Chuong 4 x

  • Published on
    01-Jul-2015

  • View
    67

  • Download
    8

Embed Size (px)

Transcript

  • 1. 1 1 Gii tch ton hc. Tp 1. NXB i hc quc gia H Ni 2007. T kho:Gii tch ton hc, gii tch, Php tch vi phn, o hm, vi phn, Cng thc Taylor, Khai trin Maclaurin, Quy tc Lhospital. Ti liu trong Th vin in t H Khoa hc T nhin c th c s dng cho mc ch hc tp v nghin cu c nhn. Nghim cm mi hnh thc sao chp, in n phc v cc mc ch khc nu khng c s chp thun ca nh xut bn v tc gi. Mc lc Chng 4 Php tnh vi phn ca hm mt bin ....................................................................... 2 4.1 o hm v cch tnh ....................................................................................................... 3 4.1.1 nh ngha o hm................................................................................................... 3 4.1.2 Cng thc i vi s gia ca hm s......................................................................... 3 4.2 Cc qui tc tnh o hm .................................................................................................. 4 4.2.1 Cc qui tc tnh o hm............................................................................................ 4 4.2.2 o hm ca hm s hp........................................................................................... 4 4.2.3 o hm ca hm s ngc....................................................................................... 6 4.2.4 o hm theo tham s................................................................................................ 7 4.2.5 o hm mt pha...................................................................................................... 7 4.2.6 o hm v cng ....................................................................................................... 9 4.2.7 o hm cc hm s s cp....................................................................................... 9 4.3 Vi phn ca hm s ........................................................................................................ 10 4.3.1 nh ngha................................................................................................................ 10 Chng 4. Php tnh vi phn ca hm mt bin L Vn Trc

2. 2 4.3.2 Cc qui tc tnh vi phn ........................................................................................... 11 4.3.3 Vi phn ca hm s hp........................................................................................... 11 4.3.4 ng dng ca vi phn............................................................................................. 12 4.4 Cc nh l c bn ca hm kh vi.................................................................................. 12 4.8.1 Cc tr a phng ................................................................................................... 12 4.5 o hm v vi phn cp cao........................................................................................... 18 4.8.1 nh ngha o hm cp cao.................................................................................... 18 4.8.2 Cc cng thc tng qut i vi o hm cp n...................................................... 18 4.8.3 Vi phn cp cao........................................................................................................ 19 4.6 Cng thc Taylor............................................................................................................ 20 4.8.1 Cng thc Taylor ..................................................................................................... 20 4.8.2 Khai trin Maclaurin................................................................................................ 22 4.7 Qui tc Lhospital kh dng v nh ......................................................................... 25 4.8.1 Dng v nh 0 0 ....................................................................................................... 25 4.8.2 Dng v dnh ...................................................................................................... 27 4.8 Kho st hm s.............................................................................................................. 30 4.8.1 Kho st ng cong cho di dng phng trnh hin.......................................... 30 4.8.2 ng cong cho di dng tham s........................................................................ 32 4.8.3 Kho st ng cong trong ta cc .................................................................... 36 4.9 Bi tp chng 4............................................................................................................. 39 Chng 4 3. 3 3 Php tnh vi phn ca hm mt bin 4.1 o hm v cch tnh 4.1.1 nh ngha o hm Gi s U l mt tp m trong , :f U v 0x U . Cho x0 mt s gia 0x nh sao cho 0x x U+ . Khi ta gi 0 0( ) ( )y f x x f x = + l mt s gia ca hm s tng ng vi s gia i s x ti im x0. Xt t s gia s gia hm s vi s gia i s. Nu t s dn n mt gii hn hu hn xc nh khi 0x , th ta ni rng hm f kh vi ti im x0, gii hn gi l o hm ca hm s ti x0 v k hiu l 0 0 0 0 ( ) ( ) ( ) lim x f x x f x f x x + = . (4.1.1) Cc k hiu y hay ( )f x l cc k hiu o hm theo Largrange, cn dy dx hay 0( )df x dx l cc k hiu theo Leibnitz v Dy hay Df(x0) l cc k hiu theo Cauchy. i khi nhn mnh bin s ly o hm, ngi ta thng vit bin thnh ch s di: 0 0, ( ), hay ( ) x x x xy f x D y D f x (4.1.2) Hm f c gi l kh vi trn U nu n kh vi ti mi im thuc U. 4.1.2 Cng thc i vi s gia ca hm s Nu hm y = f(x) kh vi ti 0 ,x U ta c th biu din s gia ca hm s 0 0 0( ) ( ) ( ) = = + y f x f x x f x nh sau. Theo nh ngha 0 0 0 ( ) lim ( ) = x f x f x x . t 0 0 ( ) ( ) = + f x f x x vi 0 khi 0 x . (4.1.3) Ta c 0 0( ) ( ) . = + f x f x x x vi 0 lim 0 x . (4.1.4) K hiu . ( ) = x x v hin nhin 0 ( ) lim 0 = x x x . Do (4.1.4) c th vit di dng 0 0( ) ( ) ( ). = + f x f x x x (4.1.5) nh l 4.1.1 Nu hm y = f(x) kh vi ti 0x U th f(x) lin tc ti x0. Chng minh: Tht vy ta c 4. 4 0 0 0( ) ( ) ( ) ( )+ = + f x x f x f x x x , suy ra [ ]0 0 0 0 0 0 0 0 0 lim ( ) ( ) lim ( ) lim ( ) lim ( ) ( ). + = + + = x x x x f x x f x f x x x f x x f x 4.2 Cc qui tc tnh o hm 4.2.1 Cc qui tc tnh o hm Trc ht ta hy nhc li cc qui tc tnh o hm bit nh l 4.2.1 Cho , :f g U , trong U l tp hp m trong R, cn f, g l hai hm kh vi ti 0x U . Khi 1 2,c c cc hm 1 2 ,c f c g+ .f g v f g (nu g(x0) 0 cng l cc hm kh vi ti im x0 v ta c cc cng thc sau: a) 1 2 0 1 0 2 0( ) ( ) ( ) ( )c f c g x c f x c g x + = + (4.2.1) b) 0 0 0 0 0( , ) ( ) ( ) ( ) ( ). ( )f g x f x g x g x f x = + (4.2.2) c) 0 0 0 0 0 02 0 0 ( ) ( ) ( ). ( ) ( ) , ( ) ( ) f x g x g x f xf x g x g g x = . (4.2.3) 4.2.2 o hm ca hm s hp nh l 4.2.2 Cho :g U V v :f V trong U, V l hai tp hp m trong , hm u=g(x) kh vi ti 0x U v hm y=f(u) kh vi ti u0=g(x0) V . Khi hm hp 0f g kh vi ti x0 v ta c cng thc 0 0 0 0( ) ( ) ( ( )) ( )f g x f g x g x = (4.2.4) hay gn hn .x u xy y u = . (4.2.5) Chng minh: Theo cng thc (4.1.5) hm f kh vi ti u0, nn ta c 0 0 0( ) ( ) ( ) ( )uf f u u f u f u u u = + = + . Mt khc hm g kh vi ti x0 nn 0 0 0( ) ( ) ( ) ( )xu g x x g x g x x x = + = + . Th u vo biu thc f ta c [ ]0 0 0 0 0 0 0 ( ) ( ) ( ) ( ) ( ) ( ) = ( ). ( ) ( ) ( ) ( ). u x u x u f u u f u f u g x x x u f u g x x f u x u + = + + + + Chia c 2 v cho x 5. 5 5 0 0 0 0 0 ( ) ( ) ( ) ( ) ( ). ( ) ( ) .u x u f u u f u x u f u g x f u x x x + = + + Ta thy do hm u lin tc ti x0 nn khi 0x th 0u v 0 0 0 0 ( ) ( ( )) ( ), ( ) ( ) ( ( )) ( ). o o f u f g x f g x f u u f u f g x f g x = = + = = = By gi ta hy vit li biu thc trn di dng: 0 0 0 0 0 0 ( ) ( ) ( ) ( ) ( ). ( ) ( ) . .u x u f g x f g x x u u f u g x f u x x u x = + + Cho 0x ta c 0 0 0 0( ) ( ) ( ( )). ( ),uf g x f g x g x = v cng thc c chng minh. V d 3: i) Ta thy 0lnx x a a e a= > nn ln ( ) ( )x x a a e = , t u = xlna, ln ( )' .ln lnu x a x e e a a a= = Do ta c cng thc sau ln( )x x a a = a vi 0a > . (4.2.6) ii) Ta c 0ln x e x = >x v Do : 1 1ln ln ( ) ( ) . . . .x x x e e x x x = = = . V ta c cng thc sau: 1 ( ) .x x = . (4.2.7) V d 4: Tnh 1 1 cos x x d I e dx + = vi 1x t 1 1 cos x u x = + 1 1 1 1 cos . . cos x u u x x d x I e e u e dx x + = = = + Li t 1 1 x v x = + ta c 2 1 1 1 2 1 1 1 1 (cos ) sin . sin sin . ( ) x x x v v v x x x x = = = + + + + Cui cng 1 1 2 1 1 2 11 cos . .sin ( ) x x x I e xx + = ++ . V d 5: Cho , :f g U trong f(x)>0, x U v tn ti ( ), ( )f x g x vi x U . 6. 6 Khi ( ) ( )ln ( ) ( )ln ( ) ( ) ( ( )) ( ( ).ln ( )) ( ) =( ( )) . ( )ln ( ) ( ). . ( ) g x g x f x g x f x g x d d d f x e e g x f x dx dx dx f x f x g x f x g x f x = = + 4.2.3 o hm ca hm s ngc nh l 4.2.4 Gi s hm f(x) kh vi lin tc trn (a,b) vi 0( )f x ( , )x a b . Khi hm f(x) n iu thc s nn c hm ngc x = g(y), : ( ( ), ( )) ( , ).g f a f b a b Khi g(y) cng kh vi ti y = f(x) v c o hm g(y) tho mn h thc: 1 ( ) ( ) g y f x = (4.2.8) hay gn hn: 1 y x x y = . (4.2.9) Chng minh: Do (g.f)(x) = x ( , )x a b Hay ( ( ))g f x x= ( , )x a b . Ly o hm hai v ng thc trn theo x ta c 1 hay 1( ( )). ( ) ( ). ( )g f x f x g y f x = = suy ra 1 ( ) ( ) g y f x = ( , )x a b . V d 6: i) Xt hm s y = arcsinx vi 1< x nn 2 1yx x = , suy ra 2 1 1 xy x = . Tng t, tao c cc cng thc sau: ii) y = arccosx vi 1< x nn ( ) (1) 0 1 f x f x > hay ( ) (1)f x f 1 Ta ni rng 1 1 1 ! (0) 0, ( ) , ( ) ( 1) 1 (1 ) n n n n f f x f x x x + + = = = + + Do 25. 25 25 2 3 1 ln(1 ) ... ( 1) ( ) 2 3 n n nx x x x x x n + = + + + trong 1 1 1 1 ( ) ( 1) . 1 (1 ) n n n n x x n x + + = + + . V d 6: Xt khai trin ( ) (1 ) , , 0f x x = + . Tng t nh trn, ta c th chng minh c rng 2( 1) (1 ) 1 ... 2! ( 1)...( 1) ( ). ! n n x x x n x x n + = + + + + + + + (4.6.20) 4.7 Qui tc Lhospital kh dng v nh Trc ht ta xt gii hn ( ) lim ( )x c f x g x trong trng hp f(x) v g(x) dn ti 0 khi x c . Trng hp c bit ca gii hn ny l: Nu nh f lin tc ti im x0, tc l 0 0 0 0lim( ( ) ( )) lim( ) 0 x x x x f x f x x x = = th ta gi gii hn 0 0 0 ( ) ( ) lim x x f x f x x x , nu tn ti, l o hm 0( )f x . Cho nn ta hy vng rng bng cch s dng nhng nh l v o hm ta c th kh c mt s