Программа и экзаменационные задачи по математическому анализу для студентов 1 курса

Fbgbkl_jkl\hh[s_]hbijhn_kkbhgZevgh]hh[jZah\ZgbyJN ey bkke_^h\Zgby g_ij_ju\ghklb y′(x ) \ lhqd_ x0 = 0 jZkk...

Author: Мешков В.З. | Половинкин И.П. | Попов А.В. | Астахов А.Т.

7 downloads 208 Views 133KB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

Fbgbkl_jkl\hh[s_]hbijhn_kkbhgZevgh]hh[jZah\ZgbyJN
NZdmevl_lijbdeZ^ghcfZl_fZlbdbbf_oZgbdb DZn_^jZ^bnn_j_gpbZevguomjZ\g_gbc

Ijh]jZffZbwdaZf_gZpbhggu_aZ^Zqb ihfZl_fZlbq_kdhfmZgZebam ^eyklm^_glh\dmjkZnZdmevl_lZIFF

KhklZ\bl_eb <AF_rdh\ BIIheh\bgdbg :<Ihidh\ :L:klZoh\

IGZqZevgu_k\_^_gbybal_hjbbfgh`_kl\bfZl_fZlbq_kdhceh]bdb 1. Fgh`_kl\Zbhi_jZpbbgZ^gbfbK\hckl\Zhi_jZpbcgZ^fgh`_kl\Zfb 2. \Zhij_^_e_gbybbo wd\b\Ze_glghklv G_h[oh^bfh_ b ^hklZlhqgh_ mkeh\b_ kms_kl\h\Zgby ij_^_eZ qbkeh\hc ihke_^h\Zl_evghklb IVIj_^_ebg_ij_ju\ghklvnmgdpbb 16. Ij_^_evgu_ ]jZgbqgu_ bahebjh\Zggu_ \gmlj_ggb_ lhqdb fgh`_kl\ L_hj_fZ ;hevpZgh-<_c_jrljZkkZ\l_jfbgZoij_^_evguolhq_dfgh`_kl\ 17. Hij_^_e_gb_ij_^_eZnmgdpbb\lhqd_ih=_cg_bihDhrbbbowd\b\Ze_glghklv 18. G_ij_ju\ghklv nmgdpbb \ lhqd_ JZaebqgu_ nhjfu aZibkb mkeh\by g_ij_ju\ghklb H^ghklhjhggb_ ij_^_eu b h^ghklhjhggyy g_ij_ju\ghklv G_h[oh^bfh_ b ^hklZlhqgh_ mkeh\b_ kms_kl\h\Zgby ij_^_eZ nmgdpbb \ lhqd_ :jbnf_lbq_kdb_ k\hckl\Z ij_^_eh\nmgdpbc 19. Ij_^_ebg_ij_ju\ghklvkmi_jihabpbbnmgdpbc 20. Djbl_jbcDhrbkms_kl\h\Zgbyij_^_eZnmgdpbb\lhqd_ 21. I_j\ucaZf_qZl_evgucij_^_e

22. bnn_j_gpbZevgh_bkqbke_gb_nmgdpbc\_s_kl\_gghci_j_f_gghc 36. Ijhba\h^gZy nmgdpbb \ lhqd_ >bnn_j_gpbjm_fZy \ lhqd_ nmgdpby G_h[oh^bfh_b^hklZlhqgh_mkeh\b_^bnn_j_gpbjm_fhklbnmgdpbb\lhqd_>bnn_j_gpbZe nmgdpbb\lhqd_G_ij_ju\ghklv^bnn_j_gpbjm_fhcnmgdpbbH^ghklhjhggb_ijhba\h^gu_ 37. =_hf_ljbq_kdbc kfuke ijhba\h^ghc b ^bnn_j_gpbZeZ nmgdpbb \ lhqd_ MjZ\g_gb_dZkZl_evghcd]jZnbdmnmgdpbb\lhqd_Nbabq_kdbckfukeijhba\h^ghc 38. Ijhba\h^gu_kmffuijhba\_^_gbybqZklgh]h^bnn_j_gpbjm_fuonmgdpbc 39. Ijhba\h^gZyh[jZlghcnmgdpbb 40. Ijhba\h^gZyb^bnn_j_gpbZekeh`ghcnmgdpbbBg\ZjbZglghklvnhjfui_j\h]h ^bnn_j_gpbZeZhlghkbl_evgh\u[hjZi_j_f_gguo 41. Ijhba\h^gu_ihdZaZl_evghceh]Zjbnfbq_kdhcbkl_i_gghcnmgdpbc 42. Ijhba\h^gu_ljb]hghf_ljbq_kdboh[jZlguoljb]hghf_ljbq_kdbonmgdpbc 43. Ijhba\h^gu_]bi_j[hebq_kdbobh[jZlguo]bi_j[hebq_kdbonmgdpbc 44. Hij_^_e_gby ijhba\h^guo \ukrbo ihjy^dh\ Ijhba\h^gu_ \ukrbo ihjy^dh\ kmffubijhba\_^_gbynmgdpbcNhjfmeZE_c[gbpZ 45. >bnn_j_gpbZeu \ukrbo ihjy^dh\ Hlkmlkl\b_ bg\ZjbZglghklb nhjfu ^bnn_j_gpbZeh\\ukrboihjy^dh\ 46. EhdZevguc wdklj_fmf nmgdpbb L_hj_fZ N_jfZ h g_h[oh^bfhf mkeh\bb ehdZevgh]hwdklj_fmfZ 47. L_hj_fZJheey 48. L_hj_fZEZ]jZg`Zhdhg_qguoijbjZs_gbyo

L_hj_fZDhrbhdhg_qguoijbjZs_gbyo JZkdjulb_g_hij_^_e_gghkl_cihijZ\bemEhiblZey Fgh]hqe_gL_cehjZNhjfmeZL_cehjZkhklZlhqgufqe_ghf\nhjf_I_Zgh NhjfmeZL_cehjZkhklZlhqgufqe_ghf\nhjf_EZ]jZg`Z NhjfmeZL_cehjZkhklZlhqgufqe_ghf\nhjf_Dhrb JZaeh`_gb_ihnhjfme_L_cehjZFZdehj_gZ we_f_glZjguonmgdpbc m x y = e , y = sin x, y = cos x, y = (1 + x ) (m ≠ 1,2 ), y = ln (1 + x ) hklZlhqgh_mkeh\b_kljh]hcfhghlhgghklb 56. L_hj_fZ h kf_g_ agZdZ ijhba\h^ghc \ hdj_klghklb lhqdb >hklZlhqgu_ mkeh\by fhghlhgghklb\hdj_klghklblhqdbehdZevgh]hwdklj_fmfZ 57. IjbagZdbwdklj_fmfZnmgdpbbbf_xs_cijhba\h^gu_\ukrboihjy^dh\ 58. hklZlhqgh_mkeh\b_\uimdehklb 59. LhqdZi_j_]b[Z>hklZlhqgh_mkeh\b_i_j_]b[Z 60. :kbfilhluMjZ\g_gb_Zkbfilhl 61. Bkke_^h\Zgb_nmgdpbc 62. NmgdpbbaZ^Zggu_iZjZf_ljbq_kdb
Ijbf_juj_r_gbyaZ^Zqij_^eZ]Z\rbokygZwdaZf_g_ihfZl_fZlbq_kdhfmZgZebam x n _keb x n = Ijbf_jGZclb lim n →∞

n n

8 −1 16 − 1

.

n n a = 1 _keb a > 1 H[hagZqbf 0 < a n = n a − 1, ⇒ a = (1 + a n ) ≥ na n J_r_gb_: IhdZ`_fqlh lim n →∞

a n

ih g_jZ\_gkl\m ;_jgmeeb Hlkx^Z ke_^m_l 0 < a n ≤ AgZqbl ih l_hj_f_ h lj_o a n = 0 , ⇒ lim n a = lim(1 + a n ) = 1 Imklv z n = n 2 , lim z n = 1 ih ihke_^h\Zl_evghklyo lim n →∞ n →∞ n→ ∞ n →∞ n

^hdZaZgghfm lim n n →∞

Hl\_l

lim x n = n →∞

( 2 ) − 1 = lim z = lim z 16 − 1 ( 2) −1 8 −1

n

n →∞ n

3 4

n →∞

3 n 4 n

(

)

(z − 1) z n2 + z n + 1 3 −1 = lim n = . − 1 n→∞ (z n − 1) z n2 + 1 (z n + 1) 4

(

)

3 . 4

sin (k!) . k =1 k ⋅ (k + 1) n

Ijbf_jBkke_^h\ZlvgZkoh^bfhklvdjbl_jbcDhrb _keb x n = ∑ J_r_gb_>ey ∀ ε > 0 b ∀ p ∈ N bf__f xn+ p − xn =

sin((n + p )!) + (n + 1)(n + 2)

sin ((n + 2)!) sin ((n + p )!) 1 1 ++ ≤ + + (n + 2)(n + 3) (n + p )(n + p + 1) (n + 1)(n + 2) (n + 2)(n + 3) 1 1 1 1 1 1 1 + = − + − ++ − = (n + p )(n + p + 1) n + 1 n + 2 n + 2 n + 3 n + p n + p +1 1 1 1 1 = − < < ε ijb Ke_^h\Zl_evgh x n + p − x n < n +1 n + p +1 n +1 n +1 +

1 1 − 1 = N (ε ) bex[hfgZlmjZevghf p BlZd ∀ ε > 0 ∃N = N (ε ) = − 1 ε ε ∀ n ≥ N ∀ p ∈ N : x n + p − x n < ε Mkeh\b_ Dhrb \uiheg_gh b ihwlhfm ^ZggZy qbkeh\Zy ∀n >

ihke_^h\Zl_evghklvkoh^blky 1

 sin x  x 2 Ijbf_jIjbf_gyyijZ\behEhiblZeygZclb lim   . x →0  x  J_r_gb_ A^_kv g_hij_^_e_gghklv \b^Z 1∞ Ijbf_gyy nhjfmem u v = e v ln u

(u > 0, v > 0) b

ihevamykv \hafh`ghklvx i_j_oh^Z d ij_^_em \ ihdZaZl_e_ nmgdpbb e x gZoh^bf 1

1

sin x

ln lim z ( x ) 2  sin x  x 2 lim  = lim e x x = e x →0 , x →0 x →0  x  x  x cos x − sin x  sin x ⋅  ln sin x  x2  = lim 1 ⋅ x ⋅ x cos x − sin x  = x = lim z (x ) = lim lim   2 0 → x →0 x →0 x x → 0 2 x sin x 2x x x2   x cos x − sin x 1 1 1 sin x 1  − x sin x  = lim = lim = − lim =− .  3 2 2 x →0 2 x →0  2 x 6 x →0 x 6 x  1

1

−  sin x  x 2 =e 6. Ke_^h\Zl_evgh lim   x →0  x  1

1

−  sin x  x 2 =e 6. Hl\_l lim   x →0  x 

Ijbf_jFh`gheb^hhij_^_eblvnmgdpbx y = f (x )\lhqd_jZaju\Z x0 lZdqlh[uhgZklZeZ 1

sin x  x 2 g_ij_ju\ghc\wlhclhqd__keb y =   , x0 = 0 ?  x  1

1

− sin x  x 2 = e 6 kf j_r_gb_ ijbf_jZ lh nmgdpby y = f (x ) [m^_l J_r_gb_ LZd dZd lim0   x→  x  1  x sin x   2 1    , x≠0 − g_ij_ju\ghc \ lhqd_ x0 = 0 _keb iheh`blv f (0) = e 6 BlZd nmgdpby y =  x   −1 e 6 , x=0

g_ij_ju\gZ\lhqd_ x0 = 0 .

 x 2 ⋅ sin x , x ≠ 0 ′ Ijbf_j GZclb ijhba\h^gmx y (x ) nmgdpbb y =  bkke_^h\Zlv y\ey_lky eb x=0 0 , y ′(x ) g_ij_ju\ghc\lhqd_ x0 = 0 ?

J_r_gb_ ?keb x ≠ 0 lh y ′(x ) = 2 x sin − cos AgZq_gb_ y ′(0) \uqbkebf ih hij_^_e_gbx 1 x

1 x

1 ∆y = lim ∆x ⋅ sin = 0 BlZd nmgdpby y (x ) ^bnn_j_gpbjm_fZ gZ \k_c qbkeh\hc ∆x → 0 ∆x ∆x → 0 ∆x 1 1  2 x sin − cos , x ≠ 0 ijyfhcb y ′(x ) =  . x x 0 , x=0 y ′(0 ) = lim

y ′(x ) Hq_\b^gh qlh >ey bkke_^h\Zgby g_ij_ju\ghklb y′(x ) \ lhqd_ x0 = 0 jZkkfhljbf lim x →0

1 = 0 ijhba\_^_gb_ [_kdhg_qgh fZehc nmgdpbb gZ h]jZgbq_ggmx _klv [_kdhg_qgh x 1 cos g_kms_kl\m_l^ey^hdZaZl_evkl\Z\hkihevah\Zlvkyhij_^_e_gb_f fZeZynmgdpby Z lim x →0 x ij_^_eZnmgdpbbih=_cg_gZyaud_ihke_^h\Zl_evghkl_c LZdbfh[jZahf y′(x ) jZaju\gZ\ lim 2 x ⋅ sin x →0

lhqd_ x0 = 0 dhlhjZyy\ey_lkylhqdhcjZaju\Zjh^Znmgdpbb y′(x ) . Ijbf_jBkke_^h\ZlvgZ^bnn_j_gpbjm_fhklvnmgdpbx y = arcsin (cos x ) . J_r_gb_ Nmgdpby y (x ) = arcsin (cos x ) hij_^_e_gZ gZ \k_c qbkeh\hc ijyfhc i_jbh^bq_kdZy k

i_jbh^hf 2π h[eZklvagZq_gbc y ≤

π ]jZnbdgZjbk 2

y π 2

y = arcsin(cos x ) −π −

π 2

π 2

−

Ijbf_gyy

ijZ\beh

y ′(x ) =

=

− sin x

1 − cos 2 x

π

0

3π 2

π 2

jbk

^bnn_j_gpbjh\Zgby

− sin x = − sgn (sin x ) , sin x

x

(x ≠ kπ ,

keh`ghc

nmgdpbb

ihemqbf

k ∈ Z ).

< lhqdZo [ = N π , N ∈Ζ ih hij_^_e_gbx h^ghklhjhggbo ijhba\h^guo bf__f \(N π + ∆[ ) − \(N π ) \ +′ = OLP

∆[ →+

∆[

=

= OLP

∆[ →+

= OLP

DUFVLQ [FRV(N π + ∆[ )] − DUFVLQ[FRV(N π )] = ∆[

∆[ →+

= OLP

[

∆[

]

[

DUFVLQ (− ) FRV(∆[ ) − DUFVLQ (− )

∆[ →+

(− ) N +

= OLP

[

DUFVLQ (FRV(N π ) ⋅ FRV(∆[ ) − VLQ (N π ) ⋅ VLQ (∆[ )) − DUFVLQ (− )

∆[ →+

N

N

]=

N

]=

(− ) N + DUFVLQ VLQ (∆[ )

OLP ∆[ →+ ∆[ DUFVLQ VLQ (∆[ ) VLQ (∆[ ) N+ ⋅ = (− ) . ∆[ VLQ (∆[ )

∆[

=

DUFVLQ[FRV (N π + ∆[ )] − DUFVLQ [FRV(N π )] N+ = − (− ) . ∆[ LZd dZd \ +′ (N π ) ≠ \ −′ (N π ) N ∈ Ζ lh \ lhqdZo [ = N π nmgdpby \([ ) = DUFVLQ (FRV [ ) g_

:gZeh]bqgh \ −′ (N π ) = ∆OLP [ →−

^bnn_j_gpbjm_fZ AZf_qZgb_ AZ^Zqm fh`gh j_rblv b ^jm]bf kihkh[hf Z bf_ggh ihdZaZlv qlh \ +′ () ≠ \ −′ () \ +′ (π ) ≠ \ −′ (π ) b\hkihevah\Zlvkyk\hckl\hfi_jbh^bqghklbnmgdpbb \([ ) . Hl\_lNmgdpby \([ ) = DUFVLQ (FRV [ ) ^bnn_j_gpbjm_fZgZ\k_cqbkeh\hcijyfhcdjhf_lhq_d [ = N π N ∈Ζ . Ijbf_jGZclb \ ( Q) ([ ) ^eyke_^mxsbonmgdpbc Z \([ ) = VLQ [ + FRV [ ; [ \([ ) = VLQ [ ⋅ VLQ [ ; \ \([ ) = [ ⋅ FRV [ . J_r_gb_Z Ihke_lh`^_kl\_gguoij_h[jZah\Zgbcihemqbf

− FRV [ \([ ) = (VLQ [ + FRV [ ) − VLQ [ ⋅ FRV [ = − VLQ [ = =− = + FRV [ π (FRV α [ )( Q) = α Q FRV α [ + πQ  lh \ ( Q) ([ ) = Q− FRV  [ + Q  . − FRV [ VLQ [ = VLQ [ − VLQ [ . [ Bf__f \([ ) = VLQ [ ⋅ VLQ [ = π Q π Q π Q ( Q)    LZddZd (VLQ α [ ) = α Q VLQ  α [ +  lh \ ( Q) ([ ) = Q− VLQ  [ +  − Q− VLQ  [ +  .      

LZd

dZd

\ Ijbf_gbf nhjfmem E_c[gbpZ (X Y) \(

)

([ ) = & Q [ (FRV [ ) + & Q ([

( Q)

)′ (FRV [ )(

Q −)

( Q)

+ & Q ([

Q

= ∑ FQN X ( Q− N ) Y( N ) iheh`b\ \ g_c X = FRV [ Y = [ N =

)″ (FRV [ )(

Q−)

=

  π Q π (Q − ) π (Q − )  = Q ⋅ [ ⋅ FRV  [ +  + [ Q ⋅ Q − FRV  [ +  + + Q (Q − ) ⋅ FRV  [ + =      

π Q π Q   πn  = Q ⋅ [ FRV [ +  + Q ⋅ Q[ ⋅ VLQ  [ +  − n(n − 1)2 n − 2 cos  2 x + =     2    π Q π Q Q(Q − )   = Q  [ −  FRV [ +  + Q ⋅ Q[ ⋅ VLQ  [ +  .      

π Q  ;  π Q π Q   VLQ  [ +  − Q− VLQ  [ +  ;      

Hl\_lZ \ ( Q) ([ ) = Q− FRV  [ +

[ \ ( Q) ([ ) = Q−

 

\ \ ( Q) ([ ) = Q  [ −

Q(Q − ) π Q π Q    FRV [ +  + Q ⋅ Q[ ⋅ VLQ  [ + .     

AZ^Zqb^eykZfhklhyl_evgh]hj_r_gby I.Ij_^_eqbkeh\hcihke_^h\Zl_evghklb [ Q _keb 1. GZclb OLP Q→∞ Q + Q Z [ Q = 0,7+0,29+...+ [ [ Q = Q

\ [ Q = ^ [ Q = ` [ Q = b

−

Q Q

−

Q

Q

−

−

Q

Q − Q +

Q

Q

Q+ Q+ Q

[Q =

Q +

;

Q

Q + Q Q −

Q

 Q +Q     Q + Q + 

] [ Q =

−

Q + + Q + ; Q +

+ Q _ [ Q = Q

Q

;

;

a [ Q = Q + Q − Q − Q ;

;

d [ Q =

(Q) QQ

GZclb\_jogbcbgb`gbcij_^_eu

;

πQ πQ [ [ Q = + VLQ ; Q πQ Q Q \ [ Q = Q − + ; ] [ Q = + ; FRV Q + Q πQ Q π Q   FRV _ [ Q =  +  ⋅ (− ) Q + VLQ ; ^ [ Q =  Q + Q

Z [ Q = DUFWJ Q ⋅ FRV

Bkke_^h\ZlvgZkoh^bfhklvdjbl_jbcDhrb Q Q Q FRV (N ) ; \ [ Q = ∑ ; Z [ Q = ∑ [ [ Q = ∑ N N= N N = N ⋅ (N + ) N=

Q

] [ Q = ∑ N = Q

` [ Q = ∑ N =

Q

VLQ N FRV N ^ [ Q = ∑ N N N=

(− ) N ⋅ N − FRV π N

Q

; a [ Q = ∑ N =

_ [ Q = ∑

(− ) N ⋅ N N

Q

N =

;

,,Ij_^_enmgdpbb

; N

FRV [ g_kms_kl\m_l >hdZaZlvqlh [OLP →+∞ I ([ ) _keb Kms_kl\m_leb OLP [ →D [

Z I ([ ) = ([ + ) ⋅ VLQ Z [ I ([ ) = VLQ [ ⋅ VLQ

, Z [

 [  [ + [ [ >   ( ) \ I [ =  Z ] I ([ ) = DUFFWJ   + [ ⋅ VLQ Z [ [  − FRV [ [ <  [   ^ I ([ ) = DUFWJ   + [ ⋅ FRV [ ⋅ FRV Z [ [

GZclbij_^_eunmgdpbc [



 [

[

π [



Z OLP  VLQ + FRV  [ OLP [ ⋅ OQ  FRV  \ OLP (FRV [ ) ( + [→ [ →∞  [ →∞    H[ − H−

] OLP [→

[

[ + VLQ ( [

  ( + [ ) [ ` OLP [→  H 

)

^ OLP [→

[

[

)

;

− FRV [ _ OLP [ FRV [ ;

[ [ ( + [ ) [ − H    OLP OLP DUFWJ [ a b   ;  [ →+∞  π [→  [ 

[ →

 DUFVLQ [  [ d OLP   e OLP + [ − [ [→  [→  [

g OLP [→

OQ VLQ

(

)

[

  [ ⋅  [ − [ +  ; f OLP [ →∞  

[ [ − ( + WJ [ ) ( + h [OLP ([ + [ ) [ i OLP →+∞ [→ OQ [

OQ [

) ;

VLQ (H [ − − ) [ k OLP l OLP ; j [OLP →+ [ → OQ ( + FRV [ ) [→ OQ [ OQ FRV [

[

m OLP [→

[

+ [ VLQ [ − FRV [

n OLP [ →

+ WJ [ − + VLQ [ [

.

,,,G_ij_ju\ghklvnmgdpbb IjbdZdhfagZq_gbbiZjZf_ljZanmgdpby f(x) , [m^_lg_ij_ju\gZ_keb (( + [ ) Q − ) [ [ ≠ Q ∈ Ν [ OQ ( + [ ) [ ≠ Z I ([ ) =  [ I ([ ) =  ; D

[ =

D

[ =

 [ [ ( − ) [ ≠ (VK [ ) [ + [ VLQ [ ≠ [ \ I ([ ) =  ] I ([ ) =  ; [ = D D [ =

GZclb lhqdb jZaju\Z nmgdpbb mklZgh\blv bo jh^ ^hhij_^_eblv nmgdpbx ih g_ij_ju\ghklb\lhqdZomkljZgbfh]hjZaju\Z

 [ − ([ + ) [ VLQ [ ≠ Z \ = [ \ =  [ \ \ = ; ([ − ) − [ VLQ π [  [ = ] \ = VJQ (VLQ [ ) ^ \ = VJQ (FRV [ ) _ \ = ; [ −[ − H  VLQ [ − [ ≠ [⋅  [ a \ = H [ b \ = H ; ` \ =  [  [ = 

VLQ

VLQ

d \ = H g \ =

[

[

e \ = [

−

⋅H

−

f \ =

[

π [  FRV    

VLQ [ h \ = VLQ [ [ −[

FRV

DUFVLQ [ ; VLQ [

;

Fh`gh eb ^hhij_^_eblv nmgdpbx y=f(x) \ lhqd_ jZaju\Z [ g_ij_ju\ghc\wlhclhqd_ Z \ = [ + [ [ = [ \ = [

−

 VLQ [    [ 

\ \ = 

[

[ = ] \ = [

⋅H

−

lZd qlh[u hgZ klZeZ

[

[ = ;

− [

[ = ;

Ijb dZdbo agZq_gbyo iZjZf_ljh\ a b b nmgdpby f(x) g_ij_ju\gZ gZ k\h_c h[eZklb hij_^_e_gby_keb ([ − ) [ ≤ [ [ ≤  Z I ([ ) = D[ + E < [ < [ I ([ ) =  ;

[ + D[ + E [ >  [ ≥  [ [ FRV([ ) VLQ [ [ ∈[− π π ] [ ≠ [ ≠ π  . \ I ([ ) = D [ = E [ = π 

IVIjhba\h^gZynmgdpbb IjZ\beZ^bnn_j_gpbjh\Zgby GZclb h^ghklhjhggb_ ijhba\h^gu_ nmgdpbb I ([ ) = [ − [ ⋅ J([ ) \ lhqd_ [ ]^_ J([ ) g_ij_ju\gZy\lhqd_ [ nmgdpbyBf__lebnmgdpby I ([ ) ijhba\h^gmx\lhqd_ [ ? 2GZclbijhba\h^gmx \ ′([ ) nmgdpbb

 [ ⋅ VLQ \([ ) =  [ 

ij b [ ≠ ij b [ =

bbkke_^h\Zlvy\ey_lkyeb \ ′([ ) g_ij_ju\ghc\lhqd_ [ =0.

GZclb qbkeZ D E D E lZd qlh[u nmgdpby y=f(x) [ueZ ^bnn_j_gpbjm_fZ gZ \k_c qbkeh\hcijyfhc

[>

π D [ + E [ >  π π  [ ∈ −  [ \ = [ ⋅ VLQ π [ [ ∈[− ] ;    D [ + E [ < − π [ <− [< H − H ≤[ ≤H ; [ >H

 D [ + E   Z \ = FRV [   D [ + E   D [ + E  \ \ = [ OQ [ D [ + E  

Bkke_^h\ZlvgZ^bnn_j_gpbjm_fhklvke_^mxsb_nmgdpbb Z \ = π − [ ⋅ VLQ [ [ \ = [ ⋅ [ \ \ = VLQ [ ; ] \ = DUFVLQ ( VLQ [ ) ^ \ = DUFVLQ ( FRV [ ) _ \ = DUFFRV ( FRV [ ) ;  π [ ⋅ FRV _keb [ ≠ [ a \ = ([ + ) ` \ =   _keb [ = 

b \ = DUFFRV

− [ d \ = + [

([ + )

− ([ − )

;

;

GZclbh^ghklhjhggb_ijhba\h^gu_\mdZaZgguolhqdZo^eyke_^mxsbonmgdpbc π [= N ∈ Ζ [ I ([ ) = VLQ [ [ = ; [ N + [ \ I ([ ) = VLQ [ [ = [ = π ] I ([ ) = DUFVLQ [ = ± . + [

Z I ([ ) = [ ⋅ FRV

Bkke_^h\ZlvgZ^bnn_j_gpbjm_fhklv\lhqd_x=0 nmgdpbx   [ − H [ − _keb [ ≠ I ([ ) =   _keb [ = 

GZclbijhba\h^gu_nmgdpbc

[

 VLQ [   + ORJ [ ;  [ 

Z \ = [ + [ [ + [ [ [ \ = FK [ H + [ [ \ \ =  [

[

] \ = ORJ [ ⋅ ORJ [ H + OQ ⋅ ORJ [ ⋅ [ ^ \ = [ H

;

DUFWJ [

_ \ = ( FRV[ )

H[

;

` \ = [ H + ( [ + ) ; 8. Bkoh^ybahij_^_e_gbyijhba\h^ghc, gZclb I ′ ( ) , _keb FWJ [

WJ [

    [ [ ⋅ FRV VLQ  [ ⋅ VLQ  [ ≠ + [ ≠ [ [ I [ =  Z I ([ ) =   [   [ = [ =  

;

 H [ − FRV [  OQ FRV [  [ ≠  [ ≠ \ I [ =  [ ] I [ =  ; [   [ = [ =  GZclb G \ ^eynmgdpbb \ = VLQ [ _keb

Z [ -g_aZ\bkbfZyi_j_f_ggZy[ [ -nmgdpbyg_dhlhjhci_j_f_gghc GZclb \ ( Q) ([ ) ^eyke_^mxsbonmgdpbc [ \ = H [ ⋅ VLQ [ \ \ = ([ + ) ⋅ H [ ; Z \ = VLQ [ + FRV [ ; ] \ = [ ⋅ (VLQ [ + FRV [ ) ^ \ = [ ORJ ( − [ ) _ \ = ([ − )⋅ [ ⋅ [ ;

[ a \ = ( + [ ) − [ b \ = VLQ [ ⋅ VLQ [ ; d \ = ([ + [ ) ⋅ FRV [ e \ = ([ + [ )(VLQ [ − FRV [ ) f \ = [ − [ .

` \ = [ FRV

V.NhjfmeZL_cehjZ Ijbf_g_gb_ijhba\h^ghcdbkke_^h\Zgbxnmgdpbc 1. JZaeh`blvihnhjfme_FZdehj_gZke_^mxsb_nmgdpbb Z I ([ ) = FRV [ + VLQ [ ^h R([ Q+ ) [ I ([ ) = H [ [ ^h R([

\ I ([ ) = OQ

+ [ −[

VLQ

^h R([ Q ) ] I ([ ) =

−

)

;

^h R( [ Q ) ;

− [

^ I ([ ) = OQ ( − [ ) ^h R([ ) _ I ([ ) = ([ +) Bkihevamyhkgh\gu_jZaeh`_gbygZclbij_^_eu Q

^h R([

[

)

.

( + [ ) − H H [ VLQ [ − [ ( + [ ) FRV [ − H [ OLP \ OLP ; Z OLP [→ [→ [→ [ [ [ π  FRV  FRV [  [ H [ + OQ ( − [ )   H [ − + [ OLP OLP ] OLP ^ _ . [→ [→ [→ [ FRV [ − VLQ [ OQ FRV [ VLQ (VLQ [ )

[

Bkke_^h\ZlvgZwdklj_fmfnmgdpbb Z \ = DUFVLQ (VLQ [ ) [ \ = ([ + ) − ([ − ) \ \ = VLQ [ + FRV [ ;

[ + [ − [ ` \ = [ OQ [ a \ = [ OQ [ b \ = DUFFRV + [ − [− d \ = [ ⋅ H e \ = VLQ [ + FRV [ f \ = [ − ⋅ H [ .

] \ = ([ + ) H [ ^ \ = ([ − ) H − [ _ \ = DUFVLQ

GZclbbgl_j\Zeu\uimdehklbblhqdbi_j_]b[Z]jZnbdZnmgdpbb Z

;

\ = [ + [ \ =

[

[ −

\ \ = H − [ ;

] \ = [ ⋅VLQ (OQ [ ) ^ \ = H DUFWJ [ _ \ = [ OQ [ ; ` \ = H − [ ⋅ VLQ [ a \ = [ ( + [ ) .

;

EBL?J:LMJ:

>_fb^h\bq;IK[hjgbdaZ^ZqbmijZ`g_gbcihfZl_fZlbq_kdhfm ZgZebam-FGZmdZ- 527 c. 2. K[hjgbdaZ^ZqihfZl_fZlbq_kdhfmZgZebamij_^_eg_ij_ju\ghklv^bnn_j_gpbjm_fhklvE> Dm^jy\p_\:>DmlZkh\<BQ_oeh\ FBRZ[mgbg-FGZmdZ- 592 c. FZl_fZlbq_kdbcZgZeba\\hijhkZobaZ^ZqZoMq_[ihkh[b_^ey \mah\<N;mlmah\GQDjmlbpdZy=GF_^\_^_\::Rbrdbg Ih^j_^<N;mlmah\Z-FDmjkfZl_fZlbq_kdh]hZgZebaZ-F