Bai Giang Tom Tat Giai Tich

Bi ging tm tt gii tch (3vht) Bin son Phm Th Hin

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Lu hnh ni b c nhn

MC LC

Phn th nht : Tm tt l thuyt..3Chng 1 : Gii hn...3I. Ni dung cn nh...31) Gii hn dy s..32) Gii hn hm s7II. Bi tp p dng..19Chng 2 : Php tnh vi phn hm mt bin.22I. Ni dung cn nh.221) o hm cp 1..222) Vi phn cp 1....283) o hm cp cao.294) Vi phn cp cao.315) ng dng...31II. Bi tp p dng...42Chng 3 : Hm nhiu bin.52I. Ni dung cn nh.521) nh ngha.522) Gii hn.533) o hm ring cp 1...534) Vi phn ton phn (Vi phn cp 1) .585) o hm ring cp cao....606) Vi phn cp cao..607)V d p dng.608) Cc tr (Hai bin)...609) Cc tr (Ba bin).66II. Bi tp p dng..69Chng 4 : Php tnh tch phn hm mt bin..77I. Ni dung cn nh.771) Nguyn hm v tch phn bt nh...772) Phng php tnh tch phn.773) Tch phn xc nh.804) ng dng.845) Lin h gia nguyn hm v tch phn xc nh.896) Tch phn suy rng loi 1..907) Tch phn suy rng loi 2....92II. Bi tp p dng...93Chng 5 : Phng trnh vi phn ..102I. Ni dung cn nh.102


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Lu hnh ni b c nhn

1) Phng trnh tch bin....1022) Phng trnh vi phn tuyn tnh cp 1.1033) Phng trnh vi phn ton phn.1094) Phng trnh vi pn tuyn tnh cp 21105) Phng trnh vi phn tuyn tnh cp 2 khuyt y.1146) Phng trnh vi phn tuyn tnh cp 2 vi h s l hng s..115II. Bi tp p dng...122Phn th hai : Mt s luyn tp..127


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Lu hnh ni b c nhn

PHN TH NHT : TM TT L THUYTChng 1 : Gii hn

I. Ni dung cn nh :1) Gii hn ca dy s :a) nh ngha :+ hay , c gi l dy s tng qut.

V d : .

+ S c gi l gii hn ca dy s nu v k hiu l.

Ch : .V d : Dng nh ngha chng minh cc gii hn sau :* .

, .

Vy : hay lim 12n

n

n .

Ch : k hiu l ly phn nguyn.

* .

, .

Vy : hay .

*2 3lim 12 1nn

n .

0 , 2 3 2 2 2 2 1 21 2 1 12 1 2 1 2 2 1 2n

n nn n n n

.

Vy : 0 01 2 2 30, 1 , : 12 2 1

nn N n n

n

hay

2 3lim 12 1nn

n .

+ Dy con : Cho l dy s, dy s l dy tng. Khi c gi l dycon ca dy .V d :

( ):

nn x f nf N R

{ ( )} { (1), (2), (3), , ( )}nx f n f f f f n nx

1 1 1 1 1, , , , ,

1 2 3n n

a { }nx 0 00, , : nn N n n x a lim nn

x a 0n N

lim 12n

n

n

0 2 2 2 2(1 )1 2 22 2 2 2

nn n

n n n n

0 02(1 )0, , : 1

2n

n N n nn

2(1 )

2

3lim 02 3nn

n

0 2 2 2 2

3 3 3 31 10

2 3 2 3 2 3 2 2 2n n n n

nn n n n n

2

0 0 210, , : 0

2 2 3n

n N n nn

2

3lim 02 3nn

n

{ }nx { } ,kn N k N { }knx{ }nx


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Lu hnh ni b c nhn

Dy l dy con ca dy s .Ch : chng t mt gii hn ca mt dy s no khng tn ti th ta a ra hai dy con.Nu gii hn ca hai dy con tin v hai gi tr khc nhau khi th gii hn khngtn ti. Ngc li, Nu gii hn ca hai dy con tin v cng mt gi tr khi thgii hn cng nhn mt gi tri .V d : Chng t gii hn sau khng tn ti.+ .

Tht vy : khi ; khi .

+ .

Tht vy : khi ; khi .

+ .

Ta c : ch nhn hai gi tr v ; v khi .

b) Tnh cht :* Mt dy hi t th dy b chn, tc l .* Nu l hai dy hi t th :

.

* iu kin cn v dy s hi t (tc l l hu hn) l ,.

* Nu cc dy s tha iu kin v th .V d : Tnh cc gii hn sau :+ .

Ta c : .

M .

Do : .

+ .

2 2 12 2 1( 1) 1 khi ; ( 1) 1 khik kk kx k x k ( 1)nnx

n a n

a

lim( 1)nn

22 ( 1) 1kkx k 2 12 1 ( 1) 1kkx k

1lim ( 1)3

n

n

n

22

21( 1) 13

kk

kx

k 2 1

2 12 1

1( 1) 13

kk

kx

k 2

22lim cos

1 3nn n

n

2

cos3n 1

2 1

2

2 11n

n n

{ }nx ,nx K n { },{ }n nx y

limlim lim lim ; lim . lim .lim ; lim , lim 0lim

nn n

n n n n n n n n nn n n n n n n n

n nn

xxx y x y x y x y y

y y

{ }nx lim n

nx a 00, n N

0 , : n p nn n p x x { },{ },{ }n n nx y z n n nx y z lim limn n

n nx z a lim nn y a

2lim cos !1nn

nn

2 2 21 cos ! 1 cos !1 1 1n n n

n nn n n

2 2lim lim 01 1n n

n n

n n

2lim cos ! 01nn

nn

3lim!

n

n n


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Lu hnh ni b c nhn

Ta c : .

M .

Do : .

+ .

Ta c : .

(Tht vy : Vi , ta c : .

Vi , ta c : .

Gi s n ng vi , tc l : .

Ta cn chng minh n ng vi , tc l : .)

.

M 3lim 0n n

.

Do : .

c) Mt s gii hn c bn :( l hng s); ; ; .

Ch :+ Tng ca cp s cng :

, l cng sai.

Trng hp c bit : .

+ Tng ca cp s nhn :

, l cng bi. .

33 3.3.3.3...3 3.3.3 3 32 30 . .! 1.2.3.4... 1.2.3 4 3 4

n nn

n n

3lim 04

n

n

3lim 0!

n

n n

1lim!nn n

!3

nn

n

1n 11 11! 1

3 3

2n 22 42! 1.2 2

3 9

n k !3

kkk 1n k

11( 1)!3

kkk

1 1 30!

3

n n

n

nn n

1lim 0!nn n

limn

C C C0, khi 0

lim, khi 0n

n

0 , khi 1lim

, khi 1n

n

qq

q

1lim 1n

ne

n

1 2 3 1 1 2 1 3 2( ) (2 ( 1) ),2 2n n nn nS a a a a a a a n d d a a a a d

( 1)1 2 32

n nn

321 2 3 1

1 2

1, 1,

1

n

n n

aaqS a a a a a q qq a a

q1lim , 1

1nnaS q

q


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Lu hnh ni b c nhn

Trng hp c bit : .

V d :

+

.

+ .

+ .

+

.

+ .

+1 2 3 4 2 1lim lim lim 1111 11

n n n

n n

nn

nn

.

+ .

(V )

+ .

12 11 , 1

1

nn xx x x x

x

1 1 ( 1 )lim 1 lim lim1 1n n n

n n n n n n n nn n n

n n n n

1 1lim lim lim

211 1 1 11 1 1n n n

n n

n n nnn n

1 11 1

55 7 15 7 0 1 177lim lim lim

5 75 7 5.0 7 755. 77 7

nn n

n n n

n n nn nn n n

n

111

2 1 011 1 1 1 111 2 22 4 884 2 2 2lim 2. 2. 2. 2 2 lim 2 lim 2 2 2 4

n

n n

n n n

3

2 2 2 2

3 3 3 3

1 1( 1)(2 1) 1 21 2 3 ( 1)(2 1)6lim lim lim lim

6 6n n n n

n n n nn n n n n n

n n n n

1 11 2 (1 0)(2 0) 2 1lim6 6 6 3n

n n

2

2 22 2lim 1 lim 1n

n

n ne

n n

13 3

1 12 2 1 1 1lim lim 1 1 lim 1 lim 13 3 3 3

nn nn n n

n n n n

n ne

n n n n e

( 1) 1 1lim lim lim 1333 1 011n n n

n n

nn

nn

2 22 1

3 32 2 2 1 2

2 2 2 22 2 2 2 2 1 2 1lim lim 1 1 lim 1 lim 1

3 3 3 3

nn

n nn n nn

n n n n

n n n n n ne

n n n n


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Lu hnh ni b c nhn

(V )

2) Gii hn ca hm s :a) nh ngha :Hm s : , trong ( ) l tp xc nh ca hms, c gi l min gi tr ca hm s. th hm s l tp hp cc im tha mn phng trnh .V d :+ Hm s c min xc nh l , min gi tr l .+ Hm s c min xc nh l , min gi tr l .

+ . + . + .

+ . + . + .

* Hm ngc : Cho l n nh (tc l , gi s ). Khi tn ti hm ngc : .

V d :

+ sin ( , [ 1, 1]) arcsiny x x R y x y hay arcsin [ 1, 1], ,2 2

y x x y .+ cos ( , [ 1, 1]) arccosy x x R y x y hay arccos [ 1, 1], 0,y x x y .+ tan , , ( , ) arctan

2 2y x x y x y hay arctan ( , ), ,2 2y x x y

.+ cot ( (0, ), ( , )) arccoty x x y x y hay arccot ( , ), (0, )y x x y .* Hm hp : .

V d : Cho .

* Hm n : Nu th ta ni phng trnh xc nh mt hm n mt bin.

V d :+ Phng trnh xc nh hai hm n .+ .* Tnh cht :

22

22

22

1 12 22 2 0lim lim lim 2333 1 011n n n

nn n n n

nn

nn

( ):

x y f xf D R

D R { sao cho ( ) c nghia}D x R y f x

( )f D R(x,y) ( )y f x

2y = x + 1 D = R f(D) = 1; + 2y = 4 x D = 2; 2 f(D) = 0;2

2

23

,

3xy D Rx

, \{ }sinxy D R Z

x

2

2

2, ( , 1) (1, )

1xy Dx

, \{0}

xey Dx

ln , (0, )xy Dx

2

24

, \{ 3, 1}4 3

x xy D Rx x

( )

:x y f x

f D R

1 2,x x D 1 1 1 2( ) ( )x x f x f x

1

1

( ): ( )

y x f yf f D D

1 2 2 3 1 3( ) ( ) ( ) [ ( )]

: , : , :x y f x y z g y x w h x g f x

f D D g D D h g f D D

22

1 2 321 1( ) 2( ), ( ) ( \{ 1}), ( )( ) [ ( )] ( )1 3

x xf x x D R g x D R g f x g f x D Rx x

( , ( )) 0, ,F x y x x y D

( )y y x

2 2 4x y 2 21 24 , 4 , [ 2, 2]y x y x x 1 yy xe


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Lu hnh ni b c nhn

+ Tnh n iu : , gi s .- Nu th hm s ng bin (tng).- Nu th hm s nghch bin (gim).V d :- Hm s ng bin trn khong , nghch bin trn khong .- Hm s ng bin vi mi .- Hm s ng bin trn khong .+ Chn, l :- Hm s c tnh cht chn, l khi min xc nh ca n c tnh i xng.- Nu th hm s l hm l, th i xng qua gc ta (O). Nu thhm s l hm chn, th i xng qua trc tung (Oy).V d :

2y x 1

1

1 0 1 x 0 1 e x

0 1

- Hm s l hm s chn, th i xng qua trc tung (Oy).- Hm s l hm s l, th i xng qua gc ta (O).- Hm s l hm s khng c tnh chn l. V min xc nh ca n khng ixng.+ Tun hon chu k : ( )f x l hm tun hon nu tn ti hng s 0L sao cho :

( ) ( ),f x L f x x D .Chu k ca mt hm tun hon ( )f x l s

0min , ( ) ( )

kT k f x k f x .

V d :+ Hm s l hm tun hon vi chu k .

+ Hm l hm tun hon vi chu k l

.

b) Gii hn :

1 2,x x D 1 2x x1 2( ) ( )f x f x1 2( ) ( )f x f x

2( )y f x x (0, )x ( ,0)x 3( )y f x x x R

( ) lny f x x (0, )x

( )y f xf( x) = f(x) f( x) = f(x)

y y 3y x y

4 lny x

1

2 1 1 2 x2( )y f x x 3( )y f x x

( ) lny f x x

y = sin(mx + n) 2T =m

2 1 cos4xy = sinx + cosx = sinx + cosx = 1 +2

2 T = =4 2


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Lu hnh ni b c nhn

+ .

+ .

V d : Dng nh ngha chng minh cc gii hn sau :+ .

, cho .

.

Theo nh ngha th .

+ .

, chn , khi nu th .

Theo nh ngha th .+ 2

3lim 9x

x .0 , cho 2 9x th :

2 222( 3) 6( 3) 3 6 3 3 3 9 3 3 9 3 3 9x x x x x x x 0 3 9 3x (V 9 3, 0 nn 9 3 0 ).

0 , chn 9 3 th x sao cho 20 3 9x x .Theo nh ngha th 2

3lim 9x

x .

+ . + . + . + . + .

+ Chng t rng khng tn ti.

Ta ly hai dy , vi .

Khi ta c :

.

Nh vy, nhng .c) Tnh cht :* Nu th :

+ .

+ .

lim ( ) { } ( ) \{ }: ( ) , khin n nx

f x x U x f x n

00lim ( ) 0, 0, : 0 ( )

x xf x a x D x x f x a

2lim (3 4) 2x

x 0 3 4 ( 2) 3( 2) 3 2 2

3x x x x

0, , 2 : 0 ( 2) 3 4 ( 2)3

x x x

2lim (3 4) 2x

x

2lim 2x

x

0 . 2 2 . 2x 2 222 2

x xx

x

2

lim 2x

x

0lim lnx

x lim ln

xx lim 0

x

x

e

x lim

x

x

e

x 2

0lim(2 cos 2 ) 1xx

e x

0

1 1lim .cosx x x

1 2{ },{ }n nx x 1 2

1 2;

2 (2 1)n nx xn n

1 2 (2 1) (2 1) (2 1)( ) 2 .cos(2 ) 2 ; ( ) .cos .cos 02 2 2 2n n

n n nf x n n n f x n 1 2{ } 0,{ } 0, khin nx x n 1 2( ) , ( ) 0, khin nf x f x n

0 0

lim ( ) , lim ( )x x x x

f x a g x b

0 0 0

lim ( ) ( ) lim ( ) lim ( )x x x x x x

f x g x f x g x a b

0 0 0

lim ( ). ( ) lim ( ). lim ( ) .x x x x x x

f x g x f x g x a b


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Lu hnh ni b c nhn

+ .

* Nu th .Ch :+ .+ Cng thc cng lng gic :

.

+ Cng thc tch thnh tng : Ly , sau chia hai v cho 2.

+ Cng thc tng thnh tch : t , ri ly .

+ Cng thc nhn i : Cho . T suy ra cng thc h bc.+ Cch nh cc gi tr ca cc gc c bit : :Radian :

Bc 1 : Ta vit 5 s :Bc 2 : Ta ly cn bc 2 :

Bc 3 : Ta chia cho 2 :

l i vi sin, cn i vi cos th ta vit ngc li.Nh vy c hai cu hi cn t ra l :- i vi vic i t ra radian vi gc c bit ln hn th sao?Tr li : Ta ch cn thc hin php ton cng hoc nhn trn cc gc c bit nh hn hocbng .V d : Nu ta cn i gc ra radian th ta lm nh sau : V nn ta ly

.

- i vi gi tr ca cc gc c bit ln hn th sao?Tr li: i vi gi tr ca cc gc c bit ln hn th ta ly gi tr ca cc gc nh hni xng qua gc v ch ti du (sin nht gi tr dng khi , nht gi tr mkhi , cos nht gi tr dng khi hoc , nht gi tr mkhi ,) khi hm nhn gi tr tng ng.

0

0 0

0

lim ( )( )lim , lim ( ) 0( ) lim ( )x x

x x x x

x x

f xf x a g x bg x g x b

0 00lim ( ) , lim ( )

x x y yf x y g y L 0 0lim [ ( )] lim ( )x x y yg f x g y L

ln ln( )b be e b

cos( ) cos .cos sin .sin (1)cos( ) cos .cos sin .sin (2)sin( ) sin .cos sin .cos (3)sin( ) sin .cos sin .cos (4)

x y x y x yx y x y x yx y x y y xx y x y y x

(1) (2); (3) (4)

2

2

A BxA x y

B x y A By

(1) (2); (3) (4)

x y sin tan2 1 cot

0 0 0 0 00 30 45 60 90 1

06 4 3 2

0 1 2 3 4 1 1 0 20 1 2 3 4 0 cos0 1 2 3 4

2 2 2 2 21 32

090

0900120 0 0 0120 90 30

4 22 6 6 3

090090 090

090 0 00 180 0 0180 360 0 00 90 0 0270 360

0 090 270


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Lu hnh ni b c nhn

V d : Nu ta mun tnh th ly gi tr ca gc i xng qua gc . Nu ta muntnh th ly gi tr ca gc i xng qua gc v i du (v cos i).iii) Mt s gii hn c bn :* Dng : .

V d :

+ .

+ .

+0 0 0

ln(2 1) ln(1 2( 1)) 2( 1)lim lim .lim 1.2.1 22( 1)

x x x

xx x x

e e e

x e x .

+ .

+ .

+

2 1221 1

2

2sin 1lim24

x

x x

.

+ .

+

.

+

0sin135 045 0900cos120 060 090

00

x

x 0 x 0 x 0 x 0 x 0 x 0

sinx ln(1+ x) tgx arcsinx arctgx e 1lim lim lim lim lim lim 1x x x x x x

2 2 23 3 3 2

2 2 2 20 0 0 0

cos 2 1 1 cos 2 3( 1) 2sinlim lim lim lim 3 2 53

x x x

x x x x

e x e x e x

x x x x

ln( )

1 1

1 1lim lim 1ln ln( )

xx x

xx x

x e

x x x

2

2 2x 0 x 0 x 0

x x x2sin 2sin .sin1 cosx 12 2 2lim lim lim

x xx x 24 .2 2

2 2 22 2 2

2 2

ln(2 ) ln(2 ) ln(2 )

2 2 2 20 0 0 0 0

2 1 1 1 ln(2 ) 1 ln(2 )lim lim lim . lim .lim 1.ln 2 ln 2ln(2 ) ln(2 )

x x xx x x

x xx x x x x

e e e

x x x x

2

1 cos(x 1) 1 cos(x 1)

2 2 2 2x 1 x 1 x 1 x 1

x 12sine 1 e 1 1 cos(x 1) 1 cos(x 1) 2lim lim . lim lim

1 cos(x 1)x 1 x 1 x 1 x 1

2 22 2

2 2 2 2

3x x3x x

3x x 3x x 2 22

2 2 2 22 2 2 2x 0 x 0 x 0 x 0

2 2 2

3(e 1) e 1e 1 (e 1)e e e 1 (e 1) 3 1 13x xxlim lim lim lim

sin 3x sin x 9sin 3x sin xsin 3x sin x sin 3x sin x 9 1 4x 9x x

2 2 2 2

22 2 20 0 0

ln(2 cos 2 ) ln(1 2 1 cos 2 ) ln(1 2 1 cos 2 ) 2 1 cos 2lim lim lim .2 1 cos 2

x x x x

xx x x

e x e x e x e x

x x xe x

2 2 2

2 2 2 20 0 0 0

ln[1 2 1 cos 2 ] 2 1 cos 2 2( 1) 1 cos 2lim .lim 1. lim lim 1.(2 2) 42 1 cos 2

x x x

xx x x x

e x e x e x

x x xe x

3 3 3

2 2 2 20 0 0 0

cos 4 cos 4 cos 4 1 1 cos 4 cos 4 1 cos 4 1lim lim lim limx x x x

x x x x x x

x x x x

3 23 3

20 0 2 3 2 3

cos 4 1 cos 4 cos 4 1cos 4 1 cos 4 1lim lim

cos 4 1 cos 4 cos 4 1x xx x xx x

x x x x x


12---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

.

* Dng : .

V d :

+ . + .

+

(V ).

+ (V ).

+ .

(V )

+ (v ).

+

.

(V

2 2

20 0 2 3 2 3

2sin (2 ) 2sin (2 )lim limcos 4 1 cos 4 cos 4 1x x

x x

x x x x x

2 2

20 0 2 3 2 3

8sin (2 ) 8sin (2 ) 8 8 4lim lim2 3 3(2 ) cos 4 1 (2 ) cos 4 cos 4 1x x

x x

x x x x x

1 x1xx 0 x

1lim 1 x lim 1 ex

2x

2 22 22 x

e1 12 x 2 xx x e

x 0 x 0lim 1 x e lim 1 x e e

22

2

2 22 2

2 2lim 1 lim 1x

x

x xe

x x

2x 2

22 22 2x 2

2(e 1) sin1 1

2 x 2 3x 2(e 1) sinx 0 x 0lim 2 1 sin lim 1 2(e 1) sin e

x

xx

xe x x

2 22 2

2 2 20 0 0

2( 1) sin 2( 1) sinlim lim lim 2 1 3x x

x x x

e x e x

x x x

2

22 2

sin (2 )1 1

2 2 4sin (2 )0 0

lim 1 sin (2 ) lim 1 sin (2 )x

xx x

x xx x e

2 2

2 20 0

sin (2 ) 4sin (2 )lim lim 4(2 )x xx x

x x

2

22 22 2

2 1 cos21 1 x

4x 2 1 cos2x 0 x 0lim 2 cos 2 lim 1 2 1 cos 2 e

x

x

e x

x xe xe x e x

2 2 2

2 2 2 20 0 0 0

2 1 cos 2 2 2 1 cos 2 2( 1) 1 cos 2lim lim lim lim 2 2 4x x x

x x x x

e x e x e x

x x x x

2 21

1 x 1 x + 12 x x

2 2x 0 x 0

x x + 1 xlim lim 1 ex + 1 x + 1

20

1 1lim 11 1 0x x

3 3 31 1 1

sin (2 ) sin (2 ) sin (2 )

0 0 0

1 tan(2 ) 1 tan(2 ) sin(2 ) tan(2 )lim lim 1 1 lim 11 sin(2 ) 1 sin(2 ) 1 sin(2 )

x x x

x x x

x x x x

x x x

3sin(2 ) tan(2 ) 1

.1 sin(2 ) 1 sin(2 ) sin (2 ) 1sin(2 ) tan(2 )2

0

sin(2 ) tan(2 ) 1lim 11 sin(2 )

x xx x x

x x

x

x xe

x e

3 3 30 0 0

1sin(2 )sin(2 ). 1sin(2 )

cos(2 )sin(2 ) tan(2 ) cos(2 )lim lim lim(1 sin(2 ))sin (2 ) (1 sin(2 ))sin (2 ) (1 sin(2 ))sin (2 )x x x

xxx

xx x x

x x x x x x


13---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

)

* .

d) nh l (kp) : Nu v th .V d :

+ Tnh .

V nn .

M .

Do theo nh l kp, ta c .

+ Chng t rng : .

Ta c : .

M . Do .

e) V cng b (VCB) :i) nh ngha : Nu khi m th c gi l VCB.V d : Khi th cc hm c gi l ccVCB.ii) So snh :+ Cho ( )f x v ( )g x l 2 VCB. Khi : Nu th ta ni v l hai VCB

tng ng. K hiu : .+ Khi v l tng ca cc VCB th khi so snh ta ly bc thp nht ca t s ( ) sosnh vi bc thp nht ca mu s ( ) so snh vi nhau.V d :

+ .

+ .

iii) Cc VCB tng ng :

2 20 0

11cos(2 ) 1cos(2 )lim lim(1 sin(2 ))sin (2 )) (1 sin(2 ))(1 cos (2 )).cos(2 )x x

xx

x x x x x

0 0

cos(2 ) 1 1 1lim lim(1 sin(2 ))(1 cos(2 )(1 cos(2 )).cos(2 ) (1 sin(2 ))(1 cos(2 )).cos(2 ) 2x xx

x x x x x x x

x x + x 1 x 1

lim arctgx = ; lim arctgx = ; lim arcsinx = ; lim arcsinx =2 2 2 2

( ) ( ) ( )f x g x h x

0 0

lim ( ) lim ( )x x x x

f x h x a 0lim ( )x x g x a

2

x 1

1lim (x + 1) sinx + 1

11 sin 1

x + 1 2 2 21( 1).(x 1) (x 1) . sin 1.(x 1)

x + 1

2 2

x 1 x 1lim (x + 1) lim (x + 1) = 0

2

x 1

1lim (x + 1) sin 0x + 1

2

1 1lim .cos 0x x x

2 2 2

1 1 1 1 11 cos 1 .cosx x x x x

2 21 1lim lim 0

x xx x 21 1lim .cos 0x x x

x ( ) 0f x ( )f x0x sin , , ln(1 ), 1,1 cos ,arcsin , , ...xx tgx x e x x arctgx

( )lim 1( )xf xg x

( )f x ( )g x( ) ( )f x g x

( )f x ( )g x ( )f x( )g x

7 5 2 7 5 2

8 6 8 60 0 0

ln(1 3 6 2 ) 3 6 2 2 2lim lim limln(1 2 3 ) 2 3 3 3

VCB VCB

x x x

x x x x x x x x x

x x x x x x x

2 2 2

3 3 30 0 0 0

3 2 cos 2 3( 1) 1 cos 2 3.2 2 6lim lim lim lim 6x x VCB VCB

x x x x

e x e x x x x

x x x x x x x


14---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

* Khi th cc hm sau y l cc VCB tng ng :

.

* V d : Tnh cc gii hn sau :

+ .

+ . + .

+ . + .

+

.

+ .

+ ln311 1 1ln(3 ) 1 ln3ln30 0 0 0lim 3 lim 1 3 1 lim 1 1 lim 1 ln 3 3xx x

VCB xx x xx x x x

x x x xx x e x x x e e

.

+

.

+

.

+

.

0x 2

sin , ,arcsin , ; ln(1 ) ; ( 1) ;(1 cos ) ; ( 1 1)2

x nx xx tgx x arctgx x x x e x x xn

2 21 1 2

1 1 1 1 1

sin( 1) 1 1 (1 )(1 )lim lim lim lim lim[ (1 )] 21 1 1 1

x xVCB VCB

x x x x x

e e x x xx

x x x x

2 2 2VCB VCB

2 2 2x 0 x 0 x 0

ln(1 + sin x) sin x xlim lim lim 1x x x

2

1 cosx VCB VCB

2 2 2x 0 x 0 x 0

xe 1 1 cosx 12lim lim lim

x x x 2

2

VCB

x 1 x 1 x 1

(x 1)1 cos(x 1) x 12lim lim lim 0

x 1 x 1 2

x0

x ln(x ) xVCB0xx 1 x 1 x 1

x 1 e 1 ln(x )lim lim lim 1x.lnx x.lnx ln(x )

2

ln(2 ) 2

3 3 3 3 30 0 0 0 0

(2 )ln(2 )2 cos 2 2 1 1 cos 2 1 1 cos 2 ln 2 22lim lim lim lim limx

xx x VCB

x x x x x

xx x e x x x

x x x x x x x x x x

0

ln 2lim ln 2VCB

x

x

x

20

VCB VCB0

2 2 2 2 2x 0 x 0 x 0 x 0 x 0

xln(cosx) ln(1 + (cosx 1)) cosx 1 (1 cosx) 12lim lim lim lim lim

x x x x x 2

22 22 21

1 1 21 VCB xx x 2ln(1 + x ) ln(1 + x )

x 0 x 0 x 0

xlim 1 + e cosx lim 1 + (e 1) + (1 cosx) lim 1 + x +2

2

32 2 313x2 2

x 0

3lim 1 + x e2

2 2

2 20

VCB VCB0

2 2 22 2 x 2 2 xx 0 x 0 x 0

x (2x) +1 + x.sinx cos2x ( 1 + x.sinx 1) (1 cos2x) 2 2lim lim lim

x sin (x )x sin (e 1) x sin (e 1)

2 2

VCB VCB

2 2 2 2x 0 x 0

5x 5x52 2lim lim

x (x ) x 2

22 22 2x x1

1 1 21 VCB VCBarcsin(x )x x 2

arcsin(e 1) arcsin(e 1)x 0 x 0 x 0

(2x)lim 2e cos2x lim 1 2(e 1) (1 cos2x) lim 1 2x2

2 2 41 1VCB 1 12 2 4x 4xx 0 x 0lim 1 4x lim 1 4x e


15---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

+

.

(V )Ch : khng phi lc no ta cng p dng v cng b c. Trong trng hp ta gp bi tonm gii hn dng hiu ca hai hm v cng b tng ng gn nhau th khng p dng vcng b c v n s b trit tiu, ta phi s dng phng php khc (phng php qui tcLHospital s c cp n chng 2).V d :

+ . + . + . + .

f) V cng ln (VCL) :i) nh ngha : Nu khi x m ( )f x th ( )f x c gi l VCL.ii) So snh :+ Cho ( )f x v ( )g x l 2 VCL. Khi : Nu ( )lim 1( )x

f xg x

th ta ni ( )f x v ( )g x l hai VCLtng ng. K hiu : .+ Khi ( )f x v ( )g x l tng ca cc VCL th khi so snh ta ly bc cao nht ca t s ( ( )f x ) sosnh vi bc cao nht ca mu s ( ( )g x ) so snh vi nhau.V d :

+ .

+ .

Ch : Nu l VCB th l VCL. Ngc li, Nu l VCL th l VCB.g) S lin tc ca hm s :i) Khi nim :

2 2 2 22 2

ln 2 ln 2ln(1 ) ln(1 )4 41

2 20 0

2 cos 2 2 cos 2lim lim 1 13 1 3 1

x x x xx x

x x

e x e x

x x

2 22 2 22

ln 2ln 2ln(1 )ln(1 ) 24 2 4

2 20 0

1 2( 1) 1 cos 2 1 32 cos 2 1 3lim 1 lim 13 1 3 1

x xx x xx

x x

e x xe x x

x x

2 2

2

2 2 2

21

ln 22

2 2 ln 2.

3 1 3 1 222 2

ln 2 ln(2 ) 12 20 0

(2 )22 3 2 14lim 1 lim 1 2

3 1 3 1 2

x xx

x x x

VCB x

x x

xx

xx

e ex x

2

2 2 20 0

2 ln 2 ln 2lim . lim ln 23 1 2 3 1x x

x

x x x

2x 2

2x 0

e 1 xlimx.sin 2x 4x

2x 0

e cosx 4xlimx

2x 0

x arctgxlimx

2 2x 0

x sin xlimx.tgx

( ) ( )f x g x

2012 1973 1993 2012

2016 1956 1975 2016 430 4 1lim lim lim 02 10x x x

x x x x

x x x x x

2 2

2 29lim lim 13x x

x x

x x

( )f x 1( )f x ( )f x1( )f x


16---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

+ Hm s c gi l lin tc ti nu .+ Hm s c gi l lin tc bn tri ca nu .+ Hm s c gi l lin tc bn phi ca nu .+ Nu hm s lin tc ti v lin tc ti th cng lintc ti .+ Hm s cp l nhng hm c xy dng t nhng hm c bn (hm a thc, m, loga, lytha, ) bi cc php ton +, , *, :, .i vi hm s cp th n lin tc trn min xc nh ca n.ii) V d :* Hm s l hm s cp.

* Hm s l hm s cp.

* Hm s khng l hm s cp.

Trong phn ny ta s xem xt s lin tc ca cc hm s khng s cp.* Xt s lin tc ca cc hm s sau :

+ .

- Vi th hm s l hm s cp nn n lin tc trn min xc nh ca n.- Xt ti x = 0.

Ta c : .

f(0) = a- Vy : Nu a = 0 th hm s f(x) lin tc ti x = 0, suy ra hm s f(x) lin tc trn R.

Nu th hm khng lin tc ti x = 0.

+ .

- Vi x > 1, x < 1 th hm s l hm s cp nn n lin tc trn min xc nh ca n.- Xt ti x = 1.Ta c : .

.

- Nu th hm s lin tc ti x = 1, suy ra hm s lin tc trn R.- Nu th hm s khng lin tc ti x = 1.

( )f x x a lim ( ) lim ( ) ( )x a x a

f x f x f a ( )f x a lim ( ) ( )

x af x f a

( )f x a lim ( ) ( )x a

f x f a ( )y f x x a ( )z g y ( )y f a [ ( )]z g f x

x a

2 2( ) 2 cos 2 sinxy f x e x x 22 2

2sin ln(2 cos 2 )( )

1

xx e xy f xx

2

2 4ln(3 1 cos 2 )

, khi 0( ) 5cos 2 , khi 0

xe xxy f x x x

x x x

2sin (2x), khi x 0f(x) = x

a , khi x = 0

x 0

2 2VCB

x 0 x 0 x 0

sin (2x) (2x)lim lim lim (4x) = 0x x

a 021 xe 1

, khi x > 1f(x) = x 1a + sin(x 1), khi x 1

x 1lim (a + sin(x 1)) = a = f(1)

21 x 2VCB

x 1 x 1 x 1 x 1

e 1 1 x (1 x)(x + 1)lim lim lim lim ( (x 1)) 2x 1 x 1 x 1

a = 2a 2


17---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

+ .

- Vi x > 0, x < 0 th hm s l hm s cp nn n lin tc trn min xc nh ca n.- Xt ti x = 0.Ta c :

.

.

f(0) = a.- Nu th hm s lin tc ti x = 0, suy ra hm s lin tc trn R.

- Nu th hm s khng lin tc ti x = 0.

* Xc nh a hm s sau lin tc trn R.

+

2

2 2 2ln(5 )

2 5ln(1 2 )

23 2cos 2

, khi 0( ) 1 3 , khi 0

a

x x xe xxf x xx

.

- Vi 0x th hm s ( )f x l s cp nn n lin tc trn min xc nh ca n. Do , hms ( )f x lin tc trn R th

0lim ( ) (0) 3x

f x f .Ta c :

2 2

2 22 2 2 2ln(5 ) ln(5 )

22 5ln(1 2 ) 2 5ln(1 2 )

2 20 0 0

3 2cos 2 3( 1) 2(1 cos 2 )lim ( ) lim lim 11 1

a a

x xx x x x

x x x

e x e x xf xx x

2

22 2 2 2 1

2 22 2

ln(5 ) ln(5 )2 2 10 1 2( 1)2 2ln (5 )1 12 6 ln(5 ) 2 22 2

2 20 0

(2 )3 2 62lim 1 lim 1 5 51 1

aa

x x x xaVCB x a

x x

xx x

xe e a a

x x

Vy, nu 2 25 3 5 9 2a a a th hm s ( )f x lin tc trn R .

2

2

x

2

ln(cosx) , khi x 0

x

cos2x e, khi x < 0f(x) =

x + 5x.tgx

a , khi x = 0

2 2 2

22

x x x VCB

2 2 2 2 2x 0 x 0 x 0 x 0

(2x) x

cos2x e cos2x 1 e 1 (1 cos2x) (e 1) 2lim lim lim limx + 5x.tgx x + 5x.tgx x + 5x.tgx x + 5x

2

2x 0

3x 1lim6x 2

2

VCB VCB

2 2 2 2x 0 x 0 x 0 x 0

xln(cosx) ln(1 + cosx 1) cosx 1 12lim lim lim lim

x x x x 2

1a =

2

1a

2


18---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

+ .

- Vi th hm s ( )f x l hm s cp nn n lin tc trn min xc nh ca n. Do , hm s ( )f x lin tc trn R th

0lim ( ) (0)x

f x f a .

Ta c :

.

- Vy vi a = 2e th hm s lin tc trn R.

+ .

- Vi th hm s ( )f x l hm s cp nn n lin tc trn min xc nh ca n. Do , hm s ( )f x lin tc trn R th

0lim ( ) (0)x

f x f a .

Ta c : .

- Vy vi th hm s lin tc trn R.

+ .

Ta c :

1 , khi 11 1( ) lim , khi 1

1 2 0 , khi 1

nn

x

f x xx

x

.

Vi th gii hn khng tn ti v 1 2, khi 21lim , 1,2,3... , khi 2 11 nn

n kk

n kx

.

Hm s khng lin tc. V1 1

1lim ( ) 1 (1) 0 lim ( )2x x

f x f f x .

+ .

Ta c : .

Hm s lin tc. V .

2 21x 2 ln(1 + x )2 x , khi x 0f(x) = a , khi x = 0

x 0

2 22 2 x2 2 11 11 VCBln(1 + x )x 2 x 2 ln(2 ) 2ln(1 + x ) ln(1 + x )x 0 x 0 x 0lim 2 + x lim 1 + 2 1 + x lim 1 + e 1 + x

2 2 2 (ln2 + 1)1 1 1VCB 2 2 2 2 (ln2 + 1)x x x (ln 2 1)x 0 x 0 x 0lim 1 + x .ln2 + x lim 1 + x (ln2 +1) lim 1 + x (ln2 +1) e 2e

2x

2e cosx

, khi x 0f(x) = x a , khi x = 0

x 0

2 2

2 22

x x VCB

2 2 2 2x 0 x 0 x 0 x 0

x 3xx

e cosx e 1 1 cosx 32 2lim lim lim limx x x x 2

3

a =21( ) lim

1 nnf x

x

1x 1lim1 nn x

1( ) lim1 nn

f xx

2

( ) lim1

nx

nxn

x x ef xe

2

2

, khi 0( ) lim 0, khi 0

1, khi 0

nx

nxn

x xx x ef x x

ex x

2

( ) lim1

nx

nxn

x x ef xe

2

0 0lim lim (0) 0x x

x x f


19---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

II. Bi tp p dng :1) Chng minh rng :a) : (0, ) , ( ) ln( 1)f R x y f x x l n nh.b) 2: [ 4, ), ( ) 4f R x y f x x x l ton nh.c) 4 3: \{ 2} \{2}, ( )

2 4xf R R x y f xx

l song nh.2) Tm min xc nh v min gi tr ca cc hm s sau :a) 2 4 8y x x . b) 21 4y x . c)

24y x . d) 23 2xy .s : a) , [4, )D R G , b) , (0,1]D R G , c) [ 2, 2], [0, 2]D G , d) , (2, )D R G .3) Tm hm ngc (c ch ra min xc nh v min gi tr) ca cc hm s sau :a) 24 ,0 2y x x . b) 1

1xyx

. c)2 5

2xy

x

. d)2

1xy

x . e)

2 3xy e .

s : a) 24y x . b) 11xy

x

. c)5 2

2xy

x

. d) 2xy

x . e)

ln( 3)2xy .

4) Cho2

2( )2

xf xx

. Tm ( )f f x . s : 22( )

6 4xf f x

x .

5) Tm ( )f x , vi2

41

1xf x

x x

. s : 21( )

2f x

x .

6) Tnh cc gii hn sau :a) . b) . c) .

d) . e) . f) .

g) . h) . i) .

j) . k) 2 2 2 2 2 2 2 23 7 11 4 1lim 1 .5 5 .9 9 .13 (4 3) (4 1)nn

n n

.

l) 3ln 2

sin 2

0

1 tan 2lim1 sin 2

x

x

x

x

. m) . n) .

o) . p)24

2 40

3 2cos cos 2limln(1 )

x

x

e x x

x x . q) 2

1lim ( 2)sin2x

xx

.

r) . t) 2 40ln(3 2cos 2 )lim

x

x

x x . u)

2

21

ln(2 )lim1xx

x .

v) 3ln 22 2 40

lim 3 2cos 2x x xx

e x . w) 21

sin ( 1)0

lim 3 2cos xex

x . x) 2 2 212 ln(1 4 )0lim sin 2x x xx e x .s : a) 1 2 ; b) 3 2 ; c) 1 2 ; d) 1 ( 1)e ; e) 4 3 ; f) ; g) : lim 2 3 2n n

nHD x x ; h) ; i) 1 16 ;

lim 1n

n n n 3 3 2 2lim 3 2n n n n n 21 3 5 (2 1)lim 2 3 4n nn n

1 2 3

1 2 3lim 2 2 2 2

n

nn

e e e e

2 2 2 2

31 3 5 (2 1)lim

n

n

n

22 4 6 2lim

2nn

n

1 3 5 2 1lim2 4 8 2nn

n

3 3 3 3

41 3 5 (2 1)lim

16nn

n

3 3 3 3

41 2 3lim

4 3 4nn

n n

5 13 35 2 3lim6 36 216 6

n n

nn

21 x

x 1

e cos(x 1)limx 1

2x

2x 0

ln(2e cosx)limx x.arcsinx

2 2 2ln 2

ln(1 3 )

20

3 cos 2lim2

x x x

x

e x

x

2 21x 2 ln(1 + x )

x 0lim e sin x

1 2


20---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

j) 3 2 ; k) 1 8 ; l) 2 ; m) ; n) 5 4 ; o) ; p) 3 ; q) 0; r) ; t) 4 ; u) 1 ; v) 8 ; w) e ; x) e ;7) Xt s lin tc ca cc hm s sau :

a) . b) .

c) . d) .

e) . f) .

g) . h) .

i) . j) .

k) . l) .

m) . n) .

o) . p) .

S :a) ; b) ; c) ;d) ; e) ; f) ; g) ; h) ;i) ;j) ; k) ; l) ; m) ; n) ; o) ; p) ;8) Xc nh a cc hm s sau lin tc trn R.

a) . b) .

2 2 2e2x

2e cos(ax)

, khi x 0f(x) = x 2 , khi x = 0

2sin (x + 1), khi x 1f(x) = x + 1

a , khi x = 1

1x xe x , khi x 0f(x) = a , khi x = 0

2 x + 3 , khi x 21 cos[a(x + 2)]f(x) =

, khi x > 2x + 2

2x 1e 1

, khi x 1f(x) = x + 1 a , khi x = 1

2x 2

2ln(e sin x)

, khi x 0f(x) = x + x.tgx a , khi x = 0

2

2sin [3(x 1)]

, khi x 1f(x) = x 1

a , khi x = 1

2x 5x + 6, khi x 3f(x) = x 3

a , khi x = 3

22

2 6sin ( 1) ln(2 cos 2 )

, khi 0( ) 6 4 , khi 0

ax xe e xxf x x x

a x

2 2

ln[5 4cos( )], khi 0

ln(1 3 )( ) 3 4 , khi 0

axx

x xf xa x

2

2 22

ln(1 )ln(1 )4 2

22 cos 2 sin 2

, khi 0( ) 3 1 3 1 , khi 0

a

x xxe x xxf x x

a x

3

2

ln[3 2cos(2 )], khi 0( ) 2

2 , khi 0

axe xxf x x x

a x

2 22

ln 2ln(1 4 )

22 cos 2

, khi 0( ) 5 1 3 1 , khi 0

x xxe xxf x x

a x

3sin 2 tan 2

, khi 0( ) , khi 0

x xxf x x

a x

22 2 2ln(2 )3 ln(1 4 )3 2cos 2 , khi 0( ) 3 , khi 0

a

x x xe x xf xa x

2

2 1

, khi 1( )1

, khi 1

x

x

ee xf xx

a x

a = 2 a = 0 2a = e a = 2 a = 2 a = 1 a = 9 a = 1 2 4a a 2 4a a 1 2a a 1 2a a 1a 4a 1 2a a 0a

2

2sin (3x)

, khi x 0f(x) = x a + 2 , khi x = 0

1x x2 + x , khi x 0f(x) = a , khi x = 0


21---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

c) . d) .

e) . f) .

g) . h) .

i) . j) .

k) . l) .

m) . n) .

o) . p) .

q) 2 2 2 2ln 22 ln3 ln(1 )3 , khi 0( ) , khi 0

x x x xx e xf xa x

. r) .

t)2 2 2

2 4s in ( )

, khi 0( ) ln(1 ) 3 , khi 0

x xe e axxf x x x

a x

. u)

22 2

2 4ln[ s in ( )]

, khi 0( ) 3 , khi 0

xe axxf x x x

a x

.

v) ; w) .

S : a) ; b) ; c,f) ; d,l) 5 2a ; e,g) 1a e ; h) 1 2a ; i,p) ; j,o) ;k) 9 4a ; m) ; n) ; q) 2a ; r) ;t,u,w) 1 2a a ; v) 2 4a a ;

2 x 1sin (e 1), khi x > 1f(x) = lnx

a + x 2 , khi x 1

2(x 1)

2

ln 2e cos(x 1), khi x 1f(x) = x 1

a , khi x = 1

1xx + 2

, khi x 0f(x) = 3x + 2 a , khi x 0

2x 1

2(e 1).ln x

, khi x > 1f(x) = x 1

a + 3x 2 , khi x 1

2 2

12 x ln(1 x )2

x + 1, khi x 0f(x) = 3x + 1

a , khi x 0

2x 2

4 2 2(e 1).ln(1 x )

, khi x 0f(x) = x + arcsin (x ) a , khi x 0

2 1x sin , khi x 0

f(x) = x a , khi x 0

2x 2

4sin(e 1).ln(cos x)

, khi x 0f(x) = x a , khi x 0

2x 2

22e cosx cos x

, khi x 0f(x) = x a , khi x 0

2

1 x.sinx cos 2x, khi x 0f(x) = x

a , khi x 0

2

2sin [a(x + 1)]

, khi x 1f(x) = (x + 1)

1 , khi x 1

2

1 cos[2a(x 3)], khi x 3(x 3)f(x) =

x + 5 , khi x < 3

2x + 2x 8

, khi x 4f(x) = x 4 ax + 6 , khi x 4

2sin [2(x 3)], khi x 3

3 x 3f(x) = a , khi x = 3

22

2 23 2 cos(2 )

, khi 0( ) ln(1 ) 3 cos 2 , khi 0

ax

x

e axxf x x x

e x x

2

2 2 2

2

ln(1 )2 ln 2 ln(1 2 )

2

2 sin (2 ), khi 0( )

6 7 cos , khi 0

a

x x x

x

xxf x

e x

a x x

2

2

2 4

3 cos(2 )ln2( )

, khi 0

3 cos 2 , khi 0

xe ax

xf x xx x

a x x

a = 7 a = 2e a = 1 a = 0 a = 1

a = 1 a = 2 1 4a a


22---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

Chng 2 : Php tnh vi phn hm mt bin

I. Ni dung cn nh :1) o hm cp 1:a) nh ngha : Nu cc gii sau tn ti th ta c nh ngha o hm.

+ ; + ; + ;

- Nu th hm s c (tn ti) o hm cp 1 ti .- Nu th hm s khng c (tn ti) o hm cp 1 ti .ii) ngha o hm :+ ngha hnh hc :

0

( ) ( )lim lim tanx A B

f a x f a ABx CB

.

Nh vy, l bng h s gc ca tip tuynvi th hm s ti im .+ ngha c hc : Vn tc tc thi ca chuyn ng

l .

V d :+ Dng nh ngha tnh o hm ca cc hm s sau :- ti .

Ta c : .

- ti .Ta c :

.

- ti .

Ta c : .

- ti .

Ta c : .

- ti .Ta c :

( ) ( )( ) limx a

f x f af ax a

( ) ( )( ) limx a

f x f af ax a

( ) ( )( ) limx a

f x f af ax a

( ) ( )f a f a x a( ) ( )f a f a x a ( )y f x

yC

T

( )f a ( )y f x ( , ( ))A a f a A B

( )s s t 0 00

( ) ( )( ) limtt

s t t s tv s t

t 0 a a x x

2( ) 1y f x x 1x 2 2

1 1 1 1 1

( ) (1) ( 1) 0 1 ( 1)( 1)(1) lim lim lim lim lim( 1) 21 1 1 1x x x x x

f x f x x x xf xx x x x

( ) sin 2y f x x 2x

2 2 2 2

2 4 2 4 2 4 2 42sin .cos 2 cos( ) (2) s in2 s in4 2 2 2 2(2) lim lim lim lim2 2 2 2

VCB

x x x x

x x x x

f x f xfx x x x

2 2

2 2 cos( 2)lim lim[2cos( 2)] 2cos 4

2x xx x

xx

29( ) xy f x e 3x

29 2

3 3 3 3 3

( ) (3) 1 9 ( 3)( 3)(3) lim lim lim lim lim[ ( 3)] 63 3 3 3

x VCB

x x x x x

f x f e x x xf xx x x x

( ) cos3y f x x 0x 2

0 0 0 0 0

( ) (0) cos3 cos0 cos3 1 (3 ) 3(0) lim lim lim lim lim 00 0 2 2

VCB

x x x x x

f x f x x x xfx x x x

2( ) ln(5 4 )y f x x 1x


23---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

.

- ti .Ta c :

.

-

3( ) 1y f x x ti 1x .Ta c :

33

3

( 1) , khi 1( ) 1 ( 1) , khi 1

x xy f x x

x x

.

3 32

1 1 1 1

( ) ( 1) ( 1) 0 ( 1)( 1 ) lim lim lim lim [ ( 1) ] 0( 1) 1 1x x x xf x f x xf x

x x x

.

3 32

1 1 1 1

( ) ( 1) ( 1) 0 ( 1)( 1 ) lim lim lim lim ( 1) 0( 1) 1 1x x x xf x f x xf x

x x x

.

V ( 1 ) ( 1 )f f nn hm s c o hm cp 1 ti 1x . Suy ra ( 1) 0f .

+ Cho hm s .

*) Xc nh a hm s f(x) lin tc trn R.- Vi th hm s l hm s cp nn n lin tc trn min xc nh ca n. Do hms lin tc trn R th ch cn hm s lin tc ti ( ).- Xt tiTa c :

.

Vy vi th hm s lin tc trn R.**) Vi gi tr a va tm c trn, hm s c o hm cp 1 ti hay khng?Xt : .

2 2

2

1 1 1 1

5 4 5 4ln ln 1 19 9( ) ( 1) ln(5 4 ) ln 9( 1) lim lim lim lim( 1) 1 1 1x x x x

x x

f x f xfx x x x

2 2

2

1 1 1 1 1

4 4 4 4ln 19 4( 1) 4( 1)( 1) 4( 1) 89lim lim lim lim lim

1 1 9( 1) 9( 1) 9 9VCB

x x x x x

x x

x x x x

x x x x

2( ) tan(4 )y f x x 1x

2

2 22

21 1 1 1

sin(4 4)( ) ( 1) tan(4 ) tan 4 4 4cos(4 ).cos 4( 1) lim lim lim lim( 1) 1 1 ( 1)cos(4 ).cos 4

VCB

x x x x

x

f x f x xxfx x x x x

2

2 2 2 21 1 1

4( 1) 4( 1)( 1) 4( 1) 8lim lim lim( 1)cos(4 ).cos 4 ( 1)cos(4 ).cos 4 cos(4 ).cos 4 cos 4x x xx x x x

x x x x x

1 cos2x 2e cos x, khi x 0f(x) = x

a , khi x 0

x 00x

0lim ( ) (0)x

f x f 0x

2 22x 21 cos2x 2 1 cos2x 2 22VCB VCB

x 0 x 0 x 0 x 0

2xx

e cos x (e 1) (1 cos x) (e 1) x 2lim lim lim limx x x x

0lim(3 ) 0x

x a = 0

0x x 0

f(x) f(0)limx 0


24---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

Ta c :

.

Vy hm s c o hm cp 1 ti .

+ Hm s c o hm cp 1 ti hay khng?.

Ta c :

- .

- .

V nn hm s khng c o hm cp 1 ti .

+ Cho hm s .

*) Xc nh a hm s f(x) lin tc trn R.- Vi th hm s l hm s cp nn n lin tc trn min xc nh ca n. Do hms lin tc trn R th ch cn hm s lin tc ti ( ).- Xt ti

Ta c : 0 .

Vy , vi th hm s lin tc trn R.**) Vi gi tr a va tm c trn, hm s c o hm cp 1 ti hay khng?Xt : .

Ta c : .

Vy, hm s c o hm cp 1 ti .b) Cc qui tc o hm :

1 cos2 2

1 cos2 2 1 cos2 2 2

2 2 20 0 0 0

cos 0cos 1 1 cos 1 cos 2 sinlim lim lim lim

0

x

x x VCB VCB

x x x x

e x

e x e x x xx

x x x x

2

22

2 20 0

(2 )32lim lim 3

x x

xx

x

x x

0x 2x 1 2

2

e cos (x 1), khi x < 1

x 1f(x) =ln x

, khi x 1x 1

1x

2

2

x 1 2

x 1 2 2VCB

22x 1 x 1 x 1 x 1

e cos (x 1) 0f(x) f(1) e 1 1 cos (x 1) 2(x 1)x 1f '(1 ) lim lim lim lim 2x 1 x 1 (x 1) x 1

2

2 2VCB

x 1 x 1 x 1 x 1 x 1

ln x 0f(x) f(1) ln (1 + x 1) (x 1) x 1x 1f '(1 ) = lim lim lim lim lim 0x 1 x 1 (x 1)(x + 1) x 1 (x + 1) x + 1

f '(1 ) f '(1 ) 1x

2x 2

3 2 x(e cos x).ln(1 + x )

, khi x 0f(x) = x + x.sin (e 1) a , khi x 0

x 00x

0lim ( ) (0)x

f x f 0x

2 2

22 2

x 2 x 2 4VCB

3 2 x 3 2 x 3 3 3x 0 x 0 x 0 x 0

x(x ).x(e cosx).ln(1 + x ) (e 1 1 cosx).ln(1 x ) 3x2lim lim lim limx x.sin (e 1) x + x.sin (e 1) x + x 4x

a = 0

0x x 0

f(x) f(0)limx 02 2

22 2

x 2 x 2 4VCB

3 2 x 4 2 2 x 4 4 4x 0 x 0 x 0 x 0

x(x ).x(e cosx).ln(1 + x ) (e 1 1 cosx).ln(1 x ) 3x 32lim lim lim limx(x x.sin (e 1)) x + x .sin (e 1) x + x 4x 4

0x


25---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

+ ; + ; + .

c) o hm hm hp :Nu hm ( )y f x c o hm ti x a v hm ( )z g y c o hm ti ( )y f a . Khi hm hp

( )( )z g f x c o hm ti x a v ( ) ( ) ( ) ( ( )). ( )z a g f a g f a f a .V d : Tnh o hm cp 1 ca cc hm s sau :+ 2 2(3 2 4)y x x .Ta t 23 2 4u x x , khi 2y u . Suy ra ( ) ( ). ( ) 2 .(6 2) 2(3 2 4)(6 2)y x y u u x u x x x x + 3 3 2.y x f x .Ta c : 3 3 2 3 2 3 2 3 2 3 2 3 2 332 1. . . 3 . . . .3y x f x f x x x x f x f x xx .+ Gi s ( ), ( )f x g x c o hm vi mi x R . Tnh o hm ca cc hm s sau :-

2 23 ( ) ( )y f x g x Ta t 2 2( ) ( )u f x g x . Khi

13 3y u u .

Suy ra 23

22 23

1 2 ( ). ( ) ( ). ( )( ). ( ) .(2 ( ). ( ) 2 ( ). ( )) .3 3 ( ) ( )

f x f x g x g xy y u u x u f x f x g x g xf x g x

.

- ( )log ( ),0 ( ) 1, ( ) 0f xy g x f x g x .Ta c :

( )ln ( )log ( )ln ( )f x

g xg x f x .

2

( ) ( )ln ( ) ln ( )ln ( ) 1 ( ) ( )( ) ( )ln ( ) ln ( ) ln ( ) ( ) ( )

g x f xf x g xg x g x f xg x f xy yf x f x f x g x f x

.- ( ( ))y f f x .Ta t ( )u f x . Khi ( )y f u . Suy ra ( ). ( ( )). ( )y f u u f f x f x .d) o hm hm ngc :Nu hm s ( )y f x c o hm hu hn ti x a v ( ) 0f a th hm s ( )x g y c o hmti y b v 1( ) ( )g b f a .V d: Tnh o hm ca cc s sau :+ 2 2

2

1 1arcsin sin cos 1 sin 1 (arcsin )

1y x

y

y x x y x y y x y xx x

.

(T 2 2 2cos sin 1 cos 1 siny y y y . V ,2 2

y nn cos 0y )

( )u v u v ( . )u v u v v u 2 , 0u u v v u vv v


26---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

+ 2 22

1 1arccos cos sin 1 cos 1 (arccos )

1y x

y

y x x y x y y x y xx x

.

(T 2 2 2cos sin 1 sin 1 cosy y y y . V 0,y nn sin 0y )+ 2 22 2

1 1 1arctan tan 1 tan 1 (arctan )

cos 1y x yy x x y x y x y x

y x x .

+ 2 22 21 1 1

arccot cot (1 cot ) (1 ) (arccot )sin 1y x y

y x x y x y x y xy x x

.e) o hm hm n :Gi s ( , ( )) 0F x y x . Khi tnh o hm ca hm ( )y f x ta o hm ( , ( )) 0F x y x theo x(coi v tri l hm hp ca x ), sau gii phng trnh thu c theo ( )y x .V d :+ Tnh o hm ca cc hm s sau :-

2 2 2 21 ( ) 1 2 2 . 0 xx y x y x y y yy

.

-

22 2 2

22ln ( ) ( ln ) 2

1

yy y y y

y

y ye xyyx xe y yx xe y x y xy e y e x yy x y xye .

+ Chng minh hm n ( )y y x xc nh bi phng trnh ln 1xy y tha mn phng trnh :2 ( 1) 0y xy y .

Ly o hm hai v ca phng trnh ln 1xy y ny, ta c :2

20 01

y yy xy y xyy y yy xy .

Suy ra :2

2 2 2( 1). 01

yy xy y yxy .

f) o hm ca hm s cho bi phng trnh tham s :Gi s hm s y ph thuc bin x thng qua mt bin trung gian t : ( ), ( )x t y t , ( , )t v trn ( , ) hm ( )x t c hm ngc 1( )t x . Khi tx

t

yyx

.V d :

+ Tnh ( )y x , vi :23cos

, 0,22sin

x tt

y t

.

Ta c : 6sin .cos 2cos 12cos 6sin .cos 3sin

t tx

t t

x t t y tyy t x t t t

.

+ Tnh ( )y x ti 1x , vi :1

3

33 2

tx

y t t

.

Ta c :1 2

12

3 .ln 3 3 33 .ln 33 3

tt t

x ttt

x y tyxy t

.


27---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

Khi 11 1 3 1tx t . Vy1 1

6( ) ( )ln 3x t

y x y x .g) Bng cc o hm c bn : ( )u u x (u l mt hm theo x ), , ,a b C l hng s.

( ) 0C 1( ) . .u u u

( ) .u ue u e ( ) . .lnu ua u a a ln uu

u

log lna uu u a (sin ) .cosu u u

(cos ) .sinu u u 2( ) cos

utgu

u

2(cot ) sinugu

u

2(arcsin )

1u

uu

2( ) 1

uarctgu

u

2(arccos )

1u

uu

2( ) 1

uarccotgu

u

2 22 .ln u a a u

u a u a

2 2ln uu u b u b

V d : Tnh y, vi :- ;

- .

- .

-

22sin ( 1)xy e .2 2 2 2 2 2 22[sin ( 1)] 2( 1) .cos( 1).sin( 1) 2.2 . .cos( 1).sin( 1)x x x x x x xy e e e e x e e e

2 2

2 . .sin[2( 1)]x xx e e .Ch : Khi gp bi ton yu cu tnh o hm ca hm c dng th ta tin hnh nhsau : Ly ln hai v ta c , tip theo ta ly o hm hai v theota c :

.

V d : Tnh o hm ca cc hm s sau :+ , 0xy x x .

ln ln( ) ln (ln ) ( ln ) ln 1 (ln 1)x x xyy x y x x x y x x x y x xy .

+ln

, 1(ln )x

x

xy xx

.

ln ln ln 2 2ln ln ln ln (ln ) ln ln(ln ) (ln ) (ln ) ( ln(ln ))(ln ) (ln )x x

x x

x x

x xy y x x x x x y x x xx x

2 3x 2 ' 3x 2 3x ' 3x 2 3xy = x e ; y' = (x ) e x (e ) 2xe 3x e 2 ' 2 '

42xy = x.sinx + arctg(x ); y' = (x.sinx) (arctg(x )) sinx x.cosx +

1 + x

' '

x 1 lnxln x x.(lnx) lnx.( x ) 2 + lnxx 2 xy = ; y' =x xx 2x x

( )( )v xy u x( )ln ln[ ( ) ] ( ).ln ( )v xy u x v x u x x

( )( ) ( )( ).ln ( ) . ( ) ( ) ( ).ln ( ) . ( )( ) ( )v xy u x u xv x u x v x y u x v x u x v x

y u x u x


28---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

ln2 1 2 1ln ln(ln ) ln ln(ln )ln (ln ) ln

x

x

y xx x y x x

y x x x x x .

+3 3

5 2 3

. 1. (4 )

x xyx x

.

Ta c :

3 3 3 5 2 3 3 5 2 33 35 2 3. 1ln ln ln . 1 ln . (4 ) ln( ) ln 1 ln ln (4 ). (4 )

x xy x x x x x x x xx x

1 2 33ln ln(1 ) ln ln(4 )3 5 2

x x x x .3 3

5 2 3

3 1 2 3 . 1 3 1 2 33( 1) 5 2(4 ) 3( 1) 5 2(4 )

. (4 )y x xyy x x x x x x x xx x

.

2) Vi phn cp 1:a) nh ngha : ;V d : Tnh dy, vi y = f(x) :+ .

Ta c : .

Vy : .

+ .

Ta c : .

Vy : .

b) Cng thc tnh gn ng : .V d : Dng vi phn cp 1 tnh gn ng biu thc sau :+ .

Xt ;

Ta c : ; ;

.

dy = f '(x)dx

2

2xy = f(x) = , x 1(x + 1)

'2 2 ' 2 2 2 ' 2 2

2 4 4 4x ( x ) (x + 1) ( x )[(x + 1) ] 2x(x + 1) ( x )[2(x + 1)] 2x(x + 1)f'(x) = (x + 1) (x + 1) (x + 1) (x + 1)

3

2xdy = dx(x + 1)

2y = ln(x + x + 1)

2 ' 2 '' 22

2 2 2 2

2x1 +(x + x + 1) x' + ( x + 1) 12 x + 1f'(x) = ln(x + x + 1)x + x + 1 x + x + 1 x + x + 1 x + 1

2

1dy = dxx + 1

f(x) = f(a + x) f(a) + f'(a). x

3 23A = ln 2 + (0,98) + (0,98) 33 2f(x) = ln 2 + x x

a = 1, x = 0,98 1 = 0,02 33 2f(1) = ln(2 + 1 1 ) = 2.ln2 1,386

'33 23 33 2 3 2

3 2 1x2 + x x 132 3 xf '(x) = f'(1) =

242 + x x 2 + x x


29---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

Vy : .

+ Bit f(u) kh vi lin tc ln cn im 1, f(1) = 2, . Dng vi phn cp 1 tnh gnng biu thc .

Xt hm s ; ; ;;

Vy : .

3) o hm cp cao :a) nh ngha :+ Nu o hm c tn ti o hm th ta c o hm cp 2 nh sau :

,f (x) f (a)f (x) = lim

x ax a ,

(n 1) (n 1)(n) f (x) f (a)f (x) = lim

x ax a

.

V d : Cho hm s .

- Xc nh a hm s lin tc trn R.- Vi gi tr a va tm c trn, hm s c tn ti khng?Gii :- V vi th hm s l hm s cp nn n lin tc trn min xc nh ca n. Do hm s lin tc trn R th .Ta c :* .

* .

* .

Vy vi th hm s lin tc trn R.

- Ta c : * .

* .

* .

3 23 13A = ln 2 + (0,98) + (0,98) 1,386 .( 0,02) 1,37624 f '(1) = 1 3A = 1,9998.f (1,9998)

3y = x.f( x ) a = 2, x = 1,9998 2 0,0002 3y(1) = 1.f( 1 ) f(1) = 23 3 3 33 3 3 7y' = f( x ) x. x.f '( x ) y'(1) = f( 1 ) 1. 1.f '( 1 ) 2

2 2 2 2

3 7A = 1,9998.f (1,9998) 2 + .( 0,0002) 1,99932

x a

f(x) f(a)f '(a) limx a

x a

f '(x) f '(a)f ''(a) limx a

3sin x, khi x 0

2. xf(x) = a , khi x = 0

f ''(0)

x 0x 0 x 0lim f(x) = lim f(x) = f(0)

f(0) = a3 3 3 2VCB

x 0 x 0 x 0 x 0

sin x sin x x xlim lim = lim lim 02. x 2.x 2x 2

3 3 3 2VCB

x 0 x 0 x 0 x 0

sin x sin x x xlim lim = lim lim 02. x 2.x 2x 2

a = 0

'3 2 3

2sin x 3cosx.sin x.( 2x) ( 2).sin xf '(x) = , khi x < 0

2x 4x

'3 2 3

2sin x 3cosx.sin x.(2x) 2.sin xf '(x) = , khi x > 0

2x 4x

3

3 3VCB

2 2x 0 x 0 x 0 x 0 x 0

sin x 0f(x) f(0) sin x x x2xf '(0 ) = lim lim lim = lim lim 0

x 0 x 2x 2x 2


30---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

* .

V nn tn ti , suy ra .

*

.

*

.

V nn khng tn ti .+ Hay : ;+

+ ,

vi

b) V d : Tnh .Ch : sin cos ; cos sin ; sin cos ; cos sin

2 2 2 2x x x x x x x x

.+ .

;

; .

+ .

.

+ .

.

+ .

t .Ta c : .

.

3

3 3VCB

2 2x 0 x 0 x 0 x 0 x 0

sin x 0f(x) f(0) sin x x x2xf '(0 ) = lim lim lim = lim lim 0

x 0 x 2x 2x 2

f '(0 ) = f '(0 ) f '(0) f '(0) = 0

2 3

2

x 0 x 0

6x.sin x.cosx 2sin x 0f '(x) f '(0) 4xf ''(0 ) = lim lim

x 0 x

2 3 3 3VCB

3 3x 0 x 0 x 0

6x.sin x.cosx 2sin x 6x .cosx 2x 3cosx + 1lim = lim lim 14x 4x 2

2 3

2

x 0 x 0

6x.sin x.cosx 2sin x 0f '(x) f '(0) 4xf ''(0 ) = lim lim

x 0 x

2 3 3 3VCB

3 3x 0 x 0 x 0

6x.sin x.cosx 2sin x 6x .cosx 2x 3cosx 1lim = lim lim 14x 4x 2

f ''(0 ) f ''(0 ) f ''(0)

(n) (n 1) (n 1) (n)y = f(x); y' = f '(x); y'' = (y')' = (f '(x)) ' = f ''(x); ...; y = (y )' = (f (x)) ' = f (x) (n) (n) (n)(f g) (x) = f (x) g (x);

n(n) 0 (n) 1 (1) (n 1) k (k) (n k) k (k) (n k)

n n n n

k = 0(f . g) (x) = C f(x).g (x) + C f (x).g (x) + +C f (x).g (x) = C f (x).g (x)

kn

n!C =k!(n k)!

(n)y

y = sin2x'

2 2 2y' = 2cos2x = 2sin(2x + ); y'' = 2sin(2x + ) 2 cos (2x + ) 2 sin (2x + )2 2 2 2

(n) n ny = 2 sin (2x + )

2xy = (x + 2)e

x x x x x x (n) xy' = e + (x + 2)e (x + 3)e ; y'' = e + (x +3)e (x + 4)e ;...; y (x + n + 2)e 11y = = (2x + 3)

2x + 3

2 3 (n) n n (n + 1)y' = 2(2x + 3) ; y'' = ( 2)2( 2)(2x + 3) ;...; y = ( 1) .n!.2 (2x + 3) 2 3xy = x e

2 3xf(x) = x ;g(x) = e2 (3) (n)f(x) = x ; f'(x) = 2x; f''(x) = 2; f (x) = = f (x) = 0

3x 3x 2 3x (n) n 3xg(x) = e ; g'(x) = 3e ; g''(x) = 3 e ; = = g (x) = 3 e


31---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

Vy : .4) Vi phn cp cao :a) nh ngha : .b) V d : Tnh :+ . + 1

1ln(2 3), ( 1) .2 . (2 3)n n n n

ny x d y dx

x

.

+ ( 2) , ( 2 )x n x ny x e d y x n e dx . + 5 51 ( 4)! 4, ( 1) .(4 5) 4! (4 5)n

n n n

n

ny d y dxx x

.5) ng dng :a) Xp x hm a thc : Cho hm s kh vi n cp ti . Khi ta c :* Cng thc Taylor :

.

* Cng thc Maclaurin : Khi a = 0 th cng Taylor c gi l cng thc Maclaurin.

* Cng thc Maclaurin ca mt s hm c bn :

+ .

V d : Vit cng thc Maclaurin ca hm s .

+ .

V d : Vit cng thc Maclaurin ca hm s :

.

+ .

V d : Vit cng thc Maclaurin ca hm s :

.

+ .

V d : Vit cng thc Maclaurin ca hm s :.

(n)(n) 2 3x 0 2 n 3x 1 n 1 3x 2 n 2 3x n 2 3x 2n n ny = x e C x .3 .e + C 2x.3 .e + C 2.3 .e 3 .e (9x + 6.n.x + n(n 1)) 2 2 n (n) ndy = f '(x)dx; d y = f ''(x)dx ; ; d y = f (x)dx

nd yn nny = sin2x; d y = sin(2x + )dx

2

( )y f x 1n x a(n) (n+1)

2 n n + 1f '(a) f ''(a) f (a) f (c)f(x) = f(a) + (x a) + (x a) + + (x a) + (x a) , c (a, x)1! 2! n! (n + 1)!

(n) (n+1)2 n n + 1f '(0) f ''(0) f (0) f (c)f(x) = f(0) + x + x + + x + x , c (0, x)

1! 2! n! (n + 1)!

2 3 n c cnx n +1 k n +1

k = 0

x x x x e 1 ey = e = 1 + + + x = x + x , c (0, x)1! 2! 3! n! (n + 1)! k! (n + 1)!

3cn3x k n + 1

k = 0

1 ey = e = (3x) + (3x) ,c (0, x)k! (n + 1)!

3 5 7 2m + 1m m + 1 2m + 2x x x x x sin(c + (m +1 ).)y = sinx = + + ( 1) ( 1) x , c (0, x)

1! 3! 5! 7! (2m + 1)! (2m + 2)!

3 2m + 1m m + 1 2m + 22x (2x) (2x) sin(2c + (m +1 ).)y = sin2x = + + ( 1) ( 1) (2x) , c (0, x)

1! 3! (2m + 1)! (2m + 2)!

2 2mm m + 1 2m + 1

cos(c + (2m +1 ). )

x x 2y = cosx = 1 + ( 1) ( 1) x , c (0, x)2! (2m)! (2m + 1)!

2 2mm m + 1 2m + 1

cos(2c + (2m +1 ). )(2x) (2x) 2y = cos2x = 1 + ( 1) ( 1) (2x) , c (0, x)

2! (2m)! (2m + 1)!

n 1n k n +1k = 1

( 1)( 2)...( k + 1) ( 1)( 2)...( n)(1 c)y = 1 + x 1 x + x ,c (0, x)k! (n + 1)!

1 1

1 221 1 1 1y = (x 3) (x 2) f (x) f (x)

x 5x + 6 (x 2)(x 3) x 3 x 2


32---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

.

Vy :

, .

b) Quy tc LHospital :+ Quy tc LHospital 1 :

Nu m v th .

V d : Tnh cc gii hn sau :

- .

-

Hay

- .

Hay .

- .

( v d ny ta ch p dng c VCB cho mu s ch khng p dng VCB c cho t s, vt s c dng l hiu ca hai hm gn bng nhau)+ Quy tc LHospital 2 :

Nu m v th .

V d : Tnh cc gii hn sau :

- .

(n) (n) (n) n (n + 1) n (n + 1) n1 2 n + 1 n + 1

1 1y f (x) f (x) = ( 1) n!(x 3) ( 1) n!(x 2) ( 1) n! (x 3) (x 2)

n

k k n + 1 n + 1k + 1 k + 1 n + 2 n + 22

k = 0

1 1 1 1 1y = ( 1) x ( 1) xx 5x + 6 3 2 c 3 c 2

c (0, x)

00

x f(x) 0, g(x) 0

x

f'(x)lim = g'(x) x

f(x)lim = g(x)

22 2 2

'0 2 x2 x x 3 x0

'x 1 L' x 1 x 1

x e ex e e 2xe + 2x elim lim lim 4ex 1 1x 1

0

2 0 2

x 0 L' x 02

2x1 +x + ln(1 + x ) 11 + xlim lim .1x + arctgx 21 +

1 + x

2 2VCB

x 0 x 0 x 0 x 0

x + ln(1 + x ) x + x x(1 + x) 1 + x 1lim lim lim lim .x + arctgx x + x 2x 2 2

0 x x

x 20

x 0 L' x 0

1e .sinx + e .cosx +

e .sinx + tgx cos xlim lim 1x + sinx 1 + cosx

x x x xVCB

x 0 x 0 x 0 x 0

e .sinx + tgx e .x + x x(e + 1) e + 1lim lim lim lim 1x + sinx x + x 2x 2

2 2 2 2 2

0x 2 x 2 x x xVBC 0

4 4 4 3 4 3 4 2x 0 x 0 L' x 0 x 0 x 0

e 1 x e 1 x 2xe 2x 2x(e 1) e 1 1lim lim lim lim limsin 3x (3x) 3 4x 3 4x 3 2x 162

x f(x) , g(x)

x

f'(x)lim = g'(x) x

f(x)lim = g(x)

x + L' x + x + x +

1lnx 2 x 2xlim lim lim lim 01 xx x

2 x


33---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

- .

- lim lim 01

x x

x L x

e e

x

hay

1lim lim lim 0x u

ux u u

e e

x u ue

.

- lim lim1

x x

x L x

e e

x

.

Ch :+ Gii hn khng p dng c qui tc LHospital. V khng tn ti.

Nhng (V m nn ).

+ Khi gp cc dng , ta a v dng hoc .

+ Khi gp bi ton yu cu tnh gii hn c dng th ta tin hnh nh sau :t , sau ly ln hai v ta c , tip theo ta ly giihn hai v ta c :

.

+ Khi xt th ta cn ch : Nu th ta p dng VCB bnh thng, cnnu th ta p dng qui tc Lhospital.V d :- Dng :

.

- Dng : .

- Dng : .t

.

x xx

xxx + L' x + x +

x

x + 2 1 + 2 .ln2lim lim lim (x + 2 )1 + 2 .ln2ln(x + 2 )

x + 2

sinlimsinx

x x

x x lim cosx x

sin1sin 1 0lim lim 1

sinsin 1 01x x

x

x x xxx x

x

sin 10 x

x x 1lim 0

x x sinlim 0

x

x

x

0 0; 0. ; 0 ; 00

( )lim ( )v xx

u x

( )( )v xy u x ( )ln ln[ ( ) ] ( ).ln ( )v xy u x v x u x

( )lim(ln ) lim ( ).ln ( ) ln lim lim ( ).ln ( ) lim lim[ ( ) ]v x Ax x x x x x

y v x u x y v x u x A y u x e

( ) ( )limx a

f x f ax a ( ) 0f a

( ) 0f a

( ) 0 00 0

x 0 x 0 L' x 0 L' x 0

1 1 sinx x cosx 1 sinx 0lim lim lim lim 0x sinx x.sinx sinx + x.cosx cosx + cosx x.sinx 2 0.0

(0. )20.

2 2 2

x 0 x 0 x 0 L' x 0 x 02 3

1ln x 2 xxlim (x .ln x 2x ) lim x (ln x 2) lim lim lim 01 2 2

x x

0(0 ) sinxx 0lim x

sinx

x 0 x 0 x 0 x 0y = x lny = sinx.lnx lim (lny) = lim (sinx.lnx) ln( lim y) = lim (sinx.lnx)

sinx

L'x 0 x 0 x 0 x 0 x 02

1lnx sinx sinxxln( lim y) = lim lim lim . 0 lim x 11 cosx x cosxsinx sin x


34---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

- Dng : .

t

Vy : .

- Cho hm s .

-) Xc nh hm s lin tc trn .- ) Vi cc gi tr va tm c trn, hm s c o hm cp 1 ti hay khng?Gii :-) Vi th hm s l hm s cp nn n lin tc trn min xc nh ca n. hms lin tc trn th .Ta c :

.

Vy, nu th hm s lin tc trn .

-) Vi th .

Ta xt : .

Vy vi th hm s c o hm cp 1 ti .

Vi th .

Ta xt :

0( ) 1x xx +lim x + 3

1 x x xx xx + x + x + x +

ln(x + 3 ) ln(x + 3 ) ln(x + 3 )y = x + 3 lny = lim (lny) = lim ln( lim y) = limx x x

x

x x x 2 x 3x

x x x 2x + x + L' x + x + L' x + L' x +

1 + 3 ln3ln(x + 3 ) 1 + 3 ln3 3 ln 3 3 ln 3x + 3ln( lim y) = lim lim lim lim lim ln3

x 1 x + 3 1 + 3 ln3 3 ln 3

1x xx +lim x + 3 3

3

3

2

cos3, khi 0( )

, khi 0

axe xxf x x x

a x

a ( )f x R

a ( )f x 0x

0x ( )f x( )f x R 2

0lim ( ) (0)x

f x f a

2

3 3

3 3 30 0 0 0 0

(3 )3cos3 1 1 cos3 32lim ( ) lim lim lim lim 3

ax ax VCB VCB

x x x x x

xax

e x e x axf x ax x x x x x x

2 2 03 3 0

3a

a a a aa

( )f x R

0a 31 cos3

, khi 0( ) 0 , khi 0

xxf x x xx

2

3

2 4 20 0 0 0

1 cos3 (3 )0( ) (0) 1 cos3 92lim lim lim lim0 0 2

VCB

x x x x

x xf x f xx x

x x x x x

0a ( )f x 90 (0)

2x f

3a 9

3cos3

, khi 0( ) 9 , khi 0

xe xxf x x xx

9

0 09 3 9 23 0 0

2 4 30 0 0 ' 0 '

cos3 9( ) (0) cos3 9( ) 9 3sin 3 9(1 3 )lim lim lim lim0 0 2 4

x

x x

x x x L x L

e x

f x f e x x x e x xx xx x x x x x


35---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

.

Vy vi th hm s c o hm cp 1 ti .

- Cho hm s .

-) Xc nh a hm s lin tc trn .- ) Vi cc gi tr a va tm c trn, hm s c o hm cp 1 ti hay khng?Gii :

- ) Vi l hm s cp, vi l hm s cp nnn lin tc trn min xc nh ca n. hm s lin tc trn R th .Ta c :

Vy, nu .

- Xt

.

+ Vi .

+ Vi .

(Ch : gii hn ny ta khng dng c v cng b)c) Kho st hm s :*) Tp xc nh : , x0 l im m lm cho hm s khng xc nh.V d :- Hm s c min xc nh l .

- Hm s c min xc nh l .


090

2' 0

81 9cos3 54 81 9 0 90lim 452 12 2 0 2

x

L x

e x x

x

3a ( )f x 0 (0) 45x f 2a x3e 1 2cosx

, x 0f(x) = 3x 3a 2 , x = 0

( )f x R

( )f x 0x 2a x3e 1 2cosx

x 0, f(x) =3x

x = 0, f(x) = 3a 2

x 0lim f(x) = f(0) = 3a 2

2 2

22

a x a x 2VCB VCB2

x 0 x 0 x 0 x 0 x 0

x3a x + 23e 1 2cosx 3(e 1) + 2(1 cosx) 3a x2lim f(x) = lim = lim lim lim = a3x 3x 3x 3x

2 2 a = 1a = 3a 2 a 3a + 2 = 0

a = 2

2

2

a x

a x

2x 0 x 0 x 0

3e 1 2cosx (3a 2)f(x) f(0) 3e 1 2cosx (3a 2)3x3xlim = lim = lim

x 0 x 0 3x

2 20 0

2 a x 4 a x 40 0

L' x 0 L' x 0

3a e + 2sinx 3(3a 2) 3a e + 2cosx 3a + 2lim lim =6x 6 6

5

a = 1 f '(0) =6

50 25

a = 2 f '(0) = =6 3

0D = R\{x }

21y =

1 + xD = R

21y =

1 x D = R\{ 1,1}lnxy =

xD = (0, + )


36---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn


*) Tnh y tm im cc tr.- Nu th hm s lun tng (ng bin) trn min xc nh ca n.- Nu th hm s lun gim (nghch bin) trn min xc nh ca n.- Nu c nghim trong min xc nh ca n v i du qua nghim y th hm s c cctr.C th :+ Nu c mt nghim n l thTH 1 x TH 2 x

0 + + 0y y

+ Nu c hai nghim n l thTH 1 x TH 2 x

+ 0 0 + 0 + 0y y

+ Nu c ba nghim (nghim n l v nghim kp l ) thTH 1 x TH 2 x

+ 0 0 0 + 0 + 0 + 0y y

- Nu th hm s lun tng (ng bin) trn min xc nh ca n.x

+

y- Nu th hm s lun gim (nghch bin) trn min xc nh ca n.

x

y

V d :- .

- .

xey =x

D = R\{0}

y' > 0y' < 0y' = 0

y' = 0 1xa > 0 1x a < 0 1x

y' y'

y' = 0 1 2x , xa > 0 1x 2x a < 0 1x 2x

y' y'

y' = 0 1 2 3x , x , x 1 3x , x 2xa > 0 1x 2x 3x a < 0 1x 2x 3x

y' y'

y' > 0

y'

y' < 0

y'

2 2 21 2

; 0 01 (1 )

xy y xx x

2

3 3 21 3

; 0, \{ 1}1 (1 )

xy y x Rx x


37---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

- .

*) Tnh tm im un ( c nghim), khong li ( ), khong lm ( ).*) Tim cn :- Tim cn ng : Nu th l tim cn ng, x0 l im m lm cho hm skhng xc nh.- Tim cn ngang : Nu th l tim cn ngang (Hm s c tim cn ngang khibc t nh hn hoc bng bc ca mu, l i vi hm hu t, cn vi hm bt k th tyvo hm s c th m c kt lun. V d : hm s c hai tim cn ngang (bn

tri, bn phi), hm s c tim cn ngang bn phi , hm s c tim cnngang bn tri .).- Tim cn xin : Nu v th l tim cn xin (Hm sc tim cn xin khi bc t ln hn bc ca mu, l i vi hm hu t, cn vi hm bt kth ty vo hm s c th m c kt lun.).*) Bng bin thin :

x 0x

yyy

*) V th:V d : Kho st v v th ca cc hm s sau :

- .

+ Tp xc nh : .

+ Tnh y : .

+ Tnh y : .

+ Tim cn :

* Tim cn ng : V nn suy ra l tim cn ng.

* Tim cn ngang : V nn suy ra l tim cn ngang.

+ Bng bin thin :

2 2

21 2 2

; 0, \{ 1}1 ( 1)

x x x xy y x Rx x

y y'' = 0 y'' < 0 y'' > 0

0x xlim f(x) = 0x = x

xlim f(x) = b y = b

2

xy =x 1 y = 1

lnxy =x

y = 0xey =

x

y = 0

x

f(x)lim = ax x

lim [f(x) ax] = b y = ax + b

2

2xy = (x + 1)

D = R \ { 1}2 2 2

4 4 42x(x + 1) 2(x + 1)( x ) 2x(x + 1) 2x 2xy' = 0 x = 0, (x = 1)(x + 1) (x + 1) (x + 1)

4 3 2 4

8 8( 4x 2)(x + 1) 4(x + 1) ( 2x 2x) 2(x + 1) (2x 1) 1y'' = 0 x = , (x = 1)(x + 1) (x + 1) 2

2

2x 1 x 1

xlim y = lim (x + 1) x = 1

2

2x x

xlim y = lim 1(x + 1) y = 1


38---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

x 0 y + 0y 0 +

0y 1 9

+ th : y

x

0

- .

+ Tp xc nh : .+ Tnh y : .

+ Tnh y : .

+ Tim cn : V nn suy ra l tim cn ngang.

+ Bng bin thin :x 0y + + 0y + 0 0 +

1y 3/4 3/4

0 0+ th : y

1

1

1

1

1

21y =

x + 1D = R

2 22xy' = 0 x = 0(x + 1)

2 2

2 42(x + 1)(1 3x ) 1y'' = 0 x =(x + 1) 3

2x x

1lim y = lim 0x + 1

y = 0

1/ 3 1/ 3

1

0 x


39---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

- .

+ Tp xc nh : .+ Tnh y :

.

+ Tnh y :

.

+ Tim cn :

* Tim cn ng : V nn suy ra l tim cn ng.

* Tim cn ngang : V nn suy ra l tim cn ngang.

+ Bng bin thin :x

y 0 +y 0 + + +

y

+ th :

-

3

2x + 4y =

x.

* Tp xc nh D = R\{0} .

*2 2 3 4 3

34 4 4

3x .x 2x.(x +4) x 8x x(x 8)y' = = = = 0 x(x 8) = 0 x = 2x x x

.

2

22x 6y = (x + 1)

D = R \ { 1}2 2 2

4 4 44x(x + 1) 2(x + 1)( 2x + 6) 4(x + 1)(x + 3) 4x 16x 12y' = 0 x = 3, (x = 1)(x + 1) (x + 1) (x + 1)

4 3 2 4

8 8( 8x 16)(x + 1) 4(x + 1) ( 4x 16x 12) 8(x + 1) (x 4)y'' = 0 x = 4, (x = 1)(x + 1) (x + 1)

2

2x 1 x 1

2x 6lim y = lim (x + 1) x = 1

2

2x x

2x 6lim y = lim 2(x + 1) y = 2

4 3 1

2 26 / 9

3 2y

4 3 2 1 0 x1

2y 226

93


40---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

*3 4 3 4 4

8 8(4x 8)x 4x (x 8x) 24xy' ' = = >0, x D

x x

.* Tim cn :

-

3

2x 0 x 0

x +4lim y = lim = x = 0x

l tim cn ng. * th : y

-

3

3x x

2x x

y x +4lim = lim =1x x y x

4lim[y x]= lim =0x

l tim cn xin.

* Bng bin thin : 3x 0 2 y + 0 +y + + +

0 2 xy

3- 2

xy =x 1 .

* Tp xc nh D = R\{ 1} .*

2 2

2 2 2 2(x 1) 2x.x (1+x )y' = = < 0, x D(x 1) (x 1)

*

2 2 2 2 22 2

2 4 2 42x(x 1) 2.2x[ (1+x )] 2x(x 1)(3+x )y' ' = = 0 2x(x 1)(3 + x ) = 0 x = 0(x 1) (x 1)

.

* Tim cn : - 2x 1 x 1xlim y = lim = x = 1

x 1 l tim cn ng. * th : y

- 2x x

xlim y = lim = 0 y = 0x 1

l tim cn ngang.* Bng bin thin :

x 1 0 1 y 1 0 1 xy + 0 +

0 y 0

0

-

2 23 3y = (x + 1) (x 1) .* Tp xc nh D = R .

*1 13 3

3 3

2 2 2 1 2 1y' = (x+1) (x 1) = . .3 3 3 3x+1 x 1

. y' > 0 vi 1 < x < 1 v y' < 0 khix < 1x > 1

.


41---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

*4 43 3

4 43 3

2 2 2 1 2 1y' ' = (x+1) + (x 1) = . + . = 0 x = 09 9 9 9(x+1) (x 1)

.

y' ' > 0 khi x > 0 v y' ' < 0 khi x < 0 .

* Tim cn : 2 22 23 3 4 2 2 4x x x 3 3 3(x+1) (x 1)lim y = lim (x+1) (x 1) = lim = 0(x+1) + (x 1) (x 1) .y = 0 l tim cn ngang. * th : y

* Bng bin thin : 3 4x 1 0 1 y + + y 0 + +

0 3 4 1 0 1 xy 0

3 4 03 4

-

2

x 2y =x + 2

.

* Tp xc nh D = R .

* 2

2

2 32 22 2 2

x(x 2)x +2

2x+2 2x+2x +2y' = = = = 0 2x +2 = 0 x = 1(x +2) x +2x +2 (x +2)

.

*

3 112 22 2 2 22

22 3 2 3

3 x 22(x +2) 2x(x +2) (2x +2) 2(x +2) (2x 3x 2)2y' ' = = = 0 2x 3x 2 = 0 1(x +2) (x +2) x2

.

* Tim cn :

-

2 2

1 22lim lim lim 12 1 2x x x

x xxyx x x

y = 1 l tim ngang bn tri.

-

2 2

1 22lim lim lim 12 1 2x x x

x xxyx x x

y = 1 l tim ngang bn phi.

* Bng bin thin : * th : yx 2 1 12 y 0 + + 1y 0 + + 0

1 1 2 1 0 12 2 xy 4 6 1 1

3 3


42---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

II. Bi tp p dng :1)

a) Cho hm s 32

cos 2, khi 0( )

6 , khi 0

Axe xxf x x x

A x

.

+ Xc nh A hm s ( )f x lin tc trn R .+ Vi gi tr ca A va tm c trn, hm c ( )f x tn ti (0)f hay khng?

b) Cho hm s 32

3 1 2cos(3 ), khi 0( )

, khi 0

Axe xxf x x x

A x

.

+ Xc nh A hm s ( )f x lin tc trn R .+ Vi gi tr ca A va tm c trn, hm c ( )f x tn ti (0)f hay khng?

c) Cho hm s 21 cos[ ( 2)]

, khi 2( 2)( ) 3 4 , khi 2

A xx

xf xA x

.

+ Xc nh A hm s ( )f x lin tc trn R .+ Vi cc gi tr ca A va tm c trn, hm c ( )f x tn ti (0)f hay khng?

d) Cho hm s 31 cos[2 ( 1)]

, khi 1( ) 3 2 5 , khi 1

A xxf x x x

x x

.

+ Xc nh A hm s ( )f x lin tc trn R .+ Vi cc gi tr A va tm c trn, hm s ( )f x c tn (0)f hay khng?s : a) 2 (0) 4, 3 (0) 13 2A f A f . b) 0 (0) 9, 3 (0) 45 2A f A f .

c) 2 (0) 0, 4 (0) 0A f A f . d) 3 (0) 0, 3 (0) 0A f A f .2) Cc hm s sau c tn ti o hm cp 1 (ti gi tr x tng ng) hay khng?

a)

21 2

3

1 .ln 1 sin ( 1), khi 1( ) 1

0 , khi 1

xe xxf x xx

. b)

2sin ( 2), khi 2

3 2( ) 0 , khi 2

xx

xf xx

.

c)1 cos[2( 1)]

, khi 1( ) 1 1 , khi 1

xe xxf x xx

. d)

1 cos[2( 1)], khi 1

2 1( ) 0 , khi 1

xx

xf xx

.

e)2

33 1 2cos

, khi 0( ) 3 6 , khi 0

xe xxf x x xx

. f)

2

5ln(2 cos 2 )

, khi 0( ) 0 , khi 0

xe xxf x x xx

.

s : a) (1) 1f . b) 1 1(2 ) , (2 )3 3

f f . c) 5(1)2

f . d) ( 1 ) ( 1 )f f . e) (0) 7f . f) (0) 4f .


43---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

3) Tnh o hm cp 1 ca cc hm s sau :a) ln(arcsin 3 )y x . b) arctan(ln )y x . c) 2(ln )xy x . d) 1 arctanln

1 arccotxyx

.

e) ln(arctan ) xy x . f)2

lglg ( 2)x

x

xyx

. g)2

22

arcsin1

xyx

. h) 2( 1). 2 2y x x .

i)2

2ln2

xyx

. j) 2ln (ln )y x . k) arctan arccoty x x . l) arcsin arccosy x x .

m) 2 2 3 2y x f x x . n) 3 21y x f x . o) 2

3

2 (3 2)1

x xy

x

. p) arctanarctan .x xy e e .s : a)

2

31 9 .arcsin 3

yx x

. b) 21

(1 ln )y x x . c)2(ln ) .(2 ln )xy x x x x .

d) 2 arctan arccot(1 )(1 arctan )(1 arccot )x xy

x x x

. e)ln

21 ln(arctan ) . .ln(arctan ) (1 )arctan

x xy x xx x x

.

f)2 2

lg 2 2lg ( 2) 2 ln lg

. ( 2).ln10.lg( 2) ln10x

x

x x x xyx x x x x

. g)

2 2 4

4( 1) 1 2 3

xyx x x

.

h)21 12 2 2 .

4 2 . 2 2

xy x xx x

. i) 22 2( 2)( 2)xy

x x

. j)2ln(ln )

lnxy

x x . k) 0y . l) 0y .

m) 2 3 2 2 2 3 2322 . 2 .3y x f x x x x f x xx . n) 2 2 3 223 . 1 . . 11 xy x f x x f xx .o)

2

3( 2) ( 3) 2 1 3

.( 1) 2 3 1x xy

x x x x

. p)arctan arctan

2 21

. . .arctan( )1 1

xx x x

x

ey e e ee x

.4) Tnh ( )y x , vi ( )y y x c xc nh bi phng trnh sau :

a) 2 ln yy x x y e . b) ln 0x yyx

. c)2 1 yx y xe y . d)

21ln 0xxyy

.

s : a) 2 ln yxy y yy y x ye . b)

1 ln .ln ln.

ln ln 1

x xx y y

y yy

x xx x

y y

. c) 221

y

y

xy eyx xe

. d)2 2

2(1 )

(1 )(1 )xy y xy

x xy .

5) Dng vi phn cp 1 tnh gn ng gi tr ca biu thc sau :a) 4ln 2 1,0024 1,0024A ; b) 2 1,9996(1,9996) .arctan 2B ; c) 54 1(16,0016)C ;d) 2 3ln 4 (2,0016) (2,0016)D ; e) 33 16 (1,9984)E ; f) 2 2 1,9984F ;s : a) 1,0024; 1; 0,0024; 1,3863x a x A . b) 1,9996; 2; 0,0004; 3,139144x a x B .c) 516,0016; 16; 0,0016; 2 .0,999875x a x C . d) 2,0016; 2; 0,0016; 2,7736x a x D .


44---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

e) 1,9984; 2; 0,0016; 2,0016x a x E . f) 1,9984; 2; 0,0016; 1,9999x a x F .6) Tnh (1)df , vi :a) 3 2( )

7xf xx

; b)3 2( ) .arctan

3xf x x ; c) 3 2( ) ln

3 2xf xx

; d) 3 3 2( ) ln 2f x x x ;s : a) 3(1)

16df dx . b) 3(1)

9 4df dx

. c) 12(1)5

df dx . d) 13(1)24

df dx .7) Chng t rng :a) Hm s 2.ln(1 )y x x tha mn phng trnh

32

24

( 1)x

x y xy yx

.

b) Hm s .arctany x x tha mn phng trnh 2 22 2( 1) 0, 0x y y y xx x .c) Hm s 2 2x xy e e tha mn phng trnh 22 8 8 xy y y e .d) Hm s ( 1) xy x e tha mn phng trnh 2 0y y y .e) Hm n ( )y y x xc nh bi phng trnh 1 xy ye tha mn h thc sau :

3 2(2 ) ( ) 2( ) 0y y x y y .f) Hm n ( )y y x xc nh bi phng trnh ln 1 0xy y tha mn h thc sau :

3 2 2. ( ) (2 ) 0y y y y y .

8) Tm ( )x thng qua ( )x . Gi s hm ( )y x l hm ngc ca hm sau :a) xy x ; b) xxy x ; c) 2xy x ; d) ln(ln ) , 0

x

x

xy xx

; e)lg

, 1(lg )x

x

xy xx

;

s : a) 1( ) (ln (x) 1)x x . b) 2( ).ln ( )( ) ( ( ).ln ( ) ( ).ln (x)) .lnx x

xx x x x x

.

c) ( ) ( )( )( ) ( ).2 .ln 2.ln (x) 2x xx

xx x

. d)2

( ).ln ( )( ) ( ).ln (x).ln(ln (x)) (x) 2 ln ( )x x

xx x x

.

e)2

( ).ln ( ).ln10( )lg ( ).ln (x) ln10.lg (x) (x).lg ( ).ln10 ( )

x xx

x x x x

.

9) Hm s3sin ( 1)

, khi 12 1( )

0 , khi 1

xx

xf xx

c tn ti ( 1)f hay khng?

s : ( 1 ) 1 1 ( 1 )f f .10) Hm s 3( ) 1f x x c tn ti ( 1)f hay khng? s : ( 1 ) 6 6 ( 1 )f f .11) Tnh ( ) ( )nf x , t suy ra cng thc Maclaurin ca hm ( )f x tng ng.a) 31( ) (2 3)f x x ; b) 5

1( )3 4

f xx

; c)3( ) xf x xe ; d) 2( ) ( 3) xf x x e ; e) 2 2( ) (2 1) (2 1)

xf xx x

;

f)4

1( )5 4

f xx

; g)1( )

3 2f x

x ; h) ( )

mf x x x ; i) ( ) 3 3x xf x ; j) 22 3( ) 6 8xf x

x x

;


45---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

s :

a)( )

( ) ( 3)3

1 ( 2)!( ) ( 1) 2 (2 3)(2 3) 2!n

n n n nnf x xx

.

1( 3) 1 ( 4) 1

30

1 ( 1) ( 2)! ( 1) ( 3)!( ) 2 (2.0 3) . 2 (2. 3) . , (0, )(2 3) 2! ! 2!( 1)!k nn

k k k n n n

k

k nf x x c x c xx k n

.

b)( ) 5 1

( ) 1 55

(5 4)1( ) ( 1) 3 (3 4)

53 4

n

n n

n n nin

if x x

x

. (

1(5 4) 1.6.11...(5 4)

n

ii n

)1

15 1 5 6

1 11 15 515

0

( 1) (5 4) ( 1) (5 4)1( ) 3 (3.0 4) . 3 (3. 4) , (0, )

5 . ! 5 .( 1)!3 4

k nk n

k nnk k n ni i

k nk

i if x x c x c x

k nx

.c) ( )( ) 3 1 1 3( ) ( 1) 3 ( 1) .3nn x n n n n xf x xe x n e .

1 1 3.0 1 1 33 1

0

( 1) 3 .0 ( 1) . .3 ( 1) 3 .c ( 1) ( 1)3( ) , (0, )! ( 1)!

k k k k n n n n cn

x k n

k

k e n ef x xe x x c xk n

.d) ( )( ) 2 1 1 2( ) ( 3) ( 1) 2 ( 1) ( 6)2nn x n n n n xf x x e x n e .

1 1 2.0 1 1 22 1

0

( 1) 2 .0 ( 1) ( 6)2 ( 1) 2 .c ( 1) ( 7)2( ) ( 3) , (0, )! ( 1)!

k k k k n n n n cn

x k n

k

k e n ef x x e x x c xk n

e) ( )( ) ( 2) ( 2)2 2 1 ( 1)!( ) ( 1) 2 (2 1) (2 1)(2 1) (2 1) 8 1!

n

n n n n nx nf x x xx x

.

( 2) ( 2)2 20

1 ( 1)!( ) ( 1) 2 (2.0 1) (2.0 1)(2 1) (2 1) 8 1!k!n

k k k k k

k

x nf x xx x

1 1 ( 3) ( 3) 11 ( 2)!( 1) 2 (2 1) (2 1) , (0, )8 1!(n 1)!n n n n nn c c x c x .

f)( ) 4 1

( ) 1 44

(4 3)1( ) ( 1) 5 (3 4)

45 4

n

n n

n n nin

if x x

x

.

11

4 1 4 51 11 14 4

140

( 1) (4 3) ( 1) (4 3)1( ) 5 (5.0 4) . 5 (5. 4) . , (0, )

4 . ! 4 .( 1)!5 4

k nk n

k nnk k n ni i

k nk

i if x x c x c x

k nx

.g)

( ) 2 1( ) 21 (2 1)!!( ) ( 1) 3 (3 2)

23 2

n n

n n n

n

nf x xx

. ( (2 1)!! 1.3.5.7...(2 1)n n )2 1 2 31

1 12 21

0

1 ( 1) (2 1)!! ( 1) (2 1)!!( ) 3 (3.0 2) . 3 (3. 2) . , (0, )2 . ! 2 ( 1)!3 2

k nk nnk k n n

k nk

k nf x x c x c xk nx

.h) ( )1 1( ) 1 1( )

1

1( ) 2n

nn nn m m m

if x x x x i x

m

.


46---------------------------------------------------------------------------------------------------------------------------------------

Lu hnh ni b c nhn

1 111

11 1

0

1 12 .0 2 .0( ) , (0, )

! ( 1)!

n nk nm m

nk nm i i

k

i im mf x x x x x c x

k n

.

i) ( )( ) ( ) 3 3 3 ln 3 ( 1) 3 ln 3nn x x x n n x nf x .0 0 1 1

1

0

3 ln 3 ( 1) 3 ln 3 3 ln 3 ( 1) 3 ln 3( ) 3 3 , (0, )! ( 1)!

n n n c n n c nnx x k n

kf x x x c x

k n

.j)

( )( ) ( 1) ( 1)

22 3 11 7( ) ( 1) ! ( 4) ( 2)

6 8 2 2

n

n n n nxf x n x xx x

.( 1) ( 1)

20

2 3 k! 11 7( ) ( 1) (0 4) (0 2)6 8 k! 2 2

nk k k k

k

xf x xx x

1 ( 2) ( 2) 1( 1)!( 1) ( 4) ( 2) , (0, )(n 1)!n n n nn c c x c x .

12) Vit cng thc Maclaurin ca cc hm s sau :a) 22 3( ) 5 6

xf xx x

; b) 2 24 1( ) (4 3) (4 1)

xf xx x

; c) ( ) cos 2f x x ; d) ( ) 4 3xf xx

;

e) ( ) sin 3f x x ; f) 2 26 1( ) (3 2) (3 1)xf x

x x

; g)3( ) ( 2) xf x x e ; h)

3( )

5 4xf xx

;

i) 51( ) (4 3)f x x ; j)2( ) s in 2 .cos3f x x x ; k) ( ) ln(3 2)f x x ; l)

5

14 5

yx

;

s : a)( 1) ( 1)

20

2 3 ( 1) ![9(0 3) 7(0 2) ]( )5 6 !

k k knk

k

x kf x xx x k

1 ( 2) ( 2)

1( 1) ( 1)![9( 3) 7( 2) ], (0, )( 1)!

n n nnn c c x c x

n

.

b)( 2) ( 2)

2 20

4 1 1 ( 1) ( 2)!4 [(4.0 3) (4.0 1) ]( ) .(4 3) (4 1) 8 1! !k k k kn

k

k

x kf x xx x k

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Bai Giang Tom Tat Giai Tich