai tuf ln(2e) = Ine + Ine V i ln(2e) = l n + Ine ^ Ine + Ine Gidi • ln500 = l n ^ ' = ln5'^ + l n ' = 31n5 + 21n2 = 3b + 2a 16 • I n — = l n ^ - ' = l n ' + l n - ' = 41n2 - 21n5 = 4a - 2b •ln6,25 = I n — = l n = 2[ln5 - l n ] = 2(b - a) 100 1, 1 , 98 , 99 '"^ o + '^^^T + + I n — + l n 99 100 = I n l - l n + l n - l n + + ln98 - ln99 + ln99 - InlOO = I n l - InlOO - l n ( ) ' = - ( l n + ln5) = - ( a + b) 44 (Bdi 44 trang 97 SGK) Gidi — ln(3 + V2 ) - 41n( ^/2 + 1) - ^ l n ( 72 - 1) logS-logl5 logl,0165 De CO dugc 20 t r i e u dong t h i p h a i sau mot t h d i gian la n = BAITAP 43 (Bdi 43 trang 97 SGK) = 2^' - So cdc chuf so cua S = A.e''^ A: so' vo'n ban dau = log 2-' - log(0,5^3) + log(0,5^32) ^ i o g l ^ | _ _ ^ i ^ g ^ 39 (Bdi 39 trang 93 SGK) 2,71828 logl,0165 17,58 (qui) Vay sau n a m , t h a n g (4 n a m , qui) ngifdi gufi se c6 i t nha't 20 t r i e u tiT so von 15 t r i e u dong ban dau) (vi sau qui h a i , ngifdi gijfi m d i n h a n dUc/c l a i ) ^= ^ ln( 72 16 |-^ln(72 J-D + If - 21n( 72 + 1)^ - ^ l n ( 72 - 1)^ 16 25 +l)'^-^ln(72 16 -1)^ = - | J [ l n ( + l ) ^ + l n ( - 1)2] 16 ;-^ln[(72 •lo +1)1(72 - l ) ^ j = - ^ l n ( l ) = 16 (Bai 45 trang 97 SGK) Gidi Ti le t a n g trufdng m i gicr ciia lohi v i khuS'n Tii cong thufc S = A.e''' 300 = lOO.e^'' ^ ^ ^ l n _ 0 - l n l 0 ^ ]n3 ^ 5 T i 1§ t a n g t r u d n g ciia loai v i k h u a n n^y la 21,97% m i gid Sau 10 gid, tii 100 v i k h u a n se c6: TiJf 100 con, de c6 200 t h i thori gian c^n t h i e t l a 200 = lOO.e^-^'^^' , =>t= In200-lnl00 0,2197 = ln2 0,2197 o-,r: o ã>,n u - ô 3,15 gift = gid p h u t 46 (Bdi 46 trang 97 SGK) Gidi PC D a o h a m c i i a h a m so m u * H a m s6 y = a" C O (3ao h a m l a y ' = a''.lna * H a m so y = e" c6 dao h a m l a y ' = e" * H a m so y = a"'"* c6 dao h a m la y ' = u'(x).a""'\lna u(x) * H a m so' y = e c6 d a o h a m la y ' = u'(x).e' D a o h a m c u a h a m so' l o g a r i t * Ham s o ' y = logaX (vdi x > 0) c6 d a o h a m y ' = xlna * H a m so' y = I n x ( v d i x > 0) c6 dao h a m y ' = — X u'(x) * Ham uix).lna sd' y = logaU(x) (vdi u(x) > 0) c6 dao h a m y ' = u'(x) * H a m so' y = Inu(x) ( v d i u(x) > 0) c6 dao h a m y ' = u(x) T i le p h a n hiiy h a n g n a m ciia Pu^'^ Ta c6: S = lO.e''^^^*'" Dac b i $ t : y = I n i x | (vdfi m o i x Suy r = ^"^ " ^"^^ « -2,84543.10^^ ~ -0,000028 24360 Vay svf p h a n h u y ciia Pu^^** AKac t i n h theo c6ng thuTC g _ ^ g-0,000028t T r o n g S va A t i n h b k n g gam, t t i n h b&ng n a m V a i 100 g a m Pu^''^, t h d i gian cAn t h i e t de p h a n h u y gam la = 10 e"'^'^^^'^^*'' ] n l - j n l O ^ 82235 (nam) -0,000028 Vay sau k h o a n g 82235 n a m t h i 10 g a m chat Pu^'*'' p h a n hiiy gam 0) t h i y ' = — y = I n I u(x) I (vdfi m o i u(x) 0) t h i y ' = — — u(x) b i e n t h i e n v a t h i c u a h a m so m u , h a m so l o g a r i t a) Ham so mu y = a' • T $ p x&c d i n h : M • D o n g b i e n t r e n K k h i a > 1; nghich bien t r e n R k h i < a < • - Do t h i D i qua d i e m (0; 1) Nhm p h i a t r e n true h o a n h N h a n true h o a n h l a t i e m can ngang \' §5 HAM SO MU VA HAM SO LOGARIT - a CAN I I \ O I DUI\ MlIC? K h a i n i $ m v e h a m so m u v a h a m so l o g a r i t * Cho a > va a - H a m so' y = a" duoc goi la h a m so' mu co so' a - H a m so y = logaX dJoc goi l a h a m so l o g a r i t co so a * H a m so' y = a'' va y = logaX l i e n tuc t a i m o i d i e m m a no duac xac d i n h • Vxo e M, l i m a" = a'^ X->Xo l i m logaX = logaXo • Vxo e (0; +oo); Dang dS thi b) Ham so y = logaX • T a p xdc d i n h la (0; +oo) ã Dong bien trĐn (0; +oo) k h i a > 1, nghich bien t r e n (0; +oo) k h i < a < • Do t h i - D i qua d i e m ( ; 0) - N k m b e n p h a i true t u n g - N h a n true t u n g la t i ^ m can diJng * M p t so g i d i h a n : • lim x-*o l n ( l + x) X , , e^-l , = 1; h m = 1 X Dang dS thi ^ 51 (Bdi 51 trang 112 SGK) Gidi a) - Ham so y = (x/2 )" xac dinh tren R - Ham so dong bien tren R (ca so J2 > 1) Do thi hkm so: - Di qua cdc diem (0; 1); (1; ^); (2; 2) - N^m phia tren true hoanh - Nhan true hoanh lam tiem can ngang BAITAP 47 (Bdi 47 ti-ang 11 SGK) Gidi a) Khi nhiet ciia nUdc la 100"C thi luc P = 760 -2258.624 Do ta c6: 760 = a 10'"°^"'' -2258.624 2258.624 hay 760 = a 10 ^'^ ; a = 760.10 -2258.624 Vgly a « 863188841,4 40*273 b) Tinh ap sua't htfi nifdc: p = a 10**"'^ - 863188841,4.10 = hm = -3e.-hm — _ g6x3x b) lim = lim x >0 +0 49 (Bai 49 trang 112 SGK) x ^ go2 x V = 863188841,4.10 SI'S" ^ 52,5 mniH = -3e 3x x-u = - = -3 Gidi X a) y = (x - De'" ^ y' = e'^ + (x - l).2e''' = e'^ (1 + 2x - 2) - (2x - l)e'^ b) y = x ' Ve'" + y' = ix'y4^^ +xl(N/e^^+l)' = x V i ^ + x l f^'" 2Ve'" +1 = 2xN/e^' D y3j H^m s6' y = >/2 + V3 nghich bien tr§n M (vi co so 2•—TV xac dinh tren [3) - Ham so nghich bien tren E (co so — < 1) '2-^1 Do thi ham so I ' 3j V 9j - Di qua cdc diem (0; 1); - Nam phia tren true hoanh - Nhan true hoanh la tiem can ngang 52 (Bdi 52STT trang 112 SGK) Loai am Gidi b) Ham so' y = 22M.624 48 (Bai 48 trang 112 SGK) Gidi e^-e^-^ , e ^ ( l e ^ ) a) lim = lim y y ^ x -1 -r I Do Idtn (L) \ NgiTSng nghe Nhac em diu 4000 Nhac manh phat tCf loa 6,8.10** Tieng may bay phan liic 2,3.10'^ NgLf5ng dau tai 10'^ 53 (Bdi 53 trang 113 SGK) Gidi , , ln(l + 3x) 31n(l + 3x) ln(l + 3x) ^ , ^ a) hm = lim = 3iim = 3.1 = x^o X "-0 3x 3x b) Vi lim , ln(l +5 x^) = 1, x^ + x') , xln(l + x') x-.ahm , ln(l nen = hm = 0.1 = 0^ x->o X "-•0 x 54 (Bdi 54 trang 113 SGK) Gidi a) y = ( x - l ) W x dB 36 dB 88 dB 124 dB 130 dB y' = (3x + (3x - 2)2(lnx)'lnx 01 2-X 2)'ln^x / 0+ (3x o^ 1- 12) (InM' 01 2= 31n^x2(3x-2)lnx = 31n + 2(3x - 2) — Inx = 31n x + X X b) y'y == V( V x U^ ^l ^.Inx^ )Mnx'+Vi^^.(lnx2)'= V2 + Vs < 1) x.lnx' = + 2NA? +1 2Vx^ +1 Inx^ + ? T l ^ ^ c) y = x l n +X y' = x ' l n -1 + x + x I n -1 + x = I n -1 + x + X (i + §6 HAM SOLUYTHLfA x7 + x d)y = InCx'' +1) 2x , [ln(x^ + l ) ] ' x - l n ( x ^ + l ) x ' y = = 55 (Bdi 55 trang 113 SGK) Gidi a) H ^ m so y = logaX nghich b i e n t r e n (0; + x ) e v i cc( so' — < e b) H ^ m so y = log«x dong b i e n t r e n (0; +ãô) VI cd so a = 3(73 - N/2) ^T^-^-^^^^ K h a i n i ^ m h a m so l u y thxia - H a m so luy thCfa la h a m so' dang y = x" (trong a la h a n g so') - H a m s6' y = x°, v6i a k h o n g nguyen xdc d i n h t r e n (0; +oo) D a o h a m c u a h a m so l u y thvta - H a m so y = x" (vdi a e - H a m so' y = u°(x) (v6i a G K) C6 dao h a m y' = a u ° ' kx).u'(x) V a i n e t ve su" b i e n t h i e n t h i h a m so l u y thtifa * H ^ m so l u y thiTa y = x" (vdi a 0) x^c d i n h t r e n (0; + x ) * H a m so dong b i e n t r e n (0; +t3o) n§'u a > v^ nghieh bien t r e n (0; neu a < • Do t h i h a m so qua ( ; 1) vdfi m o i a 56 (Bdi 56 trang 113 SGK) Gidi a) - H ^ m s6' y = log ,^ x xdc d i n h t r e n (0; +oo) Do t h i h ^ m so +x) Mot s6 dang dd thi V3 + x/2 > - H a m so d n g b i e n t r e n (0; + « ) K ) c6 dao h ^ m y' = a x " " ' O - D i qua d i e m ( l ; ) , ( ^ ^ ; l ) - Nkm bgn p h d i true t u n g - N h a n true t u n g la difdng t i | m ean diJng b) H ^ m so y = l o g j X xac d i n h t r e n (0; + x ) - H ^ m s6' nghieh b i e n t r e n (0; + ta eo bat d^ng thijfc x" > x° o p > a Do j3 = - - va a = - 2 (C2) la t h i h a m so y = x (Ci) 1^ t h i h ^ m so y = x ^ ^8 (Bdi 58 trang 117 SGK) Gidi a) y = (2x + D ' y' = 7i.(2x + D " - ^(2x + D ' = 27i.(2x + 1)"" ' b) D a t u = In'Sx Vay u' = 3.1n25x.an5x)' = S.ln^Sx ^ 5x ^^"'^"^ (Bdi 30, trang 206 SGK) Gidi + 375 a ) Tiixh — z / 6- 5- _ (3 - S) + (1 + 3N/3)i z " n § n (p = 71 - + 271 71 12 2- 12 7t - (3 - V3)i + (1 + 3^/3) 12 12j ' v/3i b) T a c6: sd(OM; O M ' ) = Sd(Ox, O M ' ) - Sd(Ox, O M ) = cp' - cp • = acgumen-— (sai khac k2n) z T r o n g (p, (p' 1^ acgumen cua z cos((p' ~(p) = va sin((p' - (p) = i i - + = ism- 12 Tt 12 27: 271 n3 cos— + i s i n — 3 r 7t ^ t ^ isn 112"" ; — 371 971 , 37t -|3 C O S — + isin— 4 71 '4 71 1— w = cos47t + isin47i = ) 71 12 371 371 • Zi = ZQE = C O S — + i s i n — 4 U Y Z2 = E M Z0.8 T A = COS r I 771^ 12j + isin 77t^ 12 J P 207 SGK) Gidi cos4(p + isin4(p = (coscp + sincp)'' = cos''(p + 4(cos"V)(isin(p) + 6(cos^(p)(i)^sin^(p + 4(cosip)i'*sin''(p + i''sin''(p = cos''(p + 4cos^(p.sin(p.i - 6cos^(psin^(p - 4.coscpsin'^(p.i + sin''(p = (cos^cp - 6cos^{psin% + sin''(p) + (4cos^cpsin(p - 4cos(psin"'ip).i Vay * S 12 cos4(p = cos''{p - 6cos^sin^(p + sin''(p sin''(p = 4cos'^(psin(p - 4cos(psinV T i n h r : r = J^ + f = , T i n h sin4(p, cos4(p iV3 v a smcp = 12 Z^ = (Bai 33, trang 207 SGK) 71 = cos — + i s i n — = (0 4 • Zn = C O S — + i s i n — (Bdi 32, trang ^ — n§n (p = - ( M p t acgumen cua w ) n n Ttt d6 w = cos — + i s i n — 4 T i n h (p: coscp = - - - 71 27t^' C O S — 7: | + k27i ( k G Z ) L + 73) = - ^ + 371 b) Bieu d i e n h i n h hoc cdc so' phufc Zo, Z i , Z + i) x/2 T i n h cp: cos(p = — , sincp = 12 cos— + i s i n — z^e« 271 (do 8^ = Gidi (Bai 31, trang 206 SGK) isin- Vay Z l a n g h i e m phUcfng t r i n h z„ - w = N e n goc i J o n g giac ( O M , O M ' ) c6 so Tinh r: r = = C O S '27t + ^ ' + i s i n '27: + ^ ' I 4, I 4; Vay z, la n g h i e m cua phifcfng t r i n h z,^ - w = ^/3 ^ N e n cp' - (P = ~ + k27i ( k e Z ) ^/2 -(1 = COS — + i s i n — 4 * (Z2)3 = (ZoE^)^ = z w = = cos.- 71 r 71 ^ 27T^ cos l l ^ >+ 971 z' + x/Si c6 = t Vay Zo 1^ m p t n g h i e m cua phifcfng t r i n h z^ - w = 10 - cos— + i s m — 10 = z n = cos — + i s i n ^ - 3^/3 + 3(1 + 3S)i 10 + 10x/3i cos— + i s i n — 3J 7t _ 271 Tif d6 e = 3+i _ [(3 - V ) + (1 + 3N/3)i](3 - i) - (Mgt a c g u m e n c u a s) ^ = — * M ' 73 — Gidi - i T i n h r: r = V s + T = u ^/3 , -1 T i n h cp: coscp = — va sincp = — rv' 4b thi - coscp - isincp = s i n - cos 2 la 2j -2sin- 2; (p 71 Gidi a) (z + + i)^ - 6(z + - i) + 13 = Dat t = z + - i Gidi phufdng trinh t^ - 6t + 13 = dang lUOng giac can tim *• Neu sin— < thi - coscp - isincp = • z+w + zw k27t, k e Z) Cp A' = - 13 = - ; ±S = ±2i 71 cos — + - + isin 2^2 2 Nen t = ± 2i * z + - i - + 2ic:>z = i la dang lifdng giac can tim * z + - i = - 2i c> z = - i V4y phuong trinh c6 hai nghi^m 1^ Zi = 3i * N e u s i n ^ = thi - coscp - isincp = = O(cosa + isina) (a e R) ON TAP CHaONG IV Gidi (2 - 3i)'' = 2' - l i + 3.2.(3i)' - ( i f = - 36i - 54 + 27i = - - 9i Phan thiic l a - , phan d o 1^-9., 63 26 X = 1K 37 (Bdi 37, trang 208 SGK) 23 — + 2xy - 2y = o 2y(x - 1) = • » , 371 - s i n — + icos cos-g-v 37t, 371 1+1 So phufc Ik so thUc \k chi k h i (Bdi 371 ^ ( - ) + (3 + 2)i ^ ( + 2)+ (2 = (x'' - y^ - 2x + 5) + (2xy - 2y).i ST: (do c o s — < 0) cos-sin n V (3 - 2i)(3 + 2i) = x^ + 2ixy - y^ - 2x - 2iy + 57t 571 (1 - iX3 + 2i) ( l - i ) ( l + i) = x^ + 2ixy + (iy)^ - 2x - 2iy + 57t (3 + 2i)(l + i) PhAn thuc 1^ — , p h i n l a — 26 ^ 26 c) ( X + iy)^ - 2(x + iy) + (x, y € R) cos- — - s i n - - i c o S g ^ 71 +3 - i 1-i b) iz + \ u-2i; DM t = -3 iz + z-2i -4=0 iz + z - 2i G i a i phiiong trinh t ' - 3t - = t = - l ,t=4 z^ = - i 3)i t = -l izr-t- -1 iz + = —z + i z-2i vdi mpi x > Nen f(x) dong bien trĐn nijfa khoang [0; + ô ) b) Do h^m so dong big'n Vdi moi X > Suy f(x) > f(0) Nghia la e" - X - > Vdy e" > X + vdi moi x > (Bdi 2, trang 211 SGK) Gidi a) f(x) = 2x^ - 12x - 10 H^m so xdc dinh tren R Su bien thien cua ham s6' a) Gidi han t a i v6 cUc l i m f ( x ) = - o o ; l i m f ( x ) = +oo X->-ac X y = — ( x - X o ) + yo hay y = — ( x - X Q ) + InXo - H^m s6' dong bien tren cdc khoang (-oo; - ) vh (2; +oo); nghich bien tren khodng (-1; 2) - H&m so dat cUc dai tai x = - , gid t r i cifc dai 1^ t{-l) = - - H^m so dat cifc tieu tai x = 2, gid t r i cUc tieu 1^ f(2) = c) Bdng bien thi§n f'(x) Xg Phifcfng trinh tiep tuyen (1) v6i (C) tai Mo 1^ b) Sii bien thi§n, cUc t r i Ta c6 f (x) = 6x2 _ 6x - 12 = 6ix^ - x - 2) x =- l fix) = o x - x - = o x =2 -00 - Tac6y'= -,y'(xo)= — X-»-MO X X = f "(x) = tai X = — va doi dau Ui a m sang diCOng fl 33^ , , * nen U —; la diem u6n cua thi l2' 2J * Giao diem cua t h i true tung (0; - ) Nhan xet: Do thi nhim U n 3 ^ la tam doi xiing l2' ) b) Tic bang bien thien ta thdy f(x) < -3 < vai moi x < Dieu chufng to phUcfng trinh f(x) = khong C O nghiem khoang (-oo; 2) * Trong nufa khoang [2; +oo) ham so lien tuc, dong bien fi:2) = - < m = 22 > Nen phucfng t r i n h ftx) = c6 nghiem nhat c) Gqi a Id nghiem thite nhat cua phUang trinh nghia Id f(a) - (a e [2; +00)) ta lai c6 f(3,5) = -3 f(3,6) = 1,23, ta c6: - < < 1,232 Do ham so' f va lien tuc va dong bien tren (2; nen 3,5 < a < 3,6 (Bdi 3, trang 211 SGK) Gidi Goi Mo(xo; yo) la mot diem bat k i cua (C) Mpt acgumen ciia z 1^ - — - cp (Bdi 1, trang 217 SGK) a) Ham so f xdc dinh 7t -1 0 f(x) +0C +0C '-30 Xn Ta chiJng minh vdi x e (0; +00), o ( x - X o ) + lnXo — - - l n — > (*) X6t h^m so g(t) = t - Int vdi t > • H^m so' xac dinh v^ lien tuc tren (0; +00) g'(t) = - i = , g'(t) = « t = -Inx > B&ng b i ^ n t h i S n t g'(t) + — — - — g(t) * Do h ^ m so n g h i c h b i e n t r e n k h o a n g (0; 1) • < t < => g(t) > g (1) => t - I n t > * Do h a m so' d6ng b i e n t r e n nufa khoang ( ; +oo) • t > => g(t) > g (1) => t - I n t > • V(Ji t = g(l) = Vay v d i m o i t > t a c6 t - I n t > hay t - - I n t > 0, v d i m o i t > — + 30 fix) = 12500 450x^ - 12500 = o x^ = 450 H a m so cifc t i e u t a i r5Vl0V o x = o + 1 - ^/6 (do X = X > 0) — *>1360 , gia t r i cUc t i l u la f(0)= - i ; f l l ) = f l l ) « 2272, fl8) * 1407 v a y chi p h i t h a p nha't la 1360 (nghin dong) Do can siJ dung 5,27 nghIa la may cho vi^c i n de diToc l a i n h i e u n h a t va dong bien t r e n k h o a n g V|y max f i x ) = ~ (Bdi trang 212 a) P(x) = m i n fix) = vk 5' ^ V6 SGK) - Gidi 4" + Ta c6 P(a) = 4" + N e n P(a) + P(b) = P(b) =^ 4^+2 4" 4' 4^4'' +2.4" +4^4^ +2.4^ 4" + 4''+2 (4" + 2)(4''+ 2) +4') 4^4'' + 2.4" + 2.4'' + 2.4 + ( ° + ' ' ) + 2(4° 4"*'' +2(4" + " ) + +4") + 2(4" +4'') 1 log, log, b ) T a t i n h l o g - i l o g = I o g - - - ^ ^ ^ = loge2 - 2 loggVe = l0g62-l0g65 = l0g6o 0 ^ 5^_5VlO^ 12500 ^ ^ g ^ 5N/10 0; - H a m so d a t ciTc t i e u t a i x = - , gia t r i ciTc t i e u y'^ 2.4°^' + 2(4' + 1407 ~ '(X) = - H a m so n g h i c h b i e n t r e n - ; f 5Vio + 6f fix) Bang bid'n t h i e n 2272 2\l(-x'+x fix) — ( x + 10) + 50x ( n g h i n dong) 3600x So l a i se u h i e u n h a t k h i chi p h i t h a p n h a t Ta t i m gid t r i nho n h a t cua h:^m so f(x) t r e n doan [ ; 8] ^ ^„ , 2500 ^„ 12500 T a c6 ftx) = + 50x + 9x H a m s6' f(x) xAc d i a h v d i m o i x > nen xdc d i n h va l i e n tuc t r e n [ ; 8] 12500 450x^-12500 fx r-TT- + 50 = 9x' 9x^ fix) x X fix) = f'(x) -f X + B a n g bien t h i g n (Bdi trang 212 SGK) Gidi Goi X l a so m a y i n sijf dung, x nguyen diicfng, < x < So ban i n ciia x indy t r o n g m o t so 3600x (bdn) C h i p h i v a n h ^ n h x m a y m p t I a n i n : 50x ( n g h i n dong) T o n g chi p h i de vain h a n h x mdy i n 50000 t d quang c&o « Gidi H a m so xdc d i n h t r e n (-2; 3) N n xac d i n h va l i e n tuc t r e n [O; 1] -2x + f'(x) = - i V(-x^ + x + 6)^ Do d6 t r e n k l i o i n g (0; + « ) , (C) n k m p h i a dU(Ji difdng t h a n g (D) X SGK) l-x^ Vay — - - I n — > , v d i m o i x > fix) = « (Bdi trang 212 Tijf d6 B = /IN' =I=I = 2,5 < Vis = A = (Bai trang 212 SGK) Gidi a) a > 0, b > thoa a^ + b^ = 7ab Suy (a + b)^ = 9ab T\i logvCa + b)^ = logvOab ^ 21og7(a + b) = logvS^ + logva + logvb => 21og7(a + b) - 21og73 = logva + logvb J a + bl = logva + logvb log,' ^ l o g v — = ^(iogTa + logvb) : ^ I b) a > 0, b > 0, a ?t v& logab = 1^ log 7b ^ log.aVb log,a + log,Vb -glogaa-llog.b ^ + v3 1+ 2 - 2N/3 - 18^/3 + 27 3l 73 l + |log,b (2 - 9V3)(2 - ) 2-9%/3 {2 + V ) ( - V ) 31-20x/3 rBdi S trang 212 SGK) a) * y = cosxe^'^l Gidi y' = (cosx)'e''""' + { c o s x ) ( e = -sinxe^*'""" + cosx 2tanx =e y = log2(sinx) (sinx)' cosx y' = sinxln2 sinxln2 b) y = e"" + 26-" y' = 4e'*'' - 26-" y" = 166^" + 26-" y'" = 64e^" - 26-" COSX - ,2tanx COS^X sinx cotx ln2 Do y ' " - 13y' - 12y = 646^" - 26"" - 13(46^" - 26-") - 12(6*" + 26-") = 646"" - 26"" - 13(4e'"' - 26-") - 12(6'" + 28"") • = 646"" - e " - 526'" + 26e-^ - 126'" - 24e" = (Bai trang 212 SGK) Gidi a) * H^m so y = 2" - T4p x^c dinh R , tap gia t r i thugc kho4ng (0; +oo) - Ham so dong bien tren K Do t h i ham so - D i qua diem (0; 1); (2; 4) - N^m tren true hoanh - Nhan true hoanh lam tiem can ngang * Ham so y = ( )" - Tap xac dinh R, tap gia t r i thupc khoang (0; +«) - Ham so dong bien tren R Do t h i ham so' - D i qua diem (0; 1) va (2; 2) - N^m tren true hoanh - Nhan true hoanh lam ti|m can ngang * Ham so y = ( )" - Tap xac dinh R, tap gia t r i thupc khoang (0; +oo) - Ham so dong bien tren R Do t h i ham so' - Di qua diem (0; 1) va (2; 3) - Nlim tren true hoanh - Nhan true hoanh lam tiem can ngang Ba t h i cAt tai (6; 1) b) Ham so' y = logsx - Tap xae dinh la (0; +oo), tap gia t r i la R yj=2 + logsx y = logaCx + 2) = log3X - Ham so dong bien tren (0; +«) Do t h i ham so' - Di qua diem (1; 0) va (3; 1) - Nkm b§n phai true tung - Nhan true tung lam ti§m can dufng * Do t h i y = + logsx suy t i i t h i ham so' y = logax hkng ph6p tinh tien theo veetP (vdi 1(0; 2)) * Do t h i y = log3(x + 2) suy tiif t h i ham so' y = loggx bkng ph6p tinh tien vector OK (vdi K ( - ; 0)) 10 (Bdi 10 trang 212 SGK) Gidi a) Sr'"'" + 81™'""= 30 Dat t = Sr'"''', t > ,1-sin^x Suy " ' " = 81' _ 81 81 8r Ta c6 phifotng trinh t + — = 30 o t^ - 30t + 81 = o t = 27 hoac t = • t = 27 « 81"" " = 27 » sinx = ± 73 = 3^ o sin^x = x = - + k27i X = 7l +k27l t = « sr-^" = r X s^"'"^" = ci> 4sin^x = » 2\8-^ = 272 sinx = ± - d) (1) 71 = - + k2n ,k e Z x = ± ^ + kn, k e Z "2 O « o » logi X = logi '11 c) 4"'«* * ^ - 6'°«^ - 2.3'°*"' o ^ UJ " = o u = -2 (loai) o t = -2 » u = — - 44 * t = - o logx = - logx = loglO-^ o X = 10-' Vay phufong t r i n h c6 nghiem la x = 10 UJi i - hoac y = - - V 2, ,u>0 ^ Vay tap nghiem ciia he phifong t n n h la S = « 4.4"'s^ - (2.3)'"^* - 2(3=')'''^.3=' = Dat t = logx Ta c6 phuong trinh: 4u^ - u - 18 = r:>X = 2-^ • y = — =>x = — (loai) • ir> 4'°^.4 - (2.3)'"'^ - S'"*"' 3'^ = 3j ISy''' + y - = o y = •y = — ^ 16 = D i l u ki?n x > Ta CO phufong t r i n h 4.2^' - 2'.3' - 18.3^' = ^ " x-3y = '3y + K l ^ 2J ^ x= ~ ^'^^^^ duac phUofng t r i n h logi X = log.l Vay phacmg t r i n h c6 hai nghiem x = ^ ^"^"^^ c> X = (thoa dieu ki^n) logi X = o ^ TCr p h u o n g t r i n h thuT n h a t x = 3y + — t h a y v a o phu'ong t r i n h thuf h a i t a < ^ X = - (thoa dieu kien) 16 Dat u = xy = - [21 t=4 — - 3t - = « t = - hoSc t = ^ „ _ fi l Do d6 (I) o logi X = - x - 3y = - o D a t t = log.x ( x > 0) t = -1 2""^'' = 2^ o logay + logax = - logaCxy) = - o xy = logi X - 31og, X + = logi X - 31ogi x - = Ta c6 phirc/ng t m h ly>0 (2) Phucmg trinh (2): logs - + ^ = ^ logs Oy) x 2 ^"^3^ o logs (log X - 31og^ X + 5\ logg 3^ o • Dieu kif n { Phucftig trinh (1): 2^8-^ = ^2 » 2\2-^^ = 2.2^ X = 7t ~ ^ + k27l V$y nghi§m phifcfng t r i n h 1^ x = ± - + k n b ) loga [logi X - 31ogi X f- 5\ Dieu k i | n x > r ^Q (1) 1 l o g - + -2—2-^og3(By) (2^ UJ - 18 = 2j 11 (Bai 11 trang 213 SGK) Gidi • a) y = log[l - log(x^ - 5x + 16)] H a m so xdc dinh k h i va chi k h i x' - x + 16>0 x' - 5x + 16 > l - l o g ( x ' - x + 16) > log(x' - x + ) < l x ' - 5x + 16 > x ' - 5x + 16 > (1) x='-5x + 16 < x < Vay tap xac dinh cua ham so la 'y = (2; 3) 13, (Bdi 13 trang b ) y = ^ l o g o , ( - x ' + X + 6) + x' +2x H ^ m s o xdc d i n h k h i v a c h i k h i (I) -x' + x + > (1) logo5(-x'+x (2) +6)>0 +2x^0 213 JSsin' X + • dx= 12 = X - G i a i ( ) logo,5(-x^ + x + ) > logo.sl -x^ + x + < l < = > x ^ - x - > o o -2 < Vay h | (I) o X < X < 1-V21 hoac X > f ( ) = - ^ hoac 2sin 14 (Bdi 14 trang I+ 213 SGK) < X < -2; 1-721 f l + N/21 Dat x = tanu dx = ; du j x ^ ( l + x*)='dx |._dx^ = ) - ^ d u = ) d u J oJ tt gg ' u + U'+l I f , ^ d + x')* + C Do X = - fu'du = — + C = 16 •' 16 b) T a c6: c o s x s i n x = s i n x c o s x b) | c o s x s i n x d x = |2sinxcos^ x d x dx 4dx x^ + x + 4x^ + x + D a t u = c o s x => d u = - s i n x d x Do Z = x t a n x - V = tanx tanxdx Tirdo - (tanxdx = dx J •'cosx D a t u = c o s x => d u = - s i n x d x tanxdx = - 2x + l u = — = x/3 u(0) = cos X •sinx Ta tinh Kiu J} ( x + +3 = - I n u I = - I n cosx V a y Z = x t a n x + I n | cosx | + C V V3 > , du = ; u(l) 2x4-1 + -pr +1 , dx V3 j = -L = S ^/3 =1 dx f I S J ^/3 oV2x + l + C dx V3 Dat du = dx dv = 4dx 2x + l ^ X f - ^ d x 'cos^x D a t u = x => 71 =u 4dx Do d6 Y = - fu'du = - — + C = - - c o s ^ x + C J 3 c) T i n h Z = Giai cos u Giai 213 SGK) + V2T D a t u = + x " => d u = 4x^dx Tinh Y = +1 dx x(0) = 0, x ( l ) = a)TinhX= 2sin 2x + C = 8C = + ^ < ^ C = + N/21 'r V a y t a p x a c d i n h c u a h a m so Ik V = (Bdi 12 trang + C = 4x - va x ^ - < < X < sin(2x + 7t/6)' dx l + %/21 va X * -2 X - X x v^ f(0) = 12 - cos 2X + ' ' T a CO fix) = (3) + 2x Giai f '(x) = Ssin^ G i a i ( ) - x ^ + X + > - < X < G i d i (3): SGK) + 1 ~ • / du ^ u' +1 ~ 2S _ d u _ u' + V3 2V3 ( n _ n ] _ 2x/3 1,3 ~ 6) n _ • ~ nsj3_ D i # n t i c h h i n h p h a n g gidri h a n bdi h a i t h i c) fx'e'dx S(x)= Dat u = => dv = e^dx => du = 2xdx V J|f(y)-g(y)ldy = j f y2 u = x Do Jxe^dx = xe" =r> V - Je^dx = xe" - e^l '0 = y = -3x^ + la nghiem r4> X = Zi = ±1 * T a x e t dau f(x) - g(x) t r e n doan [ - ; 1] - x ' + 2) ( t r o n g f(x) = - x^ va g(x) « x ) - g(x) == - X ' - ( x ' + 2) = x ' - = 2(x^ - 1) X + +00 0 + b) T u n g giao d i e m t h i x = * z, „ ^ z '0 V T r e n doan [ - ; 51 t a c6 f(y) - g(y) < 0 ^ = Tt 2-2- n Z2 - 2i) Gidi y = - 2i " = (1 + 2) + (-2 + l)i = - i Q, •2 = + 2i - (1 - 2i) = + i 1-i "~ P • 4- 1- • i i 2, : , , • ( l - i ) ( l + i) " i_ Goi M , N , P, Q, R Ih cAc d i e m bieu d i i n cdc so phufc ; Z i ; z-/, z i - Z2; Z i Zl +16 a) (N/3 Gidi - i)^ = + > / i + i ^ - (3 - 2N/3 i + i^) ^/3 i + i ^ - + N/3 i - i ^ = 4N/3 i b) (V3 + i)^ + ( v/3 - i)^ = + %/3 i + i ^ + - 73 i + i ' = + 2i^ = - = C ) ( ^/3 + i ) - ( ^/3 if = {Sf + 3{yl3fi + 3S.i^ + i^-((Sf3( 73 )^i + 3( 73 )i^ - i^) = + i - - i - + i + - i = 16i d) (73 + i)=^ _ (73 + i)''(73 + i ) ' _ (3 + 2731 + 1^)" ^ (2 + 273i)^ (73-i)^ [(73-i)(73 + i)]^ 16 (3 + 1)^ = + - Z2 (Bdi 18 trang 214 SGK) l a n g h i | m cua -4 7t — t r o n g m a t p h i n g phufc y ^ - y - 20 t r e n doan [ - ; 5] + ^ ^ z^ ^ l - i ^ (1 - i ) ( l + i ) C y ^ - y - = 0y = - ho^c y = y + 16 + i; vkx= y'-4 [(y - Ddy = (1 + 2) + (1 - 2)i _ -^ + —00 64 * zi ^ = {1 + i ) ( l + 2i) = (1 - 2) + (2 + l ) i = - + 3i S(x) = Jl-x^ - (2 - 3x^|dx = j [ - x ^ + - (-x^')] dx = j ( - x ^ + 2) dx Xet dau f(y) - g(y) = ri6 243 = (1 + i)^ = + i + i'^ = i * 2z, - -1 - 7t * Z i Z - = (1 + i ) ( l T r e n doan [ - ; 1], f(x) - g(x) < D i | n t i c h h i n h p h i n g g i d i h a n b d i h a i t h i - - - + 2X 17 (Bdi 17 trang 213 SGK) phUcfng t r i n h f(x) - g(x) + 100- b) The t i c h kho'i t r o n xoay tao t h a n h k h i quay B quanh true tung (Bdi 15 trang 213 SGK) Gidi a) H o a n l i dp giao d i e m cua h a i d6 t h i y = - x ^ « y ) - y(y) 125 Gidi V = + 2x^ = + 20)dy 7i(e' - ) 0 y 25 - (e' - e") = =e-2 -2 J(y - / a) The t i c h k h o i t r o n xoay tao t h a n h k h i quay A quanh true h o a n h = e" 0 16 (Bdi 16 trang 213 SGK) ==> du = dx dv = e^dx Z i z l = Z_tl? = i " ^ - ^ + 0 - ^ - ^ + 80 3 T a t i n h JxeMx Dat y3 ^ - ^ - + 20y - ^ fxeMx - - = e" +00 + ^ 4(1 + 73i)' _ + 273i - 16 273i - _ - ^ 73i ; '^H (Bdi 23 trang 214 S O K ) 19 (Bai 19 trang 214 SGK) Gidi a) Vofi gia t h i e t |zi = , (nghia 1& z z = 1) va z * 1, t a c6: , z+1 + z-1 z+1 z+1 z+1 z-1 z-1 +z • + z+1 Z1 • z-1 z+1 z-1 fl._^ ^ 4i(3 - iV3) ^ 4i + 4S = 73 + 1 + i-V3 (1 + i / ) a - iTS) + 3" c6, r = V3 + =2 z S coscp = —— -1 nfin = ~ f m g t acgumen cua Do d6 S Suy r a U + iV3; ( Dod6 l - i V z'-2 l + i>/3 -1 = 2'^(cosn + isin?:) = - " = - J I 571^ = cos 0, b > + Ta H ^ m so d o n g b i e n t r e n cdc k h o a n g (-x; 25 rBdf 25 trang SGK) 215 1) v a ( ; Giai 21n 1) 29 (Bai cosx = n doan ; k Z c ^ x = +^ 2 + k27t, k e Z Do "2 — ^7t^ l2j — Chpn (Bdi = - = -1 lim - = SGK) Gidi Vx^ + x.|l + = lim J - - ^ - ] vj [ ; +») h a y a^2 ^ ^2 Vay a > 33 (Bdi = lim VX^ + X + X x-»+x 215 SGK) + ^ + Gidi C h o n ( B ) - L o a i t r i r c f n g h o p ( D ) v i ( ; 1) g (d) y = x + HUdng ddn: y = x - x + ; y ' = 3x^ - SGK) 216 f r a tiT thi ham so y = log2X b^ng phe Gidi (x^ + D ' (X) = , SGK) 216 _ , 2x (x' f l)ln5 ; f ' ( l ) = ~ 21n5 = - L ln5 Gidi - H # so goc c u a t i e p t u y e n v i ( C j ) y = x - - H e so goc c u a t i e p t u y e n v d i (C2) y = (Bdi l a : y ' ( l ) = 4(1) = l a : y ' ( l ) = 2(1) = H e so goc c u a t i e p t u y e n v-jri (C3) y = - x ^ + x l a : y ' ( l ) = - ( ) + = = => -j= ; N/2 ) t h u o c d t h i hkm l o g ^a = nen => l o g ,^a = a = 72 V2 > so y = logL.x (b > 0; b 7^ 1) => logb2"2 = ^y2 => logo2 = N/2 logb2 = - v'2 33 trang C h o n (A) => l o g ,2a^^ 216 SGK) Gidi r2x' + —5—dx= 34 trang 216 / SGK) ^ 2x^ x ^ + — - dx = 3 X Gidi C h o n (D) + d u = d x ; u ( ) = a^; u(a) = a + a^ j c o s ( x + a^)dx - H $ so g6c c u a t i e p t u y e n v i ( C ) y = x^ - x + l a : y ' ( l ) = - + 3) 1) = ^/2 log^y/r" nen < b < = 1 V a y diicfng t h i n g y = x + - l a t i e m c a n x i e n c i i a t h i h a m so ( k h i x 27 rSdi 27 trang y = log22(x + ) suy D o d i e m ( N / Z ^ ; N/2 ) t h u p c t h i h a m so y = a ddn: Do d i e m (y[2^ - l i m [ y - xl = l i m (Vx^ + x - x ) = l i m X-++X 32 trang so y = Vx^ + x xac d i n h t r e n D = ( - « ; lim = l^g.ix (B) Huang nen CO y - C h o n (B) l i m y = +00 Ta Gidi (x' + l)ln5 (B) Hifdng dan H a n ^ a + b^ ^ I n a + I n b ^ > Vay X > Y ly = Y + l « X ) = l g ( x ' + 1) V a y g i a t r i n h o n h a t c u a h a m so d a cho l a - 215 > In(ab) = Ina + Inb Gidi SGK) 215 • minf(x) = f 26 trang so 31 trang Chpn ».-! 26 (Bdi thi ham (Bdi 7t 30 trang t i n h t i e n v = (-3; 71 f(x) SGK) 215 iY = y - l 71 f'(x) + b y = log22(x + 3) = log22 + log2(x + 3) 2'2 X C h o n (C) X = 71 - - + k27I T r e n m§i > I n a + I n b => I n 29 trang 30 (Bdi x = - + k27i sinx = > ab => I n C h o n ( C ) t u o n g tvt b a i x = ± - + k27i f a + b^ + b)^ > a b +«) Chon(C) H u d n g d i n f{x) = sin^x - s i n x ; f (x) = 2sinxcosx - 2cosx = 2cosx(sinx - f '(X) = (a c6: Gidi = cosudu = s i n u = s i n ( x + a^) = s i n ( a + a ) - s i n ( a ^ ) = 2cos - + 2a^ 12 a sm — < Theo de b^i: 2cos r:> 2C0S - + 2a V , sin - - sina sin — = 2sin — cos — => cos 2 + a ' ^- ± a + k27t k e 2a^ 12 2a^ = 2k7i, k Z a = cos 2a^ + a = 2k7i, k e Z Gid tri a = 7271 ia phii hap « a / J ^ra'jg- 216 SGK) Chon (C) Gidi k Hufdng d d n (in—dx, /e nguyin diCang (Ink - lnx)dx = jlnkdx - Jlnxdx = xlnk |^ = (elnk - Ink) - ! - = (e - Dink - - ~ X = (e - 1) Ink + e V k Theo gia t ' l i e t lnk + - < e - In— dx < e - :=> (e - 1) ej ^ Ink + - < — < =i> Ink < - - = ^ e e l e e Do k nguyen dirong chpn k = 1; k = mdi 36 trang 217 SGK) Gidi Chon (A) < =^ k < e Cho z = X + y i (x, y e R), z = x - iy; z" = x' + y ^ z~ = x^ + y^ Suy a = + z ^ = 2(x^ + y^) Ik so thirc z z = (x + yiXx - yi) = x^ + y^; i(z - z ) = i (0 + 2yi) = - y Vay p = z z + i(z - z ) = x^ + y^ - 2y 1& so thuc 37 (Bdi 27 trang 217 SGK) Gidi Chon (C) MUC LUC L6\6I DAU Chirong I LfNG Dl^NG D A O HAM D E K H A O SAT VA VE D O THj C U A HAM SO §1 Tinh don dieu cua hkm so §2 Cue trj cOa ham so 12 §3 Gia tri Idn nha't, gia trj nh6 nha't cua ham so 18 §4 Do thi cua ham so - Phep tinh tien he toa 30 §5 DUdng tiem can cua d6 thi ham so 34 §6 Khio sat sU bier? thien va ve thi mot so ham da thufc 44 §7 Kh3o s^t SI/ bien thi§n 58 ve,d6 thi cua moi so h^m phan thufc hau ti §8 Mot so bai toan thadng g$p ve thj 70 ON TAP CHaONG I 82 Chirong II HAI SO LUY THL/A - HAM SO MU - HAM SO LOGARIT 102 §1 Luy thCfa vdi so mu hufu ti 102 §2 Luy thCfa v6i so' mu thUc 107 Đ3 Logarit ã 109 Đ4 So e va logarit ti/nhien 115 §5 Ham so' mu va ham so logarit 116 §6 Ham so luy thiTa 121 §7 Phi/ong trinh mu va logarit 124 §8 H$ phuong trinh mu va logarit 130 §9 Bat phaong trinh mu va logarit 137 ON TAP CHLiONG II 140 Chi/Ong III N G U Y E N HAM, TICH PHAN VA C/NG DUNG 151 Hudrng ddn Xet so phCifc z = x + yi (x; y E M): z = x - yi §1 Nguyen ham Ta c6: i"''"^ = §2 M0t so phifong phap tim nguyen h^m 153 §3 Tich phan 157 §4 Mot so phiidng phap tinh tich phan 160 §5, Ung dung cua tich phan de tinh dien lich hinh phang 167 Vay a = 4xyi l a so' ao §6 ling dung tich phan de tinh the tich vat the 170 Mat khdc z z = (x + yi) (x - yi) = x' + y ' ; i(z - z ) = i(2yi) =-2xy ON TAP CHUdNG III Do Nen 22005 - z-1 = i = (x + y i ) ' = x' + 2xyi - y ^ z ' = (x - y i ) ' = x' - 2xyi - y' = V a - z ' + ( z ) ' = 4xyi (3 = z z + i(z - z ) = x' + y ' - 2y l a so thUc 38 (Bdi 38 trang 217 SGK) Gidi Chpn (B) z = X + yi (x, y e R) Chirong IV SO PHLfC §1 So phufc 151 174 184 184 §2 can bac hai cua so phufc va phuong trinh bac hai 191 Modun ciia z l a r = ^/x' + y^ §3 Dang lupng giac cua so phufc 200 Ta c6 (1 ON TAP CHL/dNG IV 208 ON TAP CUOl N A M 214 i)'z = (1 - 2i + i')(x + yi) = - i x + 2y C6 modun l a r' = ^4x^ + 4y^ = ^x^ + y ' = 2r Phdt hanh tai: m Mha sach SAO MAI 284 B a H a t - P.9 - Q.IO - T P H C M D i g n T h o a i : (08) 3927 1553 - 0903 823 701 Fax: (08) 3927 3281 E m a i l : saomaibookstore@yahoo.com Xin lien he trite tiep vdi chiing toi 936041 309739 Gia: 47.000d ... log«? ?2 log3., ,2 = (log 22' +log2l5)(log2l5-log24)-log2l5(log 222 +log,15) = (4 + log2l5)(log2l5 - 2) - log2l5. (2 + log2l5) 105 (Bdi 105, trang 133 SGK) Gidi SGK) = 21 og2l5 + log2'l5 - - 21 og2l5 +... 43, 3 12 * 6,93 • F = 120 , d - 25 ,1191ogl20 - 43, 3 12 » 8,91 • F = 140, d - 25 ,119.1ogl40 - 43, 3 12 ô 10,60 ã F = 160, d = 25 ,1191ogl60 - 43, 3 12 « 12 (Bdi 66, trang 124 SGK) a) 2" *'.5" = 20 0 » Giai 2" .2. 5"=... 4i o • y = ^ /2 x = la 4 ~2 - y^ + 2xyl = i o « 2x y = - »1 2x X= 42 — va " ^ 42 42 hoac + Zi -i Cdch 42 x = X N /2 y = 42 hoac y = -V2 V2 43i y = 42 =45 hoac + N/5 •; Z2 1-45 = khdc z2 = z + l o