Chng II : Bi Tit 2: Hm s m - Hm s lụgarit Giỏo viờn: Nguyn Phan Anh Hựng Kim tra bi c: Tớnh cỏc giỏ tr cho bng sau: x -2 x 4 x 2 -1 2 log2x HM S M.HM S LễGARIT A Mc ớch, yờu cu Hiu v bit dng nh ngha, cỏc cụng thc tớnh o hm v tớnh cht ca hm s m lụgarit Bit cỏc dng th ca hm lụgarit Bit dng c tớnh cht gii toỏn B Ni dung bi hc II Hm s lụgarit nh ngha o hm ca hm s lụgarit Kho sỏt hm s lụgarit y = log a x ( a > o, a 1) C Tin trỡnh by hc II HM S LễGARIT: 1.nh ngha: Cho s thc dng a khỏc : Hm s y = logax c gi l hm logarit c s a Vớ d : Cỏc hm s y = log x ; y = log x ; y = log y = ln x ; y = log x L nhng hm s lụgarit ln lt cú c s l : 3; ; ; e ;10 x ; PHIU HC TP Cỏc biu thc sau biu thc no l hm s lụgarit Khi ú cho bit c s : a ) y = log x d ) y = log x b) y = log x c) y = log x (2 x + 1) e) y = lnx ỏp ỏn: a ) y = log x Hm s lụgarit c s a = b) y = log x Hm s lụgarit c s a = 1/4 c) y = log x (2 x + 1) Khụng phi hm s lụgarit d ) y = log x Khụng phi hm s lụgarit e) y = lnx Hm s lụgarit c s a = e o hm ca hm s lụgarit : Ta cú nh lý sau : nh lý : Hm s y = loga x (0 < a 1) cú o hm ti mi x > ( log a x ) = x.ln a ' c bit : ( ln x ) = x ' Chỳ ý : Cụng thc o hm hm hp vi y = loga u(x) l : u' ( log a u ) = u.ln a ' PHIU HC TP : Tỡm xỏc nh ca cỏc hm s sau: a / y = log ( x + x) b / y = log 0,2 (4 x ) c / y = log x d/y= log x ỏp ỏn: a/Hm s xỏc nh x + x > hay x0 Vy TX : D=(-;-2) U (0;+ ) b/ Hm s xỏc nh x > < x < Vy TX : D=(-2;2) c/ Hm s xỏc nh Vy TX : D=(- ;3) >0 x d/ Hm s xỏc nh x > log x x 64 Vy TX : D=(0;64) U (64;+ ) Vớ d : Tớnh o hm cỏc hm s sau : y = log2(2 + sinx) Gii: ( + sin x )' cos x y' = = ( + sin x ) ln ( + sin x ) ln PHIU HC TP 3: Tớnh o hm ca cỏc hm s sau: Nhúm 1: a ) y = log ( x + x + 1) b ) y = ln( x x + 1) Nhúm 2: c ) y = log ( x +1) Nhúm 3: Nhúm 4: d ) y = ln( + x ) PHIU HC TP 3: ỏp ỏn: ( x + x + 1) ' x + a) y ' = = ( x + x + 1).ln ( x + x + 1).ln ( x x + 1) ' 2x b) y ' = = x x +1 x x +1 ( x +1) ' 2x c) y ' = = ( x +1) ln ( x +1) ln ( x + 1) ' ( x + 1) ' 2x 2 x x + x +1 = d)y ' = = = x + x + x + x + x + x + ( x + x + 1) x + 3.Kho sỏt hm s y = logax Kho sỏt v v th ca hm s logarit y = logax + Tp xỏc nh : (0 : +) + S bin thiờn o hm : y' = x.ln a Nu a > => y > => hm s ng bin trờn (0 ; +) Nu < a < => y < => hm s nghch bin trờn (0 ; +) + Tim cn : Khi a > lim+(log a x ) =; lim (log a x ) =+ x x + (log a x ) =+; lim (log a x ) = Khi < a < xlim + x + KL v tim cn : th hm s cú tim cn ng l trc tung + Bng bin thiờn : a>1 x y y + + - + 0 y = Nhn xột : th nm bờn phi trc tung Oy y a>1 -1 o x -1 -2 0< a < y=x y=3x y y=log3x x -4 -3 -2 -1 -1 -2 NHN XẫT : th hm s m y = ax v th hm s logarit y=logax i xng qua ng phõn giỏc ca gúc phn t th nht y = x CNG C : Nhc li cỏc cụng thc o hm ó hc bi Haứm soỏ logarit ( ln x ) ' = ( log a x ) ' = x x.ln a Hm s hp (ln ( log u a ) ' =uu ' u )'= u' u.ln a Nhc li bng túm tt cỏc tớnh cht ca hm s lụgarit y = logax Tp xỏc nh ẹaùo haứm (0 ; + ) y' = x ln a a > : Hm s luụn ng bin Chiu bin thiờn < a < : Hm s luụn nghch bin Tieọm caọn Tim cn ng l trc Oy th Luụn i qua im (1;0) , (a;1) V nm v phớa phi trc tung Bi tp: Cõu : Tỡm mnh sai : A B C D ( x e ) ' = (2 x + x)e ( x ln x ) ' = (2 ln x + 1).x 2x 2x ( x ) ' = 3x ln x x 2x ( log ( x + 1) ) ' = ( x + 1).ln 2 A ( x e 2x ) ' = x.e 2x + x 2e 2x = (2 x + x)e 2x B ( x ln x ) ' = x.ln x + x = (2 ln x + 1).x x x x x x C ( x ) ' = ln 2.x + x = x ( x ln + 3) 2 ( x + 1) ' 2x D ( log ( x + 1) ) ' = = ( x + 1).ln ( x + 1).ln 2 Vy : Mnh C l mnh sai Cõu Cõu : Hm s no ng bin trờn xỏc nh ca nú ? A y = 2-x B e x e x y= C D S y = log x y = log ữ x S S A) y = 2-x =(1/2)x => Hm s nghch bin trờn R e x e x e x + e x B) y = y' = > x R 2 => Hm s ng bin R C ) y = log x => Hm s nghch bin (0; + ) D ) y = log ữ=log x x => Hm s nghch bin (0; + ) HNG DN T HC NH : + Lm bi : t bi n bi SGK trang 77-78 + Bi lm thờm : Bi : Tỡm xỏc nh ca hm s : b ) y = log a) y = ln( - x + 5x 6) ữ x Bi : Tớnh o hm cỏc hm s sau : a) y = e ( cos x b) y = d ) y = ln x + x + x x +1 c) y = ( x + 1) x ) Bi : Cho hm s y = esinx CMR : y.cosx y.sinx y = Bi : Cho hm s y = x[cos(lnx)+ sin(lnx)] vi x > CMR : x2.y x.y + 2y = EM Cể BIT ? John Napier (1550 1617) ễõng ó b 20 nm rũng ró mi phỏt minh c h thng logarittme Vic phỏt minh logarithme ó giỳp cho Toỏn hc Tớnh toỏn tin mt bc di, nht l cỏc phộp tớnh Thiờn [...]... ) ' = (2 x + 2 x)e ( x ln x ) ' = (2 ln x + 1).x 2 2x 2 2x 2 ( 2 x ) ' = 3x 2 ln 2 x 3 2 x 2x ( log 2 ( x + 1) ) ' = ( x 2 + 1).ln 2 2 A ( x e 2 2x ) ' = 2 x.e 2x + x 2e 2 2x = (2 x + 2 x)e 2 2x 1 B ( x ln x ) ' = 2 x.ln x + x = (2 ln x + 1).x x x 3 x 3 x 2 x 2 C ( 2 x ) ' = 2 ln 2. x + 2 3 x = 2 x ( x ln 2 + 3) 2 2 ( x + 1) ' 2x D ( log 2 ( x + 1) ) ' = 2 = 2 ( x + 1).ln 2 ( x + 1).ln 2 2 2 Vy : Mnh... + 1).ln 2 ( x + 1).ln 2 2 2 Vy : Mnh C l mnh sai Cõu 2 Cõu 2 : Hm s no ng bin trờn tp xỏc nh ca nú ? A y = 2- x B e x e x y= 2 C D S y = log 2 x 3 1 y = log 2 ữ x S S A) y = 2- x =(1 /2) x => Hm s nghch bin trờn R e x e x e x + e x B) y = y' = > 0 x R 2 2 => Hm s ng bin R C ) y = log 2 x 3 => Hm s nghch bin (0; + ) 1 D ) y = log 2 ữ=log 2 x x => Hm s nghch bin (0; + ) HNG DN T HC NH : + Lm... hm s : 1 2 b ) y = log a) y = ln( - x + 5x 6) ữ 5 6 x Bi 2 : Tớnh o hm cỏc hm s sau : a) y = e ( cos 2 x b) y = 2 d ) y = ln x + x 2 + 1 x 1 x +1 c) y = ( x + 1) 2 x ) Bi 3 : Cho hm s y = esinx CMR : y.cosx y.sinx y = 0 Bi 4 : Cho hm s y = x[cos(lnx)+ sin(lnx)] vi x > 0 CMR : x2.y x.y + 2y = 0 EM Cể BIT ? John Napier (1550 1617) ễõng ó b ra 20 nm rũng ró mi phỏt minh c h thng logarittme... = 0 Cho x = a ==> y = 1 Nhn xột : th nm bờn phi trc tung Oy y 3 a>1 2 1 -1 o 1 x 2 3 4 5 6 7 -1 -2 0< a < 1 4 y=x y=3x y 3 2 y=log3x 1 x -4 -3 -2 -1 1 2 3 4 5 -1 -2 NHN XẫT : th hm s m y = ax v th hm s logarit y=logax i xng nhau qua ng phõn giỏc ca gúc phn t th nht y = x CNG C : Nhc li cỏc cụng thc o hm ó hc trong bi Haứm soỏ logarit ( ln x ) ' = ( log a x ) ' = 1 x 1 x.ln a Hm s hp (ln ( log u...3.Kho sỏt hm s y = logax Kho sỏt v v th ca hm s logarit y = logax + Tp xỏc nh : (0 : +) + S bin thiờn o hm : 1 y' = x.ln a Nu a > 1 => y > 0 => hm s ng bin trờn (0 ; +) Nu 0 < a < 1 => y < 0 => hm s nghch bin trờn (0 ; +) + Tim cn : Khi a > 1 lim+(log... y = 0 Bi 4 : Cho hm s y = x[cos(lnx)+ sin(lnx)] vi x > 0 CMR : x2.y x.y + 2y = 0 EM Cể BIT ? John Napier (1550 1617) ễõng ó b ra 20 nm rũng ró mi phỏt minh c h thng logarittme Vic phỏt minh ra logarithme ó giỳp cho Toỏn hc Tớnh toỏn tin mt bc di, nht l trong cỏc phộp tớnh Thiờn vn